Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 15
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In the basis-independent Dirac notation, the definition (1.97) of aHermitian-adjoint operator Ô † (t) implies the equality of the matrix elementsha|Ô †(t)|a0 i ≡ ha0 |Ô(t)|ai∗ .(1.173)Thus we can rephrase Eqs. (1.169)–(1.171) in the basis-independent formp̂ = p̂† ,x̂ = x̂† ,(1.174)Ĥ = Ĥ † .The stationary states in Eq. (1.94) have a Dirac ket representation |En i, and satisfythe time-independent operator equationĤ|En i = En |En i.(1.175)H.
Kleinert, PATH INTEGRALS271.4 Dirac’s Bra-Ket Formalism1.4.6Momentum StatesLet us now look at the momentum p̂. Its eigenstates are given by the eigenvalueequationp̂|pi = p|pi.(1.176)By multiplying this with hx| from the left and using (1.164), we find the differentialequationhx|p̂|pi = −ih̄∂x hx|pi = phx|pi.(1.177)The solution ishx|pi ∝ eipx/h̄ .(1.178)Up to a normalization factor, this is just a plane wave introduced before in Eq. (1.75)to describe free particles of momentum p.In order for the states |pi to have a finite norm, the system must be confinedto a finite volume, say a cubic box of length L and volume L3 . Assuming periodicboundary conditions, the momenta are discrete with valuespm =2πh̄(m1 , m2 , m3 ),Lmi = 0, ±1, ±2, .
. . .(1.179)Then we adjust the factor in front of exp (ipm x/h̄) to achieve unit normalization1hx|pm i = √ 3 exp (ipm x/h̄) ,L(1.180)and the discrete states |pm i satisfyZd3 x |hx|pm i|2 = 1.(1.181)|pm ihpm | = 1.(1.182)The states |pm i are complete:XmWe may use this relation and the matrix elements hx|pm i to expand any wavefunction within the box asXΨ(x, t) = hx|Ψ(t)i =mhx|pm ihpm |Ψ(t)i.(1.183)If the box is very large, the sum over the discrete momenta pm can be approximatedby an integral over the momentum space [4].Xm≈d3 pL3.(2πh̄)3Z(1.184)In this limit, the states |pm i may be used to define a continuum of basis vectorswith an improper normalization|pi ≈qL3 |pm i,(1.185)281 Fundamentalsqin the same way as |xn i was used in (1.150) to define |xi ∼ (1/ 3 )|xn i.
Themomentum states |pi satisfy the orthogonality relationhp|p0 i = (2πh̄)3 δ (3) (p − p0 ),(1.186)with δ (3) (p−p0 ) being again the Dirac δ-function. Their completeness relation readsZd3 p|pihp| = 1,(2πh̄)3(1.187)such that the expansion (1.183) becomesΨ(x, t) =d3 phx|pihp|Ψ(t)i,(2πh̄)3Z(1.188)with the momentum eigenfunctionshx|pi = eipx/h̄ .(1.189)This coincides precisely with the Fourier decomposition introduced above in thedescription of a general particle wave Ψ(x, t) in (1.83), (1.84), with the identificationhp|Ψ(t)i = f (p)e−iEp t/h̄ .(1.190)The bra-ket formalism accommodates naturally the technique of Fourier transforms. The Fourier inversion formula is found by simply inserting into hp|Ψ(t)i aRcompleteness relation d3 x|xihx| = 1 which yieldshp|Ψ(t)i ==ZZd3 x hp|xihx|Ψ(t)id3 x e−ipx/h̄ Ψ(x, t).(1.191)The amplitudes hp|Ψ(t)i are referred to as momentum space wave functions.By inserting the completeness relationZd3 x|xihx| = 1(1.192)between the momentum states on the left-hand side of the orthogonality relation(1.186), we obtain the Fourier representation of the δ-functionhp|p0 i ==ZZd3 x hp|xihx|p0 i0d3 x e−i(p−p )x/h̄ .(1.193)H.
Kleinert, PATH INTEGRALS291.4 Dirac’s Bra-Ket Formalism1.4.7Incompleteness and Poisson’s Summation FormulaFor many physical applications it is important to find out what happens to thecompleteness relation (1.148) if one restrict the integral so a subset of positions.Most relevant will be the one-dimensional integral,Zdx |xihx| = 1,(1.194)restricted to a sum over equally spaced points xn = na:NXn=−N|xn ihxn |.(1.195)Taking this sum between momentum eigenstates |pi, we obtainNXn=−Nhp|xn ihxn |p0 i =NXn=−Nhp|xn ihxn |p0 i =NX0ei(p−p )na/h̄(1.196)n=−NFor N → ∞ we can perform the sum with the help of Poisson’s summation formula∞Xe2πiµn =n=−∞∞Xm=−∞δ(µ − m).(1.197)Identifying µ with (p − p0 )a/2πh̄, we find using Eq.
(1.160):(p − p0 )a2πh̄2πh̄m.hp|xn ihxn |p i = δ−m =δ p − p0 −2πh̄aan=−∞∞X!!0(1.198)In order to prove the Poisson formula (1.197), we observe that the sum s(µ) ≡side is periodic in µ with a unit period and hasm δ(µ − m) on the right-handP∞the Fourier series s(µ) = n=−∞ sn e2πiµn . The Fourier coefficients are given byR 1/2sn = −1/2 dµ s(µ)e−2πiµn ≡ 1. These are precisely the Fourier coefficients on theleft-hand side.For a finite N, the sum over n on the left-hand side of (1.197) yieldsPNXe2πiµn = 1 + e2πiµ + e2·2πiµ + .
. . + eN ·2πiµ + ccn=−N1 − e−2πiµ(N +1)= −1 ++ cc1 − e−2πiµ= 1+!(1.199)e−2πiµ − e−2πiµ(N +1)sin πµ(2N + 1)+ cc =.−2πiµsin πµ1−eThis function is well known in wave optics (see Fig. 2.4). It determines the diffraction pattern of light behind a grating with 2N + 1 slits. It has large peaks at301 Fundamentals2πiµn in Poisson’s summation formula. In theFigure 1.2 Relevant function Nn=−N elimit N → ∞, µ is squeezed to the integer values.Pµ = 0, ±1, ±2, ±3, . . . and N − 1 small maxima between each pair of neighboring peaks, at ν = (1 + 4k)/2(2N + 1) for k = 1, .
. . , N − 1. There are zeros atν = (1 + 2k)/(2N + 1) for k = 1, . . . , N − 1.Inserting µ = (p − p0 )a/2πh̄ into (1.199), we obtainsin (p − p0 )a(2N + 1)/2h̄.hp|xn ihxn |p i =sin (p − p0 )a/2h̄n=−NNX0(1.200)Let us see how the right-hand side of (1.199) turns into the right-hand side of(1.197) in the limit N → ∞. In this limit, the area under each large peak canbe calculated by an integral over the central large peak plus a number n of smallmaxima next to it:Zn/2N−n/2Ndµsin 2πµN cos πµ+cos 2πµN sin πµsin πµ(2N + 1) Z n/2N=dµ.sin πµsin πµ−n/2N(1.201)Keeping keeping a fixed ratio n/N 1, we we may replace in the integrand sin πµby πµ and cos πµ by 1. Then the integral becomes, for N → ∞ at fixed n/N,n/2Nsin πµ(2N + 1) N →∞ n/2Nsin 2πµN−−−→dµ+dµ cos 2πµNsin πµπµ−n/2N−n/2N−n/2NZ πnZ πnN →∞N →∞ 1sin x1dxdx cos x −−−→ 1,(1.202)−−−→+π −πnx2πN −πnZn/2NZdµZwhere we have used the integral formulaZ∞−∞dxsin x= π.x(1.203)H.
Kleinert, PATH INTEGRALS311.5 ObservablesIn the limit N → ∞, we find indeed (1.197) and thus (1.205), as well as the expression (2.458) for the free energy.There exists another useful way of expressing Poisson’s formula. Consider a anarbitrary smooth function f (µ) which possesses a convergent sum∞Xf (m).(1.204)m=−∞Then Poisson’s formula (1.197) implies that the sum can be rewritten as an integraland an auxiliary sum:∞Xm=−∞f (m) =Z∞−∞dµ∞Xe2πiµn f (µ).(1.205)n=−∞The auxiliary sum over n squeezes µ to the integer numbers.1.5ObservablesChanges of basis vectors are an important tool in analyzing the physically observablecontent of a wave vector.
Let A = A(p, x) be an arbitrary time-independent realfunction of the phase space variables p and x. Important examples for such anA are p and x themselves, the Hamiltonian H(p, x), and the angular momentumL = x × p. Quantum-mechanically, there will be an observable operator associatedwith each such quantity. It is obtained by simply replacing the variables p and x inA by the corresponding operators p̂ and x̂:Â ≡ A(p̂, x̂).(1.206)This replacement rule is the extension of the correspondence principle for the Hamiltonian operator (1.92) to more general functions in phase space, converting theminto observable operators.
It must be assumed that the replacement leads to aunique Hermitian operator, i.e., that there is no ordering problem of the type discussed in context with the Hamiltonian (1.101).8 If there are ambiguities, the naivecorrespondence principle is insufficient to determine the observable operator. Thenthe correct ordering must be decided by comparison with experiment, unless it canbe specified by means of simple geometric principles.
This will be done for theHamiltonian operator in Chapter 8.Once an observable operator  is Hermitian, it has the useful property that theset of all eigenvectors |ai obtained by solving the equationÂ|ai = a|ai(1.207)can be used as a basis to span the Hilbert space. Among the eigenvectors, there isalways a choice of orthonormal vectors |ai fulfilling the completeness relationXa8|aiha| = 1.Note that this is true for the angular momentumL= x × p.(1.208)321 FundamentalsThe vectors |ai can be used to extract physical information concerning the observable A from arbitrary state vector |Ψ(t)i. For this we expand this vector in thebasis |ai:X|aiha|Ψ(t)i.(1.209)|Ψ(t)i =aThe componentsha|Ψ(t)i(1.210)yield the probability amplitude for measuring the eigenvalue a for the observablequantity A.The wave function Ψ(x, t) itself is an example of this interpretation.
If we writeit asΨ(x, t) = hx|Ψ(t)i,(1.211)it gives the probability amplitude for measuring the eigenvalues x of the positionoperator x̂, i.e., |Ψ(x, t)|2 is the probability density in x-space.The expectation value of the observable operator (1.206) in the state |Ψ(t)i isdefined as the matrix elementhΨ(t)|Â|Ψ(t)i ≡1.5.1Zd3 xhΨ(t)|xiA(−ih̄∇, x)hx|Ψ(t)i.(1.212)Uncertainty RelationWe have seen before [see the discussion after (1.83), (1.84)] that the amplitudes inreal space and those in momentum space have widths inversely proportional to eachother, due to the properties of Fourier analysis. If a wave packet is localized in realspace with a width ∆x, its momentum space wave function has a width ∆p givenby∆x ∆p ∼ h̄.(1.213)From the Hilbert space point of view this uncertainty relation can be shown to bea consequence of the fact that the operators x̂ and p̂ do not commute with eachother, but the components satisfy the canonical commutation rules[p̂i , x̂j ] = −ih̄δij ,[x̂i , x̂j ] = 0,[p̂i , p̂j ] = 0.(1.214)In general, if an observable operator  is measured sharply to have the value a inone state, this state must be an eigenstate of  with an eigenvalue a:Â|ai = a|ai.(1.215)This follows from the expansion|Ψ(t)i =Xa|aiha|Ψ(t)i,(1.216)H.
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