Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 99
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Let ψΩ (x − x0 ) be the wave functions of the trialoscillator.1 With these, we form the projectionsZΩ (x0 ) ≡Z=Z(n)(n)∗(n)dxb dxa ψΩ (xb )(xb τb |xa τa )ψΩ (xa )(n)∗dxb dxa ψΩ (xb )Z(xa ,0);(xb ,h̄β)!(n)Dxe−A/h̄ ψΩ (xa ).(5.154)If the temperature tends to zero, an optimization in the parameters x0 and Ω(x0 )should yield information on the energy E (n) of this state by containing an exponentially decreasing function(n)Z (n) ≈ e−βE .(5.155)(n)Since the trial wave functions ψΩ (x−x0 ) are not the true ones, the behavior (5.155)(n0 )contains an admixture of Boltzmann factors e−βEwith n0 6= n, which have to beeliminated. They are easily recognized by the powers of g which they carry.The calculation of (5.154) is done in the same approximation that rendered goodresults for the ground state, i.e., we approximate the part of Z (n) behaving like(5.155) as follows:x0(n)Z (n) ≈ ZΩ e−hAint /h̄iΩ ,(n)(5.156)1In contrast to the earlier notation, we now use superscripts in parentheses to indicate theprincipal quantum numbers.
The subscripts specify the level of approximation.H. Kleinert, PATH INTEGRALS4895.12 Variational Approach to Excitation Energies(n)where ZΩ denotes the contribution of the nth excited state to the oscillator partitionfunction(n)ZΩ = e−βh̄Ω(n+1/2) ,(5.157)(n)x0and hAxint0 /h̄i(n)Ω stands for the expectation of Aint /h̄ in the state ψΩ (x − x0 ). Thisapproximation corresponds precisely to the first term in the perturbation expansion(3.506) (after continuing to imaginary times).Note that the approximation on the right-hand side of (5.156) is not necessarilysmaller than the left-hand side, as in the Jensen-Peierls inequality (5.10), since themeasure of integration in (5.154) is no longer positive.For the action (5.151), the best value of x0 lies at the coordinate origin.
This(n)simplifies the calculation of the expectation hAxint0 /h̄iΩ. The expectation of x2k in(n)the state ψΩ (x) is given byDx2kE(n)Ω=h̄kn2k ,(MΩ)k(5.158)where35n2 = (n + 1/2), n4 = (n2 + n + 1/2), n6 = (2n3 + 3n2 + 4n + 3/2),241(70n4 + 140n3 + 344n2 + 280n + 105), . . . .(5.159)n8 =16After inserting (5.153) into (5.156), the expectations (5.158) yield the approximationZ(n) h̄n1 2g h̄2 n42≈ exp −β+Ω + ω22Ω4 M 2 Ω2("#).(5.160)With the dimensionless coupling constant g 0 ≡ gh̄/M 2 ω 3 , this corresponds to thevariational energiesW(n)ω2g0 ω31= h̄Ω+n2 +n .2Ω4 Ω2 4"!#(5.161)They are optimized by the extremal Ω-values(n)Ω = Ω1 =where√23ω cosh√23ω cosh13harcosh (g/g (n))13arccos(g/g (n))2 n M 2ω3.g (n) ≡ √ 23 3 n4 h̄iiforg > g (n) ,g < g (n) ,(5.162)(5.163)(n)The optimized W (n) yield the desired approximations Eappfor the excited energiesof the anharmonic oscillator.4905 Variational Perturbation TheoryFor large g, the trial frequency grows like(n)1/3gh̄4M 2 ω 3!1/3Ω≡6ñ,(5.164)ñ ≡n2 + n + 1/24n4 /3,=2n2n + 1/2(5.165)where(n)making the energy Eappgrow like(n)EBS(n)→ h̄ωκgh̄4M 2 ω 3!1/3,for large g,(5.166)with3 · 61/3(2n + 1) ñ1/3 ≈ 0.68142 × (2n + 1) ñ1/3 .(5.167)8For n = 0, this is in good agreement with the precise growth behavior to be calculated in Section 5.16 where we shall find κ(0)ex = 0.667 986 .
. . (see Table 5.9).In the limit of large g and n, this can be compared with the exact behaviorobtained from the semiclassical approximation of Bohr and Sommerfeld, whichgives the same leading powers in g and n as in (5.166), but a 3% larger prefactor 0.688 253 702 × 2 [recall (4.32)]. The exact values of E (n) /(gh̄/4M 2 ) and1 (n)/(n + 1/2)4/3 are for n = 0, 2, 4, 6, 8, 10 are shown in Table 5.4.2κ(n)with the precise numerical soluA comparison of the approximate energies Eapptions of the Schrödinger equation in natural units h̄ = 1, M = 1 in Table 5.5 showsan excellent agreement for all coupling strengths.(n)Near the strong-coupling limit, the optimal frequencies Ω1 and the approximate(n)energies E1 behave as follows [in natural units; compare (5.76) and (5.77)]:κ(n) =(n)Ω1(n)E1= 6=1/3ñg4 4/3341/3 " 1/31 31+9 4ñg(2n + 1)4#1+ ...
,(ñg/4)2/361/311++ ... .9 (ñg/4)2/31/3 "#(5.168)Table 5.4 Approach of variational energies of nth excited state to Bohr-Sommerfeldapproximation with increasing n. Values in the last column converge rapidly towards theBohr-Sommerfeld value 0.688 253 702 .
. . in Eq. (4.32).n0246810E (n) /(gh̄/4M 2 )0.667 986 259 155 777 108 34.696 795 386 863 646 196 210.244 308 455 438 771 076 016.711 890 073 897 950 947 123.889 993 634 572 505 935 531.659 456 477 221 552 442 8κ(n) /(n + 1/2)4/30.841 609 948 112 105 0010.692 125 685 914 981 3140.689 449 772 359 340 7650.688 828 486 600 234 4660.688 590 146 947 993 6760.688 474 290 179 981 43312H. Kleinert, PATH INTEGRALS4915.12 Variational Approach to Excitation EnergiesTable 5.5 Energies of the nth excited states of anharmonic oscillator ω 2 x2 /2 + gx4 /4for various coupling strengths g (in natural units). In each entry, the upper number showsthe energies obtained from a numerical integration of the Schrödinger equation, whereasthe lower number is our variational result.g/4E (0)E (1)E (2)E (3)E (4)E (5)E (6)E (7)E (8)0.10.559 1460.560 3071.769 501.773 393.138 623.138 244.628 884.621 936.220 306.205 197.899 777.875 229.657 849.622 7611.487311.440713.379013.32350.20.602 4050.604 9011.950 541.958 043.536 303.534 895.291 275.278 557.184 467.158 709.196 349.156 1311.313 211.257 313.524913.452215.822215.73280.30.637 9920.641 6302.094 642.104 983.844 783.842 405.796 575.779 487.911 757.878 2310.166510.115112.544312.473615.032814.941717.622417.50990.40.668 7730.673 3942.216 932.229 624.102 844.099 596.215 596.194 958.511 418.471 6910.963110.902813.552013.469816.264216.158819.088918.95910.50.696 1760.701 6672.324 412.339 194.327 524.323 526.578 406.554 759.028 788.983 8311.648711.580914.417714.325717.322017.202920.345220.20090.60.721 0390.727 2962.421 022.437 504.528 124.523 436.901 056.874 779.487 739.438 2512.255712.181615.183215.082818.253518.125621.454221.29740.70.743 9040.750 8592.509 232.527 294.710 334.705 017.193 277.164 649.902 619.849 1112.803912.724015.873715.765819.094518.957322.453022.28520.80.765 1440.772 7362.590 702.610 214.877 934.872 047.461 457.430 7110.282810.225713.305713.220616.505316.390719.863419.717923.365823.18800.90.785 0320.793 2132.666 632.687 455.033 605.027 187.710 077.677 3910.634910.574413.770013.680117.089416.968720.574020.420924.209124.022110.803 7710.812 5002.737 892.759 945.179 295.172 377.942 407.907 9310.963610.900014.203114.109017.634017.507621.236421.076324.995024.7996101.504 971.531 255.321 615.382 1310.347110.324416.090115.999322.408822.248429.211528.979336.436936.130144.040143.655951.986551.5221502.499 712.547 588.915 109.023 3817.437017.395227.192627.031437.938537.656249.516449.109461.820361.284274.772874.102988.314387.50591003.131 383.192 4411.187311.324921.906921.853534.182533.977947.707247.349562.281261.766077.770877.092494.078093.2307111.128110.1065005.319 895.425 7619.043419.281137.340737.247758.301657.948981.401280.7856106.297105.411132.760131.595160.622159.167189.756188.00110006.694 226.827 9523.972224.272147.017346.900073.419172.9741102.516101.740133.877132.760167.212165.743202.311200.476239.012236.799The tabulated energies can be used to calculate an approximate partition function at all temperatures:Z≈∞X(n)e−βEapp .(5.169)n=0The resulting free energies F10 = − log Z/β agree well with the previous variationalresults of the Feynman-Kleinert approximation — in the plots of Fig.
5.2, the curvesare indistinguishable. This is not astonishing since both approximations are dom(0)inated near zero temperature by the same optimal energy Eapp, while approachingthe semiclassical behavior at high temperatures. The previous free energy F1 doesso exactly, the free energy F10 to a very good approximation.4925 Variational Perturbation Theory(n)By combining the Boltzmann factors with the oscillator wave functions ψΩ (x),we also calculate the density matrix (xb τb |xa τa ) and the distribution functions ρ(x) =(x h̄β|x 0) in this approximation:(xb τb |xa τa ) ≈∞X(n)ψ (n) (xb )ψ (n)∗ (xa )e−βEapp .(5.170)n=0They are in general less accurate than the earlier calculated particle density ρ1 (x0 )of (5.94).5.13Systematic Improvement of Feynman-KleinertApproximation. Variational Perturbation TheoryA systematic improvement of the variational approach leads to a convergent variational perturbation expansion for the effective classical potential of a quantummechanical system [27].
To derive it, we expand the action in powers of the deviations of the path from its average x0 = x̄:δx(τ ) ≡ x(τ ) − x0 .(5.171)A = V (x0 ) + AxΩ0 + A0xint0 ,(5.172)The expansion readswhere AxΩ0 is the quadratic action of the deviations δx(τ )AxΩ0 =Zh̄/kB T0dτoMn[δ ẋ(τ )]2 + Ω2 (x0 )[δx(τ )]2 ,2(5.173)and A0xint0 contains all higher powers in δx(τ ):A0xint0 =Zh̄β0dτg2gg[δx(τ )]2 + 3 [δx(τ )]3 + 4 [δx(τ )]4 + . . .
.2!3!4!(5.174)The coupling constants are, in general, x0 -dependent:gi(x0 ) = V (i) (x0 ) − Ω2 δi2 ,V (i) (x0 ) ≡di V (x0 ).dxi0(5.175)For the anharmonic oscillator, they take the valuesg2 (x0 ) = M[ω 2 − Ω2 (x0 )] + 3gx20 ,g3 (x0 ) = 6gx0 ,g4 (x0 ) = 6g.(5.176)Introducing the parameterr 2 = 2M(ω 2 − Ω2 )/g,(5.177)H. Kleinert, PATH INTEGRALS5.13 Systematic Improvement of Feynman-Kleinert Approximation . . .493which has the dimension length square, we write g2 (x0 ) asg2 (x0 ) = g(r 2/2 + 3x20 ).(5.178)With the decomposition (5.172), the Feynman-Kleinert approximation (5.32) tothe effective classical potential can be written asW1Ω (x0 ) = V (x0 ) + FΩx0 +1hAxint0 ixΩ0 .h̄β(5.179)To generalize this, we replace the local free energy FΩx0 by FΩx0 +∆F x0 , where ∆F x0denotes the local analog of the cumulant expansion (3.484).
This leads to the variational perturbation expansion for the effective classical potential1hAxint0 ixΩ0h̄β1 D x0 2 Ex01 D x0 3 Ex0− 2 AintAint++ ...Ω,cΩ,c2!h̄ β3!h̄3 βV eff cl (x0 ) = V (x0 ) + FΩx0 +(5.180)in terms of the connected expectation values of powers of the interaction:DAxint0 2Ex0Ω,cDEx0 3 x0AintΩ,c≡≡...DAxint0 2Ex0ΩDEx0 3 x0AintΩ− hAxint0 ixΩ0 2 ,D− 3 Axint0 2Ex0Ω(5.181)hAxint0 ixΩ0 + 2 hAxint0 ixΩ0 3 ,(5.182).By construction, the infinite sum (5.180) is independent of the choice of the trialfrequency Ω(x0 ).When truncating (5.180) after the Nth order, we obtain the approximationΩWN (x0 ) to the effective classical potential V eff cl (x0 ). In many applications, it issufficient to work with the third-order approximation1 D 0x0 Ex0A intΩh̄βDEEx0x011 D− 2 A0xint0 2+ 3 A0xint0 3.Ω,cΩ,c2h̄ β6h̄ βW3Ω (x0 ) = V (x0 ) + FΩx0 +(5.183)In contrast to (5.180), the truncated sums WN (x0 ) do depend on Ω(x0 ).
Since theinfinite sum is Ω(x0 )-independent, the best truncated sum WN (x0 ) should lie at thefrequency ΩN (x0 ) where WNΩ (x0 ) depends minimally on it. The optimal ΩN (x0 ) iscalled the frequency of least dependence. Thus we require for (5.183)∂W3Ω (x0 )= 0.∂Ω(x0 )(5.184)At the frequency Ω3 (x0 ) fixed by this condition, Eq. (5.183) yields the desired thirdorder approximation W3 (x0 ) to the effective classical potential.4945 Variational Perturbation TheoryThe explicit calculation of the expectation values on the right-hand side of (5.183)proceeds according to the rules of Section 3.18.
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