Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 44
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(1.475).The first-order contribution from the interaction reads, after a Fourier decomposition of thepotential,ZZZ2id3 Qlim eiq tb /2Mh̄ d3 yb e−iqyb /h̄V(Q)d3 yahpb |Ŝ1 |pa i = −h̄ tb −ta →∞(2πh̄)3Z Z tb Z tbi pa Q 0M 2i003dt expdt×ẏ + δ(t −t)Q y .(2.708)tD y exph̄ Mh̄ ta2ta1902 Path Integrals — Elementary Properties and Simple SolutionsThe harmonic path integral was solved in one dimension for an arbitrary source j(t) in Eq. (3.168).For ω = 0 and the particular source j(t) = δ(t0 − t)Q the result reads, in three dimensions,1i M (yb − ya )2p3 exp h̄ 2tb − ta2πih̄(tb − ta )/M1i1× exp(tb − t0 )(t0 − ta )Q2.(2.709)[yb (t0 − ta ) + ya (tb − t0 )] Q −h̄ tb − ta2MPerforming here the integral over ya yieldsi 1iQ yb exp −(tb − t0 )Q2 .exph̄h̄ 2M(2.710)The integral over yb in (2.708) leads now to a δ-function (2πh̄)3 δ (3) (Q − q), such that the exponential prefactor in (2.708) is canceled by part of the second factor in (2.710).In the limit tb − ta → ∞, the integral over t0 produces a δ-function 2πh̄δ(pb Q/M + Q2 /2M ) =2πh̄δ(Eb − Ea ) which enforces the conservation of energy.
Thus we find the well-known Bornapproximationhpb |Ŝ1 |pa i = −2πiδ(Eb − Ea )V (q).(2.711)In general, we subtract the unscattered particle term (2.707) from (2.705), to obtain a pathintegral representation for the T -matrix [for the definition recall (1.475)]:ZZ22πh̄iδ(Eb − Ea )hpb |T̂ |pa i ≡ − lim eiq tb /2Mh̄ d3 yb e−iqyb /h̄ d3 yatb −ta →∞ Z tbZZMpa i tbidt ẏ2dt V y +t −1 .(2.712)exp −× D3 y exph̄ ta2h̄ taMIt is preferable to find a formula which does not contain the δ-function of energy conservation asa factor on the left-hand side. In order to remove this we observe that its origin lies in the timetranslational invariance of the path integral in the limit tb − ta → ∞. If we go over to a shifted timevariable t → t + t0 , and change simultaneously y → y − pa t0 /M , then the path integral remainsthe same Rexcept for shifted initial and finalRtimes tb + t0 and ta + t0 . In the limit tb − ta → ∞, thet +ttintegrals tab+t00 dt can be replaced again by tab dt.
The only place where a t0 -dependence remains isin the prefactor e−iqyb /h̄ which changes to e−iqyb /h̄ eiqpa t0 /Mh̄ . Among all path fluctuations, thereexists one degree of freedom which is equivalent to a temporal shift of the path. This is equivalent toan integral over t0 which yields a δ-function 2πh̄δ (qpa /M ) = 2πh̄δ (Eb − Ea ). We only must makesure to find the relation between this temporal shift and the corresponding measure in the pathintegral. This is obviously a shift of the path as a whole in the direction p̂a ≡ pa /|pa |. The formalway of isolating this degree of freedom proceeds according to a method developed by Faddeev andPopov36 by inserting into the path integral (2.705) the following integral representation of unity:Z|pa | ∞dt0 δ (p̂a (yb + pa t0 /M )) .(2.713)1=M −∞In the following, we shall drop the subscript a of the incoming beam, writingp ≡ pa ,p ≡ |pa | = |pb |.(2.714)After the above shift in the path integral, the δ-function in (2.713) becomes δ (p̂a yb ) inside the pathintegral, with no t0 -dependence.
The integral over t0 can now be performed yielding the δ-function36L.D. Faddeev and V.N. Popov, Phys. Lett. B 25 , 29 (1967).H. Kleinert, PATH INTEGRALS2.22 Path Integral Representation of Scattering Matrix191in the energy. Removing this from the equation we obtain the path integral representation of theT -matrixZZ2phpb |T̂ |pa i ≡ ilim eiq (tb −ta )/8Mh̄ d3 yb δ (p̂a yb ) e−iqyb /h̄ d3 yaM tb −ta →∞ Z tbZZM 2Pi tbi3dt ẏdt V y+ t −1 .(2.715)exp −× D y exph̄ ta2h̄ taMAt this point it is convenient to go over to the velocity representation of the path integral (2.699).This enables us to perform trivially the integral over yb , and we obtain the y version of (2.700).The δ-function enforces a vanishing longitudinal component of yb .
The transverse component ofyb will be denoted by b:b ≡ yb − (p̂a yb )p̂/a.(2.716)Hence we find the path integral representationZpiq2 tb /2Mh̄d2 b e−iqb/h̄hpb |T̂ |pa i ≡ ilim eM tb −ta →∞ Z tbZiiM 2 h iχb,p [v]3dt v× D v expe−1 ,h̄ ta2where the effect of the interaction is contained in the scattering phaseZ tbZp1 tb00dt v(t ) .dt V b +t−χb,p [v] ≡ −h̄ taMt(2.717)(2.718)We can go back to a more conventional path integral by replacing the velocity paths v(t) byRtẏ(t) = − t b v(t). This vanishes at t = tb .
Equivalently, we can use paths z(t) with periodicboundary conditions and subtract from these z(tb ) = zb .From hpb |T̂ |pa i we obtain the scattering amplitude fpb pa , whose square gives the differentialcross section, by multiplying it with a factor −M/2πh̄ [see Eq. (1.495)].Note that in the velocity representation, the evaluation of the harmonic path integral integratedover ya in (2.708) is much simpler than before where we needed the steps (2.709), (2.710). Afterthe Fourier decomposition of V (x) in (2.718), the relevant integral is Z tb R tbZ02iiM 2− idtΘ2 (tb−t0)Q2D3 v expdt(2.719)v −Θ(tb− t0 )Q v = e 2Mh̄ ta= e− 2Mh̄ (tb −t )Q .h̄ ta2The first factor in (2.710) comes directly from the argument Y in the Fourier representation of thepotentialZ tbpV yb +t−dt0 v(t0 )Mtin the velocity representation of the S-matrix (2.705).2.22.2Improved Formulation2The prefactor eiq tb /2Mh̄ in Formula (2.717) is an obstacle to taking a more explicit limit tb −ta → ∞on the right-hand side.
To overcome this, we represent this factor by an auxiliary path integral37over some vector field w(t): R tbZZ2i tb M 2idt Θ(t)w(t)q/h̄.(2.720)eiq tb /2Mh̄ = D3 w exp −dt w (t) e tah̄ ta237See R. Rosenfelder, notes of a lecture held at the ETH Zürich in 1979: Pfadintegrale in derQuantenphysik , 126 p., PSI Report 97-12, ISSN 1019-0643, and Lecture held at the 7th Int. Conf.on Path Integrals in Antwerpen, Path Integrals from Quarks to Galaxies, 2002.1922 Path Integrals — Elementary Properties and Simple Solutions−iqb/h̄h−iq b+R tbtaidt Θ(t)w(t) /h̄The last factor changes the exponential ein (2.717) into e.
Since bR tbis a dummy variable of integration, we can equivalently replace b → bw ≡ b − ta dt Θ(t)w(t) inthe scattering phase χb,p [v] and remain withZZp2−iqb/h̄d befpb pa =limD3 wtb −ta →∞ 2πih̄ Z ∞Z M 2i23dtv −wexp iχbw ,p − 1 .(2.721)×D v exph̄ −∞2The scattering phase in this expression can be calculated from formula (2.718) with the integraltaken over the entire t-axis:Z tbZ1 ∞pχbw ,p [v, w] = −dt0 [Θ(t0 −t)v(t0 ) − Θ(t0 )w(t0 )] .(2.722)t−dt V b +h̄ −∞MtaThe fluctuations of w(t) are necessary to correct for the fact that the outgoing particle does notrun, on the average, with the velocity p/M = pa /M but with velocity pb /M = (p + q)/M .RtWe may also go back to a more conventional path integral by inserting y(t) = − t b v(t) andR tbsetting similarly z(t) = − t w(t). Then we obtain the alternative representationZZZpfpb pa =limd2 b e−iqb/h̄ d3 ya d3 zatb −ta →∞ 2πih̄ Z tbZZi h iχM 2idtẏ − ż2e bz ,p [y] − 1 ,(2.723)×D3 y D3 z exph̄ ta2withχbz ,p [y] ≡ −1h̄Ztbtapdt V b +t + y(t) − z(0) ,M(2.724)where the path integrals run over all paths with yb = 0 and zb = 0.
In Section 3.26 this pathintegral will be evaluated perturbatively.2.22.3Eikonal Approximation to Scattering AmplitudeTo lowest approximation, we neglect the fluctuating variables y(t) and z(t) in (2.724). Since theintegral Z tbZZZZiM 233323dtẏ − żd ya d za D y D z exp(2.725)h̄ ta2in (2.723) has unit normalization [recall the calculation of (2.710)], we obtain directly the eikonalapproximation to the scattering amplitudeZhipd2 b e−iqb/h̄ exp iχei(2.726)fpeib pa ≡b,p − 1 ,2πih̄withχeib ,p1≡−h̄Z∞p dt V b +t .M−∞(2.727)The time integration can be converted into a line integration along the direction of the incomingparticles by introducing a variable z ≡ pt/M .
Then we can writeZM1 ∞eiχb,p ≡ −dz V (b + p̂z) .(2.728)p h̄ −∞H. Kleinert, PATH INTEGRALS1932.23 Heisenberg Operator Approach to Time Evolution AmplitudeIf V (x) is rotationally symmetric, it depends only on r ≡ |x|. Then we shall write the potentialas V (r) and calculate (2.728) as the integralZpM1 ∞χei≡−b2 + z 2 .dz V(2.729)b,pp h̄ −∞Inserting this into (2.726), we can perform the integral over all angles between q and b using theformulaZ πi1dθ expqb cos θ = J0 (qb),(2.730)2π −πh̄where J0 (ξ) is the Bessel function, and findZhipeidb b J0 (qb) exp iχeifpb pa =b,p − 1 .ih̄(2.731)The variable of integration b coincides with the impact parameterb introduced in Eq. (1.498).
Theresult (2.731) is precisely the eikonal approximation (1.498) with χeib,p /2 playing the role of thescattering phases δl (p) of angular momentum l = pb/h̄:χeib,p = 2iδpb/h̄ (p).2.23(2.732)Heisenberg Operator Approach to Time EvolutionAmplitudeAn interesting alternative to the path integral derivation of the time evolution amplitudes ofharmonic systems is based on quantum mechanics in the Heisenberg picture. It bears a closesimilarity with the path integral derivation in that it requires solving the classical equations ofmotion with given initial and final positions to obtain the exponential of the classical action eiA/h̄ .The fluctuation factor, however, which accompanies this exponential is obtained quite differentlyfrom commutation rules of the operatorial orbits at different times as we shall now demonstrate.2.23.1Free ParticleWe want to calculate the matrix element of the time evolution operator(x t|x0 0) = hx|e−iĤt/h̄ |x0 i,(2.733)where Ĥ is the Hamiltonian operatorĤ = H(p̂) =p̂2.2M(2.734)We shall calculate the time evolution amplitude (2.733) by solving the differential equationihih̄∂t hx t|x0 0i ≡ hx|Ĥ e−iĤt/h̄ |x0 i = hx|e−iĤt/h̄ eiĤt/h̄ Ĥ e−iĤt/h̄ |x0 i= hx t|H(p̂(t))|x0 0i.(2.735)The argument contains now the time-dependent Heisenberg picture of the operator p̂.
The evaluation of the right-hand side will be based on re-expressing the operator H(p̂(t)) as a function ofinitial and final position operators in such a way that all final position operators stand to the leftof all initial ones:Ĥ = H(x̂(t), x̂(0); t).(2.736)1942 Path Integrals — Elementary Properties and Simple SolutionsThen the matrix elements on the right-hand side can immediately be evaluated using the eigenvalueequationshx t|x̂(t) = xhx t|,x̂(0)|x0 0i = x0 |x0 0i,(2.737)as beinghx t|H(x̂(t), x̂(0); t)|x̂ 0i = H(x, x0 ; t)hx t|x0 0i,(2.738)and the differential equation (2.735) becomesih̄∂t hx t|x0 0i ≡ H(x, x0 ; t)hx t|x0 0i,orhx t|x0 0i = C(x, x0 )E(x, x0 ; t) ≡ C(x, x0 )e−iRt(2.739)dt0 H(x,x0 ;t0 )/h̄.(2.740)The prefactor C(x, x0 ) contains a possible constant of integration resulting from the time integralin the exponent.The Hamiltonian operator is brought to the time-ordered form (2.736) by solving the Heisenberg equations of motiondx̂(t)dtdp̂(t)dt==i p̂(t)i h,Ĥ, x̂(t) =h̄Mii hĤ, p̂(t) = 0.h̄(2.741)(2.742)The second equation shows that the momentum is time-independent:p̂(t) = p̂(0),(2.743)so that the first equation is solved byx̂(t) − x̂(0) = tp̂(t),M(2.744)which brings (2.734) toĤ =M[x̂(t) − x̂(0)]2 .2t2(2.745)This is not yet the desired form (2.736) since there is one factor which is not time-ordered.
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