Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 43
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A comparison with (2.617) and (2.634) shows that the surfaceterm would be absent if the amplitude (xb tb |xa ta )Ã were calculated with the vectorpotential in the axially symmetric gauge (2.616). Thus Acl must be equal to theclassical action in this gauge. Indeed, the orthogonal part can be rewritten asA⊥cl=Ztbta()M dMdt(xẋ + y ẏ) +[x(−ẍ + ωLẏ) + y(−ÿ − ωL ẋ)] .2 dt2(2.647)The equations of motion areẍ = ωL ẏ,ÿ = −ωL ẋ,(2.648)reducing the action of a classical orbit toA⊥cl =tbMM(xẋ + y ẏ) =([xb ẋb − xa ẋa ] + [yb ẏb − ya ẏa ]) .ta22(2.649)The orbits are easily determined. By inserting the two equations in (2.648) intoeach other we see that ẋ and ẏ perform independent oscillations:ẍ˙ + ωL2 ẋ = 0,ÿ˙ + ωL2 ẏ = 0.(2.650)The general solution of these equations is1[(x − x0 ) sin ωL (t − ta ) − (xa − x0 ) sin ωL (t − tb ) ] + x0 , (2.651)sin ωL (tb − ta ) b1[(y − y0 ) sin ωL (t − ta ) − (ya − y0 ) sin ωL (t − tb ) ] + y0 , (2.652)y=sin ωL (tb − ta ) bx=H.
Kleinert, PATH INTEGRALS1832.18 Charged Particle in Magnetic Fieldwhere we have incorporated the boundary condition x(ta,b ) = xa,b , y(ta,b)= ya,b . The constants x0 , y0 are fixed by satisfying (2.648). This gives1ω(xb + xa ) + (yb − ya ) cot L (tb − ta ) ,221ωL=(y + ya ) − (xb − xa )cot (tb − ta ) .2 b2x0 =y0(2.653)(2.654)Now we calculateωLx [(x − xa ) + (xb − x0 ) cos ωL (tb − ta )] ,sin ωL(tb − ta ) b 0ωLx [(x − xa ) cos ωL (tb − ta ) + (xb − x0 )] ,=sin ωL(tb − ta ) a 0xb ẋb =(2.655)xa ẋa(2.656)and henceωxb ẋb − xa ẋa = ωL x0 (xb + xa ) tan L (tb − ta )2hiωL(x2b + x2a ) cos ωL (tb − ta ) − 2xb xa+sin ωL (tb − ta )ωLωL2=(t − ta ) + (xb + xa )(yb − ya ) .(xb − xa ) cot22 b(2.657)Similarly we findωyb ẏb − ya ẏa = ωL y0 (yb + ya ) tan L (tb − ta )2ωL+[(y 2 + ya2 ) cos ωL(tb − ta ) − 2yb ya ]sin ωL(tb − ta ) bωLωL2(yb − ya ) cot (tb − ta ) − (xb − xa )(yb + ya ) .=22(2.658)Inserted into (2.649), this yields the classical action for the orthogonal motionA⊥clhiM ωLcot[ωL (tb − ta )/2] (xb − xa )2 + (yb − ya )2 + ωL (xa yb − xb ya ) ,=2 2(2.659)which is indeed the orthogonal part of the action (2.645).2.18.5Translational InvarianceIt is interesting to see how the amplitude ensures the translational invariance of allphysical observables.
The first term in the classical action is trivially invariant. Thelast term reads∆A =MωL(xa yb − xb ya ).2(2.660)1842 Path Integrals — Elementary Properties and Simple SolutionsUnder a translation by a distance d,x → x + d,(2.661)this term changes byMωLMωL[dx (yb − ya ) + dy (xa − xb )] =[(d × x)b − (d × x)a ]z22causing the amplitude to change by a pure gauge transformation as(2.662)(xb tb |xa ta ) → eieΛ(xb )/ch̄ (xb tb |xa ta )e−ieΛ(xa )/ch̄ ,(2.663)with the phaseMωL h̄c[d × x]z .(2.664)2eSince observables involve only gauge-invariant quantities, such transformations areirrelevant.This will be done in 2.23.3.Λ(x) = −2.19Charged Particle in Magnetic Field plusHarmonic PotentialFor application in Chapter 5 we generalize the above magnetic system by adding aharmonic oscillator potential, thus leaving the path integral solvable.
For simplicity,we consider only the orthogonal part of the resulting system with respect to themagnetic field. Omitting orthogonality symbols, the Hamiltonian is the same as in(2.622). Without much more work we may solve the path integral of a more generalsystem in which the harmonic potential in (2.622) has a different frequency ω 6= ωL,and thus a Hamiltonian1p2+ Mω 2 x2 − ωB lz (p, x).H(p, x) =2M2The associated Euclidean actionAe [p, x] =Zτbτa(2.665)dτ [−ip · ẋ + H(p, x)](2.666)has the Lagrangian formAe [x] =Z0h̄βdτM 21ẋ (τ ) + M(ω 2 − ωB2 )x2 (τ ) − iMωB [x(τ ) × ẋ(τ )]z .22(2.667)At this point we observe that the system is stable only for ω ≥ ωB . The action(2.667) can be written in matrix notation asAcl =Z0h̄βdτ"#M dM(xẋ) + xT Dω2 ,B x ,2 dτ2(2.668)H.
Kleinert, PATH INTEGRALS1852.19 Charged Particle in Magnetic Field plus Harmonic Potentialwhere Dω2 ,B is the 2 × 2 -matrix0Dω2 ,B (τ, τ ) ≡−∂τ2 + ω 2 − ωB2−2iωB ∂τ2iωB ∂τ−∂τ2 + ω 2 − ωB2!δ(τ − τ 0 ).(2.669)Since the path integral is Gaussian, we can immediately calculate the partitionfunction1−1/2Z=(2.670)2 det Dω 2 ,B .(2πh̄/M)By expanding Dω2 ,B (τ, τ 0 ) in a Fourier seriesDω2 ,B (τ, τ 0 ) =∞01 XD̃ω2 ,B (ωm )e−iωm (τ −τ ) ,h̄β m=−∞(2.671)we find the Fourier componentsD̃ω2 ,B (ωm ) =2ωm+ ω 2 − ωB2−2ωB ωm22ωB ωmωm + ω 2 − ωB2!,(2.672)with the determinants22det D̃ω2 ,B (ωm ) = (ωm+ ω 2 − ωB2 )2 + 4ωB2 ωm.(2.673)These can be factorized as2222det D̃ω2 ,B (ωm ) = (ωm+ ω+)(ωm+ ω−),(2.674)ω± ≡ ω ± ωB .(2.675)withThe eigenvectors of D̃ω2 ,B (ωm ) are1e+ = √21i!,1e− = − √21−i!,(2.676)with eigenvalues2d± = ωm+ ω 2 ± 2iωm ωB = (ωm + iω± )(ωm − iω∓ ).(2.677)√Thus the right- and left-circular combinations x± = ±(x ± iy)/ 2 diagonalize theLagrangian (2.668) toAcl =Z0h̄βdτ(M d ∗x+ ẋ+ + x∗− ẋ−2 dτiMh ∗∗+x (−∂τ − ω+ )(−∂τ + ω− )x+ + x− (−∂τ − ω− )(−∂τ + ω+ )x− .
(2.678)2 +1862 Path Integrals — Elementary Properties and Simple SolutionsContinued back to real times, the components x± (t) are seen to oscillate independently with the frequencies ω± .The factorization (2.674) makes (2.678) an action of two independent harmonicoscillators of frequencies ω± . The associated partition function has therefore theproduct form11.(2.679)Z=2 sinh (h̄βω+ /2) 2 sinh (h̄βω− /2)For the original system of a charged particle in a magnetic field discussed inSection 2.18, the partition function is obtained by going to the limit ω → ωB inthe Hamiltonian (2.665).
Then ω− → 0 and the partition function (2.679) diverges,since the system becomes translationally invariant in space. From the mnemonicreplacement rule (2.353) we see that in this limit we must replace the relevantvanishing inverse frequency by an expression proportional to the volume of thesystem. The role of ω 2 in (2.353) is played here by the frequency in front of thex2 -term of the Lagrangian (2.667). Since there are two dimensions, we must replace11β−−−→−−−→V ,2ω − ωB ω→ωB 2ωω− ω− →0 2π/M 22and thusZ −−−→ω− →0V21,2 sinh (h̄βω) λ2ω(2.680)(2.681)where λω is the quantum-mechanical length scale in Eqs.
(2.296) and (2.351) of theharmonic oscillator.2.20Gauge Invariance and Alternative Path IntegralRepresentationThe action (2.611) of a particle in an external ordinary potential V (x, t) and a vector potentialA(x, t) can be rewritten with the help of an arbitrary space- and time-dependent gauge functionΛ(x, t) in the following form:Z tb M 2 eedtẋ + ẋ(t)[A(x, t) + ∇Λ(x, t)]−V (x, t)+ ∂t Λ(x, t)A[x] =2cctae− [Λ(xb , tb ) − Λ(xa , ta )] .(2.682)cThe Λ(x, t)-terms inside the integral are canceled by the last two surface terms making the actionindependent of Λ(x, t).We may now choose a particular function Λ(x, t) equal to c/e-times the classical action A(x, t)which solves the Hamilton-Jacobi equation (1.65), i.e.,i2e1 h(2.683)∇A(x, t) − A(x, t) + ∂t A(x, t) + V (x, t) = 0.2McThen we obtain the following alternative expression for the action:Z tbi21 hedtA[x] =M ẋ − ∇A(x, t) + A(x, t)2Mcta+ A(xb , tb ) − A(xa , ta ).(2.684)H.
Kleinert, PATH INTEGRALS1872.21 Velocity Path IntegralFor two infinitesimally different solutions of the Hamilton-Jacobi equation, the difference betweenthe associated action functions δA satisfies the differential equationv · ∇δA + ∂t δA = 0,(2.685)where v is the classical velocity fieldhiev(x, t) ≡ (1/M ) ∇A(x, t) − A(x, t) .c(2.686)The differential equation (2.685) expresses the fact that two solutions A(x, t) for which the particleenergy and momenta at x and t differ by δE and δp, respectively, satisfy the kinematic relationδE = p · δp/M = ẋcl · ∇δA. This follows directly from E = p2 /2M . The so-constrained variationsδE and δp leave the action (2.684) invariant.A sequence of changes δA of this type can be used to make the function A(x, t) coincide withthe action A(x, t; xa , ta ) of paths which start out from xa , ta and arrive at x, t.
In terms of thisaction function, the path integral representation of the time evolution amplitude takes the form(xb tb |xa ta ) =eiA(xb ,tb ;xa ,ta )/h̄x(tb )=xbZx(ta )=xa×expDx(2.687) Z tbi2 ie1 hM ẋ − ∇A(x, t; xa , ta ) + A(x, t)dth̄ ta 2Mcor, using v(x(t), t),(xb tb |xa ta ) = eiA(xb ,tb ;xa ,ta )/h̄Zx(tb )=xbx(ta )=xa Z tbMi2dt (ẋ − v) .Dx exph̄ ta2(2.688)The fluctuations are now controlled by the deviations of the instantaneous velocity ẋ(t) from localvalue of the classical velocity field v(x, t). Since the path integral attempts to keep the deviationsas small as possible, we call v(x, t) the desired velocity of the particle at x and t.
Introducingmomentum variables p(t), the amplitude may be written as a phase space path integral(xb tb |xa ta ) =×eiA(xb ,tb ;xa ,ta )/h̄Zx(tb )=xbx(ta )=xa0DxZDp2πh̄ Z tb i1 2p (t),expdt p(t) [ẋ(t) − v(x(t), t)] −h̄ ta2M(2.689)which will be used in Section 18.22 to give a stochastic interpretation of quantum processes.2.21Velocity Path IntegralThere exists yet another form of writing the path integral in which the fluctuating velocities play afundamental role and which will later be seen to be closely related to path integrals in the so-calledstochastic calculus to be introduced in Sections 18.12 and 18.565. We observe that by rewritingthe path integral as Z tb ZZ tbiM 2(xb tb |xa ta ) = D3 x δ xb −xa −dt ẋ(t) expẋ −V (x) ,(2.690)dth̄ ta2tathe δ-function allows us to include the last variable xn in the integration measure of the time-slicedversion of the path integral. Thus all time-sliced time derivatives (xn+1 − xn )/ for n = 0 to N1882 Path Integrals — Elementary Properties and Simple Solutionsare integrated over implying that they can be considered as independent fluctuating variables vn .In the potential, the dependence on the velocities can be made explicit by insertingZ tbx(t) = xb −dt v(t),(2.691)tZ tx(t) = xa +dt v(t),(2.692)tax(t)=X+12Ztbtbdt0 v(t0 )(t0 − t),(2.693)wherexb + xa(2.694)2is the average position of the endpoints and (t − t0 ) is the antisymmetric combination of Heavisidefunctions introduced in Eq.
(1.312).In the first replacement, we obtain the velocity path integral Z tb Z tbZZ tbM 2i.(2.695)v −V xb − dt v(t)dtdt v(t) exp(xb tb |xa ta ) = D3 v δ xb −xa−h̄ ta2ttaX≡The measure of integration is normalized to make Z tb ZM 2idt= 1.D3 v expvh̄ ta2(2.696)The correctness of this normalization can be verified by evaluating (2.695) for a free particle.Inserting the Fourier representation for the δ-function ZZ tbZ tbd3 pidt v(t) =δ xb − xa −dt v(t) ,(2.697)expp xb − xa −(2πi)3h̄tatawe can complete the square in the exponent and integrate out the v-fluctuations using (2.696) toobtain Zd3 pip2(xb tb |xa ta ) =(t−t).(2.698)expp(x−x)−baba(2πi)3h̄2MThis is precisely the spectral representation (1.330) of the free-particle time evolution amplitude(1.332) [see also Eq.
(2.51)].A more symmetric velocity path integral is obtained by choosing the third replacement (2.693).This leads to the expression Z tbZZ tbiMdt v(t) exp(xb tb |xa ta ) =D3 v δ ∆x −dt v2h̄ ta2taZ tbZ tb1i000dt V X +dt v(t )(t − t).(2.699)× exp −h̄ ta2 taThe velocity representations are particularly useful if we want to know integrated amplitudes suchas Z tbZZMidt v2 .d3 xa (xb tb |xa ta ) =D3 v exph̄ ta2Z tbZ1 tb 0i0dtV xb −dt v(t ),(2.700)× exp −h̄ ta2 twhich will be of use in the next section.H. Kleinert, PATH INTEGRALS1892.22 Path Integral Representation of Scattering Matrix2.22Path Integral Representation of Scattering MatrixIn Section 1.16 we have seen that the description of scattering processes requires several nontriviallimiting procedures on the time evolution amplitude.
Let us see what these procedures yield whenapplied to the path integral representation of this amplitude.2.22.1General DevelopmentFormula (1.472) for the scattering matrix expressed in terms of the time evolution operator inmomentum space has the following path integral representation:ZZhpb |Ŝ|pa i ≡lim ei(Eb tb −Ea ta )/h̄ d3 xb d3 xa e−i(pb xb −pa xa )/h̄ (xb tb |xa ta ).(2.701)tb −ta →∞Introducing the momentum transfer q ≡ (pb − pa ), we rewrite e−i(pb xb −pa xa )/h̄ ase−iqxb /h̄ e−ipa (xb −xa )/h̄ , and observe that the amplitude including the exponential prefactore−ipa (xb −xa )/h̄ has the path integral representation: Z tb ZM 2idtẋ − pa ẋ − V (x) .(2.702)e−ipa (xb −xa )/h̄ (xb tb |xa ta ) = D3 x exph̄ ta2The linear term in ẋ is eliminated by shifting the path from x(t) toy(t) = x(t) −patM(2.703)leading toe−ipa (xb −xb )/h̄(xb tb |xa ta ) = eZ−ip2a (tb −ta )/2Mh̄ Z tb iM 2PD y expdt.ẏ −V y+ th̄ ta2M(2.704)3Inserting everything into (2.701) we obtainZZ2eiq tb /2Mh̄ d3 yb e−iqyb /h̄ d3 yatb −ta →∞ Z tb ZM 2pa idtt.ẏ −V y +×D3 y exph̄ ta2Mhpb |Ŝ|pa i ≡limIn the absence of an interaction, the path integral over y(t) gives simply#"Z21i M (yb − ya )3d ya p= 1,exph̄ 2 tb − ta )2πh̄i(tb − ta )/M(2.705)(2.706)and the integral over ya yieldshpb |Ŝ|pa i|V ≡0 =limtb −ta →∞eiq2(tb −ta )/8Mh̄(2πh̄)3 δ (3) (q) = (2πh̄)3 δ (3) (pb − pa ),(2.707)which is the contribution from the unscattered beam to the scattering matrix in Eq.
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