214 24 Quantum Mechanics: Foundations However the l.h.s. of (24.3) for ψ(r,t) does not necessarily agree pointwise with the r.h.s. for every r, but the identity is only valid “almost everywhere” in the following sense (so-called strong topology): (|ψ n →|ψ⇐⇒ dV |ψ n (r) − ψ(r)| 2 → 0) . The coefficients c i and c(λ)areobtainedbyscalar multiplication from the left with ψ i | and ψ λ |, i.e., c i = ψ i |ψ ,c(λ)=ψ λ |ψ . (24.4) In these equations the following orthonormalisation is assumed: ψ i |ψ j = δ i,j , ψ λ |ψ λ = δ(λ − λ) , ψ i |ψ λ =0, (24.5) with the Kronecker delta δ i,j =1fori = j; δ i,j = 0 otherwise (i.e., j δ i,j f j = f i for all complex vectors f i ), and the Dirac δ function δ(x), a so-called generalized function (distribution), which is represented (together with the limit 9 ε → 0) by a set {δ ε (x)} ε of increasingly narrow and at the same time increasingly high bell-shaped functions (e.g., Gaussians) with ∞ −∞ dxδ ε (x) ≡ 1), defined in such a way that for all “test functions” f(λ) ∈T (i.e., for all arbitrarily often differentiable complex functions f (λ), which decay for |λ|→ ∞ faster than any power of 1/|λ|) one has the property (see Part II): ∞ −∞ dλδ(λ − λ) · f(λ) ≡ f(λ ), ∀f(λ) ∈T . (24.6) This implies the following expression (also an extension of linear algebra!) for the scalar product of two vectors in Hilbert space after expansion in the basis belonging to an arbitrary observable ˆ A (consisting of the orthonormal proper and improper eigenvectors of ˆ A): ψ (1) ψ (2) = i c (1) i ∗ · c (2) i + dλ c (1) (λ) ∗ · c (2) (λ) . (24.7) For simplicity it is assumed below, unless otherwise stated, that we are dealing with a pure point spectrum, such that in (24.7) only summations appear. 9 The limit ε → 0 must be performed in front of the integral. 24.2 Measurable Physical Quantities (Observables) 215 a) However, there are important observables with a purely continuous spec- trum (e.g., the position operator ˆx with improper eigenfunctions ψ λ (x):=δ(x − λ) and the momentum operator ˆp x with improper eigenfunctions ψ λ (x):=(2π) −1/2 exp(iλ · x/); the eigenvalues appearing in (24.2) are then a(λ)=x(λ)=p(λ)=λ). b) In rare cases a third spectral contribution, the singular-continuous con- tribution, must be added, where it is necessary to replace the integral dλ by a Stieltje’s integral dg(λ) , with a continuous and monotonically nondecreasing, but nowhere differen- tiable function g(λ) (the usual above-mentioned continuous contribution is obtained in the differentiable case g(λ) ≡ λ). For a pure point spectrum one can thus use Dirac’s abstract bra-ket formalism: a) “observables” are represented by self-adjoint 10 operators. In diagonal rep- resentation they are of the form ˆ A = i a i |ψ i ψ i | , (24.8) with real eigenvalues a i and orthonormalized eigenstates |ψ i , ψ i |ψ k = δ ik , and b) the following statement is true (which is equivalent to the expansion the- orem (24.5)): ˆ 1= i |ψ i ψ i |. (24.9) Equation (24.9) is a so-called “resolution of the identity operator ˆ 1” by asumofprojections ˆ P i := |ψ i ψ i | . The action of these projection operators is simple: ˆ P i |ψ = |ψ i ψ i |ψ = |ψ i c i . Equation (24.9) is often applied as |ψ≡ ˆ 1|ψ . 10 The difference between hermiticity and self-adjointness is subtle, e.g. instead of vanishing values at the boundaries of an interval one only demands periodic behavior. In the first case the differential expression for ˆp x has no eigenfunctions for the interval, in the latter case it has a complete set. 216 24 Quantum Mechanics: Foundations 24.3 The Canonical Commutation Relation In contrast to classical mechanics (where observables correspond to arbitrary real functions f(r, p) of position and momentum, and where for products of x i and p j the sequential order does not matter), in quantum mechanics two self-adjoint operators representing observables typically do not commute. Instead, the following so-called canonical commutation relation holds 11 : [ˆp j , ˆx k ]:=ˆp j ˆx k − ˆx k ˆp j ≡ i δ jk . (24.10) The canonical commutation relation does not depend on the representa- tion (see below). It can be derived in the wave mechanics representation by applying (24.10) to an arbitrary function ψ(r) (belonging of course to the maximal intersection I max of the regions of definition of the relevant opera- tors). Using the product rule for differentiation one obtains i {∂(x k ψ(r))/∂x j − x k ∂ψ(r)/∂x j }≡ i δ jk ψ(r) , ∀ψ(r) ∈I max ⊆HR. (24.11) This result is identical to (24.10). 24.4 The Schr¨odinger Equation; Gauge Transformations Schr¨odinger’s equation describes the time-development of the wave function ψ(r,t) between two measurements. This fundamental equation is − i ∂ψ ∂t = ˆ Hψ . (24.12) ˆ H is the so-called Hamilton operator of the system, see below, which corre- sponds to the classical Hamiltonian, insofar as there is a relationship between classical and quantum mechanics (which is not always the case 12 ). This im- portant self-adjoint operator determines the dynamics of the system. Omitting spin (see below) one can obtain the Hamilton operator directly from the Hamilton function (Hamiltonian) of classical mechanics by replacing the classical quantities r and p by the corresponding operators, e.g., ˆx|ψ→x · ψ(r); ˆp x |ψ→ i ∂ ∂x ψ(r) . 11 Later we will see that this commutation relation is the basis for many important relations in quantum mechanics. 12 For example the spin of an electron (see below) has no correspondence in classical mechanics. 24.4 The Schr¨odinger Equation; Gauge Transformations 217 For example, the classical Hamiltonian H(r, p,t) determining the motion of a particle of mass m and electric charge e in a conservative force field due to a potential energy V (r) plus electromagnetic fields E(r,t)andB(r,t), with corresponding scalar electromagnetic potential Φ(r,t) and vector potential A(r,t), i.e., with B =curlA and E = −gradΦ − ∂A ∂t , is given by H(r, p,t)= (p − eA(r,t)) 2 2m + V (r)+eΦ(r,t) . (24.13) The corresponding Schr¨odinger equation is then − i ∂ψ(r,t) ∂t = ˆ Hψ = 1 2m i ∇−e ·A(r,t) 2 ψ(r,t) + {V (r)+e · Φ(r,t)}ψ(r,t) . (24.14) The corresponding Newtonian equation of motion is the equation for the Lorentz force (see Parts I and II) 13 m dv dt = −∇V (r)+e · (E + v ×B) . (24.15) Later we come back to these equations in connection with spin and with the Aharonov-Bohm effect. In (24.13) and (24.14), one usually sets A ≡ 0, if B vanishes everywhere. But this is neither necessary in electrodynamics nor in quantum mechanics. In fact, by analogy to electrodynamics, see Part II, a slightly more complex gauge transformation can be defined in quantum mechanics, by which certain “nonphysical” (i.e., unmeasurable) functions, e.g., the probability amplitude ψ(r,t), are non-trivially transformed without changes in measurable quanti- ties. To achieve this it is only necessary to perform the following simultaneous changes of A, Φ and ψ into the corresponding primed quantities 14 A (r,t)=A(r,t)+∇f(r,t) (24.16) Φ (r,t)=Φ(r,t) − ∂f(r,t) ∂t (24.17) ψ (r,t)=exp +ie · f(r,t) · ψ(r,t) (24.18) 13 With the Hamiltonian H = (p−eA) 2 2m one evaluates the so-called canonical equa- tions ˙x = ∂H/∂p x ,˙p x = −∂H/∂x, where it is useful to distinguish the canonical momentum p from the kinetic momentum mv := p − eA. 14 Here we remind ourselves that in classical mechanics the canonical momentum p must be gauged. In Schr¨odinger’s wave mechanics one has instead i ∇ψ,and only ψ must be gauged. 218 24 Quantum Mechanics: Foundations with an arbitrary real function f (r,t). Although this transformation changes both the Hamiltonian ˆ H, see (24.14), and the probability amplitude ψ(r,t), all measurable physical quantities, e.g., the electromagnetic fields E and B and the probability density |ψ(r,t)| 2 as well as the probability-current density (see below, (25.12)) do not change, as can be shown. 24.5 Measurement Process We shall now consider a general state |ψ. In the basis belonging to the observable ˆ A (i.e., the basis is formed by the complete set of orthonormalized eigenvectors |ψ i and |ψ λ of the self-adjoint ( ˆ= hermitian plus complete) operator ˆ A), this state has complex expansion coefficients c i = ψ i |ψ and c(λ)=ψ λ |ψ (obeying the δ-conditions (24.5)). If in such a state measurements of the observable ˆ A are performed, then a) only the values a i and a(λ) are obtained as the result of a single measure- ment, and b) for the probability W( ˆ A, ψ, Δa) of finding a result in the interval Δa := [a min ,a max ) , the following expression is obtained: W ( ˆ A, ψ, Δa)= a i ∈Δa |c i | 2 + a(λ)∈Δa dλ|c(λ| 2 . (24.19) In this way we obtain what is known as the quantum mechanical expectation value, which is equivalent to a fundamental experimental value, viz the aver- age over an infinitely long series of measurements of the observable ˆ A in the state ψ: ( A) ψ (i) = i a i |c i | 2 + dλa(λ)|c(λ)| 2 (ii) = ψ| ˆ A|ψ . (24.20) Analogously one finds that the operator δ ˆ A 2 := ˆ A −(A) ψ 2 corresponds to the variance (the square of the mean variation) of the values of a series of measurements around the average. 24.6 Wave-particle Duality 219 For the product of the variances of two series of measurements of the observ- ables ˆ A and ˆ B we have, with the commutator [ˆa, ˆ b]:= ˆ A ˆ B − ˆ B ˆ A, Heisenberg’s uncertainty principle: ψ| δ ˆ A 2 |ψ·ψ| δ ˆ B 2 |ψ≥ 1 4 ψ| ˆ A, ˆ B |ψ 2 . (24.21) Note that this relation makes a very precise statement; however one should also note that it does not deal with single measurements but with expectation values, which depend, moreover, on |ψ. Special cases of this important relation (which is not hard to derive) are obtained for ˆ A =ˆp x , ˆ B =ˆx with ˆ A, ˆ B = i , and for the orbital angular moments 15 ˆ A = ˆ L x and ˆ B = ˆ L y with ˆ A, ˆ B =i ˆ L z . On the other hand permutable operators have identical sets of eigenvectors (but different eigenvalues). Thus in quantum mechanics a measurement generally has a finite influ- ence on the state (e.g., |ψ→|ψ 1 ; state reduction), and two series of mea- surements for the same state |ψ, but non-commutable observables ˆ A and ˆ B, typically (this depends on |ψ!) cannot simultaneously have vanishing expec- tation values of the variances 16 . 24.6 Wave-particle Duality In quantum mechanics this important topic means that a) (on the one hand) the complex probability amplitudes (and not the prob- abilities themselves) are linearly superposed, in the same way as field amplitudes (not intensities) are superposed in coherent optics, such that interference is possible (i.e., |ψ 1 + ψ 2 | 2 = |ψ 1 | 2 + |ψ 2 | 2 +2Re(ψ ∗ 1 · ψ 2 )); whereas 15 and also for the spin momenta (see below). 16 Experimentalists often prefer the “short version” “. . . cannot be measured si- multaneously (with precise results)”; unfortunately this “shortening” gives rise to many misunderstandings. In this context the relevant section in the “Feyn- man lectures” is recommended, where it is demonstrated by construction that for a single measurement (but of course not on average) even ˆp x and ˆx can simultaneously have precise values. 220 24 Quantum Mechanics: Foundations b) (on the other hand) measurement and interaction processes take place with single particles such as photons, for which the usual fundamental conservation laws (conservation of energy and/or momentum and/or an- gular momentum) apply per single event, and not only on average. (One should mention that this important statement had been proved experi- mentally even in the early years of quantum mechanics!) For example, photons are the “particles” of the electromagnetic wave field, which is described by Maxwell’s equations. They are realistic objects, i.e. (massless) relativistic particles with energy E = ω and momentum p = k . Similarly, (nonrelativistic) electrons are the quanta of a “Schr¨odinger field” 17 , i.e., a “matter field”, where for the matter field the Schr¨odinger equation plays the role of the Maxwell equations . The solution of the apparent paradox of wave-particle duality in quan- tum mechanics can thus be found in the probabilistic interpretation of the wave function ψ. This is the so-called “Copenhagen interpretation” of quan- tum mechanics, which dates back to Niels Bohr (in Copenhagen) and Max Born (in G¨ottingen). This interpretation has proved to be correct without contradiction, from Schr¨odinger’s discovery until now – although initially the interpretation was not undisputed, as we shall see in the next section. 24.7 Schr¨odinger’s Cat: Dead and Alive? Remarkably Schr¨odinger himself fought unsuccessfully against the Copen- hagen interpretation 18 of quantum mechanics, with a question, which we paraphrase as follows: “What is the ‘state’ of an unobserved cat confined in a box, which contains a device that with a certain probability kills it immedi- ately? ” Quantum mechanics tends to the simple answer that the cat is either in an “alive” state (|ψ = |ψ 1 ), a “dead” state(|ψ = |ψ 2 ) or in a (coherently) “superposed” state (|ψ≡c 1 |ψ 1 + c 2 |ψ 2 ). With his question Schr¨odinger was in fact mainly casting doubt on the idea that a system could be in a state of coherent quantum mechanical super- position with nontrivial probabilities of states that are classically mutually 17 Relativistic electrons would be the quanta of a “Dirac field”, i.e., a matter field, where the Dirac equation, which is not described in this book, plays the role of the Maxwell equations of the theory. 18 Schr¨odinger preferred a “charge-density interpretation” of e|ψ(r)| 2 .Butthis would have necessitated an addition δV to the potential energy, i.e., classically: δV≡e 2 RR dV dV |ψ(r)| 2 |ψ(r )| 2 /(8πε 0 |r−r |), which – by the way – after a sys- tematic quantization of the corresponding classical field theory (the so-called “2nd quantization”) leads back to the usual quantum mechanical single-particle Schr¨odinger equation without such an addition, see [24]. 24.7 Schr¨odinger’s Cat: Dead and Alive? 221 exclusive (“alive” and “dead” simultaneously!). Such states are nowadays called “Schr¨odinger cat states”, and although Schr¨odinger’s objections were erroneous, the question led to a number of important insights. For exam- ple, in practice the necessary coherence is almost always destroyed if one deals with a macroscopic system. This gives rise to corresponding quantita- tive terms such as the coherence length and coherence time. In fact there are many other less spectacular “cat states”, e.g. the state describing an object which is simultaneously in the vicinity of two different places, ψ = c 1 ψ x≈x 1 + c 2 ψ x≈x 2 . Nowadays one might update the question for contemporary purposes. For example we could assume that Schr¨odinger’s proverbial cat carries a bomb attached to its collar 19 , which would not only explode spontaneously with a certain probability, but also with certainty due to any external interaction process (“measurement with interaction”). The serious question then arises as to whether or not it would be possible to verify by means of a “quantum measurement without interaction”, i.e., without making the bomb explode, that a suspicious box is empty or not. This question is treated below in Sect. 36.5; the answer to this question is actually positive, i.e., there is a possibility of performing an “interaction-free quantum measurement”, but the probability for an interaction (⇒ explosion), although reduced considerably, does not vanish completely. For details one should refer to the above-mentioned section or to papers such as [31]. 19 If there are any cat lovers reading this text, we apologize for this thought exper- iment. 25 One-dimensional Problems in Quantum Mechanics In the following we shall deal with stationary states. For these states one can make the ansatz : ψ(r,t)=u(r) · e −i Et . As a consequence, for stationary states, the expectation values of a constant observable ˆ A is also constant (w.r.t. time): ψ(t)| ˆ A|ψ(t)≡u| ˆ A|u . The related differential equation for the amplitude function u(r) is called the time-independent Schr¨odinger equation. In one dimension it simplifies for vanishing electromagnetic potentials, Φ = A ≡ 0, to: u = −k 2 (x)u(x) , with k 2 (x):= 2m 2 (E − V (x)) . (25.1) This form is useful for values of x where E>V(x) , i.e., for k 2 (x) > 0 . If this not the case, then it is more appropriate to write (25.1) as follows u =+κ 2 (x)u(x) , with κ 2 (x):= 2m 2 (V (x) −E) . (25.2) For a potential energy that is constant w.r.t. time, we have the following general solution: u(x)=A + exp(ik · x)+A − ·exp(−ik · x)and u(x)=B + exp(+κ · x)+B − · exp(−κ · x) . Using cos(x):=(exp(ix)+exp(−ix))/2and sin(x):=(exp(ix) − exp(−ix))/(2i) 224 25 One-dimensional Problems in Quantum Mechanics we obtain in the first case u(x)=C + · cos(κ ·x)+C − · sin(κ) , where C + = A + + A − and C − =i(A + − A − ) . The coefficients A + , A − etc. are real or complex numbers, which can be determined by consideration of the boundary conditions. 25.1 Bound Systems in a Box (Quantum Well); Parity Assume that V (x)=0 for |x|≥a, whereas V (x)=−V 0 (< 0) for |x| <a. The potential is thus an even function, V (x) ≡ V (−x) , ∀x ∈R, cf. Fig. 25.1. Therefore the corresponding parity is a “good quantum number” (see below). Fig. 25.1. A “quantum well” potential and a sketch of the two lowest eigenfunc- tions. A symmetrical quantum-well potential of width Δx =2anddepthV 0 =1 is shown as a function of x. In the two upper curves the qualitative behavior of the lowest and 2nd-lowest stationary wave functions (→ even and odd parity) is sketched, the uppermost curve with an offset of 0.5 units. Note that the quantum mechanical wave function has exponential tails in the external region which a clas- sical bound particle never enters. In fact, where the classical bound particle has a point of return to the center of the well, the quantum mechanical wave function only has a turning point, i.e., only the curvature changes sign . was in fact mainly casting doubt on the idea that a system could be in a state of coherent quantum mechanical super- position with nontrivial probabilities of states that are classically mutually 17 Relativistic. the observable ˆ A are performed, then a) only the values a i and a( λ) are obtained as the result of a single measure- ment, and b) for the probability W( ˆ A, ψ, a) of finding a result in the interval a :=. with a question, which we paraphrase as follows: “What is the ‘state’ of an unobserved cat confined in a box, which contains a device that with a certain probability kills it immedi- ately? ” Quantum