An Introduction to Mathematical Modeling: A Course in Mechanics by J. Tinsley Oden Copyright © 2011 John Wiley & Sons, Inc. i ^ H ^ ^ ^ CHAPTER 11 шш^—^^я— DYNAMICAL VARIABLES AND OBSERVABLESIN QUANTUM MECHANICS: THE MATHEMATICAL FORMALISM 11.1 Introductory Remarks We have seen that the solutions of Schrödinger's equations are wave functions Φ = Φ(ζ,ί) that have the property that |Φ| 2 = Φ*Φ is the probability density function giving the probability that the elementary particle under study is at position x at time t (actually that the particle is in the volume between x and x + dx). Moreover, the knowledge of the wave function Φ (or equivalently, the momentum wave function Φ = Φ(ρ, t)) determines completely the dynamical state of the quantum system. We shall now build on these ideas and develop the appropriate mathematical setting for an operator theoretic framework for quantum mechanics that brings classical tools and concepts to the theory. Everything we derive is applicable to functions defined on Rd (d = 1,2,3), but virtually all of the results can be demonstrated without loss of generality on R. The notation for the spatial coordinate An Introduction to Mathematical Modeling: A Course in Mechanics, First Edition. By J. Tinsley Oden © 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc. 115 116 CHAPTER 11. DYNAMICAL VARIABLES AND OBSERVABLES x will be used interchangeably with q (or x with q in M), as q is the classical notation for a generalized coordinate in "phase space" where coordinate-momentum pairs (q, p) = (q t , q2, · · ·, q/v> Pi> Рг> · · · > P/v) define dynamical states. We will frequently treat dynamical variables, such as momentum p, as operators, and when it is important to emphasize the operator character of the result, we will affix a tilde to the symbol (i.e. for momentum p, the associated operator is p = —ihj-). Thus, a function F = F(q, p) is associated with etc an operator F(q 1? q2, ...,qN, -гПщ;, ~ « ^ , · · ·, ~ih^) · The coordinates (ql5 q 2 , . . . , qN) will hereafter be understood to be Cartesian coordinates because the operator notation must represent dynamical quantities in a way that is invariant under a change (e.g. a rotation) of the coordinate axes. Indeed, while ordinary multiplication of functions is commutative, the corresponding operators may not commute, so the qi are interpreted as Cartesian coordinates to avoid ambiguity. 11.2 The Hilbert Spaces L2(R) (or L2(Rd)) and H1{R){orH1{Rd)) Since the wave function must have the property that |Ф| 2 is integrable over R (Ф is "square-integrable"), it must belong to the following space: L2(K) (or L 2 (R d )j is the space of equivalence classes [u] of measurable complex-valued functions it : R —> С equal almost everywhere on Ш (or Rd) (v G [it] =>■ v = и everywhere except on sets of measure zero) such that \u\2 = u*u is Lebesgue integrable on Ш (or Rd). Thus, и G L2(M) implies that и represents an equivalence class [u] of functions equal almost everywhere on R such that JR u2dx < oo. The space L2(R) is a complete inner product space with the inner 11.2 THE HILBERT SPACES L 2 (R) AND Я 1 (К) 117 product of two functions ф and φ in L2(R) defined by (Φ,φ) = [ Φ*φάχ, (11.1) where ф* is the complex conjugate of φ. By a "complete" space, we the mean one in which every infinite sequence of elements {UJ}^, entries of which get closer together in the metric distance of the space as j —> oo (i.e., ||itj — Uk\\ —» 0 as j , к —> oo), always converges to an element и of the space. Such sequences are called Cauchy sequences. Complete inner product spaces are called Hilbert spaces. Thus L 2 (R) is a Hilbert space. The associated norm on L2(R) is then \\ф\\ = у/Щф). (11.2) It can be shown that L2(R) is reflexive (in particular, by the Riesz theorem, for every continuous linear functional / in the dual (L2(R))', there is a unique uj e L2(R) such that f(v) = (v, Uf) and ||/||(L 2 (R))' = ||u/||. Also, L2(M) is separable, meaning that it contains countable everywhere dense sets. In other words, for any и e L2(R) and any ε > 0, there exists an infinite sequence of functions {« n }^li ш £2(М) and an integer M > 0 such that \\un — u\\ < ε for all n > M. We shall demonstrate how such countable sets can be computed given any и in the next section. We are also interested in spaces of functions that have partial derivatives in L2(Rd). This is the Sobolev space, Hl{Rd) = ive L2(Rd) and ψ- e L2(Rd); i = 1, 2 , . . . , N\. (11.3) This is also a Hilbert space with inner product, (Ф,<р)г= [ JRd {νφ*·νφ + φ*φ)άάχ, where ^ i=l dxi dxi ' г г (11.4) 118 CHAPTER 11. DYNAMICAL VARIABLES AND OBSERVABLES and ddx = dx\ dx2 · · · dxa- When we write the L2(]Rd)-inner product of a function with Нф, we regard the derivatives in Я in a generalized sense. So, (φ, Нф) = J ^ ( — ν φ * -νψ + νφ'φ) ddx, (11.5) which is well-defined for Я 1 (Rrf) functions (for smooth V). If ψ and ψ are smooth enough, an integration by parts of (11.5) gives (<Л Нф) = j = ! ψ* ( - — Δ ^ + V φ) ddx ψ*Ηφάάχ. For functions with partial derivatives of order m > 0 in L2(Rd), we analogously define spaces Hm(Rd) with inner products defined in an analogous way. 11.3 Dynamical Variables and Hermitian Operators A dynamical variable is some physical feature of the quantum system that depends upon the physical state of the system. In general, we adopt the convention that the state is described by the position coordinates q and the momentum p (or for a system with iV particles, q l7 q 2 , . . . , qN;p1? p 2 , · · ·,Рлг)· Thus, a dynamical variable is a function Q = Q(4i> 42. · ■ ■. Члг5 Pn P2> ■ ■ ·. Рлг)· Since the momentum components p^ can be associated with the operators Pj = —ih д/dqj, dynamical variables likewise characterize operators V dq} dq2 dqN ) 11.3 DYNAMICAL VARIABLES AND HERMITIAN OPERATORS 119 Analogously, one could define an operator \ σρι σρ 2 ^ΡΝ The expected value or mean of a dynamical variable Q for quantum state Ф is denoted (Q) and is denned as (Q) = f ф*£ф dq I ί Ф*Ф d9 JR ' JR _ (Φ,£Φ) (φ,φ) = <Φ,ρΦ>, (11.6) if (Ф,Ф> = 1. Any operator Л : L 2 (R) —> L2(M) is said to be Hermitian if (V,Ap) = (^,<p) W>,<^GL2(IR). (11.7) In other words, a Hermitian operator has the property / φ*Αψ dq = (Αφ)*φ dq, Jm Ju for arbitrary ф and ψ in L2( Any operator Q that corresponds to a genuine physically observable feature of a quantum system must be such that its expected value (Q) is real; i.e., is real. Therefore, for such Q we must have (φ,<2φ) = (<3Φ,φ) In other words: Уф^ЕЬ2 120 CHAPTER 11. DYNAMICAL VARIABLES AND OBSERVABLES (Technically, we reserve the word "observable" to describe dynamical variables defined by Hermitian operators which have a complete orthonormal set of eigenfunctions, a property taken up in the next section; see Theorem 11 .B.) We note that the variance of a dynamical variable is defined as °Q = (Q2)-(Q)2- (Π.8) The dynamical variable Q takes on the numerical value (Q) with certainty if and only if <7Q = 0. This observation leads us to a remarkable property of those dynamical variables that can actually be measured in quantum systems with absolute certainty, the so-called observables. For a Hermitian operator Q, we have o*Q = (Q2) - (Q)2 = (Q)2 - 2{Q) (Q) + (Q) = = ((Q-(Q))2) ({Q-(Q))*,{Q-(Q))V) = ||(Q-(Q»*H2· (11-9) Thus, if OQ = 0, we have QV = <<Э>Ф. (11.10) This is recognized as the eigenvalue problem for the Hermitian operator Q, with eigenvalue-eigenfunction pair ((Q), Ψ). Therefore, the "fluctuations" (variances) of the dynamical variable Q from its mean (Q) vanish for states Ф which are eigenfunctions of the operator Q, and the expected value (Q) is an eigenvalue of Q. We summarize this finding in the following theorem: 11.4 SPECTRAL THEORY OF HERMITIAN OPERATORS 121 Theorem 11 .A The dynamical variable Q of a quantum system possesses with certainty (with probability 1) the well-defined value (Q) if and only if the dynamical state of the system is represented by an eigenfunction Φ of the Hermitian operator Q : L2(R) —>■ L2(K) associated with Q; moreover, the expected value (Q) is an eigenvalue ofQ, and Q is then an observable feature of the system. A canonical example is the time-independent Schrödinger equation, Ηφ = Еф. Thus, as we have seen earlier, the determinate energy states are eigenvalues of the Hamiltonian. Theorem 11 .A establishes the connection of quantum mechanics with the spectral theory of Hermitian operators. We explore this theory in more detail for the case of a discrete spectrum. 11.4 Spectral Theory of Hermitian Operators: The Discrete Spectrum Returning to the eigenvalue problem (11.10), let us consider the case in which there exists a countable but infinite set of eigenvalues Ajt and eigenfunctions tpk € L2(R), к = 1,2,... for the operator Q. In this case, Q is said to have a discrete spectrum. The basic properties of the system in this case are covered or derived from the following theorem: 122 CHAPTER 11. DYNAMICAL VARIABLES AND OBSERVABLES Theorem 11.В Let (Xk^k), к = 1,2,..., denote a countable sequence ofeigenvalue-eigenfunction pairsfor the Hermitian operator Q : L2(R) -> L2' ; i.e., Q<Pk = Xk4>k, A; = 1,2, (11.11) Then l.lf<$k is an eigenfunction, so also is οψ^,ο being any constant. 2. If(pk , φ\. , . . . , φ\. are M eigenfunctions corresponding to the same eigenvalue Xk, then any linear combination of these eigenfunctions is an eigenfunction corresponding to Afe. 3. The eigenvalues are real. 4. The eigenfunctions φ^ can be used to construct an orthonormal set—i.e., a set of eigenfunctions ofQ such that 1 0 (V?fci ψτη) — акт — if к — m, if к ф va. (11.12) 5. Any state Ф G L (Ш) can be represented as a series, ф (?) = 5^<Wfc(g), (11.13) fc=l where (11.14) Cfc = ( У * , Ф ) , and by (11.13) we mean = 0. lim m-юо fc=l (11.15) 11.4 SPECTRAL THEORY OF HERMITIAN OPERATORS 123 Proof Parts 1 and 2 are trivial. In property 2, the eigenfunctions are said to have a degeneracy of order M. To show 3, we take the inner product of both sides of (11.11) by φ^ and obtain the number Xk = (ч>к,Яч>к)/(<Рк,<Рк), which is real, because Q is Hermitian. To show 4, consider <5<Л = Xkfk Q<^m = Χτηψτη, m Ф к. and Then, (ψτη, Q^Pk) = Afe (ψη, ipk) = (Q<-Pm, φ^ = Am(<^m, yjfc). Thus, (Л* - Аш) (<рт, ч>к) = 0 for m ф к, Хк ф Хт. Finally, to show 5, the eigenfunctions φ^ are assumed to be normalized: (¥>fc,¥>fc) = 1 > A; = 1,2, Thus they form an orthonormal basis for L 2 (R). Equation (11.13) is then just the Fourier representation of Φ with respect to this basis. Equation (11.14) follows by simply computing (^ т с т <Лп,<л) and using Eq. (11.12). n Various functions of the operator Q can likewise be given a spectral representation. For instance, if Q^fe = Xkipk, then QVfe = Q{Qipk) = QXk<pk = XkQ^Pk = AJkVfc, and, in general, (Q)Vfc = Afc^fc- 124 CHAPTER 11. DYNAMICAL VARIABLES AND OBSERVABLES Symbolically, (sin Q)<^fc = {Q- -y<53 H )<Pk = sin(Afc)i^fc, etc. Suppose /] bkfk, и fc=l then sm(Qu) = Y2 &fe(sin h) fkk=\ In general, if Q is a Hermitian operator with discrete (nondegenerate, for simplicity in notation) eigenvalue-eigenfunction pairs (A*, φ^) and F is any smooth function on R, we may write F(<2)u = ^òfcF(Afc)</?fc· (11.16) fc=l For Q : L2(R) -» L 2 (R), (11.16) is meaningful if the series converges; i.e. if the sequence of real numbers, ßn = J2\bkF(Xk)\\ fc=l converges in R. For F(Q) = e^, (eK, 11.5 OBSERVABLES AND STATISTICAL DISTRIBUTIONS 125 this series always converges. In particular, fc=l oo fc=l and |c fc e^ fc | = |ο^|2 —» 0 as к —» oo. 11.5 Observables and Statistical Distributions The statistical distribution of a quantity Q associated with a quantum dynamical system is established by its characteristicfimction a : Ж —> R. In general, the characteristic function is, to within a constant, the Fourier transform of the probability density function ρ associated with a random variable X: oo / ■00 e*xg(x)dx, (11.17) ρ(χ) being the probability of finding X in (x, x + dx). Thus, α(ξ) is the expected value of ειξχ: α(ξ) = (ε*χ). (11.18) The generalization of this case to a Hermitian operator Q on L2(M) reveals, as we shall see, an important property of the coefficients c^ of (11.14). First, we note that if the random variable X can only assume discrete values Xk,k = 1,2,..., and if ρι, ρ2,... are the probabilities of these values, then oo a(0 = X>e***, fe=oo with (11.19) E f c a = i· Returning now to quantum dynamics, as noted earlier, any dynamical variable Q of a quantum system characterized by a Hermitian operator Q on L2(R) with a discrete spectrum of eigenvalues λ^ is an observable, although observables may be described by operators with a continuous spectrum, noted in the next section. The reason for that term is clear 126 CHAPTER 11. DYNAMICAL VARIABLES AND OBSERVABLES from Theorem 11 .A, but the role of the characteristic function α(ξ) gives further information. For the wave function Φ representing the dynamical state of a quantum system, we have β(0 = (φ,ε'^Φ) <Φ,Φ> oo X>e*\ (11.20) fc=oo where now ί?* = Μ 2 / ( Ψ , Φ ) = |(^,Φ)|2/(Φ,Φ). (11.21) Comparing (11.19) and (11.20), we arrive at the following theorem: Theorem 11.С Let Q be an observable of a quantum dynamical system with the discrete spectrum of eigenvalues {Xk}k%i- Then the probability that Q takes on the value Xk is Qk = Ы 2 = |(у>*,Ф)| 2 , where φ^ is the eigenfunction corresponding to Xk and Φ is the normalized wave function, (Φ, Φ) = 1. Thus, the only values Q can assume are its eigenvalues A b λ 2 , . . . , and the probability that Q takes on the value Xk is |cfc|2 = |(у?*, Ф)| . We observe that the discrete probabilities Ok satisfy oo oo fe=l m=l 1 = (Ф, Ф) = ^ Σ Ck(fik, Σ Cmfrn/ к oo т fc=l fc=l oo 11.6 THE CONTINUOUS SPECTRUM 127 as required. 11.6 The Continuous Spectrum Not all dynamical variables (Hermitian operators) have a discrete spectrum; i.e., not all are observables in quantum dynamical systems. For example, the momentum operator p = —ihd/dq is Hermitian but does not have a discrete spectrum and, therefore, is not an observable (in keeping with our earlier view of the uncertainty principle for position and momentum). In particular, the eigenvalue problem for p is (cf. Messiah [37, pp. 179-190]) pU(p\q)=pU(p',q), (11.22) where U(p ,q) is the eigenfunction of p with an eigenvalue A = p , a continuous function of the variable p; i.e., the spectrum of p = — ih-^is continuous. We easily verify that U(p\q) = -^=e^h. (11.23) By analogy with the Fourier series representation (11.13) of the dynamical state ψ for discrete spectra, we now have the Fourier transform representation of the state for the case of a continuous spectrum: 1>(q) = -jL= ί φ(ρ)β*'"Η<Ιρ. (11.24) In analogy with (11.14), we have <р(р') = {и(р',д),Ф(я))- (11.25) The analogue arguments cannot be carried further. Indeed, the function (11.23) is not in L2(M), and a more general setting must be constructed to put the continuous spectrum case in the proper mathematical framework. 128 CHAPTER 11. DYNAMICAL VARIABLES AND OBSERVABLES 11.7 The Generalized Uncertainty Principle for Dynamical Variables Recall that for any dynamical variable characterized by a Hermitian operator Q, its expected value in the state Φ is <д> = (ф,д*), where Q is the operator associated with Q (Q(x,p, t) = Q(x, f J^, t)), and (Ф, Ф) = 1. Recall that its variance is <r2Q = ((Q-(Q))2) = (*dQ-(Q))2*)- (11.26) For any other such operator M, we have σ^ = < Φ , ( Μ - ( Μ » 2 Φ ) = <(M - (Μ))Φ, (M - (Μ»Ψ). Noting that for any complex number z, |z\ 2 > (Im(z)) 2 = (^(z —z*)) , it is an algebraic exercise (see exercise 1 in Set II .2) to show that (11.27) where [Q, M] is the commutator of the operators Q and M: [Q,M]=QM-MQ. (11.28) The generalized Heisenberg uncertainty principle is this: For any pair of incompatible observables (those for which the operators do not commute, or [Q, M] Φ 0), the condition (11.27) holds. We will show that such incompatible operators cannot share common eigenfunctions. The presence of the г in (11.27) does not render the right-hand side negative, as [Q, M] may also involve г as a factor. To demonstrate that (11.27) is consistent with the elementary form of the uncertainty principle discussed earlier, set Q = x and M = p = 11.7 THE GENERALIZED UNCERTAINTY PRINCIPLE FOR DYNAMICAL VARIABLES 129 (h/i) d/dx. Then, for a test state φ, we have hd-φ \χ,ρ\φ = х-г dx = Шф. h d - — (χφ) г dx Hence, 2 or ->?>(И =(Г h σχσν > - , in agreement with the earlier observation. In the case of a Hermitian (or self-adjoint) operator Q with eigenvalue-eigenvector pairs {(A^, ^)}j° = 1 ,we have seen that oo ЯФ = J^AfcCfc^fc, cfc = (φΐι,Ψ). fe=l The projection operators {Pk} defined by РкФ = {ψΐτ, Φ) ψΗ = Ck<Pk, have the property that oo oo к=\ fc=l ф = 1ф = ^2 Ск<рк = Σ Р кФ, so that, symbolically, oo ^ Pk = I = the identity. ( 11.29) fc=l For this reason, such a set of projections is called a resolution of the identity. The operators Q can thus be represented as oo Q= ^\kPk. fc=l It is possible to develop an extension of these ideas to cases in which Q has a continuous spectrum. 130 CHAPTER 11. DYNAMICAL VARIABLES AND OBSERVABLES 11.7.1 Simultaneous Eigenfunctions There is a special property of commuting operators called the property of simultaneous eigenfunctions. Suppose operators Q and M are such that ψ is an eigenfunction of both Q and M; i.e. Μφ = \ψ and Qi\) = μψ. Then QM-φ = ζ)Χφ = Χμφ = Μμφ = MQ^; so (QM - MQ)ip = [Q, М\ф = О. Thus, if operators have the same eigenfunctions, they commute. The converse holds for compatible operators with nondegenerate eigenstates. If {ψη} is a sequence of nondegenerate eigenstates of a Hermitian operator Q, which commutes with the Hermitian operator M, andQ^n = КФпАЬеп 0=(Фт,[М,$]Фп) = (фт, МС}фп) - (фт,ЯМфп) = \„(фт, Мф>п) - \т(фт, = Мфп) (К-\п)(Фт,Мфп). Since the eigenstates are nondegenerate, for n Ф m we have (ψη, Мфт) = 0, which implies that (фп, Мфт) oc δητη, or that Мфт is a constant times фт. Thus, Q and M share eigenstates. For the degenerate case, only a subset of the eigenstates of Q may possibly be eigenstates of M.

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