Quantum mechanics

Classical Physics

When our view of a moving object is obstructed by a large billboard, we instinctively believe that the object has passed behind it before reappearing This assumption reflects our understanding of physical reality, a key concept in the EPR paradox formulated by Einstein, Podolsky, and Rosen.

In a stable system, if we can accurately predict the value of a physical quantity, it indicates the existence of a corresponding element of physical reality.

1Presentations of quantum mechanics resting upon few basic principles start with

[15], which remains a cornerstone on the subject.

In many presentations, the solution to the wave equation for a free particle is often represented by the plane wave exp[i(kx−ωt)] However, manipulating the momentum and energy operators to derive this plane wave as a solution to differential equations is not entirely satisfactory This approach presents challenges, such as the fact that plane waves are not square integrable and thus problematic as wave functions Additionally, it suggests that quantum mechanics relies solely on differential formulations, which can lead to the misconception that the position wave function is the exclusive means of describing quantum states.

Fig 2.1 The trajectory of the car behind the billboard as an element of physical reality

The classical framework is built on certain preconceptions, particularly the existence of continuous time functions known as trajectories, represented as x(t) and p(t) for position and momentum, respectively Trajectories serve as a crucial connection between the physical world and its mathematical representation, enabling the formulation of Newton's second law.

The motion equation provides a continuous and deterministic forecast of a system's evolution As a second-order equation, it allows for the determination of a system's state if the position and velocity of each particle are known at a specific moment.

Maxwell's theory of electromagnetism, a key component of classical physics, is defined through fields that are specified at every point in space and time Unlike particles, these fields can be infinitely refined, allowing for precise measurements Additionally, electromagnetism is recognized as a deterministic theory, meaning it predicts outcomes with certainty based on initial conditions.

Essential assumptions in classical physics about both particles and ﬁelds are:

• The possibility of nondisturbing measurements

• There is no limit to the accuracy of values assigned to physical properties

Classical physics, as described by Schwinger, merges physical properties with their numerical representations, emphasizing that there is no separation between the two This idealization allows for nondisturbing measurements, facilitating the continuous assignment of numerical values to physical properties, which forms the foundation of mathematical representation in physics.

Such “obvious” assumptions are no longer valid in quantum mechanics. Therefore, other links have to be created between the physical world and the mathematical formalism.

2.2* Mathematical Framework of Quantum Mechanics

Classical electromagnetism states that an inhomogeneous magnetic field, oriented along an axis such as the z-axis, causes particles to deviate from their path in a direction perpendicular to that axis The degree of this trajectory bending is influenced by the characteristics of the magnetic field and the properties of the particles.

The mathematical framework of quantum mechanics reveals that the distribution of silver atoms in a magnetic field, as demonstrated by Otto Stern and Walther Gerlach in 1921, results in only two distinct lines on a screen, contrary to classical expectations of a continuous distribution This phenomenon occurs because the magnetic moment of the atoms is linked to an intrinsic angular momentum known as spin, which allows only two possible projections: ±1 Thus, the behavior of unpolarized particles in a magnetic field highlights the fundamental principles of quantum mechanics.

2π, (2.2) where h is the Planck constant It has the dimensions of classical action (energy×time) (See Table 14.1 and Sect 5.2.1 for more details.)

It's important to note that a physical quantity having only two values does not necessitate the rejection of classical physics For example, a computer operates using bits, which are classical systems that can exist in one of two states As is characteristic of classical systems, their state remains unchanged upon measurement, thereby enhancing the stability of classical computers.

Vectors on a plane provide a distinct description, as the addition of two vectors always results in another vector, unlike the sum of two states of a bit Any vector Ψ can be expressed as a linear combination, represented mathematically as Ψ = c_x ϕ_x + c_y ϕ_y, where c_x and c_y are amplitudes, and ϕ_x and ϕ_y are perpendicular unit vectors This relationship is encapsulated in the scalar product, shown by the equation Ψ|Ψ = c*_x c_x + c*_y c_y, demonstrating the orthogonality of the vectors with the relation ϕ_i | ϕ_j = δ_ij.

Fig 2.2.Representation of a vector in two-dimensions The same vector Ψ can be expressed as the sum of two diﬀerent systems of basis vectors

3Although the bits in your PC function on the basis of quantum processes (for instance, semiconductivity) they are not in themselves quantum systems.

In quantum mechanics we allow complex values of the amplitudes.

Another crucial property of the vector space is that the same vector Ψ may be expressed as a combination of other sets of perpendicular vectors η x , η y along rotated axis (Fig 2.2b) Ψ =b x η x +b y η y (2.5)

Hilbert spaces, which can be generalized to any number of dimensions, play a crucial role in quantum mechanics This article highlights key properties of these spaces, with a more detailed exploration provided in Section 2.7.

• Any vector Ψ may be expressed as a linear combination of orthonormal basis statesϕ i [as in (2.3)] Ψ i c i ϕ i ; c i = ϕ i |Ψ ≡ i|Ψ, (2.6) ϕ i |ϕ j =δ ij (2.7)

• Linear operators ˆQact on vectors belonging to a Hilbert space, transforming one vector into another Φ = ˆQΨ (2.8)

These operators obey a noncommutative algebra, as shown in Sect 2.7* for the case of rotations in three-dimensions We deﬁne the commutation operation through the symbol

[ ˆQ,R]ˆ ≡QˆRˆ − RˆQ,ˆ (2.9) where the order of application of the operators is from right to left [ ˆQRˆΨ Qˆ( ˆRΨ)].

• If the vector ˆQϕ i is proportional to ϕ i , then ϕ i is said to be an eigenvector of the operator ˆQ The constant of proportionalityq i is called the eigenvalue

• The scalar product between a vector Φ a = ˆQΨ a and another vector Ψ b is called the matrix element of the operator ˆQbetween the vectors Ψ a and Ψ b , and it is symbolically represented as 4 Ψ b |Q|Ψ a ≡ b|Q|a ≡ Ψ b |Φ a (2.11)

The matrix elements of the unit operator are the scalar products Ψ a |Ψ b ≡ a|b = Ψ b |Ψ a ∗ The norm Ψ|Ψ 1/2 is a real, positive number.

4Dirac called the symbolsa|and|athe bra and ket, respectively [15].

Basic Principles of Quantum Mechanics

Some Comments on the Basic Principles

Quantum mechanics serves as a versatile framework applicable to various systems, ranging from single particles to many-body systems and fields Rather than being a standalone physical theory, it provides a foundational structure for the development of diverse physical theories.

In this article, we propose that the vector state symbolizes our understanding of reality rather than reality itself This understanding encompasses the development of potential tests (measurements) applicable to the system and the associated probabilities of their outcomes.

We propose a novel connection between the physical world and mathematics, where physical quantities correspond to noncommuting operators State vectors are formed through operations involving these mathematical entities, and the physical world is influenced by predicting measurement outcomes, specifically the eigenvalues of these operators and their associated probabilities This intricate two-way relationship between the physical realm and mathematical formalism presents significant challenges, as highlighted by David Mermin.

Learning quantum mechanics can be challenging, particularly in grasping how its abstract formalism relates to real-world laboratory phenomena This often requires creating simplified models that approximate these complex realities.

6Notions of probability theory are given in Sect 2.8*.

We have adopted the reduction interpretation of the measurement process, a perspective historically favored by many physicists In Chapter 12, we will further explore the measurement problem The most skilled physicists possess remarkable intuition for identifying the crucial aspects of actual phenomena that need representation in abstract models, while recognizing which features are non-essential and can be disregarded.

Understanding quantum states involves clarifying common misconceptions, as it's equally important to highlight what these principles do not imply.

The state vector shares similarities with other fields that describe the physical world, yet it is fundamentally distinct from electric and magnetic fields found in electromagnetic waves Unlike these fields, which transmit momentum and energy and have changes that propagate at a finite speed dependent on the medium, the state vector operates under different principles.

A common misconception is that "energy eigenstates are the only allowed states," which likely stems from the focus on the eigenvalue equation solution and its resemblance to the accurate assertion that "energy eigenvalues are the only permitted energies."

A state vector represents a collection of classical systems, while in the Copenhagen interpretation, it is applied to a single system Importantly, none of the recognized statistical interpretations regard this ensemble as classical.

• “A state vector describes a single system averaged over some amount of time.” The state vector describes a single system at a single instant.

Measurement Process

The Concept of Measurement

When two or more systems interact, the presence of one system influences changes in the other, and this relationship is reciprocal Typically, varying initial conditions result in distinct changes, though there are exceptions to this pattern.

A measurement is a process in which a system is put in interaction with a piece of apparatus The apparatus determines the physical quantity or observable to be measured (length, weight, etc.).

Measurement involves two crucial steps: first, preparing the system for measurement by determining its initial state According to Bohr, a "phenomenon" is defined as an observation made under specific conditions, which encompasses the entire experimental setup This definition stands in contrast to the EPR interpretation of physical reality.

8See also [21], specially Sects 1.2, 2.1, and 3.6.

The second crucial step for quantum systems involves a noticeable macroscopic change in the apparatus that can be detected by a cognitive system Typically, this change is initiated by a detector located at one end of the apparatus The physical quantity's magnitude is defined when this change can be expressed in numerical terms.

Quantum Measurements

The key distinction between classical and quantum systems lies in the fact that quantum systems cannot be measured without causing irreversible changes, regardless of the sophistication of the measuring tools used This phenomenon arises from the principles discussed in Section 2.3.

When measuring a physical quantity Q in a system represented by the state Ψ, obtaining a result q j guarantees that repeating the measurement immediately will yield the same value q j with certainty This process alters the coefficients, transforming c i to δ ij, indicating that the measurement causes the system to transition into an eigenstate of the measured quantity, known as state vector reduction Exceptions arise only when the initial state is already aligned with one of the eigenvectors.

Given an initial state vector Ψ, we do not know in general to which eigenstate the system will jump Only the probabilities, represented by |c i | 2 , are ϕ y ϕ y ϕ x ϕ x ψ c y c x ψ = c x ϕ x + c y ϕ y

Fig 2.3.The reduction of the state vector as a result of a measurement

Classical systems can be influenced by measurements, but it is generally assumed that the resulting disturbance can be minimized or accurately predicted through calculations This understanding aligns with the established probabilities associated with these systems.

• Their value is always positive

• Their sum is 1 (if the state Ψ is normalized)

• The orthogonality requirement (2.7) ensures that the probability of obtaining any eigenvalue q j = q i vanishes if the initial state of the system is the eigenstateϕ i (see Table 2.1).

In quantum mechanics, the inherent indeterminacy arises from our ability to predict only the probability |c i |² of obtaining eigenvalues q i from a state vector Ψ This limitation means that additional measurements cannot enhance our knowledge of the system, as the initial state Ψ transforms into a new state ϕ i upon measurement.

According to the interpretation adopted in Sect 2.3.1, it is our knowledge of the system that jumps when we perform a measurement, rather than the physics of the system.

In the expansion (2.6), when a subset of basis states ϕ_k shares the same eigenvalue q_k = q, the probability of measuring this eigenvalue is given by k |c_k|^2 Following the measurement, the system collapses into the normalized state Ψ, represented as Ψ = (1/k) Σ |c_k|^2 Σ c_k ϕ_k.

The concept of probability implies that we must consider a large number of measurements performed on identical systems, all of them prepared in the same initial state Ψ.

The diagonal matrix element 10 is given by the sum of the eigenvalues weighted by the probability of obtaining them: Ψ|Q|Ψ i q i |c i | 2 (2.20)

The mean value, also known as the expectation value of the operator ˆQ, represents the average outcome of measurements taken on identical systems Unlike a specific measurement result q i, the mean value encompasses the overall average of all results obtained.

The uncertainty or standard deviation ∆Q in a given measurement is deﬁned as the square root of the average of the quadratic deviation:

10The matrix element is said to be diagonal if the same vector appears on both sides of the matrix element.

Some Consequences of the Basic Principles

2.5 Some Consequences of the Basic Principles

This section explores the implications of quantum principles through thought experiments, which can also serve as a foundation for generalizing these principles We examine a Hilbert space with two independent states, ϕ ±, which are eigenstates of the operator ˆS, corresponding to the eigenvalues ±1.

The equation Sϕˆ ± =±ϕ ± is satisfied, confirming the scalar products ϕ + |ϕ + = ϕ − |ϕ − = 1 and ϕ + |ϕ − = 0 Numerous physical observables can be represented by such operators, with the z-component of spin 1/2 being a prominent example discussed throughout this book in sections 2.2*, 3.1.3, 5.2, and 9.2.

To create a filter that ensures exiting particles are in a definite eigenstate, we begin by splitting a particle beam into two separate ϕ ± beams, similar to the Stern and Gerlach experiment In the second phase, each beam is directed back toward the original path, with the option to mask one beam at the halfway point This setup, illustrated in Fig 2.4a, features the ϕ − beam masked off and is referred to as a ϕ-filter, enclosed within a continuous-line box.

Any experiment requires ﬁrst the preparation of the system in some deﬁnite initial state (Sect 2.4.1) Particles leave the oven in unknown linear combinations Ψ ofϕ ± states Ψ = ϕ + |Ψ ϕ + + ϕ − |Ψ ϕ − (2.23)

The particles are collimated and travel along the y-axis, with a focus on preparing them in the filtered state ϕ + This is achieved by restricting the escape of particles in the state ϕ − from the initial filter, as illustrated in Fig 2.4b.

In the final phase of the experimental setup, an additional filter is integrated into the detector to assess the level of filtration The detector is equipped with a photographic plate that captures the incoming particles, allowing for observation by the experimentalist.

In the initial experiment, the detector is positioned directly after the first filter When the ϕ− channel is blocked in the detector, all particles successfully pass through; conversely, if the ϕ+ channel is obstructed, no particles are transmitted.

11Another example is given by the polarization states of the photon (seeSect 9.5.2 † ) Most of the two-state experiments are realized by means of such optical devices.

Quantum mechanical thought experiments, as illustrated in Fig 2.4, demonstrate fundamental principles by showcasing various components such as filters, particle state preparation, and detection methods The vertical bars represent fixed path blocking, while the slanting bars indicate adjustable paths Each experiment involves two measurements: one with the upper channel of the detector open and the lower channel blocked, and another with the opposite configuration The outcomes of these processes yield probabilities of 1 and 0 for the states |ϕ + |ϕ + and |ϕ − |ϕ +, respectively.

We examine a new set of basis states, η ±, which fulfill the orthonormality conditions: η+ |η+ = 1, η− |η− = 1, and η+ |η− = 0 It is confirmed that the operator ˆR, which satisfies the eigenvalue equation ˆRη ± = ±η ±, does not commute with ˆS To enable the detector filter to block particles in the η ± states, we must modify it accordingly When ˆR represents the spin component in the x-direction, this modification requires rotating the filter by an angle of π/2 around the y-axis The dashed boxes in the diagram indicate filters that allow particles to exit in the η ± states, referred to as η-type filters.

2.5 Some Consequences of the Basic Principles 17

A particle exiting the first filter in the stateϕ + reorients itself, by chance, within the second filter This process is expressed by expanding the statesϕ ± in the new basis ϕ ± = η+ |ϕ ± η++ η − |ϕ ± η − (2.24)

According to Principle 3, a particle will emerge from the detector ﬁlter in the stateη+ with probability |η+ |ϕ + | 2 or in the stateη − with probability

If the η − channel of the second filter is blocked, the particle can be projected into the state η + with a probability of |ϕ + |η + | 2, or it may be absorbed with a probability of 1− |ϕ + |η + | 2 = |ϕ + |η − | 2 This outcome mirrors the classical Malus law, yet it highlights the probabilistic nature of the projection process, leading to the loss of any information regarding the previous orientation ϕ +.

In the experiment depicted in Fig 2.4e, substituting the detector with the original Stern–Gerlach apparatus, as shown in Fig 5.3, highlights a fundamental difference between classical waves and quantum mechanics While classical waves can simultaneously amplify their intensities at two locations on a screen, this scenario would contradict the principle of charge conservation for electrons Quantum mechanics, in contrast, assigns probabilities to individual particles at each location, emphasizing that we are not observing classical wave behavior.

In our experiments, we observe results that significantly diverge from classical predictions by utilizing a ϕ-type detector filter and introducing an η-type filter between the first filter and the detector Particles prepared in the ϕ + state exit the second filter in the η + state, with the first filter aligning the spin along the positive z-axis and the second filter oriented along the positive x-axis The detector measures the count of particles exiting in the ϕ ± states, corresponding to spins pointing up or down in the z-direction We apply the inverse expansion of the relationship between the η and ϕ states to analyze the results.

Thus, the total amplitudes for particles emerging in one of the states ϕ ± are 14 ϕ + |η + η + |ϕ + (2.26) ϕ − |η + η + |ϕ + (2.27)

12An alternative could be that the electron chooses its path just upon entering the second ﬁlter However, this interpretation is inconsistent with the results of experiment 2.4g.

13The amplitudes in (2.24) and in (2.25) are related by ϕ + |η ± = η ± |ϕ + ∗ and ϕ − |η ± = η ± |ϕ − ∗ , according to Table 2.1.

14One reads from right to left.

Despite the annihilation of the ϕ − state fraction within the first filter, both components ϕ ± can still emerge from the detector filter This phenomenon cannot be explained by classical physics and highlights a key quantum principle: the inability to simultaneously determine two non-commuting observables A precise measurement of one observable (R) inevitably compromises the information about the other (S).

The experiment's results align with Principle 1 outlined in Section 2.3, indicating that the state vector η + encapsulates all relevant information about the system, rendering its past history irrelevant to future events This information is effectively lost due to the collision with the blocking mask positioned within the second filter.

In a repeated experiment where the mask is removed from the second filter, the total amplitude is calculated as the sum of amplitudes from two potential intermediate states In the first scenario, all particles pass through, resulting in an amplitude of 1, while in the second scenario, no particles are transmitted, yielding an amplitude of 0 This illustrates that despite opening more channels, the overall particle count decreases, highlighting the impact of the experimental setup on particle transmission.

The recent experiment produced an interference pattern typically linked to waves; however, it uniquely reveals that particles are consistently detected as discrete lumps of uniform size on a screen, with no fractional detection, highlighting their indivisible nature This phenomenon illustrates wave-particle duality, supported by established principles.

Commutation Relations and the Uncertainty Principle

random, since relative probabilities|ϕ ± |η+ | 2 are the only information that we can extract from measurements.

2.6 Commutation Relations and the Uncertainty

The commutation relation between two Hermitian operators, denoted as ˆr and ˆs, plays a crucial role in defining the precision with which their corresponding physical quantities can be simultaneously measured This relationship extends the Heisenberg uncertainty principle, traditionally applied to momenta and coordinates, to any pair of observables, highlighting that the uncertainty arises directly from their commutation relations.

One assumes two Hermitian operators, ˆR,S, and deﬁnes a third (non-ˆ Hermitian) operator ˆQsuch that

Qˆ ≡Rˆ+ iλS,ˆ (2.30) whereλis a real constant The minimization with respect toλof the positively deﬁned norm [see (2.41)]

In this section, we apply the definition of the Hermitian conjugate, as stated in equation (2.12) The relationship [ ˆR,S]ˆ + =−[ ˆR,S] arises from the Hermitian nature of the operators, as indicated in equation (2.40) By substituting the minimum value λ min into equation (2.31), we derive significant results.

The following two operators ˆr,sˆhave zero expectation value: ˆ r≡Rˆ− Ψ|R|Ψ, sˆ≡Sˆ− Ψ|S|Ψ, (2.35) and the product of their uncertainties is constrained by [see (2.21)]

Operators associated with observables can be expressed in a specific mathematical form When preparing multiple quantum systems in the same state Ψ, measuring observable r in some systems and observable s in others, the product of the standard deviations of these measurements must adhere to a defined inequality.

In the case of coordinate and momentum operators, the relation (2.16) yields the Heisenberg uncertainty relation

This relationship is fundamentally rooted in basic principles, particularly the commutation relation (2.16) It represents an inherent limitation to our knowledge that cannot be surpassed, even with advancements in experimental techniques.

When a system is in an eigenstate of the operator ˆr, measuring the observable associated with it provides the corresponding eigenvalue However, for noncommuting operators like ˆs, the value of the observable remains uncertain This scenario is exemplified by a plane wave representing a particle in free space, where the momentum can be precisely measured, yet the particle's position is distributed across all of space.

The state vector Ψ can be an eigenstate of both operators ˆr and ˆs simultaneously only if these operators commute, leading to a zero product of their uncertainties Additionally, if the operators commute and the eigenvalues of ˆs are distinct within a specific subset of states, then the matrix elements of ˆr will also be diagonal within that same subset.

Heisenberg developed the uncertainty relations to address the wave-particle paradox, highlighting that pure particle behavior necessitates localization, while wave behavior is evident only with definite momentum He proposed that these classical descriptions are mutually exclusive, rendering neither valid in intermediate scenarios Nonetheless, quantum mechanics must align with the motion of elementary particles through trajectories Heisenberg's solution involves constructing states Ψ that incorporate specific amounts of localization in both momentum and position, suggesting that a particle's motion can exhibit characteristics of both wave and particle behavior.

The apparent classical trajectory of a pion demonstrates classical motion along defined paths, but it is essential to account for variations in momentum and position This spread ensures that the amplitudes in momentum eigenstates and position eigenstates adhere to the uncertainty principles.

Figure 2.5 illustrates the capture of a pion by a carbon nucleus, allowing for the determination of particle mass, energy, and charge through measurements of track length, grain density, and scattering direction Assuming a pion kinetic energy of 10 MeV and using its mass of 139 MeV/c², the resulting momentum is calculated to be pπ = 53 MeV/c The uncertainty in the direction perpendicular to the track, estimated from the track width of approximately 1 μm, results in ∆p⊥ ≈ 10⁻⁷ MeV/c.

∆p ⊥ /p π ≈10 − 9 is too small to produce a visible alteration of the apparent trajectory.

A Hilbert space extends the concept of Euclidean three-dimensional space, allowing for operations such as the summation of vectors, represented as c a Ψ a + c b Ψ b, and the scalar product, denoted as Ψ b |Ψ a = c ab While constants c a, c b, and c ab are typically real numbers in classical space, quantum mechanics necessitates the inclusion of complex values for these constants.

Two vectors are orthogonal if their scalar product vanishes A vector Ψ is linearly independent of a subset of vectors Ψ a ,Ψ b , Ψ d if it cannot be expressed as a linear combination of them 17 (Ψ =c a Ψ a +c b Ψ b +ã ã ã+c d Ψ d ).

16Deﬁnitions of these fundamental operations is deferred to each realization of Hilbert spaces [(3.2), (3.4) and (4.1), (4.2)] In the present chapter we use only the fact that they exist and thata|b=b|a ∗ , a|a>0.

17Although the term “linear combination” usually refers only to ﬁnite sums, we extend its meaning to include also an inﬁnity of terms.

Table 2.1 Some relevant properties of vectors and operators in Euclidean and Hilbert spaces

The concepts of orthonormal basis vectors, denoted as ϕ i, allow for the complete representation of any vector Ψ as a linear combination of these basis vectors The scalar product i|Ψ represents the projection of Ψ onto the basis vector ϕ i Additionally, the relationships between the scalar products of vectors Ψ a and Ψ b, along with the square of the norm of vector Ψ, are defined in terms of the amplitudes c i The dimension ν of the associated Hilbert space indicates the number of states in a basis set, which is 3 in normal space This book explores Hilbert spaces with dimensions ranging from two to denumerable infinity.

In ordinary space, vectors are characterized by how they transform under rotation operations, denoted as ˆR η (θ), where η represents the rotation axis and θ indicates the angle It is important to note that these rotation operations are typically noncommutative, which can be demonstrated by executing two consecutive rotations of θ = π/2.

This article explores the effects of sequential rotations around the x-axis and y-axis, comparing the outcomes with those obtained by reversing the order of these rotations Additionally, it discusses how vectors in Hilbert spaces can be transformed through the action of operators, denoted as ˆQ, which adhere to a noncommutative algebra, as defined in the commutation operation outlined in equation (2.9).

In ordinary space, a dilation ˆD is an operation yielding the same vector multiplied by a (real) constant This operation has been generalized in terms

The Fourier expansion represents a well-known example of function expansion using an orthonormal basis set, specifically through the exponentials exp(ikx) This expansion forms a complete set of eigenfunctions relevant to the case of a free particle, as discussed in Section 4.3.

Fig 2.6 The ﬁnal orientation of the axes depends on the order of the rotations.

R ν here represents a rotation ofπ/2 around theν-axis of eigenvectors and eigenvalues in (2.10) In general, linear combinations of eigenvectors do not satisfy the eigenvalue equation.

2.7.1* Some Properties of Hermitian Operators

The Hermitian conjugate operator ˆQ + is deﬁned through (2.12) Similarly, we may write Ψ b |Q|Ψ a =QΨˆ a |Ψ b ∗ =Qˆ + Ψ b |Ψ a (2.38) The following properties are easy to demonstrate:

According to (2.12), the norm of the state ˆQΨ is obtained from

Assume now that the stateϕ i is an eigenstate of the Hermitian operator

Qˆ corresponding to the eigenvalueq i In this case, i|Q|i=q i i|i, i|Q|i ∗ =q ∗ i i|i,i|Q|i=i|Q|i ∗ → q i =q ∗ i (2.42)Therefore, the eigenvalues of Hermitian operators are real numbers.

Consider now the nondiagonal terms j|Q|i=q i j|i, i|Q|j ∗ =q ∗ j i|j ∗ =q ∗ j j|i (2.43)

0 = (q i −q j )j|i, (2.44) i.e., two eigenstates belonging to diﬀerent eigenvalues are orthogonal They may also be orthonormal, upon multiplication by an appropriate normalization constant, which is determined up to a phase.

The eigenvectors of a Hermitian operator form a complete set of states for a system, allowing any state function Ψ to be represented as a linear combination of the basis states ϕ i.

We deﬁne the projection operator (a theoretical ﬁlter)|ii| through the equation

|ii|ϕ j ≡ i|jϕ i =δ ij ϕ i , (2.45) which implies that i

Matrix Formalism

A Realization of the Hilbert Space

The state vector Ψ may be expressed by means of the amplitudesc i ﬁlling the successive rows of a column vector: Ψ = (c i )≡

The dimension of the Hilbert space is given by the number of rows The sum of two column vectors is another column vector in which the amplitudes are added: α B Ψ B +α C Ψ C = (α B b i +α C c i ) (3.2)

The scalar product requires the deﬁnition of the adjoint vector Ψ + , i.e., a row vector obtained from Ψ with amplitudes: Ψ + = (c ∗ a , c ∗ b , , c ∗ ν ) (3.3)

The scalar product of two vectors Ψ B and Ψ C is deﬁned as the product of the adjoint vector Ψ + B and the vector Ψ C , viz., Ψ B |Ψ C i=ν i=a b ∗ i c i , Ψ|Ψ i=ν i=a

A useful set of (orthonormal) basis states is given by the vector columns ϕ i with amplitudes c j =δ ij In such a basis, the arbitrary vector (3.1) may be expanded as Ψ =c a

All the properties listed in Table 2.1 are reproduced within the framework of column vectors.

Operators are represented by square matrices

The matrices representing physical observables are Hermitian, with the initial state j indicating the columns and the final state i indicating the rows The sequence a, b, , ν is irrelevant as long as it remains consistent across both columns and rows, ensuring that the matrix elements i|Q|j are positioned on the diagonal The construction of the matrix elements i|Q|j follows the method outlined in (2.11), and if ϕ i is part of the basic set, it plays a crucial role in this context.

A matrix multiplying a vector yields another vector, so that Ψ B = ˆQΨ C ←→b i j i|Q|jc j (3.8)

The product of two matrices is another matrix:

Sˆ= ˆQRˆ ←→ i|S|j k i|Q|kk|R|j, (3.9) which is consistent with the closure property (2.47) The multiplication of matrices is a noncommutative operation, as beﬁts the representation of quantum operators.

The Solution of the Eigenvalue Equation

In matrix form, the eigenvalue equation (2.10) reads

⎟⎠ , (3.10) which is equivalent to theν linear equations (one equation for each value ofi) j=ν j=1 i|Q|jc j =q c i (3.11)

The unknowns to be determined are the eigenvalues \( q \) and the amplitudes \( c_i \) To solve the equation, the original matrix \( (i|Q|j) \) is transformed into a diagonal form, where the diagonal elements represent the eigenvalues, expressed as \( i|Q|j = \delta_{ij} q_i \) The corresponding eigenvector for each eigenvalue is defined by the amplitudes \( c_j = \delta_{ij} \), as indicated in equation (3.5).

The linear homogeneous equations (3.11) have the trivial solution c i = 0, to be discarded The existence of additional, nontrivial solutions requires the determinant to vanish: det (i|Q|j −qδ ij ) = 0 (3.13)

This eigenvalue equation is equivalent to a polynomial equation for q Its ν roots are the eigenvalues of the operator ˆQ.

The determinant's vanishing indicates that one equation can be represented as a linear combination of the other ν−1 equations By eliminating one equation, such as the one related to the last row, and dividing the others by c a, we derive a set of ν−1 nonhomogeneous linear equations This process allows us to calculate the ratios c b /c a, c c /c a, and so forth, for each eigenvalue q, leading to the necessary normalization.

The equation can be derived by applying the expansion (2.6) to both sides of the general eigenvalue equation ˆQΨ = qΨ, resulting in j c j Qϕˆ j = q j c j ϕ j Taking the scalar product with ϕ i on both sides of this equation leads to the formulation presented in (3.11).

When multiple roots share the same eigenvalue, it is necessary to eliminate additional equations to achieve a nonhomogeneous set Equation (3.4) specifies the value of |c_a|², while the overall phase of the state vector remains arbitrary It's important to note that the relative phases within the linear combination hold physical significance, despite the overall phase being inconsequential Diagonalization produces a new set of eigenstates, η_a, which can be represented as linear combinations of the original basis states, ϕ_i, as shown in equation (3.14).

The amplitudes |a|i| are the matrix elements of a unitary matrix U The squared modulus |a|i|² represents the probability of measuring the eigenvalue q_i, linked to the eigenstate ϕ_i, when the system is in state η_a Additionally, it indicates the probability of measuring the eigenvalue r_a, associated with eigenstate η_a, when the system is in state ϕ_i.

Application to 2×2 Matrices

An example of a diagonal matrix is the matrix representing thez-component of the spin operator (Sect 5.2.2):

The eigenvectors corresponding to spin up and spin down are, respectively, ϕ ↑ z ≡ϕ (s z =¯ h/2) 1 0

Let us diagonalize a general Hermitian matrix of order two: a|Q|a a|Q|b b|Q|a b|Q|b

(a|Q|a − b|Q|b) 2 + 4|a|Q|b| 2 , (3.18) while the amplitudes of the eigenvectors are given by c b c a ±

Figure 3.1 plots the eigenvalues q ± and the initial expectation values as a function ofQ≡ a|Q|a, assuming a traceless situation (a|Q|a=−b|Q|b) and a|Q|b = 2 The eigenvalue q + is always higher than |Q|, while q − is

In a 2×2 system, the eigenvalues \( q \pm \) exhibit continuous curves as a function of \( Q \), with the half energy distance between the diagonal matrix elements represented by dotted lines that remain consistently below \(-|Q|\) This behavior indicates that the two eigenvalues repel each other and do not intersect when \( a|Q|b = 0 \) The distance \( \Delta Q = 2 + |a|Q|b|^2 - Q \) quantifies the increase in the highest eigenvalue of \( Q \) resulting from the superposition of states \( \phi_a \) and \( \phi_b \), reaching its maximum at the crossing point \( Q = 0 \).

The physical world features numerous systems characterized by two distinct states, exemplified by an electron interacting with two protons For simplification, we can disregard the motion of the protons due to their significantly greater mass compared to the electron The two states, ϕ a and ϕ b, correspond to the electron being bound to each proton, representing a hydrogen atom and a separate proton, respectively In this context, the Hamiltonian ˆH serves a role analogous to ˆQ in the relevant equations.

The stability of the ionized hydrogen molecule is rooted in quantum mechanics, stemming from the superposition of states ϕ a and ϕ b, which enables the formation of a bound state For a more in-depth discussion, refer to Section 8.4.1.

The calculations (3.18) and (3.19) are quickly made using the matrix associated with the spin component ˆS x (5.23):

3In fact, any two states suﬃciently isolated from the remaining ones may be approximated as a two-state system, for which the no-crossing rule holds See also Sect 2.5.

Equation (3.18) yields the eigenvaluess x =±¯h/2 and the eigenvectors η ↑ x = 1

The equations illustrate how the eigenstates of ˆS x can be represented as linear combinations of the eigenstates of ˆS z Notably, the significance of the relative sign is highlighted, even though the probability of measuring the z-component as either pointing up or down remains identical for both states.

(3.22) transforms the basis set of eigenvectors of the operator ˆS z into the basis set of eigenvectors of ˆS x , in accordance with (3.14):

Similarly, the operator ˆS z is transformed into the operator ˆS x [see (2.52)]:

The trace of the matrix for the spin operator S x remains invariant and equals zero under unitary transformation In this context, the eigenvalues of the operators ˆS z and ˆS x are identical, which aligns with physical expectations, as the eigenvalues hold physical significance while the coordinate system's orientation in isotropic space does not.

Harmonic Oscillator

Solution of the Eigenvalue Equation

We intend to solve (2.18) The unknowns are the eigenvalues E i and the eigenfunctionsϕ i The fundamental tool entering the present solution is the commutation relation (2.16).

We ﬁrst deﬁne the operatorsa + , a a + ≡

The operators ˆxand ˆpare Hermitian, since they correspond to physical observables Therefore the operatorsa, a + are Hermitian conjugates of each other, according to (2.39) They satisfy the commutation relations

We now construct the matrix elements (2.11) for both sides of (3.30), making use of two eigenstatesϕ i ,ϕ j : i|[H, a + ]|j= (E i −E j )i|a + |j= ¯hωi|a + |j (3.32)

We conclude that the matrix elementi|a + |j vanishes, unless the diﬀerence

The energy difference between two eigenstates, denoted as E i − E j, is a constant value of ¯hω This allows us to sequentially arrange the eigenstates linked by a creation operator a +, with each consecutive energy difference being ¯hω Consequently, we can assign an integer number n to each eigenstate.

Sincea, a + are Hermitian conjugate operators, we may also write n+ 1|a + |n=n|a|n+ 1 ∗ (3.33) Finally, we expand the expectation value of (3.31):

This is a ﬁnite diﬀerence equation in y n =|n+ 1|a + |n| 2 , of the type 1 y n −y n−1 Its solutions are

In the equation |n+ 1|a + |n| 2 = n + c, where c is a constant, the left-hand side is positive definite, necessitating a lower limit for the quantum number n, which we set to n = 0, corresponding to the ground state ϕ0 This leads to the conclusion that the matrix element 0|a + |−1 must vanish, determining the constant c as 1 Consequently, from the relation −1|a|0 = 0, it follows that aϕ0 = 0, indicating that the ground state is annihilated by the annihilation operator a.

The whole set of orthogonal eigenstates may be constructed by repeatedly applying the operatora + , the creation operator. ϕ n = 1

These states are labeled with the quantum numbern They are eigenstates of the operator ˆn=a + a, the number operator, with eigenvaluesn: ˆ nϕ n = 1

The factor 1/√ n! ensures the normalization of the eigenstates.

In order to ﬁnd the matrix elements of the operators ˆx and ˆp, we invert the deﬁnition in (3.29): ˆ x ¯h

, (3.39) and obtain the nonvanishing matrix elements n+ 1|x|n=n|x|n+ 1 ¯h

Fig 3.2 Harmonic oscillator potential and its eigenvalues All energies are given in units of ¯hω The dimensionless variableu=x/x chas been used

Substitution of (3.39) into the Hamiltonian yields

, (3.42) where the operator ˆnhas the quantum number n(=0,1,2, ) as eigenvalues. The Hamiltonian matrix is thus diagonal, with eigenvalues E n represented in Fig 3.2 n|H|n=E n = ¯hω n+1 2

Creation and annihilation operators play a crucial role in many-body quantum physics and quantum field theory These operators facilitate the representation of the creation and annihilation of various particles, including phonons, photons, and mesons, making them essential tools in advanced quantum mechanics.

Quantum mechanics offers a clear derivation of matrix element properties through the fundamental commutation relation, applicable to any scenario involving two operators that meet this relation, particularly when the Hamiltonian is quadratic in these operators.

Some Properties of the Solution

In the following we use this exact, analytical solution of the harmonic oscillator problem to deduce some relevant features of quantum mechanics 4 The

While many examples in these notes may suggest that most quantum problems are analytically solvable, it is important to recognize that this is not the case Most quantum challenges necessitate a deep understanding of physics for approximation and often require substantial computational resources Further exploration of the spatial aspects of the harmonic oscillator problem will be addressed in Section 4.2.

The classical equilibrium position of x=p=0 contradicts the uncertainty principle, as it suggests that both coordinate and momentum can be precisely determined at the same time By substituting ∆x in equation (3.26) with the expression in (3.27), we arrive at the concept of zero-point energy as described in equation (3.43).

The minimum energy of a harmonic oscillator is quantified as 2¯hω, a concept rooted in quantum mechanics that predates its formal establishment In 1924, Roger Mulliken demonstrated that incorporating this minimum energy leads to improved alignment with experimental data comparing the vibrational spectra of molecules composed of different isotopes of the same element The concept of zero-point energy has diverse applications, including elucidating the intermolecular Van der Waals forces and exploring the significant implications of electromagnetic vacuum effects as represented by the ground state of infinite harmonic oscillators.

• By using the closure property (2.47) and the matrix elements (3.40) and (3.41), one obtains the matrix element of the commutator [ˆx,p]:ˆ n|[x, p]|m=n|x|n+ 1n+ 1|p|m+n|x|n−1n−1|p|m

The matrix elements of the operators ˆx 2 and ˆp 2 may be constructed in a similar way:

2 , (3.46) which implies the equality between the kinetic energy and potential expectation values (virial theorem).

Applying the deﬁnition of the root mean square deviation ∆Q given in (2.21), the product ∆x∆pyields

The uncertainty principle highlights the fundamental relationship between the commutation relations of two operators and the resulting uncertainties in measuring their associated physical quantities This connection underscores the inherent limitations in precision when assessing these measurements.

5The procedure is only expected to yield correct orders of magnitude It is a peculiarity of the harmonic oscillator that the results are exact.

In quantum mechanics, the invariance under the parity transformation (x → −x) is significant, as it indicates that both kinetic energy and harmonic oscillator potential energy remain unchanged by this transformation This principle is encapsulated in the commutation relation, highlighting the fundamental symmetry in quantum systems.

The relationship between the operators ˆH and ˆΠ allows for the simultaneous determination of their eigenvalues In this context, the eigenstates of the harmonic oscillator Hamiltonian also serve as eigenstates of the parity operator ˆΠ The eigenvalues of ˆΠ are defined by the requirement that ˆΠ² must equal the single eigenvalue π² = 1, indicating that the system remains unchanged after two applications of the parity transformation Consequently, the operator ˆΠ has two eigenvalues: π = ±1 The associated eigenfunctions are classified as either even functions, which remain invariant under the parity transformation (π = 1), or odd functions, which change sign (π = -1) This property is exemplified in the harmonic oscillator, where the operators a⁺ and a change sign under the parity transformation, resulting in the parity of the state labeled by the quantum number n being represented as Πˆϕₙ = (−1)ⁿϕₙ.

1 Find the eigenvalues and verify the conservation of the trace after diagonalization.

2 Find the eigenvector corresponding to each eigenvalue.

3 Check the orthogonality of states corresponding to diﬀerent eigenvalues.

4 Construct the unitary transformation from the basic set of states used in (3.5) to the eigenstates of this matrix.

1 Calculate the eigenvalues as a function of the real numbers a, c.

2 Show that the odd terms in c vanish in an expansion in powers of c

3 Show that the linear term does not disappear if|c| |a|.

Problem 3.Which of the following vector states are linearly independent? ϕ 1 i

Problem 4.Consider the two operators

2 Determine whether or not the operators commute.

3 If so, obtain the simultaneous eigenvectors of both operators.

Problem 5.Consider a unit vector with components cosβ and sinβ along thez- andx-axes, respectively The matrix representing the spin operator in this direction is written as ˆS β = ˆS z cosβ+ ˆS x sinβ.

1 Find the eigenvalues of ˆS β using symmetry properties.

3 Find the amplitudes of the new eigenstates in a basis for which the operator ˆS z is diagonal.

Problem 6.Ifaanda + are the annihilation and creation operators deﬁned in (3.29), show that [a,(a + ) n ] =n(a + ) (n−1)

1 Calculate the energy of a particle subject to the potential V(x) =V 0+ cˆx 2 /2 if the particle is in the third excited state.

2 Calculate the energy eigenvalues for a particle moving in the potential

1 Express the distance x c as a function of the mass M and the restoring parameterc used in Problem 7.

2 If cis multiplied by 9, what is the separation between consecutive eigenvalues?

3 Show thatx c is the maximum displacement of a classical particle moving in a harmonic oscillator potential with an energy of ¯hω/2.

Problem 9.Evaluate the matrix elements n+η|x 2 |n and n+η|p 2 |n in the harmonic oscillator basis, forη= 1,2,3,4:

1 Using the closure property and the matrix elements (3.41),

2 Applying the operators ˆx 2 and ˆp 2 , expressed in terms of thea + , a, on the eigenstates (3.37).

3 Find the ration+ν|K|n/n+ν|V|n(ν = 0,±2) between the kinetic and the potential energy matrix elements Justify the diﬀerences in sign on quantum mechanical grounds.

Problem 10.Calculate the expectation value of the coordinate operator for a linear combination of harmonic oscillator states with the same parity.

1 Construct the normalized, linear combination of harmonic oscillator states Ψ =c 0ϕ 0 +c 1ϕ 1 for which the expectation value Ψ|x|Ψ becomes maxi- mized.

2 Evaluate in such a state the expectation values of the coordinate, the momentum and the parity operators.

In certain chemical bonds, nature utilizes the property of electrons extending from atoms in a manner akin to the linear combination known as Ψ This phenomenon is referred to as hybridization.

Problem 12.Verify the normalization of the states (3.37).

The Schr¨ odinger Realization of Quantum Mechanics

Section 4.1 presents the fundamental principles of quantum mechanics through position wave functions, leading to the derivation of the time-independent Schrödinger equation, which explicitly reveals the spatial dimensions involved in quantum problems.

In Section 4.2, the harmonic oscillator problem is revisited, allowing readers to compare two different interpretations of quantum mechanics by examining the results from this section alongside those discussed in Section 3.2.

In Section 4.3, we explore solutions to the Schrödinger equation without external forces, highlighting normalization challenges These issues are addressed by examining particles in an infinitely deep square well or along a large circumference, as discussed in Section 4.4.1 The practical applications of these solutions are illustrated through scenarios such as the step potential and square barrier, detailed in Sections 4.5.1 and 4.5.2, which serve as simplified models for scattering experiments Additionally, free-particle solutions are utilized in the context of the finite square well bound-state problem (Section 4.4.2), periodic potentials (Section 4.6), and the tunneling microscope application (Section 4.5.3).

Time-Independent Schr¨ odinger Equation

Probabilistic Interpretation of Wave Functions

Information may be extracted from the wave function through the probability density (2.59) [27]: ρ(x) =|Ψ(x)| 2 (4.11)

The probability of ﬁnding the particle in the intervalL 1 ≤x≤L 2is given by the integral L 2

In particular, the probability of ﬁnding the particle anywhere must equal one:

1 = Ψ|Ψ, (4.13) which implies that the wave function should be normalized.

We now discuss how this probability changes with time t We therefore allow for a time dependence of the wave function 3 [Ψ = Ψ(x, t)]: d dt

According to the time-dependent Schrödinger equation, we can substitute i¯hΨ with ˆ˙ HΨ, leaving us with only the kinetic energy contribution, as the potential terms cancel out.

3The time dependence of the wave function is discussed in Chap 9 We anticipate the result here because the notion of probability current is needed in the next few sections.

∂x = 0, (4.16) where we have deﬁned the probability current j(x, t)≡ − i¯h

Equation (4.16) represents a continuity equation akin to those utilized in hydrodynamics for mass conservation Visualize a long prism aligned along the x-axis, defined by two square areas, A, at positions x = L1 and x = L2 This framework allows us to examine the changes in the probability of locating a particle within the prism.

L 1 ρdx , is equal to the diﬀerence between the ﬂuxes leaving and entering the prism, viz.,A[j(L 2)−j(L 1)] (see Fig 4.1).

The probability density and probability current provide spatial dimensions to the Schrödinger interpretation of quantum mechanics, which is particularly valuable in chemistry In this context, the distribution of electron density within atoms correlates with the increasing chemical affinities of elements.

The expression for the probability current underscores the need to use complex state vectors in quantum mechanics, since the current vanishes for real wave functions.

Here we may continue the list of misconceptions that prevail in quantum mechanics [20]:

The probability current j(x) indicates the velocity of a particle at position x, which may misleadingly suggest that particles consist of distinct parts with specific positions and momenta However, it is important to clarify that particles are not composed of individual segments.

• “For any energy eigenstate, the probability density must have the same symmetry as the Hamiltonian.” This statement is correct in the case of j 2 j 1 dρ /dt

Fig 4.1 Conservation of probability density The rate of change within a certain interval is given by the ﬂux diﬀerences at the boundaries of the interval

The Harmonic Oscillator Revisited

Solution of the Schr¨ odinger Equation

Rewriting equations using dimensionless coordinates is beneficial as it eliminates cumbersome constants and allows solutions to be applicable to a broader range of cases In this context, the coordinate \( x \) and energy \( E \) are normalized by their respective characteristic values, leading to the definitions \( u = \frac{x}{x_c} \) and \( e = \frac{E}{\hbar \omega} \).

The Schr¨odinger equation thus simpliﬁed reads

2u 2 ϕ n (u) =e n ϕ n (u) (4.21) This equation must be supplemented with the boundary conditions ϕ n (±∞) = 0 (4.22)

The eigenfunctions and eigenvalues are of the form ϕ n (x) =N n exp

TheH n are Hermite polynomials 4 of degreen= 0,1,2, The eigenfunctions and eigenvalues are also labeled by the quantum number n Up to a phase, the constantsN n are obtained from the normalization condition (4.13)

Since the Hamiltonian is a Hermitian operator, the eigenfunctions are orthogonal to each other and constitute a complete set of states: n|m ∞

The solutions corresponding to the lower quantum numbers are displayed in Table 4.1 and Fig 4.2.

Table 4.1 Solutions to the harmonic oscillator problem for small values ofn.P n is deﬁned in (4.28) n e n H n N n π 1/4 x 1/2 c P n (%)

4The reader is encouraged to verify that the few cases listed in Table 4.1 are correct solutions Use can be made of the integrals

The figure illustrates the probability densities of a harmonic oscillator potential, comparing quantum mechanical (continuous lines) and classical (dashed lines) representations These densities are plotted against the dimensionless distance, u, for various quantum numbers n, specifically 0, 1, 2, and 5 Additionally, vertical lines indicate the classical amplitudes, xn, highlighting the differences between quantum and classical behavior in harmonic oscillators.

Spatial Features of the Solutions

The following features arise from the spatial dimension associated with the Schr¨odinger formulation:

Probability density in quantum mechanics reveals the presence of nodes, particularly in states other than n = 0, which challenges the classical idea of a particle following a continuous trajectory Unlike classical particles that traverse every point in a potential, quantum particles cannot be located at these nodes This phenomenon parallels stationary wave patterns found in organ pipes, where the ends of the pipe dictate sound behavior, similar to how boundary conditions influence the quantum harmonic oscillator.

In comparing classical and quantum mechanical probability densities, it is essential to note that the classical probability of locating a particle is inversely related to its speed, expressed as \( v = \sqrt{\frac{2E}{M} - \omega^2 x^2} \) for a harmonic oscillator Consequently, we can define the classical probability density as \( P_{\text{clas}} = \frac{\omega}{\pi v} \).

The probability of locating a particle within the classically allowed interval of −x_n to x_n is 100% Here, x_n is defined as x_c(2n + 1)^(1/2) for a particle with energy ¯hω(n + 1/2) In classical mechanics, the probability density exhibits a minimum near the origin and increases as the particle nears the interval's boundaries Conversely, the quantum mechanical density distribution for the ground state shows an opposite trend However, as the quantum number n increases, the quantum density distribution approaches the classical limit.

The tunnel effect occurs when a particle is detected within the interval \( x_c \leq x \leq \sqrt{} \), despite classical physics suggesting it would possess negative kinetic energy and imaginary momentum outside this range This phenomenon challenges classical assumptions, as it implies a precise determination of both the particle's location and momentum, which contradicts the uncertainty principle inherent in quantum mechanics.

In the region beyond the classically allowed area, specifically within the interval defined by 2x c, the probability density diminishes from N 0 2 /e at x = x c to N 0 2 /e², representing a decrease by a factor of e⁻¹ This measurement of the particle within this interval serves as a reliable indicator of the uncertainty in its position.

According to the Heisenberg principle, the minimum uncertainty in the determination of the momentum is

The relationship ∆p≥1.22√ ¯ hωM indicates that the uncertainty in kinetic energy exceeds (∆p)²/2M ≥ 3/4 ¯hω Additionally, as the potential energy rises from ¯hω/2 to ¯hω within the same interval, it remains inconclusive regarding the potential for an imaginary momentum value, thereby leaving open the possibility of the particle's penetration into the classically forbidden region.

• The probability of ﬁnding the particle in the classically forbidden region is

5This is a manifestation of the correspondence principle, which was extensively used by Bohr in the old quantum theory.

Free Particle

The probability for the ground state reaches a maximum of 16%, but this value diminishes as the quantum number increases, aligning with the trend towards classical behavior observed at higher energy levels (refer to Table 4.1).

If there are no forces acting on the particle, the potential is constant:

V(x) =V 0 Let us assume in the ﬁrst place that the energy E≥V 0 In such a case the Schr¨odinger equation reads

2M d 2 ϕ k (x) dx 2 = (E−V 0 )ϕ k (x) (4.29) There are two independent solutions to this equation, namely ϕ ± k (x) =Aexp(±ikx), k 2M(E−V 0) ¯ h (4.30)

The parameter k labeling the eigenfunction is called the wave number and has dimensions of a reciprocal length The eigenvalues of the energy and the momentum are

In contrast to the harmonic oscillator, where eigenvalues are discrete, free particles exhibit continuous eigenvalues for both momentum and energy The de Broglie relation, p = ħk = h/λ, links momentum with the particle's wavelength The probability density remains constant throughout space, represented by ρ ± k (x) = |A|², aligning with the Heisenberg uncertainty principle, which states that precise momentum determination (∆p = 0) results in complete position uncertainty (∆x = ∞) Additionally, the probability current is expressed as j ± k (x) = -iħ.

These results pose normalization problems, which may be:

• Solved by applying more advanced mathematical tools

• Taken care of through the use of tricks, as in Sect 4.4.1

• Circumvented, by looking only at the ratios of the probabilities of ﬁnding the particle in diﬀerent regions of space (Sects 4.5.1 and 4.5.2)

Fig 4.3 Real component, imaginary component, and modulus squared of a plane wave as functions of the dimensionless variable u = kx, where u is measured in radians

Since there are two degenerate solutions, 6 the most general solution for a given energyE is a linear combination Ψ(x) =A + exp(ikx) +A − exp(−ikx) (4.35)

In the scenario where the energy E is less than or equal to the potential V₀, classical mechanics offers no clear interpretation However, the harmonic oscillator problem discussed in Section 4.2.2 suggests that we should not dismiss this quantum situation The general solution for this case is expressed as a linear combination: Ψ(x) = B₊ exp(κx) + B₋ exp(−κx), where κ is defined as κ = −ik √(2M(V₀ − E)/ℏ).

The general solution exhibits divergence at infinity, indicated by |Ψ| approaching infinity as x approaches ±∞ This characteristic suggests that the solution (4.36) is applicable only when at least one extreme is unattainable For example, if the potential V 0 exceeds the energy E for x greater than a, it is necessary to set B to zero.

6Two or more solutions are called degenerate if they are linearly independent and have the same energy.

7The only diﬀerence between the two solutions (4.35) and (4.36) is whether k is real or imaginary.

One-Dimensional Bound Problems

Inﬁnite Square Well Potential Electron Gas

The potential in this case isV(x) = 0 if|x| ≤a/2 andV(x) =∞for|x| ≥a/2

To address the infinite discontinuities in the Schrödinger equation, it is necessary to ensure that the wave function is continuous and that the first derivative exhibits a finite discontinuity at the boundaries of the potential This leads to the condition Ψ(±a/2) = 0, as the wave function must vanish outside the classically allowed interval Additionally, the eigenfunctions of the Hamiltonian can be assigned definite parity by utilizing solutions where A+ = A− for even-parity states and A+ = −A− for odd-parity states Consequently, the eigenfunctions are expressed as ϕ even n (x) = 2A cos(k n x) and ϕ odd n (x) = 2A sin(k n x) within the well, while remaining zero outside, ultimately satisfying the boundary conditions.

2,2, , (4.38) where the half-integer values correspond to the even solutions and the integer values to the odd ones The eigenvalues of the energy are

Fig 4.4.Inﬁnite square well potential The energiesE n (continuous lines) and wave functionsϕ n (x) (dotted curves) are represented for the quantum numbersn= 1, 2,and 3

Readers should verify that the quantum characteristics linked to the harmonic oscillator problem are also reflected in the infinite square well scenario, as discussed in Sections 3.2.2 and 4.2.2 However, it is important to note that the tunnel effect is not applicable in this case due to the infinite discontinuity in the potential.

Increasing the size of the box allows the infinite potential well to effectively model the potential binding electrons in metals In the one-dimensional electron gas model, noninteracting electrons are confined to a large segment that significantly exceeds the dimensions of typical experimental setups.

Standing waves are not ideal for analyzing charge and energy transport by electrons, as their associated probability current is zero In metal theory, running waves, represented as exp(±ikx), are preferred By conceptualizing the endpoint at x=a/2 as connected to x=−a/2, the segment effectively becomes a circumference of length a Consequently, an electron that reaches the end of the well exits the metal and re-enters at the opposite end This model becomes more accurate as the radius a/2π increases, leading to the boundary condition Ψ(x) = Ψ(x+a), or k n a.

The total number of states matches that of standing waves, as the half-integer and integer numbers are balanced by the presence of two degenerate states, ±n The eigenfunctions and energy eigenvalues are represented as ϕ n (x) = 1.

As mentioned in Sect 4.3, these functions are also eigenfunctions of the momentum operator ˆp with eigenvalues ¯hk n Although the momenta (and the energies) are discretized, the gap

∆k= 2π/a (4.42) between two consecutive eigenvalues becomes smaller than any prescribed interval, if the radius of the circumference is taken to be suﬃciently large.

In quantum mechanics, the use of sums over intermediate states is common For wave functions of the type (4.41), this process can be simplified by converting the sums into integrals This transformation utilizes the length element in the integrals, represented as (a/2π)dk, as indicated in equation (4.42).

In Section 4.6, the model is extended to incorporate a periodic crystal structure, while Section 7.4.1 presents calculations using the electron gas model for the three-dimensional scenario.

Finite Square Well Potential

The potential reads V(x) = −V 0 < 0 if |x| < a/2 and V(x) = 0 for other values of x Here we consider only bound states, with a negative energy

In the case of the harmonic oscillator, the potential remains unchanged under the parity transformation \( x \rightarrow -x \), leading us to anticipate that the eigenfunctions will exhibit either even or odd symmetry Consequently, the solution provided in equation (4.35) is valid within the region where \( |x| \leq a/2 \).

Invariance under the parity transformation simplifies the calculation of boundary conditions to the position x = a/2 The wave function for the region to the right of this point is described by (4.36), with B + set to 0 and κ defined as (1/¯h)(2M E) 1/2.

To ensure the Schrödinger equation is valid at all points in space, both the wave function and its first derivative must be continuous, even at points with finite potential discontinuities The relationship between the continuity conditions for the wave function and its first derivative leads to the eigenvalue equation κ k = tanka.

In this analysis, we reach a limit where Equation (4.44) requires numerical solutions or a graphical method for resolution, as illustrated in Fig 4.5 The equation that establishes the value of k is equivalent to the stated equation.

E=V 0 −¯h 2 k 2 /2M Therefore, we obtain the ratio κ k 2M V 0 ¯ h 2 k 2 −1, (4.45) and (4.44) becomes

2¯h 2 θ 2 −1 = tanθ , (4.46) where θ ≡ ka/2 The function tanθ increases from zero to inﬁnity in the interval 0 ≤ θ ≤ π/2, while the left-hand side decreases from inﬁnity to a

As the value of θ increases within a specified interval, a finite value is reached where two curves intersect, indicating the lowest eigenvalue A similar reasoning applies to the subsequent roots of the equation The nth root is determined within the interval ranging from (n−1)π to (n−1/2)π.

Fig 4.5 Graphical determination of the energy eigenvalues of a ﬁnite square well potential The intersections of the continuous curve

(13/θ) 2 −1 with the dashed curvescorrespond to even-parity solutions, while those with thedotted curves correspond to the odd ones The valueM V 0 a 2 /¯h 2 = 338 is assumed

Unlike the harmonic oscillator case, the number of roots is limited, since (4.46) requires θ≤θ max , where θ max M V 0 a 2

There is a set of odd solutions that satisfy an equation similar to (4.44), namely

Unlike the classical case, the probability density is not constant in the interval

|x| ≤a/2 Moreover, there is a ﬁnite probability of ﬁnding the particle outside the classically allowed region However, the solutions tend toward the classical behavior as nincreases.

The spectrum of normalizable (bound) states is always discrete, while states with finite amplitude at infinity belong to a continuous spectrum, particularly for positive energy values.

One-Dimensional Unbound Problems

One-Step Potential

The one-step potential is defined as V(x) = 0 for x < 0 and V(x) = V0 > 0 for x > 0, illustrating an electron's movement along a conducting wire interrupted by a short gap As the electron crosses this gap, it experiences a change in potential.

In classical mechanics, a particle rebounds at x=0 and cannot enter the region x≥0 However, in quantum mechanics, this restriction is lifted For x≤0, the solution is represented as a superposition of an incoming wave and a reflected wave.

V 0 = 0 Equation (4.36) holds forx≥0 This last solution cannot be rejected, since it does not diverge on the right half-axis if we impose the boundary conditionB + = 0.

The two continuity requirements imply that

Fig 4.6.One-step potential Subscriptsaandblabel wave functions corresponding to energiesE a = 3V 0 /4 andE b = 5V 0 /4, respectively

The total wave function is given by Ψ a (x) = 1

For the condition x ≤ 0, the solution illustrates the superposition of an incident wave and a reflected wave, resulting in a standing wave due to equal amplitudes This standing wave features nodes positioned according to the relationship tan(kx) = k/κ for x ≤ 0 Additionally, the probability currents of the incident and reflected waves are represented as j I = −j R = ¯hk.

The reflection coefficient is defined as the absolute value of the ratio between reflected and incident currents In the present case,

The mutual cancellation between the two probability currents is correlated with the real character of the wave function (4.51).

The tunneling effect occurs for x ≥ 0, allowing particles to penetrate into the forbidden region over a distance approximately equal to ∆x = 1/κ This penetration is associated with uncertainties in both momentum and kinetic energy.

2M ≈V 0 −E a , (4.54) respectively The consequences of these uncertainties parallel those discussed in Sect 4.2.2.

The classical solution illustrates an incident particle that is fully transmitted but at a reduced velocity In quantum mechanics, for positions where x ≤ 0, the solution is represented by (4.35) with V0 = 0, indicating the presence of both an incident and a reflected wave For x ≥ 0, this solution remains applicable, but the wave number is defined as kb = √(2M(Eb − V0)/ħ) Notably, there is no incident wave approaching from the right, as there is no potential barrier to reflect the particle.

4.5 One-Dimensional Unbound Problems 59 the amplitude of the transmitted wave exp(ik b x) The continuity of the wave function and its ﬁrst derivative atx= 0 requires that

Using these equations, we may express the amplitudes of the reﬂected and transmitted waves as proportional to the amplitude of the incident wave, so that Ψ b (x) ⎧⎪

The probability currents associated with the incident, reﬂected, and transmitted waves are j I = ¯hk

|A + | 2 , respectively In this case we also deﬁne a transmission coeﬃcientT ≡j T /j I

(k+k b ) 2 , (4.58) and we ﬁnd thatR+T = 1 as expected, since the current should be conserved in the present case.

The phenomenon of particle bounce can be likened to a beam of light transitioning between two media with varying indices of refraction, where at least some degree of reflection occurs.

Note that the wave functions (4.51) and (4.56) may be obtained from each other through the substitutionk b (E) = iκ(E).

Square Barrier

The potential is given by V(x) = 0 (|x| > a/2) and V(x) =V 0 (|x| < a/2)

(Fig 4.7) We only consider explicitly the caseE≤V 0 Classically, the particle can only be reﬂected atx=−a/2.

Forx≤ −a/2 and for x≥a/2, the solution to the Schr¨odinger equation again takes the form (4.35), with the same value ofkfor both regions (V 0= 0).

Fig 4.7.Square barrier and associated wave function HereE= 3V 0 /4

For regions where \( x \geq a/2 \), only the transmitted wave \( C e^{ikx} \) exists, while in the intermediate region \( -a/2 \leq x \leq a/2 \), the solution follows equation (4.36) In this scenario, we cannot dismiss either wave component due to their divergent behavior at infinity, resulting in five amplitudes The continuity conditions at the boundaries yield four equations, allowing us to express the remaining amplitudes in terms of the incident wave amplitude \( A^+ \) Additionally, we can calculate the currents for the incident beam \( j_I \), reflected beam \( j_R \), transmitted beam \( j_T \), and the beam within the barrier \( j_B \), along with the reflection and transmission coefficients \( R \) and \( T \), where \( j_I = \bar{h}k \).

For values ofκa >1 the transmission coeﬃcient displays an exponential decay

The tunnel effect, a key phenomenon in quantum mechanics, is evident in the transmission through potential barriers, as explored in the context of harmonic oscillators and one-step potentials This effect plays a significant role in processes such as alpha decay of nuclei and the functioning of tunneling microscopes.

The analysis of a potential well is similar to that of a square barrier, requiring the use of a different solution for the region inside the well This scenario involves incident, reflected, and transmitted waves, with reflection and transmission coefficients that total one.

Scanning Tunneling Microscope

The scanning tunneling microscope (STM), developed in the 1980s by Gerd Binnig and Heinrich Rohrer, utilizes a sharp conducting probe positioned close to a metal sample In this setup, electrons move freely within the metal, following the electron gas model and filling levels up to the Fermi energy A potential barrier forms at the metal's surface, enabling electrons to tunnel between the probe tip and the sample The tunneling current is proportional to the transmission coefficient, increasing exponentially as the distance between the tip and surface decreases The probe tip is mounted on a piezoelectric tube, which facilitates precise movements by applying voltage, allowing for a detailed scan of the surface.

An effective value of κ can be estimated by substituting the difference V₀ - E in equation (4.36) with the average of the sample and tip work functions, W, which represents the minimum energy required to remove an electron from a metal, approximately 4 eV Consequently, this leads to an estimation of κ at around 2 Å⁻¹, indicating the relationship between the tip potential and the work function barrier.

To ensure the presence of vacant electron states in the sample for tunneling electrons to occupy, a small voltage difference (V ts) must be applied between the tip and the sample This setup allows the device to detect changes in distance at the sub-angstrom scale, highlighting its sensitivity and precision.

The Scanning Tunneling Microscope (STM) is utilized in both industrial applications and fundamental research to capture atomic-scale images of metal surfaces and various materials Additionally, it can generate minuscule tunnel currents, particularly when biological materials are applied as thin films on conductive substrates.

A crystal is composed of a periodic arrangement of N positive ions, with electrons navigating the electric field created by these ions The potential experienced by an electron in a one-dimensional crystal is illustrated in Figure 4.9, showing that V(x+d) = V(x) This section focuses on the key characteristics of single-particle eigenstates within this potential.

In classical mechanics, an electron can be bound to a single ion, preventing its transfer to another ion However, in quantum mechanics, this binding is only guaranteed when the distance between the ions is significantly large.

In scenarios where an electron is bound to a single atom within an array, the N states form an orthogonal set that is N times degenerate However, as the distance between atoms decreases to realistic levels, this degeneracy is anticipated to break, leading to the distribution of energy eigenvalues across a band.

The Bloch theorem mathematically represents the wave function of a particle in a periodic potential as ϕ k (x) = exp(ikx)u k (x), where k is a real number and u k (x+d) = u k (x) indicates that u k (x) is a periodic function.

Fig 4.9 Periodic potential The upper part of the ﬁgure represents a realistic potential The lower one mocks this potential as successive square wells

To simplify the analysis of potential energy, we replace the realistic potential with a periodic array of square well potentials, emphasizing its periodic nature rather than its specific shape In this model, the potential V(x) is defined as V(x) = -V0 for 0 ≤ x ≤ b and V(x) = -V1 for b ≤ x ≤ d, allowing us to apply our knowledge of solving square well problems effectively.

Moreover, V(x+d) =V(x) We denote the energy by−E, withE >0 We assume that the electron is bound to the crystal for negative energy values.

According to Sect 4.3, the wave functions in the interval nd≤x≤(n+ 1)d are Ψ(x) "

A + exp(ik b x) +A − exp(−ik b x), nd≤x≤nd+b ,

2M(E−V 1) (4.63) Thus the periodic function u k (x) is of the form u k (x) ⎧⎨

B +exp[(κ b −ik)x] +B − exp[−(κ b + ik)x], nd+b≤x≤(n+ 1)d

(4.64) The periodicity ofu k requires u k (x) =A +exp[i(k b −k)(x−d)] +A − exp[−i(k b +k)(x−d)], (4.65) for

The continuity conditions for the wave function result in four linear equations for the amplitudes A ± and B ± at the boundaries x = b and x = d For the system to be solvable, the determinant of the coefficients of these amplitudes must equal zero This requirement leads to the equation f(E) = cos(kd), where f(E) is defined as κ²b - k b².

The procedure is completely parallel to the previous case except for the fact that the wave function is also of the form (4.35) in the interatomic space b≤x≤d, withk c = (1/¯h)

The allowed values of E fall into bands satisfying the condition |f(E)| ≤ 1. Figure 4.10 represents the function f(E), encompassing the two regions

I and II, for the parametersV 1 =V 0 /2,d= 4bandb= ¯h

2/M V 0 Equation (4.66) remains unchanged ifkis increased by a multiple of 2π/d.

We therefore conﬁnek to the interval

We now apply the periodic boundary conditions discussed in Sect 4.4.1 The length of the circumference isa=N d Therefore, exp(ik n N d) = 1, k n =2πn

The equation 2N (4.70) incorporates the constraints outlined in (4.69), indicating that the number of possible values for k corresponds to the number of ions present in the array This finding aligns with the notion that associating the electron with each ion represents a viable solution to the problem discussed earlier in this section.

Fig 4.10.Available intervals of energy (bands), obtained with the periodic square potential of Fig 4.9

Problem 1.Using Table 4.1, verify that:

1 The operator a(3.29) annihilates the ground state wave functionϕ 0

2 The operator a + , applied toϕ 1 (x) yields√

2ϕ 2 (x) Hint: Express the operatorsa, a + as diﬀerential operators.

Problem 2.Assume an inﬁnite square well such thatV(x) = 0 in the interval

0< x < aandV(x) =∞for the remaining values ofx:

1 Calculate the energies and wave functions.

2 Compare these results with those obtained in the text centering the well at the origin and explain the agreement on physical grounds.

3 Do the wave functions obtained in the ﬁrst part have a deﬁnite parity?

Problem 3.Relate the minimum energy for a particle moving in a square well to the Heisenberg uncertainty principle.

Problem 4.Find the eigenvalue equations for a particle moving in a potential well such thatV(x) =∞for|x| ≥a/2,V(x) =V 0 ≥0 for−a/2≤x≤0, and

Problem 5.Estimate the error if we use (4.43) in the calculation of k E k Hint: Recall that n=ν n=0 n 2 =ν

Problem 6.Assume a free electron gas conﬁned to a one-dimensional well of lengtha:

1 Obtain the density of statesρ(E) as a function of energy.

2 CalculateρforE= 1 eV anda= 1 cm

Problem 7.Consider a square well such thatV(x) =∞forx a/2:

1 Write down the equation for the eigenvalues.

2 Compare this equation with the one obtained for the ﬁnite square well in Sect 4.4.2.

3 For V 0 → ∞, show that the wave function for the ﬁnite well satisﬁes the condition that it vanishes atx=a/2 and does not penetrate the classically forbidden region.

Problem 8.Calculate the number of even-parity states (EPS) and odd-parity states (OPS) for a potential of depthV 0centered at the origin, if the parameter θ= a ¯ h

2 lies in the intervals: (0, π/2), (0, π), (0,3π/2), and (0,2π).

Problem 9.Calculate the transmission and reﬂection coeﬃcients for an electron with a kinetic energy of E= 2 eV coming from the right The potential isV(x) = 0 forx≤0 andV(x) =V 0 = 1 eV forx≥0.

Problem 10.The highest energy of an electron inside a block of metal is

5 eV (Fermi energy) The additional energy that is necessary to remove the electron from the metal is 3 eV (work function):

1 Estimate the distance through which the electron penetrates the barrier, assuming that the width of the (square) barrier is much greater than the penetration distance.

2 Estimate the transmission coeﬃcient if the width of the barrier is 20 ˚A.

Problem 11.Obtain the transmission coeﬃcients of a potential barrier in the limits κa1 andκa1.

Problem 12.Estimate the sensitivity to the distance tip–sample in a STM, assuming that a relative variation of 1% in the current can be detected and κ= 2 ˚A −1

1 Show that the eigenfunction of the Hamiltonian of a periodic potential is not an eigenfunction of the momentum operator.

2 Why is it not a momentum eigenstate?

3 Give an expression for the expectation value of the momentum.

Problem 14.In the presence of interactions, it is sometimes useful to mock the spectrum by the one of a free particle (4.31) with an eﬀective mass Obtain the value of M eﬀ at the extremes of the intervals allowed by (4.66).

Hint: Expand both sides of (4.66) and add the resulting expression ∆E(k 2 ) to the kinetic energy (4.31).

Problem 15.A linear combination Ψ(x) of momentum eigenstates (4.41) representing a localized particle is called a wave packet Choose as amplitudes c p =ηexp[−p 2 /α 2 ]

1 Obtain the value ofη such that the normalization condition p |c p | 2 = 1 is satisﬁed.

3 Obtain the matrix elements Ψ|x|Ψ and Ψ|x 2 |Ψ

4 Obtain the matrix elements Ψ|p|Ψ and Ψ|p 2 |Ψ

Hint: Replace sums by integrals as in (4.43).

This chapter, along with Chapter 6, focuses on single-particle problems in three-dimensional space Chapter 5 explores quantum angular momentum from both matrix and differential equation perspectives, highlighting the differences in results that underscore the significance of spin as a fundamental quantum observable Additionally, the chapter addresses the addition of angular momenta.

The commutation relation (2.16) is straightforwardly generalized to the three-dimensional case:

In classical physics, angular momentum is a physical, observable vector L that plays an important role, since it is a conserved quantity in the absence of external torques τ :

As in the case of the Schr¨odinger equation, we quantize the problem by substituting ˆ p i → −i¯h ∂

∂x i (5.3) into the classical expression (5.2) One obtains the commutation relations [ ˆL x ,Lˆ y ] = i¯hLˆ z , [ ˆL y ,Lˆ z ] = i¯hLˆ x , [ ˆL z ,Lˆ x ] = i¯hLˆ y , (5.4)

Eigenvalues and Eigenstates

Matrix Treatment

In this article, we define quantum angular momentum through the commutation relations (5.4), which also encompass the quantum version of orbital angular momentum (5.2) Additionally, these relations incorporate other forms of angular momentum that arise solely from quantum mechanics We denote the operator components as ˆJ i, which adhere to these established relations.

[ ˆJ i ,Jˆ j ] = i¯h ijk Jˆ k , (5.6) where ijk is the Levi–Civita tensor, 1 whatever their origin may be We use the notation ˆL i for angular momentum operators associated with orbital motion (5.2).

The commutation relations allow for the simultaneous determination of the modulus squared of angular momentum and one projection, specifically ˆJ 2 and ˆJ z, but not two projections at once This leads to the construction of eigenfunctions that are common to these operators The selection of the z-component is arbitrary due to the isotropic nature of space, which means there are no preferred directions in angular momentum.

The procedure for solving this problem closely follows the matrix treatment of the harmonic oscillator (Sect 3.2.1) It is given in detail in Sect 5.4* The following results are obtained:

• The eigenvalue equations for the operators J ˆ 2 and ˆJ z can be written as

J ˆ 2 ϕ jm = ¯h 2 j(j+ 1)ϕ jm , Jˆ z ϕ jm = ¯hmϕ jm , (5.7) where the possible values of the quantum numbersj, mare

2, , (5.8) withmincreasing in units of one.

The maximum value of the quantum number m is j, and since j² is less than j(j + 1), the maximum projection of angular momentum is always less than its total magnitude, except when j equals 0 Consequently, the angular momentum vector cannot be fully aligned with the z-axis This observation aligns with the non-commutativity noted in equation (5.6), as complete alignment would lead to the disappearance of the components Ĵx and Ĵy, resulting in the simultaneous determination of the physical quantities associated with Ĵz.

Figure 5.1 illustrates the potential orientations of the angular momentum vector for j = 5/2, suggesting that angular momentum precesses around the z-axis However, this depiction is misleading, as it implies that the angular momentum vector's endpoint follows a circular path, which contradicts the principles of quantum uncertainty relations.

• The operators ˆJ x and ˆJ y display nondiagonal matrix elements within the basis (5.7), namely,

1 ijk = 1 ifi, j, kare cyclical (as fori=z,j=x,k=y); otherwise ijk =−1 (as fori=z,j=y,k=x).

Fig 5.1 Possible orientations of aj= 5/2 angular momentum vector j m |J x |jm=δ j j δ m (m ± 1) ¯ h

• None of the operators ˆJ x ,Jˆ y ,Jˆ z ,Jˆ 2 connect states with diﬀerent values of the quantum numberj.

• In analogy with (4.7), the unitary operator associated with rotations is

The rotation is defined by the axis of rotation, indicated by the vector direction α, and the size of the rotation angle α The operator ˆJ i acts as the generator of rotations around the i-axis.

Treatment Using Position Wave Functions

Orbital angular momentum is crucial in analyzing systems with spherical symmetry, such as atoms and nuclei In these cases, spherical coordinates are employed, where the coordinates are defined as x = r sin θ cos φ, y = r sin θ sin φ, and z = r cos θ The volume element in spherical coordinates is expressed as dxdydz = r² sin θ dr dθ dφ.

In these coordinates, the orbital angular momentum operators read

The detailed treatment of the orbital angular momentum operator is given in Sect 5.5* The results of such an approach are as follows:

• The simultaneous eigenfunctions of the operators ˆL 2 ,Lˆ z are called spherical harmonics and denoted byY lm l (θ, φ) They satisfy the eigenvalue equations

Lˆ 2 Y lm l = ¯h 2 l(l+ 1)Y lm l , (5.13) ΠYˆ lm l = (−1) l Y lm l , where

−l≤m l ≤l , l= 0,1,2, (5.14) and ˆΠ is the parity operator 2 (3.49).

• Using the expressions (5.12), one may construct the matrix elements of the operators ˆL x ,Lˆ y One obtains the same form as the matrix elements in (5.9), with the replacementj→l,m→m l

• The spherical harmonics constitute a complete set of single-valued basis states on the surface of a sphere of unit radius: Ψ(θ, φ) lm l c lm l Y lm l (5.15)

They can be visualized as vibrational modes of a soap bubble.

• Figure 5.2 displays the projection of some spherical harmonics on the (x, z) plane The protruding shapes have important consequences in the construction of chemical bonds.

2For the three-dimensional case, the parity operation is written as r → −r or equivalently,r→r,θ→π−θ,φ→π+φ.

Spin

Stern–Gerlach Experiment

A particle with a magnetic moment (à) in a magnetic field (B) experiences a torque (τ) When this particle is rotated by an angle (dθ) around the direction of the torque, its potential energy (U) increases The relationship is defined by the equation τ = à × B, and the change in potential energy can be expressed as dU = àBsin(θ)dθ Consequently, the potential energy as a function of the angle is given by U(θ) = -àBcos(θ), indicating that the potential energy decreases with the alignment of the magnetic moment and the magnetic field.

Classical electromagnetism states that an electric current generates a magnetic moment that is proportional to the area enclosed by the current When the current is produced by a charged particle with charge \( e \) and velocity \( v \) moving in a circular path of radius \( r \), the magnetic moment can be expressed as \( \mu = iA = ev \).

Thus the magnetic moment due to the orbital motion is proportional to the orbital angular momentum In vector and operator notation, ˆ à l = e

|e| g l à B ¯ h L ˆ , (5.18) where à B ≡ |e|¯h/2M is called the Bohr magneton (Table 14.1) andg l = 1 is the orbital gyromagnetic ratio.

The presence of a magnetic field causes a shift in a particle's energy, which is directly proportional to the angular momentum component aligned with the magnetic field, known as the Zeeman effect While classical physics describes this change as a continuous function based on angular momentum orientation, quantum mechanics reveals that the projections of angular momentum are quantized.

Therefore, an orbital angular momentum should give rise to an odd number (2l+ 1) of energy eigenstates.

For a uniform magnetic ﬁeld, there is no net force acting on the magnetic dipole However, if the ﬁeld has a gradient in thez-direction, the net force is

Figure 5.3 is a sketch of the experimental set-up used by Stern and Gerlach

Silver atoms are heated in an oven and released through a small opening The resulting beam is collimated and deflected by a perpendicular nonuniform magnetic field Ultimately, a visible deposit forms on a glass plate positioned at a distance from the deflection area.

The nuclear contributions to the magnetic moment can be disregarded since the nuclear magneton is approximately 2,000 times smaller than the Bohr magneton, which factors in the proton mass rather than the electron mass Additionally, 46 out of 47 electrons create a spherically symmetric electron cloud that possesses no net angular momentum, indicating that the total angular momentum of the silver (Ag) atom is attributed solely to the last electron The Stern–Gerlach experiment, illustrated in Fig 5.4, did not yield the expected classical continuous pattern or the orbital quantum mechanical results that would show separation into an odd number of terms; instead, the beam was divided into only two distinct beams, consistent with an angular momentum of j = s = 1/2.

Fig 5.3.Sketch of the Stern–Gerlach experimental arrangement

In a letter to Bohr, Gerlach presented two figures illustrating his experimental findings, noting that the magnetic field was insufficient at the beam's extremes For comparison, the figure on the left was generated without a magnetic field (Reproduced with permission from Niels Bohr Archive, Copenhagen)

Spin is a crucial quantum observable, integral to quantum information systems based on two-state configurations Modern advancements in spin detection and manipulation have significantly evolved since the era of Stern and Gerlach, enabling the handling of individual spins This progress fuels optimism for spintronics, which aims to utilize the spin degree of freedom in electronic circuits, akin to how charge is currently employed.

Spin Formalism

Three years following the Stern–Gerlach experiment, George Uhlenbeck and Samuel Goudsmit introduced a new quantum number to define the state of electrons and various other fundamental particles.

[31] It labels the two projections of the spin, by then a new physical entity representing an intrinsic angular momentum.

Since the spin is a pure quantum observable, only the matrix treatment formalism is possible If s= 1/2, then the basis set of states is given by the two-component vectors (3.16) [32] ϕ (z) ↑ ≡ϕ 1

The following representation of the spin operators reproduces (5.7) and (5.9) forj= 1/2:

4 I, i=x, y, z , (5.22) where theσ i are called the Pauli matrices: σ x 0 1

They all square to the unit matrixI The two-component states satisfy the eigenvalue equation:

In the context of quantum mechanics, the Pauli matrices are utilized to construct the matrix (i|Q|j), with solutions derived as shown in previous equations Each spin possesses a specific magnetic moment, represented by the equation ˆ à s = g s à ν ¯ h S ˆ, where the gyromagnetic ratio varies: g s = 2.00 for electrons, g s = 5.58 for protons, and g s = −3.82 for neutrons The constant à ν represents the negative Bohr magneton for electrons and the nuclear magneton for protons and neutrons, with corresponding values for proton charge and mass The total magnetic moment operator is expressed as ˆ à = à ˆ s + à ˆ l = à ν ¯ h g s S ˆ + g l L ˆ.

Obviously, g l = 0 for neutrons The quantal magnetic moment is not always proportional to the angular momentum.

The eigenstates of the operator ˆS x are derived through a unitary transformation of the eigenvectors of ˆS z, representing a specific instance of a broader transformation that aligns the spin s = 1/2 operator with a spatial direction defined by the angles β and φ The operator ˆS βφ can be expressed as the scalar product of the spin vector ˆS and a unit vector oriented along the selected direction.

Sˆ βφ = sinβcosφSˆ x + sinβsinφSˆ y + cosβSˆ z

2 cosβ sinβexp(−iφ) sinβexp(iφ) −cosβ

The same two eigenvalues ±¯h/2 are obtained upon diagonalization As explained in Sect 3.1.3, this is a consequence of space isotropy The diagonalization also yields the state vectors

Addition of Angular Momenta

, (5.28) while the rotational unitary transformation acting on states (5.21) is

U βφ cos β 2 sin β 2 exp(−iφ) sin β 2 exp(iφ) −cos β 2

The factor of 1/2 multiplying the angle β highlights the impact of rotations on j = 1/2 objects This effect can be confirmed by examining the amplitudes in equation (3.21) when transforming from the z to the x eigenstates.

A qubit, short for "quantum bit," represents an arbitrary linear combination of spin up and spin down states, as described in (5.28) This concept is fundamental to the field of quantum computation, discussed in Section 10.6.

Consider two angular momentum vector operators, J ˆ 1 and J ˆ 2 They are independent vectors, i.e., [ J ˆ 1 ,J ˆ 2 ] = 0 Therefore, the product states are simultaneous eigenstates of the operators ˆJ 1 2 ,Jˆ z1 ,Jˆ 2 2 , and ˆJ z2: ϕ j 1 m 1 j 2 m 2 =ϕ j 1 m 1 ϕ j 2 m 2 (5.30)

The eigenstates (2j 1+ 1)(2j 2+ 1) form a complete basis for the quantum numbers j 1, m 1, j 2, and m 2 However, this basis may not be the most practical choice A more useful alternative could be a basis defined by the quantum numbers related to the total angular momentum Jˆ, as illustrated in Fig 5.5.

Since the components ˆJ x ,Jˆ y ,Jˆ z also satisfy the commutation relations (5.4) and (5.6), there must exist another basis set made up from eigenstates of the operators ˆJ 2 and ˆJ z Since the commutation relations

[ ˆJ 2 ,Jˆ 1 2 ] = [ ˆJ 2 ,Jˆ 2 2 ] = [ ˆJ 2 ,Jˆ z ] = [ ˆJ 1 2 ,Jˆ z ] = [ ˆJ 2 2 ,Jˆ z ] = 0 (5.32) vanish, the new set of basis states may be labeled by the quantum numbers j 1 , j 2 , j, m ϕ j 1 j 2 jm ≡ ϕ j 1 ϕ j 2 j m (5.33)

The two basis sets (5.30) and (5.33) are equally legitimate According to Sect 2.7.2*, there is a unitary transformation connecting the two basis: ϕ j 1 j 2 jm m 1 m 2 c(j 1 m 1;j 2 m 2;jm)ϕ j 1 m 1 j 2 m 2 (5.34)

The quantum numbersj 1 , j 2 are valid for both sets and they are not therefore summed up in (5.34) The sum over m 1 , m 2 is restricted by the addition of projections m=m 1 +m 2 (5.35)

In quantum mechanics, the relationship between the modulus of the sum of two vectors, j1 and j2, follows a similar principle to classical vectors Specifically, the modulus j of the sum is constrained by the inequality j1 + j2 ≥ j ≥ |j1 − j2| This indicates that the modulus of the combined vectors lies between the total of their individual moduli and the absolute value of their difference.

The quantum number \( j \) is classified as an integer when both \( j_1 \) and \( j_2 \) are integers or half-integers, while \( j \) is a half-integer if only one of the components is an integer The amplitudes \( c(j_1 m_1; j_2 m_2; j m) \), known as Wigner or Clebsch–Gordan coefficients, are real numbers that adhere to specific symmetry relations, expressed as \( c(j_1 m_1; j_2 m_2; j m) = (-1)^{j_1 + j_2 - j} c(j_1, -m_1; j_2, -m_2; j, -m) \).

The inverse transformation is ϕ j 1 m 1 ϕ j 2 m 2 j=j 1 +j 2 j=|j 1 −j 2 | c(j 1 m 1 ;j 2 m 2 ;jm) ϕ j 1 ϕ j 2 j m=m 1 +m 2 (5.38)

The example of the summation of an angular momentumj 1 with the spin j 2 =s 2 = 1/2 is given in Sect 5.6* (see also Fig 5.5).

Fig 5.5 Coupling of two vectors withj 1 = 5/2 andj 2 = 1/2 may yield a vector with j = 3, m= 2 (dashed lines) The superposition (5.34) has two components,withm 1= 3 2 , m 2= 1 2 andm 1= 5 2 , m 2=− 1 2 , respectively

The expansions (5.34) and (5.38) are fundamentally geometric, allowing for their validity even when state vectors are substituted with operators that possess angular momentum quantum numbers This leads to the derivation of selection rules for the matrix elements j f m f |O λà |j i m i of any operator.

Oˆ λà , whereλ, àare angular momentum labels According to (5.38), the state

The expansion of Oˆ λà ϕ j i m i in terms of eigenstates of total angular momentum operators requires that j f and m f meet specific constraints Matrix elements of spherically symmetric operators typically vanish unless the initial and final states share identical angular momentum quantum numbers, a condition often applicable to the Hamiltonian Furthermore, the product of parities must satisfy the equation π f π O π i = +1.

In the context of quantum mechanics, the operators \( \hat{J}^± \) serve a function analogous to the creation and annihilation operators \( a^+ \) and \( a \) found in the harmonic oscillator framework Notably, the operator \( \hat{J}^- \) is the Hermitian conjugate of \( \hat{J}^+ \), which is expressed in the relationship \( \langle jm | \hat{J}^+ | jm \rangle = \langle jm | \hat{J}^- | jm \rangle^* \) By utilizing the established commutation relations, we can derive further important relationships pertinent to these operators.

The commutation relations for angular momentum operators are defined by the equations [ˆJ z, Jˆ +] = ¯hJˆ + and [ˆJ +, Jˆ −] = 2¯hJˆ z The matrix elements of these relations indicate that the expression jm |[J z, J +]|jm equals ¯h(m − m), while jm |J + |jm results in ¯hjm |J + |jm This demonstrates that the operator ˆJ + increases the projection of angular momentum by one unit of ¯h, whereas ˆJ − decreases it.

The expectation value of (5.42) yields jm|[J + , J − ]|jm=jm|J + |j(m−1)j(m−1)|J − |jm

= 2¯h 2 m , (5.44) where (5.40) has been used The solution to this ﬁrst-order diﬀerence equation in |j(m+ 1)|J + |jm)| 2 is

The left-hand side of the equation is positive, which restricts the permissible values of m to those that ensure the right-hand side remains positive Consequently, the matrix element between the last allowed eigenstate ϕ_j(m_max) and the first rejected eigenstate ϕ_j(m_max + 1) must equal zero Here, m_max represents the positive solution to the equation c = m(m + 1) By assigning the quantum number j = m_max, we can determine the value of the constant c as c = j(j + 1) Thus, the equation j(m + 1)|J + |jm) = ¯h holds true.

(j−m)(j+m+ 1), (5.46) where the positive value for the square root is chosen by convention We verify the vanishing of the matrix elements connecting admitted and rejected states: j(j+ 1)|J + |jj=j(−j)|J + |j(−j−1) = 0 (5.47)

As the quantum number m increases in increments of one unit between -j and j, the permissible values of the quantum numbers j and m are outlined in equation (5.8) The matrix elements related to the operators ˆJ x and ˆJ y can be derived from equations (5.40) and (5.46), while the addition of the squares of these matrices results in the diagonal matrix elements of ˆJ 2, as indicated in equation (5.7).

5.5* Details of the Treatment of Orbital

5.5.1* Eigenvalue Equation for the Operator ˆ L z

The eigenvalue equation for the operator ˆL z is

The solution is proportional to exp(il z φ/¯h), leading to the requirement of 3 Ψ(φ + 2π) Ψ(φ), which indicates the presence of discrete eigenvalues for l z = ¯hm l (where m l = 0, ±1, ±2, ) Consequently, the orthonormal set of eigenfunctions for the operator ˆL z is defined as ϕ m l (φ) = 1.

5.5.2* Eigenvalue Equation for the Operators ˆ L 2 , L ˆ z

We try a function of the form Ψ(θ, φ) =P lm l (θ) exp(im l φ) It follows that, in the eigenvalue equation for the operator ˆL 2 :

• We can make the replacement d 2 /dφ 2 → −m 2 l

• We may drop the exponential exp(im l φ) from both sides of the equation

3This quantization procedure is similar to the one applied in (4.40).

5.5 Details of the Treatment of Orbital Angular Momentum 79

We obtain a diﬀerential equation depending on the single variableθ:

The solutions to this equation for m l = 0 may be expressed as polynomials

P l (cosθ) of order l in cosθ, called Legendre polynomials (l = 0,1,2, ). Each P l gives rise to the 2l+ 1 associated Legendre functions P lm l (θ) with

|m l | ≤l All of them are eigenfunctions of the operator L ˆ 2 with eigenvalue ζ=l(l+ 1)¯h 2

The simultaneous eigenfunctions of the operators L ˆ 2 and ˆL z are called spherical harmonics:

Y lm l (θ, φ) =N lm l P lm l (θ) exp(im l φ), (5.51) whereN lm l are constants chosen to satisfy the orthonormalization equation: l m l |lm l π

Table 5.1 presents the spherical harmonics for lower l values, while Table 5.2 shows the traditional symbolic representations used in literature, which are primarily of historical significance The coupling of two spherical harmonics with zero angular momentum, oriented differently in space, is influenced by the angle α12 formed between these orientations, as expressed in the relevant equation.

Table 5.1 Spherical harmonics corresponding to the lowest values ofl

Table 5.2.Equivalence between quantum numberland symbolic letters l symbol

The application of equation (5.34) is illustrated by the scenario involving the second angular momentum with a spin of j² = s = 1/2 In this case, the summation comprises two terms, which correspond to the two possible values of the spin projection, mₛ = ±1/2.

According to (5.36), there are also two values for the total angular momentum j=j 1 ±1/2. ϕ (j 1 =j+ 1

This article discusses the coupling of orbital motion with electron spin, specifically in Section 6.2 The eigenstates, represented as ϕ j 1 m 1, correspond to the spherical harmonics Y l 1 m l1 (5.51) Notably, equation (5.55) remains applicable regardless of the type of angular momentum j 1.

Central Potentials

Coulomb and Harmonic Oscillator Potentials

In this section we discuss the solutions to the eigenvalue equation for two central potentials: the Coulomb potential−Ze 2 /4π 0 rand the three-dimensional harmonic oscillator potentialM ω 2 r 2 /2.

To effectively analyze the three-dimensional linear harmonic oscillator, it is essential to estimate the orders of magnitude for the involved quantities, as previously established in equation (3.28) This estimation remains applicable due to the separability of the harmonic Hamiltonian into three Cartesian coordinates, allowing the validity of (3.28) for each individual coordinate Additionally, when considering the Coulomb potential, we can apply the Heisenberg uncertainty principle, which indicates that the momentum squared is approximately three times the square of the uncertainty in momentum, satisfying the relation p² ≈ 3(∆pₓ)² ≥ 3ħ².

Therefore, the radiusr m is obtained by minimizing the lower bound energy

9 E H , (6.6) where the Bohr radius a 0 and the ground state energy of the hydrogen atom

E H are given in Tables 6.1 and 14.1.

The solutions of the Schr¨odinger equation for the Coulomb and the harmonic oscillator potentials are shown in Table 6.1 The corresponding details

1This is another application of the separation of variables method for solving partial diﬀerential equations.

Table 6.1 Solutions to the Coulomb and harmonic oscillator problems problem Coulomb harmonic oscillator characteristic length a 0= 4π 0¯h 2 /Me 2 x c ¯ h/M ω wave function R n r l (u)Y lm l (θ, φ) R n r l (u)Y lm l (θ, φ) u=Zr/a 0 u=r/x c radial quantum n r = 0,1, n r = 0,1, numbers principal quantum n=n r +l+ 1 = 1,2, N = 2n r +l= 0,1, . numbers energies Z 2 E H /n 2 ¯hω(N+ 3/2)

E H=−e 2 /8π 0 a 0 degeneracy n 2 (N+ 1)(N+ 2)/2 are outlined in Sect 6.4* The following comments stem from the comparison between the solutions for these two potentials:

• In both cases the radial factorR n r l (r) may be expressed as a product of an exponential decay, a power of u, u l , and a polynomial of degree n r

• The radial factoru l decreases the radial density|R n r l | 2 r 2 for small values of uand increases it for large values It is a manifestation of centrifugal eﬀects due to rotation of the particle.

• Both potentials display a higher degree of degeneracy than is required by spherical invariance.

In a harmonic oscillator potential, all degenerate states share the same value of (−1) l = (−1) N, indicating that they possess identical parity However, this parity uniformity does not apply to the Coulomb potential, where states can exhibit degeneracy despite having different even and odd values of l.

The energies of the Coulomb potential are illustrated in Fig 6.1, while the harmonic potential energies follow a similar pattern as shown in Fig 3.2 Notably, the eigenvalues of the Coulomb potential exhibit an accumulation point at E ∞ = 0, whereas the eigenvalues of the harmonic oscillator are equidistant.

We have veriﬁed the commonly made statement that the Schr¨odinger equation is exactly soluble for the two central potentials treated in this section.

The two Schrödinger equations are interconnected through a straightforward transformation of the independent variable, changing r to r² This relationship holds true when the energy and potential strength are exchanged, along with a rescaling of the orbital angular momentum.

[33] Thus, the Schr¨odinger equations corresponding to the Coulomb and three-dimensional harmonic oscillator potential constitute only one soluble quantum mechanical central problem, not two. u

Fig 6.1 Coulomb potential and its eigenvalues The dimensionless variable u=r/a 0 has been used

The harmonic oscillator potential can be expressed in Cartesian coordinates, allowing for the derivation of degeneracies that should be verified against the values in Table 6.1 While the Coulomb potential is crucial for describing atomic spectra, the three-dimensional harmonic oscillator serves a similar purpose for nuclear spectra, highlighting an interesting parallel despite the distinct constituents and interactions in each system.

Spin–Orbit Interaction

One may incorporate the spin degree of freedom into the present treatment. The degeneracies displayed in Table 6.1 are thus doubled.

The analysis in Section 5.3 identifies two complete sets of wave functions that account for spins of 1/2 The first set is represented as ϕ nlm l sm s = R n r l Y lm l ϕ sm s, while the second set is denoted as ϕ nlsjm = R n r l m l + m s = m c(lm l ; sm s ; jm)Y lm l ϕ sm s The Clebsch–Gordan or Wigner coefficients relevant to these functions are detailed in Section 5.6 Additionally, the first set is characterized by quantum numbers lm l and sm s, which define the modulus and z-projection of the orbital angular momentum.

Spin-orbit interaction involves the interplay between orbital angular momentum and spin momentum In this context, the magnitudes of both orbital angular momentum and spin are preserved as significant quantum numbers, accompanied by jm, which relates to the modulus and z-projection of the total angular momentum, represented as J ˆ= L ˆ+ S ˆ.

The Coulomb interaction is the dominant force within an atom, effectively explaining many atomic behaviors However, experimental spectra reveal minor energy shifts related to the quantum number j Additionally, a weaker force arises from the interaction between the magnetic moment of electron spin and the magnetic field generated by the electron's orbital motion.

Vˆso=v so S ã ˆ L ˆ , (6.9) where we have approximated the radial factor by a constantv so.

When considering an electron's perspective, the charged nucleus appears to orbit around it, generating a magnetic field due to its movement This phenomenon leads to an interaction between the electron's spin magnetic moment and the magnetic field produced by the nucleus.

Hyperfine interactions, stemming from the coupling between nuclear and electron spins, play a crucial role in the splitting of the hydrogen atom's ground state Despite being smaller in magnitude, these interactions have significant astrophysical implications that are not accounted for by other interactions.

In this section, the radial term \( R_{nrl} \) can be omitted, as the spin-orbit interaction does not influence the radial component of the wave function Additionally, the spin-orbit interaction adheres to the established commutation relations.

[ ˆV so ,Lˆ 2 ] = [ ˆV so ,Sˆ 2 ] = [ ˆV so ,Jˆ 2 ] = [ ˆV so ,Jˆ z ] = 0, (6.10) while [ ˆV so ,Lˆ z ] = 0, [ ˆV so ,Sˆ z ] = 0 Bearing in mind this property, diﬀerent procedures – already developed in these notes – may be applied in order to incorporate the interaction (6.9).

The spin-orbit interaction, which commutes with the angular momentum operator \( \hat{J}_z \), does not create diagonal interactions among the eigenstates of the angular momentum projections Consequently, this interaction conserves the total projection \( m = m_l + m_s \), resulting in 2x2 matrices that can be diagonalized as outlined in Section 3.1.3.

2 The spin–orbit interaction is diagonal within the set of eigenstates (6.8). This constitutes a signiﬁcant advantage The diagonal matrix elements are the eigenvalues, which may be obtained through calculation.

When constructing interactions with quantum variables, two key criteria are often employed: simplicity and invariance under transformations such as rotations, parity, and time reversal The interaction described in equation (6.9) meets these essential criteria and can also be derived from the nonrelativistic limit of the Dirac equation.

Due to the spin–orbit interaction, the two states with j ± =l ± 1/2 become displaced by an amount proportional to the values appearing on the right-hand side of (6.12).

Some Elements of Scattering Theory

Boundary Conditions

In this study, we analyze the scattering of an incident particle by a central, finite-sized potential The asymptotic boundary condition necessitates that the wave function at large distances is represented as a combination of an incident plane wave propagating along the z-axis and an outgoing spherical wave, as illustrated in Figure 6.2 Specifically, as the distance r approaches infinity, the wave function can be expressed as Ψ(r, θ) = A exp(ikz) + (1/r) f(k, θ) exp(ikr).

The wave number is denoted as 2M E/¯ and the amplitude of the scattered wave in the polar direction θ is represented by f k (θ) The spherical wave includes a factor of 1/r, ensuring that |Ψ(r)| 2 is proportional to 1/r 2 to maintain probability conservation The azimuth angle φ is not included due to the axial symmetry of the problem Additionally, Expression (6.13) generalizes the boundary conditions previously discussed in Section 4.5 to a three-dimensional context.

A scattering experiment involves a projectile produced by a source, which is then collimated and accelerated before colliding with a target as a plane wave Upon impact, the projectile scatters into a spherical wave, forming a solid angle that makes an angle θ with the direction of incidence.

6.3 Some Elements of Scattering Theory 89

Expansion in Partial Waves

The free particle problem, similar to a three-dimensional harmonic oscillator, has solutions in both Cartesian and polar coordinates In spherical coordinates, the Hamiltonian solutions with V(r) = 0 are expressed as ϕ (1) lm l (r, θ, φ) = j l (kr)Y lm l (θ, φ) and ϕ (2) lm l (r, θ, φ) = n l (kr)Y lm l (θ, φ), where j l and n l represent Bessel and Neumann functions, respectively These eigenstates form a complete set, allowing us to construct a general linear combination that asymptotically approaches the desired function Notably, the function exp(ikz) can be expanded as exp(ikz) = √.

The second term on the right-hand side of equation (6.13) can be expressed using the Hankel function of the first kind, which asymptotically represents an outgoing spherical wave, as shown in equation (6.34) Consequently, the most comprehensive and valid linear combination is given by Ψ(r, θ) = A l=∞ l=0.

Here c l , a l are complex amplitudes, which may be expressed in terms of δ l , the (real) phase shift of thel-partial wave: c l =√ πi l (2l+ 1) 1/2 (a 2 l −1), a l = exp(iδ l ) (6.17)

We notice thatf k (θ) is provided by the second term in the ﬁrst line of (6.16). Replacing the Hankel function by its asymptotic representation one gets f k (θ) =−i

Cross-Sections

The ratio of scattered flux in the direction θ to the incident flux along the polar axis is represented by |f(θ)|²/r² The differential cross-section, which quantifies the number of particles that emerge per unit incident flux, per unit solid angle, and per unit time, is defined as σ(θ) = |f(θ)|² = π k², with the summation running from l=0 to l=∞.

The total cross-section is the integral over the whole solid angle σ= 2π π

The phase shifts δl in scattering by a rigid sphere of radius a are derived using continuity equations at the boundary r = a of the central potential Specifically, the relationship is expressed as tanδl = jl(ka) / nl(ka) In the limit as ka approaches zero, this simplifies to tanδl = (ka)^(2l+1).

If ka= 0, all the partial wave contributions vanish except for l = 0, due to the k 2 appearing in the denominator of the cross-sections (6.19) and (6.20).

The scattering process exhibits spherical symmetry, resulting in a total cross-section that is four times greater than the area observed in classical head-on collisions This quantum phenomenon is also evident in optics and is typical of long-wavelength scattering The total surface area of the sphere, represented by σ, suggests that the waves interact with the entire area.

Some features of scattering theory deserve to be stressed:

In classical mechanics, the closest distance of approach to the z-axis for a particle with orbital angular momentum \( \bar{h}l \) and energy \( E \) is represented by \( l/k \) A classical particle will not be scattered if \( l > ka \) This concept is mirrored in quantum mechanics, where the first significant maximum of the spherical Bessel function \( j_l(kr) \) occurs at approximately \( r = l/k \) Consequently, for \( l > ka \), this maximum aligns with the region where the potential energy is zero, indicating that the upper limit of \( l \) is on the order of \( ka \).

The calculation of the probability current using the wave function can lead to interference terms throughout the entire space, which are unrealistic due to the assumption of an infinite plane wave for the incident beam In real-world scenarios, the beam is collimated, resulting in a clear distinction between the incident plane wave and the scattered wave, except in the forward direction Additionally, in most experimental setups, the collimator's opening is sufficiently large to prevent any significant effects from the uncertainty principle related to collimation.

6.4 Solutions to the Coulomb and Oscillator Potentials 91

• Interference in the forward direction between the incident plane wave and the scattered wave gives rise to the important relation σ=4π k Im f k (0)

The attenuation of the transmitted beam, indicated by Im f k (0), is directly proportional to the total cross-section σ The optical theorem expressed in equation (6.23) is widely applicable and extends beyond the confines of scattering theory.

In scattering experiments, the analysis is conducted in the center-of-mass coordinate system, necessitating the use of the projectile-target reduced mass and the energy associated with relative motion to calculate the value of k Additionally, a geometric transformation is required to relate the scattering angles θ and θ lab, as the two reference frames are in motion relative to each other at the center of mass's velocity.

6.4* Solutions to the Coulomb and Oscillator Potentials

The hydrogen atom can be simplified from a two-body problem to a one-body problem by adopting the center of mass frame, allowing for the use of reduced mass for relative motion For simplicity, the motion of the nucleus is often disregarded due to its significantly greater mass compared to the electron.

Using dimensionless variables, such as in equation (4.21), is advantageous for simplifying calculations, particularly in the context of the hydrogen atom, where the Bohr radius serves as the natural length (refer to Table 14.1) Therefore, one can express the variable as u = Zr/a₀ The solution to the radial equation (6.3) is then derived from this formulation.

R n r l (r) =N nl (Z/na 0) 3/2 exp(−u/n)u l L n r (u), (6.24) where L n r (u) are polynomials of degree n r = 0,1,2, (associate Laguerre polynomials) TheN nl are normalization constants such that

The ionization or binding energy, denoted as energyZ 2 |E H |/n 2, represents the energy required to detach an electron from the Z atom in the n state Figure 6.3 illustrates the probability density in relation to the radial coordinate for the n = 1 and n = 2 states, as detailed in Table 6.2 The probability density expression incorporates a factor of r 2, which is linked to the volume element (5.11).

Figure 6.4 combines the angular distribution associated with the spherical harmonics of Fig 5.2 with the radial densities appearing in Fig 6.3.

The Bohr radiusa 0 may be compared with the expectation value of the coordinaterin the ground state of the hydrogen atom According to Table 6.2,one gets

Fig 6.3.Radial probability densities of the Coulomb potential

Table 6.2 Radial dependence of the lowest solutions for the Coulomb potential and the three-dimensional harmonic oscillator

6.4 Solutions to the Coulomb and Oscillator Potentials 93

Figure 6.4 illustrates probability density plots of various hydrogen atomic orbitals, where the density of dots indicates the likelihood of locating the electron in specific regions This visualization aids in understanding electron distribution within hydrogen atoms (Reproduced with permission from University Science Books)

There are also positive energy, unbound solutions to the Coulomb problem. They are used in the analysis of scattering experiments between charged particles.

In the harmonic oscillator, the dimensionless length is given by the ratio u=r/x c, as in (4.20) The radial eigenfunctions are

The conﬂuent hypergeometric function F(−n r , l+ 3/2, u 2 ) represents a polynomial of order n r in u 2, where n r can take on values of 0, 1, 2, and so on Radial probability densities are illustrated in Fig 6.5, and the normalization constants N N l ensure that the equation (6.25) remains valid Additionally, the energy eigenvalues are provided within this context.

Using procedures similar to those applied for the linear harmonic oscillator, we may calculate the expectation values of the square of the radius and of the

Fig 6.5.Radial probability densities of the harmonic oscillator potential momentum We thus verify the virial theorem (3.46) once again:

The lowest energy solutions are given on the right-hand side of Table 6.2. Useful deﬁnite integrals are

6.5* Some Properties of Spherical Bessel Functions

The spherical Bessel functions j l (kr) [and the Neumann n l (kr)] satisfy the diﬀerential equation

2M j l (kr) (6.30) Their asymptotic properties for large arguments are ρ→∞ lim j l (ρ) =1 ρsin ρ−1

Table 6.3 Lowest spherical Bessel functions l j l n l

3 ρ 3 −1 ρ cosρ− 3 ρ 2 sinρ while for small arguments they are ρ lim → 0 j l (ρ) = ρ l

(2l+ 1)!!, lim ρ → 0 n l (ρ) =−(2l−1)!! ρ l+1 (6.32) The spherical Hankel functions are deﬁned by h (+) l (ρ) =j l (ρ) + in l (ρ), h ( l − ) (ρ) =j l (ρ)−in l (ρ) (6.33) Due to (6.31), these have the asymptotic expressions ρ→∞ lim h (+) l (ρ) = (−i) l+1 ρ exp(iρ), lim ρ→∞ h (−) l (ρ) = (i) l+1 ρ exp(−iρ) (6.34) The ﬁrst threej l s andn l s are given in Table 6.3.

Problem 1.Calculate the diﬀerence in the excitation energy ofn= 2 states between hydrogen and deuterium atoms.

Hint: Use the reduced mass instead of the electron mass.

1 Assign the quantum numbers nlj to the eigenstates of the Coulomb problem withn≤3.

2 Do the same for the three-dimensional harmonic oscillator withN ≤3.

1 Obtain the degeneracy of a harmonic oscillator shell N, including the spin.

2 Obtain the average value|L 2 | N of the operator ˆL 2 in anN shell.

3 Calculate the eigenvalues of a harmonic oscillator potential plus the interaction [see (7.16)]

4 Give the quantum numbers of the states with minimum energy for a given shellN.

1 Find the energy and the wave function for a particle moving in an inﬁnite spherical well of radiusawithl= 0.

2 Solve the same problem using the Bessel functions given in Sect 6.5*.

1 Find the values of r at which the probability density is at a maximum, assuming then= 2 states of a hydrogen atom.

2 Calculate the mean value of the radius for the same states.

Problem 6.Solve the harmonic oscillator problem in Cartesian coordinates. Calculate the degeneracies and compare them with those listed in Table 6.1.

1 Find the ratio between the nuclear radius and the average electron radius in then= 1 state, for H and for Pb Use R nucleus ≈1.2A 1/3 F, A(H) Z(H) = 1,A(Pb) = 208 andZ(Pb) = 82.

2 Do the same for a muon (M à = 207M e).

3 Is the picture of a pointlike nucleus reasonable in all these cases?

Problem 8.Replacer 2 → sin the radial equation of a harmonic oscillator potential Find the changes in the constantsl(l+ 1),M ω 2 andE that yield the Coulomb radial equation.

Hint: Make the replacement R(r) → s 1/4 Φ(s) and construct the radial equation usings≡r 2 as variable.

Problem 9.The positronium is a bound system of an electron and a positron (the same particle as an electron but with a positive charge) Their spin–spin interaction energy may be written as ˆH =aS ˆ e ã S ˆ p, where e and p denote the electron and positron, respectively.

1 Obtain the energies of the resultant eigenstates (see Problem 11 of Chap 5).

2 Generalize (6.12) to the product of two arbitrary angular momenta J ˆ 1 ã J ˆ 2.

Problem 10.Calculate the splitting between the 2pstates withm= 1/2 of a hydrogen atom in the presence of spin–orbit coupling and a magnetic ﬁeld

3 As a function of the ratioq= 2à B B z /¯h 2 v so

Problem 11.Calculate the current associated with the spherical wave

Aexp(ikr)/r and show that the ﬂux within a solid angle dΩ is constant.

Problem 12.A beam of particles is being scattered from a constant potential well of radius aand depth V 0 Calculate the diﬀerential and the total cross- section in the limit of low energies.

Hint: Consider only thel= 0 partial waves.

1 Obtain the internal logarithmic derivative (times a) forr=a(see Prob- lem 4).

2 Obtain the external logarithmic derivative (times a) forr=ain the low energy limit.

1 What is the analog of spherical symmetry in a two-dimensional space? Find the corresponding coordinates.

2 Write down the operator for the kinetic energy in these coordinates and ﬁnd the degeneracy inherent in potentials with cylindrical symmetry.

3 Find the energies and degeneracies of the two-dimensional harmonic oscillator problem.

4 Verify that the function ϕ n = 1 x c √ πn!exp(−u 2 /2)u n exp(±inφ) is an eigenstate of the Hamiltonian (u=ρ/x c ).

So far, we have discussed only one-particle problems We now turn our attention to cases in which more than one particle is present.

In the ﬁrst place we stress the fact that if ˆH = ˆH(1) + ˆH(2), where

Hˆ(1) and ˆH(2) refer to diﬀerent degrees of freedom (in particular, to diﬀerent particles), and if ˆH(1)ϕ a (1) =E a ϕ a (1) and ˆH(2)ϕ b (2) =E b ϕ b (2), then ϕ ab (1,2) =ϕ a (1)ϕ b (2)

In quantum physics, the Heisenberg indeterminacy principle prevents the distinction between identical particles unless they are sufficiently separated To address the quantum behavior of these particles, a new principle known as the Pauli Principle is introduced Particles are classified into two categories: fermions and bosons.

In this chapter, we explore many-body problems in the He atom that can be described using an independent-particle approach, focusing on central potentials in atomic and nuclear physics, as well as electron gas and periodic potentials in solid-state physics We utilize methods from previous sections to address fermion challenges, while phonon interactions in lattices and boson condensation are approached through generalized harmonic oscillator solutions Instead of providing a comprehensive overview of these many-body fields, we illustrate the quantum formalism with pertinent applications, highlighting groundbreaking discoveries such as quantum dots, Bose–Einstein condensation, and quantum Hall effects, which are pivotal to future technologies.

The concept of creation and annihilation operators is extended to many- body boson and fermion systems in Sect 7.8 †

The Pauli Principle

Let us now consider the case of two identical particles, 1 and 2 Two particles are identical if their interchange, in any physical operator, leaves the operator invariant:

[ ˆP 12 ,Q(1,ˆ 2)] = 0, (7.2) where ˆP 12 is the operator corresponding to the interchange process 1 ↔ 2.

The eigenstates of the operator ˆQ can also be simultaneous eigenstates of the operator ˆP 12 The operator ˆP 12 2 has a single eigenvalue of 1, reflecting the system's invariance under the interchange of particles twice Consequently, the operator ˆP 12 has two eigenvalues, ±1, categorizing the eigenstates as symmetric (+1) or antisymmetric (−1) with respect to particle interchange.

In quantum mechanics, we analyze two orthogonal single-particle states, ϕ p and ϕ q, which can also serve as eigenstates of a Hamiltonian By distributing two particles across these states, we generate four distinct two-body states The symmetric combinations of these states are represented as Ψ (+) pp = ϕ p (1)ϕ p (2), Ψ (+) qq = ϕ q (1)ϕ q (2), and Ψ (+) pq = 1.

, (7.5) while the antisymmetric state is Ψ (−) pq = 1

Entangled states, as defined in equations (7.5) and (7.6), cannot be expressed merely as a combination of the tensor product of the state vectors of particle 1 and particle 2, as discussed in Section 10.2.

The average distance between two entangled identical particles is pq|( r 1 −r 2) 2 |pq 1/2 ( ± ) p|r 2 |p+q|r 2 |q −2p|r|pq|r|q ∓2|p|r|q| 2 1/2 ,

In equation (7.7), the subscripts (±) indicate symmetric and antisymmetric states, where the initial three terms represent the average "classical" distance derived from state functions like ϕ p (1)ϕ q (2) This classical distance can be reduced for entangled particles in symmetric states, while it increases for those in antisymmetric states Consequently, symmetry creates correlations among identical particles, even without any residual interacting forces.

This article expands on the construction of symmetric and antisymmetric states for ν identical particles, utilizing the permutation operator ˆP b, which represents one of the ν! possible permutations It is demonstrated that this operator can be expressed as a product of two-body permutations ˆP ij While the decomposition into two-body permutations is not unique, the parity of the total number of permutations, denoted as η b, remains consistent.

Acting with the operator ˆS on a state of ν identical particles produces a symmetric state, whilst acting with ˆAproduces an antisymmetric state.

A new quantum principle has to be added to those listed in Chap 2:

Principle 4 There are only two kinds of particles in nature: 1 bosons described by symmetric state vectors, and fermions described by antisymmetric state vectors.

In a totally symmetric Hamiltonian, eigenstates can be classified based on their behavior when two particles are interchanged, resulting in either symmetry or antisymmetry This principle restricts the possible states, leading to the conclusion that the only observable two-body fermion state in nature is defined by equation (7.6).

All known particles with half-integer values of spin are fermions (electrons, muons, protons, neutrons, neutrinos, etc.) All known particles with integer spin are bosons 2 (photons, mesons, etc.).

Every composite object possesses a total angular momentum, often interpreted as the spin of the composite particle, determined by the addition rules outlined in Section 5.3 If this spin is a half-integer, the object behaves as a fermion; conversely, a composite system with an integer spin functions as a boson For example, helium-3, which consists of two protons and one neutron, is classified as a fermion, while helium-4, comprising two protons and two neutrons, is identified as a boson, despite both isotopes sharing identical chemical properties.

Let us distribute ν identical bosons into a set of single-particle states ϕ p and denote by n p the number of times that the single-particle state pis

For the past two decades, it has been recognized that while the traditional classification of particles as bosons or fermions applies in three dimensions, two-dimensional systems allow for the existence of anyons, which represent a spectrum of intermediate states In certain scenarios, such as the fractional quantum Hall effect, anyons can be manifested in surface layers that are just a few atoms thick.

Pauli demonstrated the intricate relationship between spin and statistics, highlighting the complexities of quantum field theory Despite Feynman's challenge for a straightforward proof of the spin-statistics theorem, a conclusive answer remains elusive The occupation numbers, denoted as n p, play a crucial role in this context To formulate the symmetrized ν-body state vector, we begin with the product Ψ pq r (1,2, , ν).

The state vector is symmetrized using the operator ˆS, resulting in the final state Ψ n p ,n q , ,n r (1,2, , ν) = NSΨˆ pq r (1,2, , ν), with N as a normalization constant and occupation numbers indicating the states In this framework, there are no limitations on the number of bosons occupying a single-boson state For example, in a two-particle scenario, the symmetric state vectors can be represented by equations (7.3), (7.4), and (7.5).

In the context of fermions, the state can be characterized by occupation numbers, which must be either zero or one to adhere to the antisymmetrization principle This principle is crucial as it enforces Pauli's exclusion principle, stating that if an electron occupies a state defined by specific quantum numbers, no additional electrons can occupy that same state Consequently, the antisymmetric state function for ν fermions can be represented as a Slater determinant, ensuring compliance with these fundamental quantum mechanical rules.

The permutation of two particles involves swapping two columns, resulting in a sign change It is essential that all single-particle states are distinct; if they are identical, the rows become equal, leading to a vanishing determinant.

A widely used representation of the states (7.9) and (7.11), in terms of creation and annihilation operators, is given in Sect 7.8 †

The ability to occupy a single symmetric state with multiple bosons leads to phase transitions, which have significant theoretical and conceptual implications, particularly evident in the phenomenon of Bose–Einstein condensation Moreover, the effects observed in fermions present even more remarkable consequences, which will be discussed later in this chapter.

Two-Electron Problems

In examining the helium atom, we initially overlook the interaction between its two electrons The lowest energy single-particle states available for these electrons are represented by the wave function ϕ 100 1.

2 m s states, where we use the same representation as in (6.7).

This article discusses the interplay of four angular momenta, comprising two orbital angular momenta and two spin angular momenta Initially, the two orbital angular momenta (L̂ = L̂₁ + L̂₂) and the two spin angular momenta (Ŝ = Ŝ₁ + Ŝ₂) are coupled Following this, the total angular momentum is derived by adding the total orbital and total spin angular momentum, resulting in the equation Ĵ = L̂ + Ŝ.

The spin component of the state vector can exhibit either spin 1 or spin 0 By applying the coupling method outlined in equation (5.55) with j1 = j2 = 1/2, we derive the corresponding states χs ms Among the three two-spin states with spin 1, all are symmetric, whereas the state with spin 0 is characterized as antisymmetric Specifically, the symmetric state is represented as χ1 1 (1,2) = ϕ ↑ (1)ϕ ↑ (2), while the antisymmetric state is expressed as χ1 0 (1,2) = 1.

We now consider diﬀerent occupation numbers for the two electrons.

In a system where two electrons occupy the lowest orbital (ϕ 100 1 2 m s), their spatial wave functions are identical, resulting in a spatially symmetric state vector Due to the exclusion principle, this configuration prohibits the existence of a symmetric spin state with spin 1 Consequently, the only viable state for these electrons is the entangled state with zero spin.

2 One electron occupies the lowest levelϕ 100 1

In quantum mechanics, the difference in radial wave functions allows for the construction of both symmetric and antisymmetric spatial states, each with angular momentum l = 0 According to the Pauli exclusion principle, there are two permissible total states: one combines a symmetric spatial part with an antisymmetric spin state, and the other does the reverse The previously overlooked interaction between electrons disrupts the degeneracy of these states; specifically, electrons in a spatially antisymmetric state experience less Coulomb repulsion due to being farther apart, resulting in a lower energy compared to those in a symmetric state.

3There is an alternative coupling scheme in which the orbital and spin angular momenta are ﬁrst coupled in order to yield the angular momentum of each particle:

J ˆ i = L ˆ i + S ˆ i (i = 1,2), as in (6.8) Subsequently, the two angular momenta are coupled together: J ˆ= J ˆ 1+ J ˆ 2 The two coupling schemes give rise to two diﬀerent sets of basis states.

3 One electron occupies the lowest level ϕ 100 1 2 m s and the other the level ϕ 21m l 1

In this case, readers are encouraged to approach it as an exercise, applying the same method as in the prior example, while considering that the orbital angular momentum is now non-zero.

Periodic Tables

The nuclear center's attraction, proportional to \( Z^2 \), enables a central-field description for multi-electron atomic systems Although the Hamiltonian incorporates the Coulomb repulsion between electrons, which is weaker and proportional to \( e^2 \), each electron experiences \( Z-1 \) repulsions By adjusting the central field, we can effectively account for these interactions in our approximations.

According to the Pauli exclusion principle, electrons in occupied energy levels cannot scatter to other occupied levels When electrons do scatter to empty levels, they must overcome the energy gap between their current occupied level and the highest filled state, which diminishes the effectiveness of the residual interaction.

• The electric ﬁelds created by electrons lying outside a radiusr tend to cancel for radiusr < r , due to the well-known compensation between the ﬁeld intensity (∝1/r 2 ) and the solid angle (∝r 2 ).

Although the optimum choice of the single-particle central potential constitutes a diﬃcult problem, it is simple to obtain the behavior at the limits r lim → 0 V(r) =− Ze 2

Electrons near the nucleus experience the full strength of the nuclear field, while those further away are influenced by the shielding effect of the other Z−1 electrons The potential at intermediate distances can be qualitatively assessed through interpolation.

The energy eigenvalues of this eﬀective potential are also qualitatively reproduced by adding the term

In the context of the Coulomb potential, the centrifugal term \(\hbar^2 l(l+1)/2M r^2\) prevents electrons in high orbital quantum number states from approaching the nucleus, reducing their exposure to the stronger attraction at smaller radii Consequently, the energy levels \(E_{nl}\) are categorized by their orbital quantum number, diverging from a simple \(1/r\) dependency These energy levels, illustrated in Fig 7.1 and following the nomenclature of Table 5.2, form clusters known as shells, where levels are closely spaced.

In a closed shell, all magnetic substates are occupied.

Fig 7.1 Electron shell structure The ﬁgure gives a rough representation of the order of single-electron levels Numbers to the right indicate the number of electrons in closed shell atoms

The ground state of an atom is established by filling single-particle states until all Z electrons are allocated Closed shells, which have zero orbital and spin angular momenta, are stable systems without loose electrons or holes This stability accounts for the unique properties of noble gases in the Mendeleev chart, corresponding to atomic numbers Z = 2, 10, 18, 36, 54, and 86 The angular momenta, stability, chemical bond characteristics, and overall chemical properties are influenced by the outer electrons in the last unfilled shell, highlighting the significant implications of the Pauli principle.

The electron configuration of an atom with multiple electrons is determined by the filling of unfilled shell states For example, in a magnesium atom (Z = 12), the lowest energy configuration is 4 (3s) 2 Additionally, the configurations (3s)(3p) and (3p) 2 are energetically similar.

In atomic systems, the total single-particle angular momentum j is often not specified due to the relatively weak spin-orbit coupling compared to electron repulsions However, in heavier elements and inner shells, the quantum numbers (l, j) regain significance.

Let us now consider the nuclear table A nucleus hasAnucleons, of which

In nuclear physics, N represents neutrons and Z denotes protons, leading to the classification of nuclei: isobars share the same mass number A, isotones have the same number of neutrons N, and isotopes possess the same number of protons Z Despite the absence of a fundamental attraction from a nuclear center and the complexity of internuclear forces, the Pauli exclusion principle remains significant Consequently, the shell model serves as a foundational framework for understanding most nuclear properties For systems characterized by short-range interactions, a realistic central potential is typically observed.

The Coulomb principal quantum number is the first in a series of quantum designations, followed by the orbital angular momentum as outlined in Table 5.2, with the exponent indicating the count of particles associated with these quantum numbers In nuclear physics, the probability density exhibits a Woods–Saxon shape, necessitating the inclusion of a strong spin–orbit interaction at the surface, which differs in sign from atomic interactions Additionally, a central Coulomb potential is present for protons.

The empirical values of the parameters are [36] v 0

Here a= 0.67 F represents the skin thickness andR=r 0 A 1/3 is the nuclear radius, withr 0= 1.20 F The resulting shell structure is shown in Fig 7.2.

The nuclear shell structure, depicted in Fig 7.2, illustrates the arrangement of single-nucleon energy levels, identified by their quantum numbers N and l, j The figure also indicates the total number of nucleons present in closed shell systems, providing a clear overview of nuclear organization.

Fig 7.3 Comparison of the Woods–Saxon and harmonic oscillator potentials [36]

Nucleons within a Woods–Saxon potential experience a similar effect to that of a harmonic oscillator potential It is essential to incorporate an attractive term, as the Woods–Saxon potential energetically favors nuclear single-particle states located near the surface more than the harmonic oscillator potential Consequently, this leads to a more straightforward effective potential.

The equation L² - L²ₙ can be utilized for bound nucleons, where ¯hω is defined as 41 MeV A⁻¹/³, with specific parameters for protons and neutrons: c = 0.13ω/¯h, d = 0.038ω/¯h for protons, and d = 0.024ω/¯h for neutrons The symbol L²ₙ represents the average value of L² in an N-oscillator shell The eigenstates are identified by the quantum numbers Nljmτ, with the new quantum number τ being 1/2 for neutrons and -1/2 for protons.

The lowest shell (N = 0) is occupied by four nucleons—two protons and two neutrons—resulting in the highly stable alpha particle Similar to electrons, closed nuclear shells do not influence the characteristics of low-lying excited states For a stable configuration akin to noble gases, both nucleons must fill closed shells, which occurs in nuclear systems with Z = N = 2 and Z = N = 8.

Unlike hydrogen, describing heavier atoms and nuclei using a central field is only a semi-quantitative approximation, as one-body terms cannot fully replace two-body interactions This approximation becomes more accurate in systems with one additional particle or hole compared to a closed shell configuration.

Motion of Electrons in Solids

Electron Gas

In a basic model of metal, electrons move independently, with the crystalline lattice's electrostatic attraction preventing their escape at the surface The findings from Section 4.4.1 regarding the electron gas can be extended to three dimensions, where the wave states are represented as the product of three one-dimensional solutions.

The volume is V = a 3 The allowed k values constitute a cubic lattice in which two consecutive points are separated by the distance 2π/a(4.40) k n i = 2π a n i , n i = 0,±1,±2, , i=x, y, z (7.18) The energy of each level is k = ¯h 2 |k| 2

To establish the ν-electron ground state, we begin by placing two electrons at the lowest energy level (k x = k y = k z = 0) and progressively filling higher energy levels as they become available In systems with a significant number of electrons, the occupied states form a spherical region in k-space, defined by the Fermi momentum (k F) and associated with the Fermi energy (E F = ¯h² k F² / 2M) At zero energy, all levels with |k| ≤ k F are filled pairwise, while those above remain vacant In the large-volume limit, the energy levels are closely spaced, allowing us to approximate summations with integrals that resemble a specific volume element.

An electron gas is defined by its Fermi temperature, T_F, which is calculated using the Fermi energy (F) and the Boltzmann constant (k_B) When the temperature (T) is significantly lower than T_F, the electron gas exhibits properties akin to those at absolute zero Additionally, the density of states per unit volume for energy levels below a certain threshold is represented by n(ε) = 2.

7.4 Motion of Electrons in Solids 109 respectively At the Fermi energy, the value of these quantities is n F= 1 3π 2 k 3 F , ρ F=3n F

For the Na typical case:n F ≈2.65×10 22 electrons cm − 3 ,k F ≈0.92×10 8 cm − 1 ,

This article examines the thermal properties of an electron gas, noting that if electrons followed classical mechanics, each would acquire energy proportional to k_B T as the temperature increases from absolute zero to T Consequently, the total thermal energy per unit volume of the electron gas is approximately u_cl = n_F k_B T Additionally, the specific heat at constant temperature remains constant and independent of temperature.

The Pauli exclusion principle restricts the majority of electrons from acquiring energy, allowing only those with an initial energy k, where F − k < k B T, to potentially gain energy Approximately, the quantity of these electrons can be expressed as ρ F k B T = 3n F.

The total thermal energy and speciﬁc heat per unit volume are u=ρ F(k B T) 2 , (7.26)

The speciﬁc heat is proportional to the temperature and is reduced by a factor

The Fermi-Dirac distribution describes the probability of an electron occupying a specific energy state Utilizing this distribution, the total energy per unit volume can be expressed as \( u = \frac{1}{2\pi^2} \).

0 ρ( )η( )k 2 dk , (7.28) which is a better approximation than (7.26) Upon integration, one obtains results similar to (7.27).

Although the electron gas model explains many properties of solids, it fails to account for electrical conductivity, which can vary by a factor of 10 30 between good insulators and good conductors.

A qualitative understanding of conductors and insulators can be derived from the band model, where the energies of electrons are organized into allowed bands due to their motion in a periodic ion array Each band comprises 2N levels, with N representing the number of ions and the factor of 2 accounting for electron spin Following the Pauli principle, electrons fill these single-particle states, with the last filled band termed the valence band When an electric field is applied, electrons in the valence band cannot be accelerated by small fields, as they would merely occupy already filled states within the same band This behavior renders the valence band inert, meaning it does not influence thermal or electrical properties A solid with only filled bands acts as an insulator, with insulation quality improving as the energy gap (∆E) between the upper valence band and the next empty band widens.

Electrons in partially filled bands can readily absorb energy from an electric field, forming what is known as a conduction band At absolute zero temperature (T = 0), this principle holds true; however, in insulators at this temperature, thermal motion increases electron energy by kBT As temperature rises, some electrons from the valence band can transition to the conduction band, characterizing the material as a semiconductor The conductivity of this system is influenced by temperature and varies according to the equation exp(−∆E/kBT).

The existence of conductors, semiconductors, and insulators stems from the Pauli exclusion principle When electrons transition to the conduction band, they create empty states known as holes in the valence band Electrons in the valence band can move to fill these holes, resulting in the formation of additional holes and generating a current This current is driven by the movement of valence band electrons, while the holes themselves carry a positive charge, representing the absence of electrons.

Semiconductors play a crucial role in the modern electronics industry, serving as essential components in electronic circuits and optical applications Their unique ability to significantly change electrical conductivity through external stimuli, such as voltage and photon flux, as well as through the introduction of specific impurities via doping, makes them indispensable in advanced technology.

Until now, we have considered ions as stationary at their positions \( R_i \), which creates the crystal lattice structure This assumption arises from the significant mass difference between ions \( M_I \) and electrons \( M \).

7.4 Motion of Electrons in Solids 111 for a small ﬂuctuation u i in the coordinate r i = R i +u i of the i ion (Born–Oppenheimer approximation) For the sake of simplicity, we make the following approximations:

• A linear, spinless chain ofN ions separated by the distanced;

In the ion-ion potential, only terms up to quadratic order in the fluctuations of ion positions are retained, as linear terms cancel out due to the equilibrium condition Additionally, the constant equilibrium term is excluded from consideration.

• Only interactions between nearest neighbors are considered.

In the context of quantum mechanics, the creation and annihilation operators, denoted as \(a_i^+\) and \(a_i\), are defined for each site \(i\) These excitations exhibit bosonic behavior due to the possibility of making multiple identical displacements at each site Additionally, the parameter \(\omega\) in the ion-ion potential can be understood as the frequency of each oscillator when it is not influenced by interactions with other oscillators.

As in classical physics, the coupled oscillators may become uncoupled by means of a linear transformation

The uncoupling procedure is described at the end of the section The resultant amplitudes λ ki , à ki and new frequenciesω k are λ ki = 1 2 ω ω k N + ω k ωN exp[ikr i ] à ki = 1 2 ω ω k N − ω k ωN exp[ikr i ] ω k = 1

√2ω k d , (7.31) where the frequenciesw k are proportional to the wave numberk If a cyclic chain is assumed [r N +1 =r 1 , see (4.4.1)] k=k n = 2π n k

In a crystal, the presence of not just electrons and ions, but also extended periodic boson structures known as phonons is significant The phonon states and energies are described by the equation Ψ = Π k ϕ n k = Π k, highlighting the complex interactions within the crystal lattice.

These lattice vibrations have consequences on many properties of crystals.

The specific heat can be analyzed through the linear relationship between frequency and momentum, represented as \( w_k = \alpha k \), which is applicable in three-dimensional systems like sound waves When the thermal frequency, defined as \( \frac{k_B T}{\hbar} \), is sufficiently small, we can disregard the occupancy of other finite frequency modes To calculate the total phonon energy per unit volume \( V \), we substitute the phonon occupancies \( n_k \) with the thermal occupancy \( \eta_k \) derived from the Bose–Einstein distribution, leading to the expression for phonon energy as \( u_{\text{phonon}} = \hbar \alpha \).

Therefore, the phonon contribution to the speciﬁc heat at smallT (well below room temperature) is proportional toT 3

The Uncoupling of the Hamiltonian

The amplitudesλ ki , à ki and the frequenciesw k are determined by solving the harmonic oscillator equation

[ ˆH, γ k + ] = ¯hω k γ k + (7.35) Using the last line of (7.29) with this equation, one obtains

7.4 Motion of Electrons in Solids 113

Perturbation Theory

The procedure mirrors celestial mechanics, where a comet's trajectory is initially determined by considering only the sun's gravitational pull, with the smaller influences of planets incorporated in subsequent approximations.

The Hamiltonian ˆH is divided into two components for analysis: the first term, ˆH 0, represents a solvable problem that closely approximates the original issue, while the second term, ˆV, is identified as the perturbation.

In our analysis, we introduce a perturbation term multiplied by a constant λ, which is less than 1, to monitor the order of magnitude of various expansion terms in the underlying theory While λ serves this purpose, it holds no physical significance and is replaced by 1 in the final expressions We proceed to solve the eigenvalue equation accordingly.

HˆΨ n =E n Ψ n (8.3) by expanding the eigenvalues and the eigenstates in powers of λand succes- sively considering all terms corresponding to the same power ofλin (8.3):

The terms independent ofλyield (8.2) The terms proportional toλgive rise to the equation

We begin by calculating the scalar product of ϕ(0)n with the states on both sides of equation (8.5) The left-hand side is zero due to equation (8.2), leading us to the first-order correction to the energy.

Therefore, the leading order term correcting the unperturbed energy is the expectation value of the perturbation.

Next, we take the scalar product withϕ (0) p , (p=n), so that

E p (0) −E n (0) ϕ (0) p |Ψ (1) n =−ϕ (0) p |V|ϕ (0) n (8.7) Using the statesϕ (0) p as basis states, we expand Ψ (1) n p=n c (1) p ϕ (0) p , c (1) p = ϕ (0) p |V|ϕ (0) n

The still missing amplitude c (1) n is determined from the normalization condition: since both Ψ n and ϕ (0) n are supposed to be normalized to unity, the terms linear inλare

Therefore, the ﬁrst-order coeﬃcientc (1) n disappears, since we can make it real by changing the (arbitrary) phase ofϕ (0) n

Equations (8.6) and (8.8) outline the first-order changes in energies and state vectors based on the matrix elements of the perturbation relative to the zero-order states For perturbation theory to converge effectively, it is essential that the condition |c (1) p |² ≤ 1 is satisfied, indicating the matrix element of the perturbation between states.

Perturbation theory is applicable only when the interaction between two states is weaker than the unperturbed distance separating them If there are non-vanishing matrix elements between degenerate states, perturbation theory is not suitable, and alternative methods such as variational approaches or diagonalization must be employed The second-order energy correction can be calculated accordingly.

Rayleigh–Schrödinger perturbation theory initially appears simple, but its complexity increases with higher-order perturbations due to the growing number of contributing terms To simplify the process, one can sum partial series of terms, as demonstrated in Brillouin–Wigner perturbation theory, which involves replacing the unperturbed energy.

E n (0) of the statenby the exact energyE n in the denominators For the case of the energy expansion, one obtains

In the Rayleigh–Schrödinger perturbation theory, the last term is identified as a third-order term, which is absent in the Brillouin–Wigner expansion due to its incorporation through the denominator replacement of the second-order term While this reduction in the number of terms may seem advantageous, it can lead to a decrease in the convergence of the perturbation expansion, reflecting the nature of the partial summations involved Additionally, the Brillouin–Wigner series may contain various powers of λ across its multiple terms.

Feynman's elegant formulation of perturbation theory employs diagrams that encapsulate both mathematical precision and a clear depiction of the underlying processes A notable achievement in this theory is the calculation of the Lamb shift, which signifies the energy difference E 2p 1.

2 in the hydrogen atom, to six signiﬁcant ﬁgures, using quantum electrodynamics (see [52], p 358).

The ground state energy of the He atom is calculated using perturbation theory in Sect 8.3.

1One can prove that the Brillouin–Wigner expansion does not contain terms in which the stateϕ (0) n appears in the numerator as an intermediate state.

Variational Procedure

This method can be viewed as the inverse of the perturbation technique, where a trial state Ψ is proposed instead of relying on a fixed set of unperturbed states This trial state can be expressed using the Hamiltonian's eigenstates, represented as Ψ = Σ c_E ϕ_E, where the results are derived from the relation Ψ|H|Ψ.

The equation |c E | 2 = E 0 illustrates that E 0 represents the ground state energy The state Ψ is influenced by a specific parameter, and by minimizing the expectation value of the Hamiltonian concerning this parameter, one can derive an upper limit for the system's ground state energy.

The extremum property of energy ensures that if the trial wave function has an error of approximately δ, the variational energy estimate will only be inaccurate by about δ² This means that even a reasonable initial guess for the wave function can yield a reliable energy estimate.

The ground state energy obtained in ﬁrst-order perturbation theory

E 0 (0) + ϕ (0) 0 |V|ϕ (0) 0 is an expectation value of the total Hamiltonian, and is thus equivalent to a nonoptimized variational calculation.

Ground State of the He Atom

The three-body problem can be simplified to a two-body problem by focusing on a significantly massive nucleus However, the remaining challenge is complex due to the Coulomb repulsion potential (V) between the two electrons The overall Hamiltonian is represented as ˆH 0 + V.

Here r 12=| r 1 − r 2 |is the distance between the electrons.

The ground state energy of a helium atom, which contains two electrons, can be analyzed by considering the independent movement of each electron in the Coulomb potential created by the nucleus This energy is proportional to Z², leading to an unperturbed energy of 8E_H, where E_H represents the energy of an electron in a hydrogen atom The antisymmetrized two-electron state vector for the helium atom's ground state is elaborated in Section 7.2, and the first-order energy correction is discussed in Section 8.6*.

2The requirement of normalization is explicitly satisﬁed by minimizing Ψ|H|Ψ/Ψ|Ψ

Molecules

Intrinsic Motion Covalent Binding

In this article, we outline the procedure for analyzing the molecular hydrogen ion, H₂⁺ Figure 8.1 depicts the two protons, labeled 1 and 2, along with the electron By assuming that the protons remain at rest, we simplify the Hamiltonian, facilitating our calculations and analysis.

In this study, we analyze the hydrogen ion, defined by the expression R = |R1 - R2| While exact numerical solutions can be derived from the Schrödinger equation using elliptical coordinates, employing a variational approach to approximate the solution proves to be more insightful.

If the distanceR is very large, the two (degenerate) solutions describe a

H atom plus a dissociated proton The two orbital wave functions are ϕ 1 =ϕ 100 (|r−R 1 |), ϕ 2 =ϕ 100 (|r−R 2 |) (8.17)

Note that such wave functions are orthogonal only for very large values ofR.

In fact, their overlap is 1|2 = 1 forR= 0.

The requirement of antisymmetry between the two protons must be taken into account As in Sect 7.2, the spin of the two protons may be coupled to

1 (symmetric spin states) or to 0 (antisymmetric spin states) The corresponding spatial wave functions should thus be antisymmetric or symmetric, respectively: ϕ ∓ = ϕ 1 ∓ϕ 2

2(1∓ 1|2 ) (8.18) The energy to be minimized with respect to the distanceRis

R →∞ E ± =E 100 (8.20)Since the matrix element in the third line of (8.19) is positive, we conclude that the energy corresponding to the spatially symmetric wave function lies lowest.

Fig 8.2.Lowest energies of the hydrogen ion as a function of the distance between protons

Figure 8.2 illustrates the two curves, revealing that only the energy associated with ϕ + shows a minimum This phenomenon can be attributed to the accumulation of electron density between the two nuclei, which facilitates the screening of Coulomb repulsion This interaction is characterized as covalent binding, as further discussed in Section 3.1.3.

Vibrational and Rotational Motions

In this discussion, we examine a diatomic molecule characterized by two distinct masses, M1 and M2 We begin by applying the established method of separating the relative and center of mass operators to analyze the system effectively.

M g P ˆ 2 , P ˆ g= P ˆ 1+ P ˆ 2 (8.21) The inversion of deﬁnitions (8.21) yields the kinetic energy

Here M g = M 1+M 2 is the total mass and à ≡ M 1 M 2 /M g is the reduced mass.

When the potential energy V(R) is solely a function of the distance between ions, the center of mass behaves like a free particle This scenario has been previously addressed in Section 4.3 The kinetic energy related to the relative motion can be represented in spherical coordinates, as detailed in equation (6.1), by making the appropriate substitutions.

Let us split the relative Hamiltonian into rotational and vibrational contributions, viz.,

We now assume that the interactions between the ions stabilize the system at the relative distanceR 0 The diﬀerencey=R−R 0will be such that|y| R 0.

The Hamiltonian for the rotational motion may be approximated as (see Fig 8.3)

The eigenfunctions in quantum mechanics are identified using the quantum numbers l and m_l To determine the energies, the operator L^2 is substituted with its eigenvalues, given by ¯h^2 l(l + 1) Notably, the photon energy associated with transitions between adjacent states rises linearly with the quantum number l.

If the stabilization at R ≈ R₀ is sufficiently effective, we can extend the radial coordinate range from 0 to -∞, as the wave function diminishes for negative R values Additionally, the R² factor in the volume element can be removed from the integrals by substituting Ψ(R) with Φ(R)/R Consequently, the radial Schrödinger equation is transformed into a linear equation similar to those discussed in Chapter 4.

Finally, the Taylor expansion of the potential around the equilibrium positionR 0 , and the replacement of the coordinateRbyy =R−R 0 , yield the harmonic oscillator Hamiltonian discussed in Sects 3.2 and 4.2 (see Fig 8.3)

Fig 8.3 Vibration (dashed line), rotation (dotted lines) and translation of the center of massG(continuous line) in a diatomic molecule

The vibrational states are equidistant from each other (Fig 3.2) The photon spectrum displays the single frequency:

Characteristic Energies

The intrinsic motion of electrons, along with the vibrational and rotational movements of nuclei, is linked to distinct characteristic energies The energy levels associated with intrinsic transitions are comparable to the excitation energies found in atoms.

The potential energy associated with vibrational motion is influenced by the Coulomb interaction, similar to the interparticle distances observed in a system characterized by a Bohr radius (a₀) Consequently, this energy is expected to be of the same order of magnitude as the intraparticle energy (E_intr).

Rotational energies are given by (8.26)

The mass ratio of electrons to protons is approximately 1/2,000, which indicates that the energy transitions between vibrational states fall within an intermediate energy range, distinct from those associated with intrinsic electron properties.

Molecular spectra exhibit vibrational states layered over intrinsic excitations and rotational states superimposed on vibrational states Allowed transitions between these states, as depicted in Figure 8.4, are represented by dotted lines The electromagnetic radiation linked to transitions among intrinsic, vibrational, and rotational states is observed in the visible, infrared, and radiofrequency regions of the optical spectrum.

As the energies of the rotational and vibrational excitations increase, the approximations become less reliable:

• Terms that are functions ofy will appear in the rotational Hamiltonian, coupling the rotational and vibrational motion

• Higher-order terms in the Taylor expansion of the potential become relevant

Approximate Matrix Diagonalizations

When perturbation theory is not applicable, diagonalization becomes essential, especially in the presence of degenerate or closely lying states, such as when adding multiple particles to a closed shell in atomic or nuclear systems Utilizing the symmetries of the Hamiltonian can reduce the size of the matrix that needs to be diagonalized Additionally, if the focus is solely on the ground state and its neighboring states, the problem can be simplified by considering only those states that are energetically close to the ground state.

Incorporating contributions from excluded states into the matrix elements of the Hamiltonian can enhance the diagonalization process This can be effectively achieved through the application of folded diagram techniques.

(a generalization of Rayleigh–Schr¨odinger perturbation theory) [53] or the Bloch–Horowitz procedure (an extension of the Brillouin–Wigner expansion) [54].

An alternative procedure consists in simplifying the expressions for the matrix elements This includes eliminating many of them (see Problem 14).

In such cases, good insight is required in order to avoid distorting the physical problem.

8.5.1 † Approximate Treatment of Periodic Potentials

This example demonstrates the relationship between exact diagonalization and perturbation theory, which can be utilized in more complex scenarios We analyze the same problem discussed in Section 4.6, focusing on the scenario of a small periodic potential.

The free-particle Hamiltonian is defined as H₀ = (1/2M) ˆp², serving as the zero-order Hamiltonian The unperturbed energy levels are illustrated in Fig 8.5a, plotted against the wave number k When the potential V(x) exhibits periodicity, such that V(x) = V(x + d), applying a Fourier transform to the potential produces significant results.

Therefore the nonvanishing matrix elements of the perturbation are k |W n |k=v n ifk −k= 2πn d , (8.34)

The article discusses bands in periodic potentials, illustrating the unperturbed parabolic energies as a function of the wave vector \( k \) In the first graph, the eigenvalue \( E^- \) is presented for the interval \( 0 \leq |k| \leq \pi/d \), while the eigenvalue \( E^+ \) is shown for \( \pi/d \leq |k| \) in the second graph The Hamiltonian matrix is also characterized in this context.

In our analysis, we focus on the nondiagonal terms, which, despite their small magnitude, cannot be addressed through perturbation theory due to their connection between degenerate states Specifically, the state with wavevector k = π/d shares the same unperturbed energy as the state with k = -π/d Therefore, it is essential to perform a diagonalization of the degenerate or quasidegenerate states, necessitating that we set the determinant ¯ h² to zero.

⎠ (8.37) are plotted as a function ofkin Fig 8.5b There are no states in the interval (1/2M) (¯hπ/d) 2 −v 1 ≤E≤(1/2M) (¯hπ/d) 2 +v 1 A gap of size 2v 1 appears in the spectrum, pointing to the existence of two separate bands.

In the region |k| ≈ π/d, the remaining nondiagonal terms v n may be treated as a perturbation, if they are suﬃciently small Unfortunately, they usually are not so in realistic cases.

8.6* Matrix Elements Involving the Inverse of the Interparticle Distance

Although the integrals involved may be found in tables, we calculate them explicitly as a quantum mechanical exercise The inverse of the distance between two particles may be expanded as

3We disregardv 0 since only aﬀects the zero-point energy.

The Legendre polynomial of order l, denoted as P l, is a function of the angle α 12 formed by the two vectors r 1 and r 2 This polynomial can be represented by coupling two spherical harmonics with zero angular momentum.

Next, we evaluate matrix elements such as

The angular integrals restrict the values oflin the summation (see Problem 5 in Chap 5) If at least one of the particles is in an s state, only one l term survives

, if both particles are insstates one obtains the value (8.14) forn 1=n 2= 1.

Quantum textbooks often overlook the issue of quantization with constraints, despite significant advancements in this field over the past 30 years This topic is crucial not only for gauge field theories but also for applications in quantum mechanics, particularly in describing many-body systems in moving reference frames Additionally, it holds conceptual significance regarding the properties of Hilbert spaces.

In this presentation we use two-dimensional rotations to exemplify a description made in terms of an overcomplete set of degrees of freedom This overcompletness requires the existence of constraints.

Intrinsic coordinates refer to a system's measurements within a rotating frame of reference The motion of this moving frame, in relation to the laboratory, is characterized by collective coordinates.

• There is an overcomplete set of angular variables (intrinsic + collective) describing transformations (rotations) of the system.

• The rotations of the system are generated (5.10) by the intrinsic angular momentum ˆL There is also a collective angular momentum ˆI, the generator of rotations of the moving frame.

The classical equations that define momenta through the partial derivatives of the Lagrange function L are unsolvable in this scenario because the function lacks information about the reference frame For example, when considering a single particle moving along a circumference with radius r₀, the Lagrange function can be represented in terms of the angular velocities ˙α and ˙φ.

Here α= tan −1 (y/x) and J =M r 2 0 is the moment of inertia From the equations

∂φ˙ , one obtains the orbital angular momentumLand the constraintf = 0:

Fig 8.6 Intrinsic (x, y) and laboratory (x lab , y lab ) coordinates of a generic point

P The two sets of coordinates are related by a rotation

Equation (8.43) highlights that rotating a particle through a specific angle in a moving frame yields results equivalent to rotating the moving frame in the opposite direction This relationship serves as a mechanical analogy for gauge invariance.

• Our aim is to quantize this classical model The following commutation relations hold:

The artificial enlargement of the vector space leads to the emergence of both physical and unphysical states and operators This situation is encapsulated by the quantum mechanical conditions, which state that fˆϕ ph = 0 and fˆϕ unph = 0.

[ ˆf ,Oˆph] = 0, [ ˆf ,Oˆunph] = 0, (8.45) where the labels ph and unph indicate physical and unphysical states or operators Except in simple cases, this separation is by no means a trivial operation.

• Since the problem displays cylindrical symmetry, there is no restoring force in the intrinsic angular direction and perturbation theory cannot be applied.

8.7.2 † Outline of the BRST Solution

To optimize the problem, one might typically reduce the number of variables to the original count using constraints However, recent advancements have focused on expanding the number of variables and incorporating a more robust symmetry.

The collective subspace is given by the eigenfunctions of the orbital angular momentum in two dimensions (5.35): ϕ m (φ) = 1

The collective coordinate φ, initially presented in Section 8.7.1 as a byproduct of the moving frame concept, is now recognized as a genuine degree of freedom.

The problem at hand has a single real degree of freedom, represented by the collective angle, while all other degrees are unphysical Consequently, a trade-off occurs where the intrinsic coordinate α is shifted to the unphysical subspace In the BRST (Becchi–Rouet–Stora–Tyutin) procedure, this subspace is integrated with auxiliary fields, ensuring that the effects of unphysical degrees of freedom on physical observables cancel out Furthermore, the degree of freedom α gains a finite frequency through its interaction with other spurious fields, making perturbation theory applicable.

Unfortunately, we cannot go beyond this point here An elementary presentation of the quantum mechanical BRST method is given in the second of references [56].

1 Obtain the expression for the second-order correction to the energy in perturbation theory and show that this correction is always negative for the ground state.

2 Calculate the second-order correction to the eigenstate.

Problem 2.Assume that the zero-order Hamiltonian and the perturbation are given by the matrices:

1 Calculate the ﬁrst-order perturbation corrections to the energies.

2 Calculate the second-order perturbation corrections to the energies.

3 Obtain the ﬁrst-order corrections to the vector states.

4 Obtain the second-order corrections to the vector states.

5 Expand the exact energies in powers of c and compare the results with those obtained in perturbation theory:

1 Calculate the ﬁrst- and second-order corrections to the ground state energy of a linear harmonic oscillator if a perturbation V(x) = kx is added, and compare with the exact value.

2 Do the same if the perturbation is V(x) =bx 2 /2.

1 Calculate the lowest relativistic correction to the ground state energy of a linear harmonic oscillator Hint: Expand the relativistic energy

2 Obtain the order of magnitude of the ratio between the relativistic correction and the nonrelativistic value of the ground state energy ifM = 2M p and ¯hω= 4.0 10 −3 eV (a molecular case).

Problem 5.Obtain the vector state up to second order in the Brillouin–Wigner perturbation theory Compare with the results (8.8) and Problem 1.

Problem 6.Show that the Brillouin–Wigner perturbation theory already yields the exact results (3.18) in second order, for a Hamiltonian of the form (3.17).

E a − b|H|b and include the diagonal termp|V|pin the unperturbed energyE p (0)

The Time Principle

This article shifts focus from static situations to the time-dependence of the state vector, introducing a new principle The time-dependent Schrödinger equation is solved precisely for simple spin cases and through perturbation theory Additionally, the concept of transition probability provides a physical interpretation of nondiagonal matrix elements and facilitates the presentation of the energy-time uncertainty relation.

Chapter 1 highlighted the instability of the hydrogen atom as a key reason for the emergence of quantum mechanics, emphasizing that a comprehensive understanding of quantum mechanics must address this fundamental issue To achieve this, an introduction to quantum electrodynamics is essential, covering crucial concepts such as induced and spontaneous emission, laser optics, selection rules, and mean lifetime.

At time t, the system is described by the time-dependent state vector Ψ(t), which evolves according to the equation Ψ(t) = U(t, t)Ψ(t) for t > t The evolution operator U(t, t) is unitary and approaches the identity operator as t approaches t Consequently, for a small time increment ∆t, the state vector updates to Ψ(t + ∆t) = U(t + ∆t, t)Ψ(t).

A new quantum principle must be added to those stated in Chaps 2 and 7.

Principle 5 The operator yielding the change of the state vector over time is proportional to the Hamiltonian

• The time evolution of the system is determined by the ﬁrst-order linear equation i¯h∂

This is called the time-dependent Schr¨odinger equation It is valid for a general state vector, and it is independent of any particular realization of quantum mechanics.

• The evolution is deterministic, since the state vector is completely deﬁned once the initial state is ﬁxed (quantum indeterminacy pertains to measurement processes).

• The evolution is unitary (i.e., the norm of the states is preserved).

• The evolution of the system is reversible.

• If [ ˆH(τ 1),H(τˆ 2)] = 0, the evolution operator is

For a time-independent Hamiltonian that satisfies the eigenvalue equation \( \hat{H}\phi_i = E_i \phi_i \), the solution to the differential equation can be obtained using the separation of variables method This leads to the expression \( \phi_i(t) = f(t)\phi_i \), where \( i\hbar \frac{df}{dt} = E_i f \) The solution for \( f \) is given by \( f = \exp\left(-\frac{iE_i t}{\hbar}\right) \) Consequently, the complete time-dependent wave function is expressed as \( \phi_i(t) = \phi_i \exp\left(-\frac{iE_i t}{\hbar}\right) \).

The solutions of a time-independent Hamiltonian are anticipated to remain constant over time; however, this condition is applicable only up to a phase factor, aligning with the findings presented in equation (9.8).

The constant of proportionality−i/¯hchosen in (9.4) insures that the time- dependent wave function for a free particle with energy E = ¯hω is a plane wave, as expected [see (4.30)]. ϕ ±k (x, t) =Aexp i(±kx−ωt)

Time-Dependence of Spin States

Larmor Precession

To begin with we give a simple but nontrivial example of a solution to (9.5).

We use as Hamiltonian the interaction (5.16), with the magnetic ﬁeld directed along thez-axis The evolution operator (9.6) is given by

Hˆ z =−ω L Sˆ z , (9.12) where ω L ≡à ν g s B/¯h is called the Larmor frequency [see (5.25)] We have used (5.22) in the expansion of the exponential in (9.12).

The time evolution is given by Ψ(t) c ↑ (t) c ↓ (t)

If the system's state is an eigenstate of the operator ˆS z at t=0, it will remain in that state indefinitely, making equation (9.13) a specific instance of equation (9.8) Conversely, if the spin is initially aligned in the positive x-direction, the initial values are c ↑ = c ↓ = 1/√2.

The probability of ﬁnding the system with spin aligned with thex-axis (or in the opposite direction) is cos 2 (ω L t/2) [or sin 2 (ω L t/2)].

1This evolution is valid only for the Hamiltonian basis Therefore, the expression Ψ(t) i c i φ i exp(−iq i t/¯h) makes no sense if φ i , q i are not eigenstates and eigenvalues of the Hamiltonian.

The expectation values of the spin components are Ψ|S x |Ψ =¯h

The spin undergoes precession around the z-axis, aligned with the magnetic field, at the Larmor frequency ω L in the negative direction (from x to -y) It does not align with the z-axis, indicating that a definite projection of angular momentum along this axis is not established This scenario illustrates true precession, which results in the uncertainty of S z.

Ift1/ω L, we speak of a transition from the initial stateϕ S x =¯ h/2 to the ﬁnal stateϕ S x = − ¯ h/2 with the probabilityω L 2 t 2 /4 In this case, the probability per unit time is linear in time.

If the z-direction is substituted by the x-direction in the Hamiltonian

Magnetic Resonance

We now add a periodic ﬁeld along the x- andy-directions, of magnitudeB and frequencyω, to the constant magnetic ﬁeld of magnitudeBpointing along thez-axis The Hamiltonian reads

Hˆ =−à s BSˆ z −à s B cosωtSˆ x −sinωtSˆ y

Due to the non-commuting nature of this Hamiltonian at different times, we cannot utilize the evolution operator Instead, we must solve the differential equation for the amplitudes c_i(t) While an analytical solution can be derived for any frequency ω, the optimal effect is achieved when ω matches the Larmor frequency ω_L For the purpose of the following derivation, we will assume ω is equivalent to sB.

We try a solution of the form (9.13), but with time-dependent amplitudes. b ↑ (t) b ↓ (t) exp[iωt/2]c ↑ (t) exp[−iωt/2]c ↓ (t)

9.2 Time-Dependence of Spin States 151

The probability of a spin flip is illustrated in Fig 9.1, where the horizontal axis represents the ratio of ω to ω L, and the vertical axis shows the corresponding probabilities A clear resonant behavior is observed when ω equals ω L The parameters for the graphs are as follows: for (a) tω L = 4 and ω/ω L = 1/10, for (b) tω L = 4 and ω/ω L = 1/2, and for (c) tω L = 2 and ω/ω L = 1/2 Additionally, comparing the last two graphs highlights the complementary relationship between time and energy, as referenced in equation (9.34) and Section 9.4.3.

The findings confirm that a spin can flip its orientation; a spin pointing up will ultimately point down, and vice versa The probabilities of the initial spin being maintained or flipped are illustrated in Figure 9.1.

For an arbitrary relation between ω andω L , the probability of a spin ﬂip is given by

Magnetic resonance occurs when the frequency ω closely matches the natural frequency ω_L, resulting in significant effects from a weak magnetic field B In this resonance condition, the interaction with the sinusoidal field cannot be considered a minor perturbation, as the required condition |ω| |ω−ω_L| is not met near resonance.

Magnetic resonance plays a crucial role in aligning spins and has diverse applications across various fields of physics, including the measurement of magnetic moments in elementary particles and the analysis of condensed matter properties Additionally, it serves as a vital tool in quantum computing and medical diagnostics.

Sudden Change in the Hamiltonian

We consider a time-dependent Hamiltonian ˆH such that ˆH = ˆH 0 fort < 0 and ˆH = ˆK 0fort >0, where ˆH 0and ˆK 0 are time-independent Hamiltonians.

We know how to solve the problem for these two Hamiltonians:

The system is initially in the stateϕ i exp(−iE i t/¯h) Fort >0, the solution is given by the superposition Ψ k c k φ k exp(−i k t/¯h), (9.23) where the amplitudes c k are time-independent, as is ˆK 0

The solution must be continuous in time in order to satisfy a diﬀerential equation Therefore, at t= 0, ϕ i k c k φ k −→c k = φ k |ϕ i (9.24)

The transition probability is given by

Time-Dependent Perturbation Theory

Transition Amplitudes and Probabilities

The coupled equations presented in (9.27) are as challenging to solve as those in (9.5), necessitating a perturbation approach Similar to the method outlined in Section 8.1, we introduce the unphysical parameter λ (where 0 ≤ λ ≤ 1) to multiply the perturbation ˆV(t) and expand the amplitudes as c k (t) = c (0) k + λc (1) k (t) + λ²c (2) k (t) + (9.28).

We impose the initial condition that the system be in the state ϕ (0) i (t) at t = 0 This condition is enforced through the assignment c (0) k = δ ki , which accounts for terms independent of λin (9.27).

At time \( t = 0 \), a perturbation is applied to the system, and the objective is to determine the probability of finding the system in a different unperturbed eigenstate \( \phi (0)_{k} \) at time \( t \) The linear terms in the perturbation parameter \( \lambda \) lead to the equation \( \dot{c}^{(1)}_{k} = -i \hbar \langle k | V | i \rangle \exp(i \omega_{ki} t) \) Consequently, the transition amplitudes can be expressed as \( c^{(1)}_{k}(t) = -i \hbar t \).

The transition probability between the initial state i and the ﬁnal state k, induced by the Hamiltonian ˆV(t), is given in ﬁrst-order of perturbation theory as

Constant-in-Time Perturbation

In the interval (0, t), the matrix elements of the perturbation k|V|i are time-independent and vanish otherwise The first-order amplitude is expressed as c (1) k =−k|V|i ¯ hω ki [exp(iω ki t)−1], while the transition probabilities are outlined in equations (9.30) and (9.31).

This result is common to many ﬁrst-order transition processes We therefore discuss it in some detail:

In the case where the final states ϕ k form a continuous set, the transition probability is proportional to the function f(ω) = (4/ω²) sin²(ωt/2) The highest peak occurs at ω = 0, with a height proportional to t², while the subsequent peak at ω ≈ 3π/t is significantly smaller, reduced by a factor of approximately 1/20 Consequently, most transitions occur within the central peak, which is indicative of resonance, while the secondary peaks correspond to diffraction processes.

The total probability is derived by integrating across frequencies, assuming the matrix element remains constant within the primary peak's frequency interval By approximating the peak's surface as an isosceles triangle with a height of 2 and a half-base of 2π/t, we find that the total probability increases linearly over time, indicating a constant probability per unit time interval.

The energy of an excited atomic state can be determined by the frequency of the photon emitted during its de-excitation This concept alters the understanding of eigenvalues for unstable states, moving away from the traditional sharp definition.

Fig 9.2.The functionf(ω) as a function of the frequencyω

9.4 Time-Dependent Perturbation Theory 155 energy, the existence of the spread is associated with an indeterminacy in the energy on the order of

The uncertainty principle highlights a fundamental relationship between energy and time, revealing inherent limitations in measuring these variables simultaneously A comparable uncertainty arises when determining the energy of excited states through electromagnetic radiation absorption This time-energy relationship was previously foreshadowed in the discussion accompanying Fig 9.1.

• Nondiagonal matrix elementsk|V|iacquire physical meaning, since they can be measured through transition rates.

• If there is a continuum of ﬁnal states, we are interested in summing up the probabilities over the setK of these ﬁnal states (k∈K):

P i→k (1) ρ(E k )dE k , (9.35) where ρ(E k ) is the density of the ﬁnal states 2 Assuming that both

|k|V|i| 2 and ρ(E k ) remain constant during the interval ∆E, and that most of the transitions take place within this interval, then

(9.36) The expression for the transition per unit time is called the Fermi golden rule: dP i→K (1) dt = 2π ¯ h |k|V|i| 2 ρ(E k ) (9.37)

Mean Lifetime and Energy–Time

The transition probability dP/dt for a single system has been established, but when considering N similar systems, such as N atoms, it becomes impossible to predict the decay of an individual system If dP/dt remains time-independent, the overall rate of change can be expressed as dN/dt = -N(dP/dt).

2The density of states is given in (7.21) for the free particle case A similar procedure is carried out for photons in (9.53).

The constant τ, known as the mean lifetime, represents the duration needed for a population to decrease by a factor of 1/e This value serves as an indicator of the uncertainty ∆ regarding the timing of decay events.

In analogy with (2.37), the energy–time uncertainty relation adopts the form

A short mean lifetime implies a broad peak, and vice versa.

In Chapter 1, we highlighted that a key indicator of the early twentieth-century crisis in physics was the classical instability observed in the motion of electrons orbiting the nucleus.

Quantum electrodynamics (QED) serves as a crucial extension of quantum mechanics, demonstrating its ability to address complex problems in the field This article provides a concise introduction to the principles and significance of QED.

This article begins by examining the electromagnetic field in a charge-free environment, specifically focusing on light waves It derives a quadratic expression for energy using canonical variables and proceeds to quantize the theory by substituting these variables with operators that adhere to established relations The discussion then shifts to the interaction between particles and the electromagnetic field, culminating in the resolution of the resulting time-dependent problem through perturbation theory.

9.5.1 † Classical Description of the Radiation Field

In the absence of charges, the classical electromagnetic vector potential A ( r, t) satisﬁes the equation

The vector A can be expressed as the sum of transverse and longitudinal components, with the longitudinal component incorporated into the particle Hamiltonian due to its role in the Coulomb interaction, which does not generate a radiation field The transverse component, A_t(r, t), adheres to the equation div A_t = 0.

It may be expanded in terms of a complete, orthonormal set A λ ( r ) of functions of the coordinates:

9.5 Quantum Electrodynamics for Newcomers 157 where we assume a large volume L 3 enclosing the ﬁeld Inserting (9.43) in (9.41) and separating variables yields the two equations d 2 dt 2 c λ +ω λ 2 c λ = 0, (9.45)

The equation 2 A λ + ω λ^2 c^2 A λ = 0 introduces ω λ as a separation constant The solution to the oscillator equation is expressed as c λ = η λ exp(−iω λ t), where η λ remains constant over time A solution to this equation can be derived from the three-dimensional generalization of the previous equations It is important to note that periodic boundary conditions are applied in this context.

There are two independent directions of polarization v λ , since (9.42) implies v λ ã k λ = 0.

We construct the electric ﬁeld

The total ﬁeld energy is expressed as

0 ω λ a λ has been made Note that since the vector ﬁeld has dimension km s −1 C −1 , the amplitudesc λ have dimension km 5/2 s −1 C −1 and the amplitudesa λ have dimension 1.

9.5.2 † Quantization of the Radiation Field

We derived an expression for the energy of the radiation field that is quadratic in the amplitudes \( a^*_\lambda \) and \( a_\lambda \) To quantize the system, we substitute these amplitudes with creation and annihilation operators \( a^+_\lambda \) and \( a_\lambda \), which adhere to the specified commutation relations This process leads us to the formulation of the Hamiltonian.

• The radiation field is made up of an infinite number of oscillators The state of the radiation field is described by all the occupation numbersn λ

• The oscillators are of the simple, boson, harmonic type introduced in (3.29) and used in Sects 7.4.3 † and 7.8 † , if the quantum radiation ﬁeld is in a stationary state without residual interactions.

• In agreement with Einstein’s 1905 hypothesis, each oscillator has an energy which is a multiple of ¯hω λ The energy of the ﬁeld is the sum of the energies of each oscillator.

The radiation field, being a function defined across all points in space and time, inherently requires an infinite number of canonical variables for its complete description However, by confining the field within a finite volume, L³, we can effectively convert this infinite complexity into a countable infinity.

In a scenario where there is no interaction between particles and the radiation field, vector states can be expressed as products of two distinct Hilbert subspaces The energy, denoted as E b,n 1 ,n 2 , , is derived from the combined contributions of both particle and radiation components, as indicated in equation (7.1): Ψ b,n 1 ,n 2 , = ϕ b (particles) × Π λ 1.

• The number of states up to a certain energyn(E λ ) and per unit interval of energyρ(E λ ) for each independent direction of polarization are 4 n(E λ ) =L 3 k λ 3

• The Hermitian, quantized, vector potential reads

3We ignore the ground state energy of the radiation ﬁeld.

The expressions in this section have been developed using a method akin to that used for equation (7.21) Unlike (7.21), which included a factor of 2 due to spin, this factor is not required in equation (9.53) However, it is reinstated in equation (9.59) to account for the two polarization directions.

The polarization states of photons can be analyzed independently of their wavelength and direction of motion, requiring only two basis states for a complete description, similar to spin A calcite crystal serves as an analogue to the Stern–Gerlach device, where a monochromatic light beam passing through the crystal generates two parallel beams with the same frequency but perpendicular polarization axes This concept has transitioned from theoretical quantum experiments to practical laboratory applications using polarized photons.

9.5.3 † Interaction of Light with Particles

In the presence of an electromagnetic ﬁeld, the momentum ˆ p of the particles 5 is replaced in the Hamiltonian by the eﬀective momentum [46] ˆ p−→p ˆ −eA ˆ t , (9.55)

The various processes can be classified based on the fine structure constant α, whose small value ensures the convergence of perturbation theory In the linear, lowest-order processes, only the perturbation term ˆV is needed, leading to transitions in the unperturbed system of particle plus radiation These transitions occur by altering the particle's state while simultaneously increasing or decreasing the number of field quanta by one unit, resulting in either emission or absorption processes.

We utilize the perturbation theory outlined in Section 9.4.2 to analyze the continuous spectrum of the radiation field This approach allows us to derive the transition probability per unit time, as indicated in equation (9.37) Additionally, it is important to note that energy conservation for the total system is upheld in accordance with the time-energy uncertainty relation.

According to (9.54) and (9.56), the matrix elements of the perturbation read b(n λ + 1)|V|an λ =K λ

√n λ + 1, b(n λ −1)|V|an λ =K λ √ n λ , (9.57) whereK λ is given by

We have neglected the exponential within the matrix element, on the basis of the estimate: k λ r ≈ ω λ a 0 /c =O(10 − 4 ), for ¯hω λ ≈1 eV The third line is derived using the relation ˆp= (iM/¯h)[ ˆH,x] (Problem 9 of Chap 2).ˆ

We next work out the product appearing in the golden rule (9.37):

3c 2 |b|r|a| 2 , (9.59) where we have summed over the two ﬁnal polarization directions and averaged over them.

9.5.4 † Emission and Absorption of Radiation

The transition probabilities per unit time are given by dP an (1) λ →b(n λ −1) dt = 2α|ω λ | 3

3c 2 |b|r|a| 2 (¯n λ + 1), (9.61) for absorption and emission processes, respectively (Fig 9.3) Here ¯n λ is the average number of photons of a given frequency.

Fundamental consequences can be extracted from these two equations:

The likelihood of a photon being absorbed is directly related to the intensity of the radiation field, denoted as ¯n λ This relationship aligns with expectations In contrast, the probability of emission involves two components: the first is influenced by the radiation field's intensity, known as induced emission, while the second component, which is independent of field intensity, enables the atom to transition from an excited state to a lower energy state in a vacuum, referred to as spontaneous emission.

Fig 9.3.The absorption process (9.60) (left) and the emission process (9.61) (right) of electromagnetic radiation Labelsa,bdenote particle states

Conceptual Framework

In a restricted Hilbert space limited to pure basis states, a single qubit can exist in only two states For n qubits, there are 2^n orthogonal product vectors in a 2^n dimensional space, which is the foundation for classical computation Within this framework, linear combinations of vectors are prohibited, allowing only permutations between the basis states, unless the dimensionality of the space is altered.

Quantum mechanics enables the existence of superpositions Ψ (n) of basis vectors ϕ (n) i, characterized by complex amplitudes c i The only constraint on quantum operations is unitarity, ensuring that the norm of the state remains preserved Consequently, classical states and operations represent a minuscule subset compared to the vast array of quantum states and operations This abundance of quantum information unlocks numerous possibilities, allowing for various interference effects and significantly faster calculations through parallel processing of state vector components As a result, the time required to solve complex problems can escalate either exponentially or polynomially with their complexity.

By taking advantage of interference and entanglement, a problem with an exponential increase in a classical computer may be transformed into a problem with a polynomial increase in the quantum case.

Despite the vast information contained in the state vector Ψ(n), extracting meaningful data proves challenging The primary method of obtaining information is through measurement, which connects Ψ(n) to a single probability |ci|² Consequently, the focus is on creating transformations that result in a state Ψ(n) where only a few amplitudes ci remain non-zero.

A quantum process begins with the system prepared in an initial state, followed by a measurement that results in a final state Measurements serve as transformations that yield classical information while irreversibly altering the system's state In contrast, quantum transformations occurring between measurements are unitary operations, which deterministically and reversibly modify the system's state A quantum algorithm is essentially a sequence of these unitary operations, often represented as quantum gates.

This chapter highlights distinctive aspects of quantum mechanics, particularly focusing on the concept of entanglement Entanglement is crucial for understanding the validity and interpretation of quantum mechanics, as discussed in subsequent sections of the text Over recent years, it has gained significant attention in the field.

20 years, a central tool in the emerging ﬁeld of quantum information.

Another important feature, also alien to classical physics, is the no-cloning theorem (Sect 10.3 † ).

This article focuses on recent advancements in quantum information, utilizing the foundational knowledge of quantum mechanics that readers are expected to possess Key examples include quantum cryptography, teleportation, and quantum computation, which will be explored in detail in the respective sections.

Applications in quantum mechanics utilize two-dimensional Hilbert spaces, as discussed in Section 5.2.2 Specifically, for a single qubit, the basis states can be represented using two-component column vectors, exemplified by the state vector ϕ = [1, 0].

Qubits can be implemented using particles with spin s = 1/2, the two polarization states of photons, or isolated energy levels While the optimal choice for computational tasks remains uncertain, photons are favored for communication applications.

Entanglement

The Bell States

A complete set of basis states for the two-spin system may be either constructed as products of the states (10.1), or represented by four-component column vectors ϕ (2) 0 =ϕ 0 ϕ 0 ⎛

In quantum computing, the first qubit is typically designated as the control qubit, while the second qubit serves as the target qubit A general quantum state can be expressed in a superposition format as Ψ(2)c = 1.

The Bell states constitute speciﬁc examples of entangled pairs: ϕ B 0 ≡ 1

• Since Bell states are orthonormal, any two-qubit state may be expressed as a linear combination of these states.

• The Bell states are eigenstates of the product operators ˆS z (1) ˆS z (2) and

Sˆ x (1) ˆS x (2) (see Problem 1) These product operators are included among the interactions in the controlling Hamiltonian (10.22), used to manipulate qubits.

The successive introduction of product interactions divides any two-qubit system into the four Bell channels, analogous to how magnetic field interactions split the channels related to a single qubit.

• The two spins in a product operator must be simultaneously measured,since detection of a single spin destroys the entanglement.

Quantum Cryptography

The no-cloning theorem states that a particle's state cannot be copied onto another particle without altering the original state, distinguishing it from classical mechanics For instance, if we have two qubits in pure states ϕ(1) and η(2), a unitary evolution can be represented as ϕ(1)ϕ(2) = Uϕ(1)η(2) If this copying process also applies to another state, φ(1)φ(2) = Uφ(1)η(2), the scalar product of these two processes leads to the equation ϕ|φ 2 = η ϕ|U + U|φ η = ϕ|φ, highlighting the inherent limitations of state duplication in quantum mechanics.

The equation presents two solutions, 0 and 1, indicating that either ϕ equals φ or they are mutually orthogonal, which establishes that a general quantum cloning device is unachievable Furthermore, even with the allowance of nonunitary cloning devices, cloning nonorthogonal pure states remains infeasible unless one accepts a finite loss of fidelity in the cloned states These findings also extend to general qubits.

Traditional methods for securing cryptographic key distribution rely heavily on human factors, making their security challenging to evaluate Consequently, these methods have largely been supplanted by cryptosystems, where cryptographic keys are conveyed using binary sequences of 0s and 1s The current security of these systems hinges on the inability of classical computers to efficiently factor large numbers However, this security may be threatened by the rise of quantum computing, prompting ongoing research into more secure key transmission methods This section demonstrates that quantum key distribution is theoretically unbreakable, with its security rooted in fundamental principles of quantum mechanics.

The BB84 protocol involves an encoder, known as Alice, who sends particles in specific eigenstates, labeled as ϕ 0, ϕ 1, η 0, and η 1 The decoder, referred to as Bob, uses a Stern–Gerlach apparatus that can be aligned along either the z- or x-direction For example, if Alice transmits three qubits polarized in the directions ↑z, ↑x, and ↓x, and Bob aligns his apparatus to the z-direction for the first qubit and to the x-direction for the last two, he will accurately detect the qubits in channels ϕ 0, η 0, and η 1, which are considered good qubits due to the matching orientations Conversely, if Alice sends a qubit in the ϕ 0 state while Bob's apparatus is oriented in the x-direction, he will detect either η 0 or η 1 with equal probability, categorizing these as bad qubits where the sender and receiver are misaligned.

Alice records the eigenvalue and orientation of each qubit she sends, while Bob randomly selects his orientations and communicates them to her With this information, Alice identifies the good qubits that will be used to encode the message Notably, their communication can occur over an open phone, as it contains no information that could be useful to an eavesdropper.

In the scenario where an eavesdropper, known as Eve, utilizes a Stern–Gerlach apparatus, she is unable to clone and resend messages to Bob; however, she can detect a qubit and retransmit it through the same channel Despite her ability to eavesdrop, both Alice and Bob will be aware of her interference For example, if Alice sends a qubit in the ϕ 1 state while Eve's apparatus is oriented in the x-direction and Bob's in the z-direction, Eve's measurement will project the qubit into one of the η channels, consequently increasing the likelihood that Bob will detect the qubit in the ϕ 0 channel.

Bob randomly selects a subset of good qubits and shares his measurement results with Alice publicly If Alice notices discrepancies between her records and Bob's message, it indicates potential interference However, if there are no discrepancies, the remaining good qubits ensure a perfect secret between Alice and Bob Quantum cryptography relies on the principle that measuring quantum states alters them, unless the observer is aware of the observables that can be measured without disturbance Consequently, Eve cannot succeed in her attempts to intercept the communication without knowing the shared basis between Alice and Bob.

Commercial equipment for bank transfers by means of quantum cryptography is available, within city boundaries.

Teleportation

Alice and Bob are separated by a macroscopic distance, with Alice's particle initially in the state Ψ c = c 0ϕ 0 + c 1ϕ 1 Their goal is to place Bob’s particle in the same state without physically transporting it or transmitting any classical information regarding the amplitudes Both Alice and Bob begin with one of two qubits prepared in a Bell state, specifically ϕ B 0 Consequently, Alice possesses two qubits: one in the state Ψ c and the other in the Bell state, leading to a three-qubit system that can be expressed in a coherent manner.

10.5 Teleportation 173 Bell analyzer entangled pair

2 bits of classical information single qubit transformation Ψc Ψc ϕ B0

Fig 10.2 Scheme illustrating the process of teleportation Dashed lines represent entangled qubits in the stateϕ B 0 , while thedotted lineindicates that classical information is transmitted Ψ (3) = Ψ c ϕ B 0

In this scenario, Bob has explicitly separated the qubit he received from the Bell state, while the two qubits held by Alice are represented in terms of Bell states.

Alice filters her two qubits into a specific Bell state while Bob's qubit is simultaneously projected into a defined state, although he overlooks its connection to the initial state Ψ c To reconstruct the original qubit, Bob must know the Bell state into which the system has collapsed Alice must communicate this information to Bob through conventional means, adhering to the speed limit of light.

Alice can construct the state Ψ c by filtering the spin and sharing the alignment axis information with Bob, enabling him to filter the particle in the same direction This method still offers advantages in teleportation, confirming the benefits of the process.

• The teleported state Ψ c might not be known by Alice If she attempts to measure it, the state of the qubit could be changed.

In quantum teleportation, Bob gains full knowledge of Alice's qubit, but this comes at the cost of the original qubit being destroyed, illustrating the no-cloning theorem.

The qubit Ψ c is defined by the amplitudes c 0 and c 1, with transmission time increasing as precision requirements grow Quantum experiments yield discrete results, while initial entanglement facilitates the conversion of discrete Bell state information into continuous qubit state information.

• The information is transmitted no matter how far distant and how long ago the entanglement took place.

Quantum teleportation was discovered in 1993 [63] It was observed for the ﬁrst time in 1997 with entangled photons [64].

The factorization procedure exemplifies the untapped potential of quantum computers, highlighting their ability to challenge the security of commonly used encryption methods Currently, the security of these codes relies on the difficulty classical computers face in efficiently breaking down a large number N into its prime factors.

Quantum calculations necessitate the use of specialized algorithms and universal quantum gates The factorization algorithm is detailed in Section 10.6.1, while commonly used quantum gates are presented in Section 10.7.

The factorization of a number into its prime components relies on the property that if a number \( a \) is coprime with \( N \) (meaning they share no common factors), we can define the function \( f_a(N) \equiv a^J \mod N \) This function exhibits at least two significant properties that are crucial for understanding its behavior in modular arithmetic.

• It is periodic For instance, ifa = 2, N, the successive values of the functionf are 1,2,4,8,1,2, and so on Thus, the period P = 4.

• Provided that P is even, the greatest common divisors of the pairs (a P /2 + 1, N) and (a P /2 −1, N) are factors of N In the present example, they are 5 and 3, respectively.

The complexity of calculating the period with a classical computer is comparable to that of other factorization algorithms; however, there is a notable quantum algorithm developed by Peter Shor that significantly enhances this process.

A quantum register of size n consists of a collection of qubits, enabling the storage of binary information represented by the numbers J = 0, 1, , (2^n - 1) For example, a 2-qubit register can hold the values 0, 1, 2, and 3 In quantum operations, we utilize a control register on the left and a target register on the right, initiating processes with both registers in a defined state.

2The contents of this section have been mainly extracted from [65].

10.6 Quantum Computation 175 state withJ = 0 [all qubits in theϕ 0 state (10.33)]: Ψ (n) 1 =ϕ (n) 0 ϕ (n) 0 (10.12)

• Load the control register with the integer series (10.34): Ψ (n) 2 = 2 − n/2

• Select a value fora(coprime toN) and for each value ofJ replaceϕ (n) 0 in the target register byϕ f aN (J) , as in (10.35): Ψ (n) 3 = 2 −n/2

Forn= 4 and the previous example, Ψ (n) 3 takes the value

To measure the target register, we obtain a value χ = f aN (J), which eliminates information about other values As outlined in equation (2.19), we only keep the terms ϕ (n) J in the control register that are multiplied by ϕ (n) χ, resulting in Ψ (n) 4 P.

⎠ϕ (n) χ , (10.16) where we have assumed that 2 n /P is an integer, 3 as in (10.15).

• The residueqmust be eliminated in order to ﬁnd the period To do so we perform a Fourier transform on the control register [see (10.36)]: Ψ (n) 5 =U FT (ctrl) Ψ (n) 4

The Fourier transform constitutes a very eﬃcient quantum operation, since it takes full advantage of quantum parallel processing The last step in

3The procedure can be extended if this is not the case.

The equation (10.17) depends on the elimination of the factor \( r < \frac{2n}{P} \) and \( r = 0 \) when \( K \) is either zero or an integer multiple of \( \frac{2n}{P} \), given that \( P \) is an integer divisor of \( 2n \) Consequently, the Fourier transform represented in equation (10.15) results in \( c_{i0} \phi^{(4)}_0 + c_{i4} \phi^{(4)}_4 + c_{i8} \phi^{(4)}_8 + c_{i12} \phi^{(4)}_{12} \phi^{(4)} \chi \) (10.19), where all indices in the control register are multiples of the period.

• Repeat the operation until the period becomes established.

The numberν of bit operations required to factor the number N with a classical computer is expected to increase withN no less rapidly than ν(N) = exp

, (10.20) where L = log 2 N is essentially the number of bits required to represent

N The number ν q of universal quantum gates needed to implement Schor’s algorithm has been estimated to be ν q (N) =L 2 (log 2 L) (log 2 log 2 L) (10.21)

The factorization process shifts from an exponential time complexity to a polynomial one, significantly enhancing efficiency With an estimated 300 microseconds per gate operation, factoring a 309-digit number using Shor’s algorithm may take weeks, comparable to classical computing estimates from 2009 based on projected improvements In stark contrast, factoring a 617-digit number could take months for a quantum computer, while a classical computer would require approximately 60 million years.

Recent advancements in experimental techniques have enabled the manipulation of spins in two-level quantum systems and the implementation of quantum gates However, scaling up to a large-scale quantum computer presents significant challenges, particularly due to decoherence caused by interactions with the surrounding environment As of now, the notable achievements in quantum computation have been restricted to the factorization of the number 15 into its prime components, 3 and 5.

Two-Slit Experiments

Since Thomas Young established the wave nature of light in 1801 (using candles as sources of light), two-slit interference experiments have become

1Intensive use has been made of [68] and [69].

Measurements and Alternative Interpretations

Old Quantum Theory (1900 ≤ t ≤ 1925)

Radiation

We now construct the matrix elements (2.11) for both sides of (3.30), making use of two eigenstatesϕ i ,ϕ j : i|[H, a + ]|j= (E i −E j )i|a + |j= ¯hωi|a + |j (3.32)

We conclude that the matrix elementi|a + |j vanishes, unless the diﬀerence

The energy difference between two eigenstates, denoted as E_i - E_j, is a constant value of ℏω This allows us to sequentially arrange the eigenstates linked by the operator a⁺, with each pair of consecutive energies differing by ℏω Additionally, this relationship enables us to assign an integer number n to each eigenstate.

Sincea, a + are Hermitian conjugate operators, we may also write n+ 1|a + |n=n|a|n+ 1 ∗ (3.33) Finally, we expand the expectation value of (3.31):

This is a ﬁnite diﬀerence equation in y n =|n+ 1|a + |n| 2 , of the type 1 y n −y n−1 Its solutions are

In the equation |n+ 1|a + |n| 2 = n + c, where c is a constant, the left-hand side is positive definite, indicating that the quantum number n has a lower limit, which we can set to n = 0, corresponding to the ground state ϕ0 Consequently, the matrix element 0|a + |−1 must vanish, determining the constant c to be 1 Thus, according to the relation −1|a|0 = 0, we find that aϕ0 = 0, demonstrating that the ground state is annihilated by the operator a, known as the annihilation operator.

The whole set of orthogonal eigenstates may be constructed by repeatedly applying the operatora + , the creation operator. ϕ n = 1

These states are labeled with the quantum numbern They are eigenstates of the operator ˆn=a + a, the number operator, with eigenvaluesn: ˆ nϕ n = 1

The factor 1/√ n! ensures the normalization of the eigenstates.

In order to ﬁnd the matrix elements of the operators ˆx and ˆp, we invert the deﬁnition in (3.29): ˆ x ¯h

, (3.39) and obtain the nonvanishing matrix elements n+ 1|x|n=n|x|n+ 1 ¯h

Fig 3.2 Harmonic oscillator potential and its eigenvalues All energies are given in units of ¯hω The dimensionless variableu=x/x chas been used

Substitution of (3.39) into the Hamiltonian yields

, (3.42) where the operator ˆnhas the quantum number n(=0,1,2, ) as eigenvalues. The Hamiltonian matrix is thus diagonal, with eigenvalues E n represented in Fig 3.2 n|H|n=E n = ¯hω n+1 2

Creation and annihilation operators play a crucial role in many-body quantum physics and quantum field theory These operators facilitate the representation of the creation and destruction of various particles, including phonons, photons, and mesons, making them essential tools in the study of quantum systems.

Quantum mechanics offers a clear derivation of the matrix element properties through the fundamental commutation relation, applicable to any scenario involving two operators that adhere to this relation, particularly when the Hamiltonian is quadratic in these operators.

3.2.2 Some Properties of the Solution

In the following we use this exact, analytical solution of the harmonic oscillator problem to deduce some relevant features of quantum mechanics 4 The

While many examples in this article may suggest that most quantum problems are analytically solvable, it is important to recognize that this is not the case Most quantum challenges necessitate a deep understanding of physics for approximation and often require significant computational resources The exploration of the spatial aspects of the harmonic oscillator problem will be addressed in Section 4.2.

The classical equilibrium position at x=p=0 contradicts the uncertainty principle, as it suggests that both coordinate and momentum can be precisely determined at the same time By substituting ∆x in equation (3.26) with the expression in (3.27), we derive the concept of zero-point energy as outlined in equation (3.43).

The minimum energy of a harmonic oscillator, represented as 2¯hω, is a fundamental quantum effect that predates the formal establishment of quantum mechanics In 1924, Roger Mulliken demonstrated that incorporating this concept significantly improved the correlation with experimental data on vibrational spectra of molecules composed of different isotopes of the same element The implications of zero-point energy extend to various phenomena, including the explanation of intermolecular Van der Waals forces and theories regarding the substantial effects of electromagnetic vacuum in the context of infinite harmonic oscillators.

• By using the closure property (2.47) and the matrix elements (3.40) and (3.41), one obtains the matrix element of the commutator [ˆx,p]:ˆ n|[x, p]|m=n|x|n+ 1n+ 1|p|m+n|x|n−1n−1|p|m

The matrix elements of the operators ˆx 2 and ˆp 2 may be constructed in a similar way:

2 , (3.46) which implies the equality between the kinetic energy and potential expectation values (virial theorem).

Applying the deﬁnition of the root mean square deviation ∆Q given in (2.21), the product ∆x∆pyields

The inequality illustrates the uncertainty principle, highlighting the close relationship between the commutation relation of two operators and the uncertainties involved in measuring their corresponding physical quantities.

5The procedure is only expected to yield correct orders of magnitude It is a peculiarity of the harmonic oscillator that the results are exact.

In quantum mechanics, the invariance under the parity transformation (x → -x) is crucial, as it indicates that both kinetic energy and harmonic oscillator potential energy remain unchanged This fundamental property is highlighted by the relevant commutation relation, underscoring the significance of parity in the behavior of quantum systems.

The relationship between the operators ˆH and ˆΠ allows for the simultaneous determination of their eigenvalues In this context, the eigenstates of the harmonic oscillator Hamiltonian also serve as eigenstates of the parity operator ˆΠ The eigenvalues of ˆΠ are defined by the requirement that ˆΠ² has a single eigenvalue of π² = 1, indicating that the system remains unchanged after two applications of the parity transformation Consequently, the operator ˆΠ has two eigenvalues: π = ±1 The corresponding eigenfunctions are classified as either even functions, which remain unchanged under the parity transformation (π = 1), or odd functions, which change sign (π = -1) This behavior is exemplified in the harmonic oscillator, where the operators a⁺ and a change sign under parity, leading to the relation Πˆϕₙ = (-1)ⁿϕₙ for states labeled by the quantum number n.