Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 84
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In the case of a purely hyperbolic behaviorone speaks of a hard chaos, which is simpler to understand. The semiclassical approximation is based precisely on those orbits of a system which are exceptional ina chaotic system, namely, the periodic orbits.The expression (4.229) also serves to obtain the semiclassical density of states inD-dimensional systems via Eq.
(4.187). In D dimensions the paths, with vanishinglength contribute to the partition function the classical expression [compare (4.204)].Application of semiclassical formulas has led to surprisingly simple explanationsof extremely complex experimental data on highly excited atomic spectra whichclassically behave in a chaotic manner.For completeness, let us also state the momentum space representation of thesemiclassical fixed-energy amplitude (4.139). It is given by the momentum spaceanalog of (4.212):(2πh̄)D X0(pb |pa )E = √|D̃S |1/2 eiS(pb ,pa ;E)/h̄−iπν /2 ,D−1p2πih̄8(4.232)M.V.
Berry and M. Tabor, J. Phys. A 10 , 371 (1977), Proc. Roy. Soc. A 356 , 375 (1977).H. Kleinert, PATH INTEGRALS4.9 Quantum Corrections to Classical Density of States403where S(pb , pa ; E) is the Legendre transform of the eikonalS(pb , pa ; E) = S(pb , pa ; E) − pb xb + pa xa ,(4.233)evaluated at the classical momenta pb = ∂pb S(pb , pa ; E) and pa = ∂pa S(pb , pa ; E).The determinant can be brought to the form:∂2S1det − ⊥ ⊥ ,DS =|ṗb ||ṗa |∂pb ∂pa!(4.234)where p⊥a is the momentum orthogonal to ṗa .This formula cannot be applied to the free particle fixed-energy amplitude (3.216)for the same degeneracy reason as before.4.9Quantum Corrections to Classical Density of StatesThere exists a simple way of calculating quantum corrections to the semiclassicalexpressions (4.201) and its D-dimensional generalization (4.209) for the density ofstates.
To derive them we introduce an operator δ-function δ(E − Ĥ) via the spectralrepresentationXδ(E − En )|nihn|,(4.235)δ(E − Ĥ) ≡nwhere |ni are the eigenstates of the Hamiltonian operator Ĥ. The δ-function (4.235)has the Fourier representation [recall (1.193)]δ(E − Ĥ) =Z∞−∞dt −i(Ĥ−E)t/h̄e.2πh̄(4.236)Its matrix elements between eigenstates |xi of the position operator,ρ(E; x) = hx|δ(E − Ĥ)|xi =Z∞−∞dt iEt/h̄ehx|e−iĤt/h̄ |xi,2πh̄(4.237)define the quantum-mechanical local density of states.
The amplitude on the righthand side is the time evolution amplitudehx|e−iĤt/h̄ |xi = (x t|x 0),(4.238)which can be represented by a path integral as described in Chapter 2. In thesemiclassical limit, only the short-time behavior of (x t|x 0) is relevant.4.9.1One-Dimensional CaseFor a one-dimensional harmonic oscillator, the short-time expansion of (4.238) caneasily be written down.
For times short compared to the period 1/ω, we expand the4044 Semiclassical Time Evolution Amplitudeamplitude (2.168) at equal initial and final space points x = xa = xb as follows in apower series of t ≡ tb − ta :1−i Mω 2 x2 t/h̄2(x tb |x ta ) = qe2πih̄t/M(i t3t2Mω 4 x2 + . . . ,1 + ω2 −12h̄ 24)(4.239)This expansion is valid for an arbitrary smooth potential V (x) if the exponentialprefactor containing the harmonic potential is replaced bye−iV (x)t/h̄ ,(4.240)whereas ω 2 and Mω 4 x2 are substituted as follows:1 00V (x),M1 0→[V (x)]2 .Mω2 →Mω 4 x2(4.241)(4.242)Hence:1−iV (x)t/h̄e(x tb |x ta ) = q2πih̄t/M(t2i t31+V 00 (x) −[V 0 (x)]2 + . .
. . (4.243)12Mh̄ 24M)Inserting this into (4.237) yields the local density of states1dtqe−i[V (x)−E]t/h̄2πh̄ 2πih̄t/Mρ(E; x) =Z∞×(t2i t3001+V (x) −[V 0 (x)]2 + . . . .12Mh̄ 24M−∞)(4.244)For positive E − V (x), the integration along the real axis can be deformed intothe upper complex plane to enclose the square-root cut along the positive imaginary t-axis in the anti-clockwise sense. Setting t = iτ and using the fact that thediscontinuity across a square root cut produces a factor two, we haveρ(E; x) =2Z0∞11 τ3τ 2 00dτ−[E−V (x)]τ /h̄qV (x)−[V 0 (x)]2 + .
. . .e1−2πh̄ 2πh̄τ /M12Mh̄ 24M)((4.245)The first term can easily be integrated for E > V (x), and yields the classical localdensity of states (4.200):ρcl (E; x) =1MM1q.=πh̄ 2M[E − V (x)]πh̄ p(E; x)(4.246)In order to calculate the effect of the correction terms in the expansion (4.245),we observe that a factor τ in the integrand is the same as a derivative h̄d/dV appliedH.
Kleinert, PATH INTEGRALS4.9 Quantum Corrections to Classical Density of States405to the exponential. Thus we find directly the semiclassical expansion for the densityof states (4.245), valid for E > V (x):d2h̄2d3h̄2 00V (x) 2 −[V 0 (x)]2 3 + . . . ρcl (E; x).ρ(E; x) = 1 −12M24MdVdV)((4.247)Inserting (4.246) and performing the differentiations with respect to V we obtain1ρ(E; x) =πh̄s1h̄2 00 31MV (x)−1/22 [E − V (x)]12M4 [E − V (x)]5/2)h̄2102 15−+ . . . .(4.248)[V (x)]24M8 [E − V (x)]7/2(Note that the proceeds in powers of higher gradients of the potential; it is a gradientexpansion.The integral over (4.248) yields a gradient expansion for ρ(E).
The second termcan be integrated by parts which, under the assumption that V (x) vanishes at theboundaries, simply changes the sign of the third term, so that we find1ρ(E) =πh̄4.9.2sM2Z1h̄2151dx++ ...[V 0 (x)]21/224M8 [E − V (x)]7/2[E − V (x)]().(4.249)Arbitrary DimensionsIn D dimensions, the short-time expansion of the time evolution amplitude (4.243)takes the form1−iV (x)t/h̄(x tb |x ta ) = qDe2πih̄t/M(i t3t2∇2 V (x) −[∇V (x)]2 + .
. . .1+12Mh̄ 24M)(4.250)Recalling the iη-prescription on page 116, according to which the singularity at t = 0has to be shifted slightly into the upper half plane by replacing t → t − iη, we usethe formula9Z∞−∞1aν−1 −aηdtitae ,e=Θ(a)2π (it + η)νΓ(ν)(4.251)and obtain the obvious generalization of (4.247):d2h̄2d3h̄2∇2 V (x) 2 −[∇V (x)]2 3 + . . . ρcl (E; x), (4.252)ρ(E; x) = 1 −12M24MdVdV()where ρcl (E; x) is the classical D-dimensional local density of states (4.210).
THeway this appears here is quite different than in the earlier calssical calculation9I.S. Gradshteyn and I.M. Ryzhik,op. cit., Formula 3.382.6. The formula is easily derived byR∞expressing (it + η)−ν = Γ−1 (ν) 0 dτ τ ν−1 e−τ (it+η) .4064 Semiclassical Time Evolution Amplitude(4.205), which may be expressed in terms of the local momentum (4.208) as a momentum integralρcl (E; x) =ZdD pδ[E − H(p, x)] =(2πh̄)DMdD pδ[p − p(E; x)]. (4.253)D(2πh̄) p(E; x)ZTo see the relation to th present appearance in (4.252) we insert the Fourier decomposition of the leading term of the short-time expansion of the time evolutionamplitudeZdD p −i[p2 /2M +V (x)]t/h̄e(4.254)(x t|x 0)cl =(2πh̄)Dinto the integral representation (4.237) which takes the formZρ(E; x) =∞−∞dt2πh̄ZdD p −i[p2 −p2 (E;x)]t/2M h̄e.(2πh̄)D(4.255)By doing the integral over the time first, the size of the momentum is fixed tothe local momentum p2 (E; x) resulting in the original representation (4.253).
Theexpression (4.210) for the density of states, on the other hand, corresponds to firstintegrating over all momenta. The time integration selects from the result of thisthe correct local momenta p2 (E; x).This generalizes (4.248) toM D/21ρ(E; x) =[E − V (x)]D/2−12Γ(D/2)2πh̄i1h̄2 h 2∇ V (x)−[E − V (x)]D/2−312MΓ(D/2 − 2))1h̄22D/2−4[∇V (x)][E − V (x)]+ ... .+24MΓ(D/2 − 3)((4.256)When integrating the density (4.256) over all x, the second term in the curlybrackets can again be converted into the third term changing its sign, as in (4.249).The right-hand side can easily be integrated for all pure power potentials.
This willbe done in Appendix 4A.4.9.3Bilocal Density of StatesIt is useful to generalize the local density of states (4.237) and introduce a bilocaldensity of states:ρ(E; xb , xa ) = hxb |δ(E − Ĥ)|xa i ==Z∞−∞Z∞−∞dt iEt/h̄e(xb t|xa 0).2πh̄dt iEt/h̄ehxb |e−iĤt/h̄ |xa i2πh̄(4.257)H. Kleinert, PATH INTEGRALS4.9 Quantum Corrections to Classical Density of States407The semiclassical expansion requires now the nondiagonal version of the short-timeexpansions (4.250). For the one-dimensional harmonic oscillator, the expansion(4.239) is generalized to22 21(xb tb |xa ta ) = qeiM (xb −xa ) /2th̄ e−iM ω x̄ t/2h̄2πih̄t/Mi t3i tt2Mω 4 x̄2 −(x − xa )2 Mω 2 + . . . ,× 1 + ω2 −12h̄ 24h̄ 24 b()(4.258)where x̄ = (xb +xa )/2 is the mean position of the two endpoints. In this expansion wehave included all terms whose size is of the order t3 , keeping in mind that (xb − xa )2is of the order h̄ in a finite amplitude.
Going to D dimensions and performing thesubstitutions (4.241) and (4.242), this expansion is generalized to1iM (xb −xa )2 /2th̄−iV (x̄)t/h̄(xb tb |xa ta ) = qDe2πih̄t/M(4.259)t2i t3i t× 1+∇2 V (x̄)−[∇V (x̄)]2 −[(x −xa )∇]2 V (x̄)+. . . .12Mh̄ 24Mh̄ 24 b()For a derivation without substitution trick in (4.241) and (4.242) see Appendix 4B.Inserting this amplitude into the integral in Eq. (4.257), we obtain the bilocaldensity of statesρ(E; xb , xa ) =Z∞−∞dt1iM (xb −xa )2 /2th̄ −i[V (x̄)−E]t/h̄eqDe2πh̄ 2πih̄t/Mi t3i tt2∇2 V (x̄) −[∇V (x̄)]2 −[(x −xa )∇]2 V (x̄) + .
. . . (4.260)× 1+12Mh̄ 24Mh̄ 24 b()The first term in the integrand is simply the time evolution amplitude of the freeparticle in a constant potential V (x̄) which has the Fourier decomposition [recall(1.330)]:ZdD p ip(xb −xa )/h̄ −iH(p,x̄)t/h̄(xb tb |xa ta )cl =ee.(4.261)(2πh̄)DIndeed, inserting this into (4.257), and performing the integration over time, we findρcl (E; xb , xa ) =ZdD pip(xb −xa )/h̄.D δ(E − H(p, x̄))e(2πh̄)(4.262)Decomposing the momentum integral into radial and angular parts as in (4.206), wecan integrate out the radial part as in (4.197), whereas the angular integral yieldsthe following function of R = |xb − xa |:Zdp̂ eip(xb −xa )/h̄ = SD (pR/h̄),(4.263)4084 Semiclassical Time Evolution Amplitudewhich is a direct generalization of the surface of a sphere in D dimensions (4.207).It reduces to it for p = 0.
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