Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 83
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. . .(4.184)This condition agrees precisely with the Bohr-Sommerfeld rule (4.25) for semiclassical quantization. At the poles, one hasZ̃QM (E) ≈ t(E)ih̄.S (En )(E − En )0(4.185)Due to (4.85), the pole terms acquire the simple formZ̃QM (E) ≈ih̄.E − En(4.186)From (4.183) we derive the density of states defined in (1.579). For this we usethe general formula1disc Z̃QM (E),(4.187)ρ(E) =2πh̄3964 Semiclassical Time Evolution Amplitudewhere disc ZQM (E) is the discontinuity ZQM (E + iη) − ZQM (E − iη) across thesingularities defined in Eq. (1.325).
If we equip the energies En in (4.186) with theusual small imaginary part −iη, we can also write (4.187) asρ(E) =1ReZ̃QM (E).πh̄(4.188)Inserting here the sum (4.183), we obtainρ(E) =or∞t(E) Xcos{n[S(E)/h̄ − π]}πh̄ n=1(4.189)∞Xt(E)ρ(E) =−1 +ein[S(E)/h̄−π] .2πh̄n=−∞!(4.190)With the help of Poisson’s summation formula (1.197), this goes over into∞t(E) t(E) Xδ[S(E)/h̄ − 2π(n + 1/2)].+ρ(E) = −2πh̄h̄ n=−∞(4.191)The right-hand side contains δ-functions which are singular at the semiclassicalenergy values (4.184).
Using once more the relation (4.85), the formula δ(ax) =a−1 δ(x) leads to the simple expressionρ(E) = −∞Xt(E)+δ(E − En ).2πh̄ n=−∞(4.192)This result has a surprising property: Consider the spacing between the energy levels∆En = En − En−1 = 2πh̄∆En∆Sn(4.193)and average the sum in (4.192) over a small energy interval ∆E containing severalenergy levels. Then we obtain an average density of states:ρav (E) =t(E)S 0 (E)=.2πh̄2πh̄(4.194)It cancels precisely the first term in (4.192). Thus, the semiclassical formula (4.183)possesses a vanishing average density of states. This cannot be correct and weconclude that in the derivation of the formula, a contribution must have been overlooked. This contribution comes from the classical partition function. Within theabove analysis of periodic orbits, there are also those which return to the point ofdeparture after an infinitesimally small time (which leaves them with no time tofluctuate). The expansion (4.183) does not contain them, since the saddle pointapproximation to the time integration (4.176) used for its derivation fails at shorttimes.
The reason for this failure is the singular behavior of the fluctuation factor∝ 1/(tb − ta )1/2 in (4.96).H. Kleinert, PATH INTEGRALS4.7 Semiclassical Quantum-Mechanical Partition Function397In order to recover the classical contribution, one simply uses the short-timeamplitude in the form (2.341) to calculate the purely classical contribution to Z(E):Zcl (E) ≡Zdpih̄.2πh̄ E − H(p, x)Zdx(4.195)This implies a classical contribution to the density of statesρcl (E) ≡Zdx ρcl (E; x),(4.196)which is a spatial integral over the classical local density of statesρcl (E; x) ≡Zdpδ[E − H(p, x)].2πh̄(4.197)The δ-function in the integrand can be rewritten asδ(E − H(p, x)) =M[δ(p − p(E; x)) + δ(p + p(E; x))] ,p(E; x)(4.198)where p(E; x) is the local momentum associated with the energy Ep(E; x) =q2M[E − V (x)],(4.199)which was defined in (4.3), except that we have now added the energy to the argument, to have a more explicit notation. It is then trivial to evaluate the integral(4.197) and (4.196) yielding the classical local density of states1M,πh̄ p(E; x)(4.200)M11=t(E),πh̄ p(E; x)2πh̄(4.201)ρcl (E; x) =and its integralρcl (E) =Zdxwhich coincides precisely with the average classical density of states ρav (E) in(4.194).
This contribution ensures that the total density of states consisting ofthe sum of (4.190) and (4.201), has on the average the correct classical value.Note that by Eq. (4.194), the eikonal S(E) is related to the integral over theclassical density of states ρcl (E) by a factor 2πh̄:S(E) =ZE−∞dE ρcl (E).(4.202)Recalling the definition (1.583) of the number of states up to the energy E we seethatS(E) = 2πh̄N(E),(4.203)which shows that the Bohr-Sommerfeld quantization condition (4.184) is the semiclassical version of the completely general equation (1.584).3984 Semiclassical Time Evolution AmplitudeThe D-dimensional generalization of the classical partition function (4.195) readsZcl (E) ≡ZDd xZih̄dD p,2πh̄ E − H(p, x)(4.204)and of the density of states (4.197):ρcl (E; x) ≡ZdD pδ[E − H(p, x)].(2πh̄)D(4.205)To do the momentum integral we separate it into radial and angular parts,dD p=(2πh̄)DZZdp pD−1Zdp̂.(4.206)The angular integral yields the surface of a unit sphere in D dimensions:Z2π D/2.dp̂ = SD =Γ(D/2)(4.207)The δ-function δ(E − H(p, x)) can again be rewritten as in (4.198), which selectsthe momenta of magnitudep(E; x) =q2M[E − V (x)].(4.208)ρcl (E) ≡Z(4.209)Thus we finddD x ρcl (E; x),where ρcl (E; x) is the classical local density of states.ρcl (E; x) = SDpD (E; x)M12M{2M[E − V (x)]}D/2−1, (4.210)2D =2 D/2 Γ(D/2)p (E; x) (2πh̄)(4πh̄ )generalizing expression (4.200).
The number of states with energies between E andE + dE in the volume element dD x is dEd3 xρcl (E; x).4.8Multi-Dimensional SystemsFor completeness we state some features of the semiclassical results which appearwhen generalizing the theory to D dimensions. For a detailed derivation see the richliterature on this subject quoted at the end of the chapter.In the multi-dimensional case, the Van Vleck-Pauli-Morette determinant (4.118)is written in the formF (xb , xa ; tb − ta ) = √1D2πih̄detD [−∂xi ∂xj A(xb , xa ; tbba1/2− ta )]e−iπν/2 ,(4.211)H.
Kleinert, PATH INTEGRALS4.8 Multi-Dimensional Systems399where ν is the Maslov-Morse index.The fixed-energy amplitude becomes the sum over all periodic orbits:7(xb |xa )E = √12πih̄D−1Xp0|DS |1/2 eiS(xb ,xa ;E)/h̄−iπν /2 ,(4.212)where S(xb , xa ; E) is the D-dimensional generalization of (4.61) and DS the (D +1) × (D + 1)-determinant:D+1DS = (−1)det∂2S∂xb ∂xa∂2S∂xb ∂E∂2S∂E∂xa∂2S∂E∂E.(4.213)The factor (−1)D+1 makes the determinant positive for short trajectories. The indexν 0 differs from ν by one unit if ∂ 2 S/∂E 2 = ∂t(E)/∂E is negative.In D dimensions, the Hamilton-Jacobi equation leads to∂2S∂2S∂H·= ẋb ·= 0,∂pb ∂xb ∂xa∂xb ∂xa(4.214)instead of (4.148).
Only the longitudinal projection of the D×D-matrix ∂ 2 S/∂xb ∂xaalong the direction of motion vanishes now. In this direction∂2Sẋb ·= 1,∂xb ∂E(4.215)so that the determinant (4.213) can be reduced to∂2S1det − ⊥ ⊥ ,DS =|ẋb ||ẋa |∂xb ∂xa!(4.216)instead of (4.150). Here x⊥b,a denotes the deviations from the orbit orthogonal toẋb,a , and we have used (2.283) to arrive at (4.216).As an example, let us write down the D-dimensional generalization of the freeparticle amplitude (4.155). The eikonal is obviously√S(xa , xb ; E) = 2ME|xb − xa |,(4.217)and the determinant (4.216) becomesDS =7M (2ME)(D−1)/2.2E |xa − xb |D−1M.C. Gutzwiller, J.
Math. Phys. 8 , 1979 (1967); 11 , 1791 (1970); 12 , 343 (1971).(4.218)4004 Semiclassical Time Evolution AmplitudeThus we find(xb |xa )E =s(2ME)(D−1)/4 i√2M E|xb −xa |/h̄1Me.2E (2πih̄)(D−1)/2 |xa − xb |(D−1)/2(4.219)For D = 1, this reduces to (4.155).Note that the semiclassical result coincides with the large-distance behavior(1.356) of the exact result (1.352), since the semiclassical limit implies a large momenta k in the Bessel function (1.352).When calculating the partition function, one has to perform a D-dimensionalintegral over all xb = xa . This is best decomposed into a one-dimensional integralalong the orbit and a D − 1 -dimensional one orthogonal to it.
The eikonal functionS(xa , xa ; E) is constant along the orbit, as in the one-dimensional case. Whenleaving the orbit, however, this is no longer true. The quadratic deviation of Sorthogonal to the orbit is∂ 2 S(x, x; E)1(x − x⊥ ),(x − x⊥ )T⊥⊥2∂x ∂x(4.220)where the superscript T denotes the transposed vector to be multiplied from theleft with the matrix in the middle. After the exact trace integration along the orbitand a quadratic approximation in the transversal direction for each primitive orbit,which is not repeated, we obtain the contribution to the partition functionZsc = 2 ∂ S(x , x ; E) 1/2ba⊥⊥∞X∂xb ∂xaxb =xa =xein[S(E)/h̄−iπν/2] ,t(E) ∂ 2 S(x, x; E) 1/2n=1⊥⊥ ∂x ∂x(4.221)where ν is the Maslov-Morse index of the orbit.
The ratio of the determinants isconveniently expressed in terms of the determinant of the so-called stability matrixM in phase space, which is introduced in classical mechanics as follows:Consider a classical orbit in phase space and vary slightly the initial point, mov⊥ing it orthogonally away from the orbit by δx⊥a , δpa .
This produces variations at⊥the final point δx⊥b , δpb , related to those at the initial point by the linear equationδx⊥bδp⊥b!=A BC D!δx⊥aδp⊥a!≡Mδx⊥aδp⊥a!.(4.222)The 2(D − 1) × 2(D − 1)-dimensional matrix is the stability matrix M. It can beexpressed in terms of the second derivatives of S(xb , xa ; E).
These appear in therelation!!!−a −bδp⊥δx⊥aa=,(4.223)bTcδp⊥δx⊥bbH. Kleinert, PATH INTEGRALS4.8 Multi-Dimensional Systems401where a, b, and c are the (D − 1) × (D − 1)-dimensional matrices∂2S⊥,∂x⊥a ∂xaa=b=∂2S⊥,∂x⊥a ∂xbc=∂2S⊥.∂x⊥b ∂xb(4.224)From this one calculates the matrix elements of the stability matrix (4.222):A = −b−1 a,B = −b−1 ,C = bT − cb−1 a,D = −cb−1 .(4.225)The stability properties of the classical orbits are classified by the eigenvalues ofthe stability matrix (4.222). In three dimensions, the eigenvalues are given by thezeros of the characteristic polynomial of the 4 × 4 -matrix M:A−λ BP (λ) = |M − λ| =CD−λ− b−1 a − λ −b−1= Tb − cb−1 a−cb−1 − λThe usual manipulations bring this to the form.(4.226)1 −a − λb−1 −b−1 a − λ −b−1 T=P (λ) =2T0 b +λ−λ |b| b + (a + c)λ + λ b1 T2 =b + (a + c)λ + λ b .|b|(4.227)Precisely this expression appears, with xb = xa , in the prefactor of (4.221) if this isrewritten as ∂ 2 S 1/2 ∂x⊥ ∂x⊥ ba xb ≈xa =x.(4.228)1/2 ∂2S∂S ∂S∂ 2 S ⊥ ⊥ +2⊥ +⊥ ∂x ∂x∂x⊥∂x⊥bbb ∂xaa ∂xa xb =xa =xDue to (4.225), this coincides with P (1)−1/2 .
The semiclassical limit to the quantummechanical partition function takes therefore the simple formZsc (E) = t(E)1eiS(E)−iπν/2.P (1)1/2 1 − eiS(E)−iπν/2(4.229)The energy eigenvalues lie at the poles and satisfy the quantization rules [compare(4.25), (4.184)]S(En ) = 2πh̄(n + ν/4).(4.230)The eigenvalues of the stability matrix come always in pairs λ, 1/λ, as is obviousfrom (4.227).
For this reason, one has to classify only two eigenvalues. These mustbe either both real or mutually complex-conjugate. One distinguishes the followingcases:1. elliptic,ifλ = eiχ , e−iχ , with a real phase χ 6= 0,4024 Semiclassical Time Evolution Amplitude2. direct parabolic,inverse parabolic,3.λ = 1,λ = −1,ifdirect hyperbolic,inverse hyperbolic,λ = e±χ ,λ = −e±χ ,if4. loxodromic,λ = eu±v .ifIn these cases,P (1) =2Y(λi − 1)(1/λi − 1)(4.231)i=1has the values1.4 sin2 (χ/2),2.03.− 4 sinh2 (χ/2)4.4 sin[(u + v)/2] sin[(u − v)/2].or4,or4 cosh2 (χ/2),Only in the parabolic case are the equations of motion integrable, this being obviously an exception rather than a rule, since it requires the fulfillment of the equationa + c = ±2b. Actually, since the transverse part of the trace integration in the partition function results in a singular determinant in the denominator of (4.229), thiscase requires a careful treatment to arrive at the correct result.8 In general, a system will show a mixture of elliptic and hyperbolic behavior, and the particle orbitsexhibit what is called a smooth chaos.
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