Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 27
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Kleinert, PATH INTEGRALS2.1 Path Integral Representation of Time Evolution Amplitudes93At this point we make the important observation that a momentum variable pn insidethe time-sliced version (2.17) at the time tn can be generated by a derivative p̂n ≡−ih̄∂xn . In the continuum limit (2.18), this becomes an operator p(t) = −ih̄∂x(t) .Its commutator with x(t) is[p̂(t), x(t)] = −ih̄,(2.19)which is the famous equal-time canonical commutation rule of Heisenberg.This observation forms the basis for deriving, from the path integral (2.14), theSchrödinger equation for the time evolution amplitude.2.1.3Schrödinger Equation for Time Evolution AmplitudeLet us split from the product of integrals in (2.14) the final time slice as a factor, sothat we obtain the recursion relation(xb tb |xa ta ) ≈Z∞−∞wheredxN (xb tb |xN tN ) (xN tN |xa ta ),(2.20)dpn (i/h̄)[pb (xb −xN )−H(pb ,xb ,tb )]e.(2.21)−∞ 2πh̄The momentum pb inside the integral can be generated by a differential operatorp̂b ≡ −ih̄∂xb .
The same is true for any function of pb , so that the Hamiltonian canbe moved before the momentum integral as follows:(xb tb |xN tN ) ≈Z∞dpn ipb (xb −xN )/h̄ −iH(−ih̄∂xb ,xb,tb )/h̄δ(xb −xN ).e=e−∞ 2πh̄(2.22)Inserting this back into (2.20) we obtainZ−iH(−ih̄∂xb ,xb ,tb )/h̄(xb tb |xN tN ) ≈ e∞(xb tb |xa ta ) ≈ e−iH(−ih̄∂xb ,xb ,tb )/h̄ (xb tb −|xa ta ),(2.23)i11 h −iH(−i∂xb ,xb ,tb )/h̄e− 1 (xb tb |xa ta ).[(xb tb +|xa ta ) − (xb tb |xa ta )] ≈(2.24)orIn the limit → 0, this goes over into the differential equation for the time evolutionamplitudeih̄∂tb (xb tb |xa ta ) = H(−ih̄∂xb , xb , tb )(xb tb |xa ta ),(2.25)which is precisely the Schrödinger equation (1.298) of operator quantum mechanics.2.1.4Convergence of Sliced Time Evolution AmplitudeSome remarks are necessary concerning the convergence of the time-sliced expression(2.14) to the quantum-mechanical amplitude in the continuum limit, where thethickness of the time slices = (tb − ta )/(N + 1) → 0 goes to zero and the number942 Path Integrals — Elementary Properties and Simple SolutionsN of slices tends to ∞.
This convergence can be proved for the standard kineticenergy T = p2 /2M only if the potential V (x, t) is sufficiently smooth. For timeindependent potentials this is a consequence of the Trotter product formula whichreadse−i(tb −ta )Ĥ/h̄ = limN →∞e−iV̂ /h̄ e−iT̂ /h̄N +1.(2.26)For c-numbers T and V , this is trivially true. For operators T̂ , V̂ , we use Eq. (2.9)to rewrite the left-hand side of (2.26) ase−i(tb −ta )Ĥ/h̄ ≡ e−i(T̂ +V̂ )/h̄N +1≡ e−iV̂ /h̄ e−iT̂ /h̄ e−i2 X̂/h̄2N +1.The Trotter formula implies that the commutator term X̂ proportional to 2 doesnot contribute in the limit N → ∞. The mathematical conditions ensuring thisrequire functional analysis too technical to be presented here (for details, see theliterature quoted at the end of the chapter).
For us it is sufficient to know thatthe Trotter formula holds for operators which are bounded from below and thatfor most physically interesting potentials, it cannot be used to derive Feynman’stime-sliced path integral representation (2.14), even in systems where the formulais known to be valid. In particular, the short-time amplitude may be differentfrom (2.13). Take, for example, an attractive Coulomb potential V (x) ∝ −1/|x|for which the Trotter formula has been proved to be valid.
Feynman’s time-slicedformula, however, diverges even for two time slices. This will be discussed in detail inChapter 12. Similar problems will be found for other physically relevant potentialssuch as V (x) ∝ l(l + D − 2)h̄2 /|x|2 (centrifugal barrier) and V (θ) ∝ m2 h̄2 /sin2 θ(angular barrier near the poles of a sphere). In all these cases, the commutatorsin the expansion (2.10) of X̂ become more and more singular. In fact, as we shallsee, the expansion does not even converge, even for an infinitesimally small . Allatomic systems contain such potentials and the Feynman formula (2.14) cannot beused to calculate an approximation for the transition amplitude. A new path integralformula has to be found. This will be done in Chapter 12. Fortunately, it is possibleto eventually reduce the more general formula via some transformations back to aFeynman type formula with a bounded potential in an auxiliary space.
Thus theabove derivation of Feynman’s formula for such potentials will be sufficient for thefurther development in this book. After this it serves as an independent startingpoint for all further quantum-mechanical calculations.In the sequel, the symbol ≈ in all time-sliced formulas such as (2.14) will implythat an equality emerges in the continuum limit N → ∞, → 0 unless the potentialhas singularities of the above type.
In the action, the continuum limit is withoutsubtleties. The sum AN in (2.15) tends towards the integralA[p, x] =Ztbtadt [p(t)ẋ(t) − H(p(t), x(t), t)](2.27)under quite general circumstances. This expression is recognized as the classicalcanonical action for the path x(t), p(t) in phase space. Since the position variablesH.
Kleinert, PATH INTEGRALS952.1 Path Integral Representation of Time Evolution AmplitudesxN +1 and x0 are fixed at their initial and final values xb and xa , the paths satisfythe boundary condition x(tb ) = xb , x(ta ) = xa .In the same limit, the product of infinitely many integrals in (2.14) will be calleda path integral . The limiting measure of integration is written aslimN →∞N ZYn=1∞−∞dxn" NY+1 Z∞−∞n=1#dpn≡2πh̄Zx(tb )=xbx(ta )=xa0DxZDp.2πh̄(2.28)By definition, there is always one more pn -integral than xn -integrals in this product.While x0 and xN +1 are held fixed and the xn -integrals are done for n = 1, . .
. , N, eachpair (xn , xn−1 ) is accompanied by one pn -integral for n = 1, . . . , N +1. The situationis recorded by the prime on the functional integral D 0 x. With this definition, theamplitude can be written in the short form(xb tb |xa ta ) =Zx(tb )=xbx(ta )=xaD0xZDp iA[p,x]/h̄e.2πh̄(2.29)The path integral has a simple intuitive interpretation: Integrating over all pathscorresponds to summing over all histories along which a physical system can possiblyevolve. The exponential eiA[p,x]/h̄ is the quantum analog of the Boltzmann factore−E/kB T in statistical mechanics.
Instead of an exponential probability, a pure phasefactor is assigned to each possible history: The total amplitude for going from xa , tato xb , tb is obtained by adding up the phase factors for all these histories,(xb tb |xa ta ) =XeiA[p,x]/h̄ ,(2.30)all histories(xa ,ta ) ; (xb ,tb )where the sum comprises all paths in phase space with fixed endpoints xb , xa inx-space.2.1.5Time Evolution Amplitude in Momentum SpaceThe above observed asymmetry in the functional integrals over x and p is a consequence of keeping the endpoints fixed in position space. There exists the possibilityof proceeding in a conjugate way keeping the initial and final momenta pb and pafixed. The associated time evolution amplitude can be derived going through thesame steps as before but working in the momentum space representation of theHilbert space, starting from the matrix elements of the time evolution operator(pb tb |pa ta ) ≡ hpb |Û(tb , ta )|pa i.(2.31)The time slicing proceeds as in (2.2)–(2.4), with all x’s replaced by p’s, except inthe completeness relation (2.3) which we shall take asZ∞−∞dp|pihp| = 1,2πh̄(2.32)962 Path Integrals — Elementary Properties and Simple Solutionscorresponding to the choice of the normalization of states [compare (1.186)]hpb |pa i = 2πh̄δ(pb − pa ).(2.33)In the resulting product of integrals, the integration measure has an opposite asymmetry: there is now one more xn -integral than pn -integrals.
The sliced path integralreads(pb tb |pa ta ) ≈"ZNYN Z ∞dpn Ydxn2πh̄ n=0 −∞#∞−∞n=1Ni X[−xn (pn+1 − pn ) − H(pn , xn , tn )] .× exph̄ n=0()(2.34)The relation between this and the x-space amplitude (2.14) is simple: By takingin (2.14) the first and last integrals over p1 and pN +1 out of the product, renamingP +1them as pa and pb , and rearranging the sum Nn=1 pn (xn − xn−1 ) as followsN+1Xn=1pn (xn − xn−1 ) = pN +1 (xN +1 − xN ) + pN (xN − xN −1 ) + . .
.. . . + p2 (x2 − x1 ) + p1 (x1 − x0 )= pN +1 xN +1 − p1 x0−(pN +1 − pN )xN − (pN − pN −1 )xN −1 − . . . − (p2 − p1 )x1= pN +1 xN +1 − p1 x0 −NX(pn+1 − pn )xn ,(2.35)n=1the remaining product of integrals looks as in Eq. (2.34), except that the lowestindex n is one unit larger than there. In the limit N → ∞ this does not matter,and we obtain the Fourier transform(xb tb |xa ta ) =Zdpb ipb xb /h̄e2πh̄Zdxb e−ipb xb /h̄Zdpa −ipa xa /h̄e(pb tb |pa ta ).2πh̄(2.36)dxa eipa xa /h̄ (xb tb |xa ta ).(2.37)The inverse relation is(pb tb |pa ta ) =ZIn the continuum limit, the amplitude (2.34) can be written as a path integral(pb tb |pa ta ) =Zp(tb )=pbp(ta )=paD0p ZDxeiĀ[p,x]/h̄,2πh̄(2.38)whereĀ[p, x] =Ztbtadt [−ṗ(t)x(t) − H(p(t), x(t), t)] = A[p, x] − pb xb + pa xa .(2.39)H. Kleinert, PATH INTEGRALS972.1 Path Integral Representation of Time Evolution Amplitudes2.1.6Quantum Mechanical Partition FunctionA path integral symmetric in p and x arises when considering the quantummechanical partition function defined by the trace (recall Section 1.17)ZQM (tb , ta ) = Tr e−i(tb −ta )Ĥ/h̄ .(2.40)In the local basis, the trace becomes an integral over the amplitude(xb tb |xa ta ) with xb = xa :ZQM (tb , ta ) =∞Z−∞dxa (xa tb |xa ta ).(2.41)The additional trace integral over xN +1 ≡ x0 makes the path integral for ZQMsymmetric in pn and xn :Z∞−∞dxN +1N ZYn=1∞−∞dxn" NY+1 Z∞−∞n=1NY+1dpn=2πh̄n=1#"ZZ∞−∞#dxn dpn.2πh̄(2.42)In the continuum limit, the right-hand side is written aslimN →∞NY+1n=1"ZZ#dxn dpn≡2πh̄∞−∞IDxDp,2πh̄Z(2.43)and the measures are related byZ∞−∞dxax(tb )=xbZx(ta )=xaD0xDp≡2πh̄ZIDxDp.2πh̄Z(2.44)HThe symbolindicates the periodic boundary condition x(ta ) = x(tb ).
In themomentum representation we would have similarlyZ∞−∞dpa2πh̄Zp(tb )=pbp(ta )=paD0p2πh̄ZDx ≡IDp2πh̄ZDx ,(2.45)with the periodic boundary condition p(ta ) = p(tb ), and the same right-hand side.Hence, the quantum-mechanical partition function is given by the path integralZQM (tb , ta ) =IDxZDp iA[p,x]/h̄e=2πh̄IDp2πh̄ZDxeiĀ[p,x]/h̄.(2.46)In the right-hand exponential, the action Ā[p, x] can be replaced by A[p, x], sincethe extra terms in (2.39) are removed by the periodic boundary conditions. In thetime-sliced expression, the equality is easily derived from the rearrangement of thesum (2.35), which shows thatN+1Xn=1pn (xn− xn−1 )xN+1 =x0= −NX(pn+1n=0− pn )xn pN+1 =p0.(2.47)982 Path Integrals — Elementary Properties and Simple SolutionsIn the path integral expression (2.46) for the partition function, the rules of quantum mechanics appear as a natural generalization of the rules of classical statisticalmechanics, as formulated by Planck.
According to these rules, each volume elementin phase space dxdp/h is occupied with the exponential probability e−E/kB T . In thepath integral formulation of quantum mechanics, each volume element in the pathQphase space n dx(tn )dp(tn )/h is associated with a pure phase factor eiA[p,x]/h̄ .
Wesee here a manifestation of the correspondence principle which specifies the transition from classical to quantum mechanics. In path integrals, it looks somewhatmore natural than in the historic formulation, where it requires the replacement ofall classical phase space variables p, x by operators, a rule which was initially hardto comprehend.2.1.7Feynman’s Configuration Space Path IntegralActually, in his original paper, Feynman did not give the path integral formula inthe above phase space formulation. Since the kinetic energy in (2.7) has usually theform T (p, t) = p2 /2M, he focused his attention upon the HamiltonianH=p2+ V (x, t),2M(2.48)for which the time-sliced action (2.15) becomesNA =N+1Xn=1p2pn (xn − xn−1 ) − n − V (xn , tn ) .2M"#(2.49)It can be quadratically completed toNA =N+1Xn=1"2x − xn−1pn − n−M2MM x − xn−1+ n22#− V (xn , tn ) .
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