quantum mechanics

Chapter 1 Contents 1Fundamentals of Quantum Mechanics 1 1.1 Introduction 1 1.1.1 Why should we study quantum mechanics? 1 1.1.2 What is quantum theory? 2 1.2 Pachycephalic Quantum Mechanics 3 1.2.1 Schr¨odinger equation for a particle 3 1.2.2 Normalization of the wave function 5 1.2.3 Distinction between the classical wave and the matter wave 6 1.2.4 Statistical Interpretation of the Wave Function: Born’s postulate 6 1.2.5 Particle Flux and Probability Conservation 7 1.3 The Many Faces of a Quantum State 9 1.3.1 Fourier transforms and Dirac’s delta function 9 1.3.2 Transformation from the position space to the momentum space . 11 1.3.3 Momentum wave function and probability distribution 12 1.3.4 The momentum operator 13 1.3.5 State representation in terms of the energy eigenstates 14 1.3.6 Meaning of the expansion coefficients 15 1.4 State Vectors 17 1.4.1 Concept of a state vector 17 1.4.2 Representation of a state vector 17 1.4.3 General properties of the state vectors 18 1.5 Observables As Hermitian Operators 19 1.5.1 Definition of a Hermitian conjugate 19 1.5.2 Examples of Hermitian conjugates 20 1.5.3 Hermitian operator 21 1.5.4 Corollary 22 1.5.5 Implication of the corollary 22 1.6 Matrix Representation of a Physical Observable 22 1.6.1 Hermitian matrix 23 1.6.2 Product of operators 24 1.6.3 Expectation value 25 1.6.4 Examples of a continuous basis set 25 1.7 Eigenvalues and Eigenstates of a Physical Observable 25 1.7.1 Definition 25 1.7.2 Properties of an eigenstate of an operator A 26 1.7.3 Theorem 26 i 1.7.4 Orthogonality theorem 27 1.7.5 Gram-Schmidt orthogonalization procedure 27 1.7.6 Physical meaning of eigenvalues and eigenstates 28 1.7.7 Eigenstate expansion and probability distribution 28 1.7.8 Important examples of eigenstates 30 1.8 Commutative Observables and Simultaneous Measurements 32 1.8.1 Commutation bracket 32 1.8.2 Commutative operators 32 1.8.3 Theorem 1 32 1.8.4 Example 33 1.8.5 Theorem 2 34 1.8.6 Implications of the theorems 35 1.9 The Uncertainty Principle 36 1.9.1 The Schwartz inequality 36 1.9.2 Proof of the uncertainty principle 37 1.9.3 Applications of the uncertainty principle 38 1.10 Examples 41 1.10.1 The Gaussian wave packet 41 1.10.2 Fourier transform of the Yukawa potential 45 1.10.3 Interference and beat 46 1.10.4 Constant of motion 48 1.10.5 The inversion symmetry 49 1.10.6 The Virial Theorem 51 1.10.7 Translational Symmetry Group 53 1.11 Problems 58 Chapter 1 List of Figures 1.1 The real part of a Gaussian wave function and its envelope. 42 1.2 Probability density from a Gaussian wave function. 42 1.3 Propagation of a Gaussian wave packet from left to right. 43 iv LIST OF FIGURES Chapter 1 Fundamentals of Quantum Mechanics If a man will begin with certainties, he shall end in doubts; But if he will be content to begin with doubts, he shall end in certainties —Francis Bacon, Advancement of Learning. 1.1 Introduction 1.1.1 Why should we study quantum mechanics? Quantum mechanics used to be the province of atomic, molecular, nuclear, and particle physics. In the last four decades, a wide range of development in basic science in astro- physics, cosmology, quantum optics, condensed matter, chemistry, and materials science and rapid progress in device technology, such as transistors, lasers, magnetic resonance imaging, scanning tunneling microscope, optical tweezers and the Hubble telescope, have made quantum mechanics the fundamental pinning of much of our civilization. Even the remarkable development of the classical nonlinear dynamics in the 20th century was rooted in the appreciation of the conceptual and methodology progress in quantum sta- tistical physics and quantum field theory. The current development of nanoscience in physics, chemistry, biology and materials science elevates the importance of mesoscopic physics, a meeting ground of the microscopic and the macroscopic, where not only one must understanding quantum mechanics but one must also have a clear comprehension of its influence on the macroscopic outcome. Schr¨odinger’s cat is no longer merely part of the gedanken parlor games of the fundamentalists in quantum mechanics. Einstein- Podolsky-Rosen paradox has evolved into “teleporting”, quantum computing and cryp- 1 2 Chapter 1. Fundamentals of Quantum Mechanics tography. The availability of lasers and of nanostructures of semiconductors has led to experimental demonstrations of simple quantum mechanical processes which used to be subjects of theoretical arguments and only whose consequences in atoms or molecules are observed. We are no longer content with merely investigating quantum processes in nature. We now strive to trap atoms, to fabricate designer nanostructures and to control the outcome of the quantum processes. These are today the many reasons why an educated person should understand quantum mechanics. It is even more so the case for a physical scientist or an engineer. 1.1.2 What is quantum theory? Quantum theory consists in states, observables, and time evolution. In this chapter, we set up the framework of the quantum theory starting with the familiar wave mechanics governed by the Schr¨odinger equation. We shall adopt the axiomatic approach of taking the Schr¨odinger equation as given and follow Born in giving the wave function a definite meaning. Via the various representations of the state in terms of the position, momentum and energy, we abstract the state as a vector in a space of infinite dimension, independent of any representation. In classical mechanics, every dynamical property of a system is a function of the positions and momenta of the constituent particles and of time. Hence, a dynamical property is an observable quantity. In quantum theory, we have a prescription to trans- late a classical property to an operator acting on a wave function. The outcome of a measurement of a property can only be predicted statistically unless the system is in an eigenstate of the operator associated with the property. Some pairs of properties, such as the position and momentum in the same direction, cannot be measured simultaneously with arbitrarily small uncertainties, thus obeying the uncertainty principle. Other pairs are not restricted by the uncertainty principle. In this chapter, we consider the general theory of the physical observables. We wish to gain a clear picture of what happens after the measurement of a property. It will also be possible to decide which pair of observables is restricted by the uncertainty principle and which pair is not. The time evolution of the state or the observables will be studied in the next chapter. 1.2. Pachycephalic Quantum Mechanics 3 The simplicity of the structure of quantum theory belies the rich texture and the depth of the theory, the multitude of microscopic phenomena within its grasp, and the subtlety of the connection to the macroscopic world. The latter are the topics of the rest of the course. 1.2 Pachycephalic Quantum Mechanics Pachy — from the Greek word pachys, meaning thick. Cephalic —pertaining to the head. Thus, pachycephalosaurus is the name given to a dinosaur with a skull bone nine inches thick. The moniker “Pachycephalic Quantum Mechanics” imitates the old course popu- larly known as “Bonehead English”. Youhave perhaps seen an attempt to establish wave mechanics in an introductory course. On the way, you might have gone through a lot of arguments purporting to show the reasonableness of the extrapolations from classical mechanics. Such an exercise is valuable in giving physical meaning to the new quantities and equations. For a second course, we can adopt a simpler route to quantum mechanics. Table 1.1 gives a recipe, with one column listing the ingredients in classical mechanics and another column transcribing them to quantum mechanics. One may take the attitude that no amount of arguing about the reasonableness of the procedure is as conclusive as applying the clear recipe to various systems and comparing the results to observation. A loftier treatment than the recipe approach is ‘axiomatic’ quantum mechanics. It sets down axioms or postulates and derive the Schr¨odinger equation from them. Such an approach will likely obscure the physical picture of the wave mechanics. Although we shall not have an exposition of axiomatic quantum mechanics, it is comforting to know of its existence. You can get a flavor of it from the book [1] in the bibliography at the end of the chapter. 1.2.1 Schr¨odinger equation for a particle For simplicity, consider a point particle with mass m. Extension to a system of many particles will be done later. Associated with the particle is a wave function Ψ(r, t) from which we shall deduce the properties of the particle. The time evolution of the wave 4 Chapter 1. Fundamentals of Quantum Mechanics Table 1.1: Table of properties in classical mechanics and corresponding ones in quantum mechanics. Property classical quantum Position rr State path r(t)or wave function Ψ(r, t), probability phase space (p, r) density at r = |Ψ(r, t)| 2 Momentum p operator on wave function  P = ¯h i ∇ Energy E operator i¯h ∂ ∂t Potential energy V (r) V (r) Hamiltonian H = p 2 2m + V (r) H = − ¯h 2 2m ∇ 2 + V (r) as an operator on wave function Equation of motion dr dt = ∂H ∂p HΨ(r, t)=i¯h ∂ ∂t Ψ(r, t), i.e., dp dt = − ∂H ∂r − ¯h 2 2m ∇ 2 Ψ+V Ψ=i¯h ∂Ψ ∂t Property A = A(r, p, t)Operator A Aψ n = α n ψ n , Ψ=  n c n ψ n Probability of finding A to be α n = |c n | 2 function is given by the Schr¨odinger equation in Table 1.1: i¯h ∂ ∂t Ψ(r, t)=  − ¯h 2 2m ∇ 2 + V (r)  Ψ(r, t). (1.2.1) We note some features of the Schr¨odinger equation: 1. It is a linear and homogeneous partial differential equation. In other words, each term contains exactly one power of the wave function Ψ(r, t)orits derivatives. If Ψ 1 and Ψ 2 are two solutions, then any linear combination of them: Ψ=a 1 Ψ 1 + a 2 Ψ 2 (1.2.2) with constants a 1 and a 2 is also a solution. Thus, the matter wave, just like the electromagnetic wave, obeys the superposition principle. That is, two waves can 1.2. Pachycephalic Quantum Mechanics 5 be combined to make another wave. The interference and diffraction phenomena follow immediately. 2. It is a first-order differential equation in time. If the wave function is specified at any instant for all positions, then it is completely determined at all times. 3. It should satisfy the correspondence principle. In the classical limit (where ¯h is unimportant), it is possible to find solutions approaching the Newtonian mechanics. 4. The classical wave equation has real coefficients. The complex representation for the solution is just a convenience. The Schr¨odinger equation has an imaginary coefficient and so the solution is in general complex. 1.2.2 Normalization of the wave function Consider the integral over all space N =  d 3 r|Ψ(r, t)| 2 (1.2.3) where d 3 r denotes the volume element dxdydz. If N =1,the wave function is said to be normalized. If N is finite, the wave function is said to be square-integrable. An integrable wave function is trivially normalized by dividing it with the square root of the integral N. Some wave functions are not square-integrable, e.g., the plane wave. There are at least a couple of ways to deal with them. One way is the so-called box normalization. Take the particle to be in an extremely large box. We are interested in the interior of the box and the boundary condition and the shape of the box are immaterial. For example, consider the plane wave in one dimension. Let the wave function be confined in the interval (−L/2, L/2) where L is enormous compared with the wavelength. Then the plane wave can be normalized by choosing the constant C to be L −1/2 .Weshall see a second way later. 6 Chapter 1. Fundamentals of Quantum Mechanics 1.2.3 Distinction between the classical wave and the matter wave It might be tempting to conclude that wave mechanics is like the classical theory of waves and that the particle nature can be completely explained in terms of the latter. It is, therefore, important to point to a crucial difference between the classical wave and the quantum wave. The classical wave, say the electromagnetic wave, can be widespread spatially. It is possible to make a measurement of the wave at a small locality hardly disturbing the wave elsewhere. Now consider a matter wave representing an electron. The wave can also be widespread so that there can be diffraction. Is it possible that the wave represents the structure of the electron spatially? One can trap an electron in a small locality whereupon there must be no electron wave outside the locality. This is the crucial difference from the classical wave. It also means that the wave cannot represent the spatial structure of the electron. 1.2.4 Statistical Interpretation of the Wave Function: Born’s postulate The classical electromagnetic wave is a measure of the electric or magnetic field. What property of the material particle does the matter wave represent? We have seen that a classical interpretation of the wave as the actual structure of the material particle runs into difficulties. Born suggested that the wave function should be a measure of the probability of finding the particle at r and t. More precisely, ρ(r, t)=|Ψ(r, t)| 2 (1.2.4) is the probability density, i.e., the probability of finding the particle in a small volume d 3 r at time t is ρ(r, t)d 3 r. This definition has the following desirable properties: 1. ρ(r, t)isalways a real positive number. 2. ρ is large where Ψ is large and small where Ψ is small. 3. If the wave function is normalized (or box normalized),  ρ(r, t)d 3 r =1 (1.2.5) [...]... degenerate states with additional quantum numbers such as the angular momentum quantum numbers and include the symmetry considerations of the Hamiltonian For a general state represented by the wave function (1.3.32), let us calculate the mean energy, E= H = Ψ|H|Ψ d3 r Ψ∗ (r)H = ψn (r)cn n = En d3 r Ψ∗ (r)ψn (r)cn n En |cn |2 = n (1.3.34) 16 Chapter 1 Fundamentals of Quantum Mechanics Similarly, the mean... problem The foregoing results are extended by replacing the quantum number n by the continuous variable E and a set of quantum numbers denoted by λ which distinguishes the states with the same energy: Hψλ (r, E) = Eψλ (r, E) (1.3.37) In the case of a spherically symmetric potential, for example, the quantum numbers λ stand for the angular momentum quantum numbers , m, which will be derived in Chapter 4... not orthogonal |ψj = |ψj − |ψi ψi |ψj , ψi |ψi is orthogonal to |ψi , and is also an eigenstate with the same eigenvalue (1.7.18) 28 Chapter 1 Fundamentals of Quantum Mechanics 1.7.6 Physical meaning of eigenvalues and eigenstates In quantum mechanics, we represent an observable property by a Hermitian operator Because the systems in which we are now interested are microscopic, a measurement of the... transform of the wave function 32 Chapter 1 Fundamentals of Quantum Mechanics 1.8 Commutative Observables and Simultaneous Measurements According to Heisenberg’s uncertainty principle, a conjugate pair of dynamic variables in the classical mechanics sense, such as x and px , cannot be measured simultaneously to arbitrary accuracy in the quantum regime A more convenient criterion to determine if a... expansion coefficients holds for any physical property 17 1.4 State Vectors 1.4 State Vectors Dirac [4] developed quantum theory in terms of the concept of the state vector and was able to use it to demonstrate the equivalence between Schr¨dinger’s wave mechanics and o Heisenberg’s matrix mechanics We follow the path of the wave function for a state and make an abstraction of the state as a vector 1.4.1... can also choose a complete set of basis states |q , where q denotes a set of quantum numbers such as x, y, z, or px , py , pz , or E, λ Then, |Ψ = dq|q q|Ψ (1.4.1) The inner product between the two state vectors, q|Ψ , is the probability amplitude of the state |Ψ being found in state |q 18 Chapter 1 Fundamentals of Quantum Mechanics To make the vector nature of the state representation more obvious,... function Ψ(r) 12 Chapter 1 Fundamentals of Quantum Mechanics 1.3.3 Momentum wave function and probability distribution By the operator form for the momentum given in the Table 1.1, we see that a plane wave with wave-vector k is an eigenstate of the momentum, 1 1 h ¯ h h eip·r/¯ = p eip·r/¯ , ∇ 3/2 3/2 i (2π¯ ) h (2π¯ ) h (1.3.15) i.e., the plane wave represents a quantum state which carries a definite momentum... Fundamentals of Quantum Mechanics We have gone full circle from the momentum operator to the momentum value p and back 1.3.5 State representation in terms of the energy eigenstates Energy eigenstates and eigenvalues The time development of the wave function of a particle obeys the Schr¨dinger equation o i¯ h ∂ Ψ(r, t) = HΨ(r, t), ∂t (1.3.25) where H is the Hamiltonian of the particle In classical mechanics, ... (1.2.11) the equation of continuity For electric charges or fluid, the equation of continuity is a consequence of the conservation of charges or matter Equation (1.2.11) is the quantum mechanical analog 1.3 The Many Faces of a Quantum State The wave function Ψ(r, t) which represents the state of a particle is a function of position and time It gives us a measure of the probability distribution of the... transform of the function ψ(x) as given by Eq (1.3.1), then +∞ ψ(x) = −∞ dk ˜ √ eikx ψ(k) 2π Lemma The Fourier transform of a Gaussian function is another Gaussian (1.3.2) 10 Chapter 1 Fundamentals of Quantum Mechanics Proof of the lemma is given by putting a Gaussian function ψ(x) = √ 1 −x2 /4σ2 e 2σ 2 (1.3.3) with a constant σ into Eq (1.3.1) and evaluating the integral by completing the square in the . Contents 1Fundamentals of Quantum Mechanics 1 1.1 Introduction 1 1.1.1 Why should we study quantum mechanics? 1 1.1.2 What is quantum theory? 2 1.2 Pachycephalic Quantum Mechanics 3 1.2.1 Schr¨odinger. evolution of the wave 4 Chapter 1. Fundamentals of Quantum Mechanics Table 1.1: Table of properties in classical mechanics and corresponding ones in quantum mechanics. Property classical quantum Position. understand quantum mechanics. It is even more so the case for a physical scientist or an engineer. 1.1.2 What is quantum theory? Quantum theory consists in states, observables, and time evolution.