Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 28
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(2.50)The momentum integrals in (2.14) may then be performed using the Fresnel integralformula (1.334), yieldingZ∞−∞"x − xn−1i dpnpn − M nexp −2πh̄h̄ 2M2 #and we arrive at the alternative representation1(xb tb |xa ta ) ≈ q2πh̄i/MNYn=1where AN is now the sumNA =N+1Xn=1"M2Zdxn∞−∞1=q,2πh̄i/Mqxn − xn−12πh̄i/M2 exp#− V (xn , tn ) ,i NA ,h̄(2.51)(2.52)(2.53)H.
Kleinert, PATH INTEGRALS992.1 Path Integral Representation of Time Evolution AmplitudesFigure 2.1 Zigzag paths, along which a point particle explores all possible ways ofreaching the point xb at a time tb , starting from xa at ta . The time axis is drawn fromright to left to have the same direction as the operator order in Eq. (2.2).with xN +1 = xb and x0 = xa . Here the integrals run over all paths in configurationspace rather than phase space. They account for the fact that a quantum-mechanicalparticle starting from a given initial point xa will explore all possible ways of reachinga given final point xb .
The amplitude of each path is exp(iAN /h̄). See Fig. 2.1 fora geometric illustration of the path integration. In the continuum limit, the sum(2.53) converges towards the action in the Lagrangian form:A[x] =ZtbtadtL(x, ẋ) =ZtbtaM 2ẋ − V (x, t) .dt2(2.54)Note that this action is a local functional of x(t) in the temporal sense as defined inEq. (1.27).2For the time-sliced Feynman path integral, one verifies the Schrödinger equationas follows: As in (2.20), one splits off the last slice as follows:(xb tb |xa ta ) ≈=Z∞−∞Z ∞−∞dxN (xb tb |xN tN ) (xN tN |xa ta )d∆x (xb tb |xb −∆x tb −) (xb −∆x tb −|xa ta ),(2.55)where12("i M(xb tb |xb −∆x tb −) ≈ qexp h̄ 22πh̄i/M∆x2A functional F [x] is calledR local if it can be written as an integralultra-local if it has the form dtf (x(t)).− V (xb , tb )R#).(2.56)dtf (x(t), ẋ(t)); it is called1002 Path Integrals — Elementary Properties and Simple SolutionsWe now expand the amplitude in the integral of (2.55) in a Taylor series1(xb −∆x tb −|xa ta ) = 1 − ∆x ∂xb + (∆x)2 ∂x2b + .
. . (xb , tb −|xa ta ).2(2.57)Inserting this into (2.55), the odd powers of ∆x do not contribute. For the evenpowers, we perform the integrals using the Fresnel version of formula (1.336), andobtain zero for odd powers of ∆x, andZd∆x∞−∞q2πh̄i/M(∆x)2n(iMexp h̄ 2∆x2 )h̄= iM!n(2.58)for even powers, so that the integral in (2.55) becomes"ih̄ 2∂ + O(2 )(xb tb |xa ta ) = 1 + 2M xb#i1 − V (xb , tb ) + O(2 ) (xb , tb −|xa ta ). (2.59)h̄In the limit → 0, this yields again the Schrödinger equation. (2.23).In the continuum limit, we write the amplitude (2.52) as a path integral(xb tb |xa ta ) ≡Zx(tb )=xbx(ta )=xaDx eiA[x]/h̄ .(2.60)This is Feynman’s original formula for the quantum-mechanical amplitude (2.1).
Itconsists of a sum over all paths in configuration space with a phase factor containingthe form of the action A[x].We have used the same measure symbol Dx for the paths in configuration spaceas for the completely different paths in phase space in the expressions (2.29), (2.38),(2.44), (2.45).
There should be no danger of confusion. Note that the qextra dpn integration in the phase space formula (2.14) results now in one extra 1/ 2πh̄i/Mfactor in (2.52) which is not accompanied by a dxn -integration.The Feynman amplitude can be used to calculate the quantum-mechanical partition function (2.41) as a configuration space path integralZQM ≈IDx eiA[x]/h̄ .(2.61)As in (2.43), (2.44), the symbol Dx indicates that the paths have equal endpointsx(ta ) = x(tb ), the path integral being the continuum limit of the product of integralsHIqDx ≈NY+1 Z ∞n=1−∞dxnq2πih̄/M.(2.62)There is no extra 1/ 2πih̄/M factor as in (2.52) and (2.60), due to the integrationover the initial (= final) position Hxb = xa representing the quantum-mechanicaltrace.
The use of the same symbol Dx as in (2.46)should not cause any confusionRsince (2.46) is always accompanied by an integral Dp.H. Kleinert, PATH INTEGRALS1012.1 Path Integral Representation of Time Evolution AmplitudesFor the sake of generality we might point out that it is not necessary to slice thetime axis in an equidistant way. In the continuum limit N → ∞, the canonical pathintegral (2.14) is indifferent to the choice of the infinitesimal spacingsn = tn − tn−1 .(2.63)The configuration space formula contains the different spacings n in the followingway: When performing the pn integrations, we obtain a formula of the type (2.52),with each replaced by n , i.e.,1(xb tb |xa ta ) ≈ q2πh̄ib /M(i× exph̄NYn=1dxn∞−∞q2πih̄n /M(xn − xn−1 )2− n V (xn , tn )2n"N+1XMn=1Z#).(2.64)To end this section, an important remark is necessary: It would certainly bepossible to define the path integral for the time evolution amplitude (2.29) withoutgoing through Feynman’s time-slicing procedure as the solution of the Schrödingerdifferential equation [see Eq.
(1.305))]:[Ĥ(−ih̄∂x , x) − ih̄∂t ](x t|xa ta ) = −ih̄δ(t − ta )δ(x − xa ).(2.65)If one possesses an orthonormal and complete set of wave functions ψn (x) solvingthe time-independent Schrödinger equation Ĥψn (x)=En ψn (x), this solution is givenby the spectral representation (1.320)(xb tb |xa ta ) = Θ(tb − ta )Xnψn (xb )ψn∗ (xa )e−iEn (tb −ta )/h̄ ,(2.66)where Θ(t) is the Heaviside function (1.301). This definition would, however, runcontrary to the very purpose of Feynman’s path integral approach, which is to understand a quantum system from the global all-time fluctuation point of view. Thegoal is to find all properties from the globally defined time evolution amplitude, inparticular the Schrödinger wave functions.3 The global approach is usually morecomplicated than Schrödinger’s and, as we shall see in Chapters 8 and 12–14, contains novel subtleties caused by the finite time slicing.
Nevertheless, it has at leastfour important advantages. First, it is conceptually very attractive by formulatinga quantum theory without operators which describe quantum fluctuations by closeanalogy with thermal fluctuations (as will be seen later in this chapter). Second,it links quantum mechanics smoothly with classical mechanics (as will be shown inChapter 4).
Third, it offers new variational procedures for the approximate study ofcomplicated quantum-mechanical and -statistical systems (see Chapter 5). Fourth,3Many publications claiming to have solved the path integral of a system have violated this ruleby implicitly using the Schrödinger equation, although camouflaged by complicated-looking pathintegral notation.1022 Path Integrals — Elementary Properties and Simple Solutionsit gives a natural geometric access to the dynamics of particles in spaces with curvature and torsion (see Chapters 10–11). This has recently led to results wherethe operator approach has failed due to operator-ordering problems, giving rise toa unique and correct description of the quantum dynamics of a particle in spaceswith curvature and torsion. From this it is possible to derive a unique extensionof Schrödinger’s theory to such general spaces whose predictions can be tested infuture experiments.42.2Exact Solution for Free ParticleIn order to develop some experience with Feynman’s path integral formula we consider in detail the simplest case of a free particle, which in the canonical form reads(xb tb |xa ta ) =Zx(tb )=xb0Dxx(ta )=xaZ"Dpiexp2πh̄h̄tbZtap2dt pẋ −2M!#,(2.67)and in the pure configuration form:(xb tb |xa ta ) =Zx(tb )=xbx(ta )=xaDx expih̄ZtbtadtM 2ẋ .2(2.68)Since the integration limits are obvious by looking at the left-hand sides of the equations, they will be omitted from now on, unless clarity requires their specification.2.2.1Direct SolutionThe problem is solved most easily in the configuration form.
The time-sliced expression to be integrated is given by Eqs. (2.52), (2.53) where we have to set V (x) = 0.The resulting product of Gaussian integrals can easily be done successively usingformula (1.334), from which we derive the simple ruleZi M (x00 − x0 )21i M (x0 − x)2qexpexpdx qh̄ 2Ah̄ 2B2πih̄A/M2πih̄B/M0"1#"i M (x00 − x)2=qexp,h̄ 2 (A + B)2πih̄(A + B)/M"1##(2.69)which leads directly to the free-particle amplitudei M (xb − xa )2q(xb tb |xa ta ) =exp.h̄ 2 (N + 1)2πih̄(N + 1)/M1"#(2.70)After inserting (N + 1) = tb − ta , this readsi M (xb − xa )2q(xb tb |xa ta ) =exp.h̄ 2 tb − ta2πih̄(tb − ta )/M14"#(2.71)H.
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