Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 95
Текст из файла (страница 95)
. , xm̄ ) =kB Th̄Z0h̄/kB Tdτ Va2m̄x0 +m̄−1X(xm em=1−iωm τ!+ c.c.)−M 22Ω (x0 , . . . , xm̄ )am̄,2(5.68)and a smearing square width of the potentiala2m̄==∞12kB T X2M m=m̄ ωm + Ω2m̄−11kB Th̄Ω2kB T Xh̄Ωcoth−1−.2 + Ω2M Ω2 2kB T2kB TM m=1 ωm(5.69)For the partition function alone the additional work turns out to be not very rewarding since itrenders only small improvements. It turns out that in the low-temperature limit T → 0, the freeenergy is still equal to the optimal expectation of the Hamiltonian operator in the Gaussian wavepacket (5.49).4665 Variational Perturbation TheoryNote that the ansatz (5.7) [as well as (5.67)] cannot be improved by allowing the trial frequency Ω(x0 ) to be a matrix Ωmm0 (x0 ) in the space of Fourier components xm [i.e., by usingPP2∗m |xm | ].
This would also lead to an exactly integrablem,m0 Ωmm0 (x0 )xm xm0 instead of Ω(x0 )trial partition function. However, after going through the minimization procedure one would fallback to the diagonal solution Ωmm0 (x0 ) = δmm0 Ω(x0 ).5.7Effective Classical Potential for AnharmonicOscillator and Double-Well PotentialFor a typical application of the approximation method consider the Euclidean actionA[x] =Z0h̄/kB Tg 4M 22 2dτẋ + ω x + x .24(5.70)Let us write 1/kB T as β and use natural units with M = 1, h̄ = kB = 1. We haveto distinguish two cases:a) Case ω 2 > 0, Anharmonic OscillatorSetting ω 2 = 1, the smeared potential (5.30) is according to formula (3.876):x20 g 4 a2 3 2 2 3g 4Va2 (x0 ) =+ x0 ++ gx0 a + a .24224(5.71)Differentiating this with respect to a2 /2 gives, via (5.37),Ω2 (x0 ) = [1 + 3gx20 + 3ga2(x0 )].(5.72)This equation is solved at each x0 by iteration together with (5.24),"#βΩ(x0 )βΩ(x0 )1coth−1 .a (x0 ) =222βΩ (x0 )2(5.73)An initial approximation such as Ω(x0 ) = 0 is inserted into (5.73) to find a2 (x0 ) ≡β/12, which serves to calculate from (5.72) an improved Ω2 (x0 ), and so on.
Theiteration converges rapidly. Inserting the final a2 (x0 ), Ω2 (x0 ) into (5.71) and (5.32),we obtain the desired approximation W1 (x0 ) to the effective classical potentialV eff cl (x0 ). By performing the integral (5.39) in x0 we find the approximate freeenergy F1 plotted as a function of β in Fig. 5.2. The exact free-energy values areobtained from the known energy eigenvalues of the anharmonic oscillator.
They areseen to lie closely below the approximate F1 curve. For comparison, we have alsoplotted the classical approximation Fcl = −(1/β) log Zcl which does not satisfy theJensen-Peierls inequality and lies below the exact curve.In his book on statistical mechanics [3], Feynman gives another approximation,called here F0 , which can be obtained from the present W1 (x0 ) by ending the iteration of (5.72), (5.73) after the first step, i.e., by using the constant nonminimalH.
Kleinert, PATH INTEGRALS4675.7 Effective Classical Potential for Anharmonic OscillatorFigure 5.2 Approximate free energy F1 of anharmonic oscillator as compared with theR∞√exact energy Fex , the classical limit Fcl = −(1/β) log −∞(dx/ 2πβ)e−βV (x) , as well asan earlier approximation F0 = −(1/β) log Z0 of Feynman’s corresponding to F1 for thenonoptimal choice Ω = 0, a2 = β/12.
Note that F0 , F1 satisfy the inequality F0,1 ≥ F ,while Fcl does not.variational parameters Ω(x0 ) ≡ 0, a2 (x0 ) ≡ h̄2 β/12M. This leads to the approximation1 h̄2 β 001Va2 (x0 ) ≈ V (x0 ) +V (x0 ) +2 12M8h̄2 β12M!2V (4) (x0 ) + . . . ,(5.74)referred to as Wigner’s expansion [4]. The approximation F0 is good only at highertemperatures, as seen in Fig. 5.2. Just like F1 , the curve F0 lies also above theexact curve since it is subject to the Jensen-Peierls inequality.
Indeed, the inequality holds for the potential W1 (x0 ) in the general form (5.32), i.e., irrespective ofthe minimization in a2 (x0 ). Thus it is valid for arbitrary Ω2 (x0 ), in particular forΩ2 (x0 ) ≡ 0.In the limit T → 0, the free energy F1 yields the following approximation for theground state energy E (0) of the anharmonic oscillator:(0)E1 =1Ω 3 g+ +.4Ω44 4Ω2(5.75)4685 Variational Perturbation TheoryThis approximation is very good for all coupling strengths, including the strongcoupling limit. In this limit, the optimal frequency and energy have the expansionsΩ1 = 1/3 "g461/311+ ...+ 1/3 4/32 3 (g/4)2/3#,(5.76)and 1/3 " 4/3g434#11+ 7/3 1/3+ ...2 3 (g/4)2/3# 1/3 "1g≈+ ...
.0.681420 + 0.137584(g/4)2/3(0)E1 (g) ≈(5.77)The coefficients are quite close to the precise limiting expression to be calculated inSection 5.15 (listed in Table 5.9).b) Case ω 2 < 0: The Double-Well PotentialFor ω 2 = −1, we slightly modify the potential by adding a constant 1/4g, so that itbecomes1x2 g 4(5.78)V (x) = − + x + .244gThe additional constant ensures a smooth behavior of the energies in the limit g → 0.Since the potential possesses now two symmetric minima, it is called the double-wellpotential . Its smeared version Va2 (x0 ) can be taken from (5.71), after a sign changein the first and third terms (and after adding the constant 1/4g).Now the trial frequencyΩ2 (x0 ) = −1 + 3gx20 + 3ga2 (x0 )(5.79)can become negative, although it turns out to remain always larger than −4π 2 /β 2,since the solution is incapable of crossing the first singularity in the sum (5.24)from the right.
Hence the smearing square width a2 (x0 ) is always positive. ForΩ2 ∈ (−4π 2 /β 2 , 0), the sum (5.24) gives∞12 X2β m=1 ωm + Ω2 (x0 )!β|Ω(x0 )|1β|Ω(x0 )|=cot−1 ,22βΩ2 (x0 )a2 (x0 ) =(5.80)which is the expression (5.73), continued analytically to imaginary Ω(x0 ). Theabove procedure for finding a2 (x0 ) and Ω2 (x0 ) by iteration of (5.79) and (5.80) isnot applicable near the central peak of the double well, where it does not converge.There one finds the solution by searching for the zero of the function of Ω2 (x0 )H. Kleinert, PATH INTEGRALS5.7 Effective Classical Potential for Anharmonic Oscillatorf (Ω2 (x0 )) ≡ a2 (x0 ) −1[1 + Ω2 (x0 ) − 3gx20 ],3g469(5.81)with a2 (x0 ) calculated from (5.80) or (5.73).
At T = 0, the curves have for g ≤ gctwo symmetric nontrivial minima at ±xm withxm =where Eq. (5.79) becomesThese disappear forvuu 1 − 3ga2tg,Ω2 (xm ) = 2 − 6ga2 (xm ).4g > gc =9s2≈ 0.3629 .3(5.82)(5.83)(5.84)The resulting effective classical potentials and the free energies are plotted inFigs. 5.3 and 5.4.Figure 5.3 Effective classical potential of double well V (x) = −x2 /2 + gx4 /4 + 1/4gat various g for T = 0 and T = ∞ [where it is equal to the potential V (x) itself].
The> 0.4, but not ifquantum fluctuations at T = 0 smear out the double well completely if g ∼g = 0.2.It is useful to compare the approximate effective classical potential W1 (x) withthe true one V eff cl (x) in Fig. 5.5. The latter was obtained by Monte Carlo simulationsof the path integral of the double-well potential, holding the path average x̄ =4705 Variational Perturbation TheoryFigure 5.4 Free energy F1 in double-well potential (5.78), compared with the exact freeenergy Fex , the classical limit Fcl , and Feynman’s approximation F0 (which coincides withF1 for the nonminimal values Ω = 0, a2 = β/12).(1/β) 0β dτ x(τ ) fixed at x0 . The coupling strength is chosen as g = 0.4, where theworst agreement is expected.(0)In the limit T → 0, the approximation F1 yields an approximation E1 for theground state energy.
In the strong-coupling limit, the leading behavior is the sameas in Eq. (5.77) for the anharmonic oscillator.Let us end this section with the following remark. The entire approximationprocedure can certainly also be applied to a time-sliced path integral in which thetime axis contains N + 1 discrete points τn = n, n = 0, 1, . . . N. The only changein the above treatment consists in the replacementR2ωm→ Ωm Ω̄m =1[2 − 2 cos(ωm )].2(5.85)Hence the expression for the smearing square width parameter a2 (x0 ) of (5.24) isreplaced bya2 (x0 ) =mYmax hi2kB T∂2Ω+Ω(x)Ωlog0m mM ∂Ω2 (x0 )m=1H. Kleinert, PATH INTEGRALS5.7 Effective Classical Potential for Anharmonic Oscillator471Figure 5.5 Comparison of approximate effective classical potential W1 (x0 ) (dashedcurves) and W3 (x0 ) (solid curves) with exact V eff cl (x0 ) (dots) at various inverse temperatures β = 1/T .
The data dots are obtained from Monte Carlo simulations using 105 configurations [W. Janke and H. Kleinert, Chem. Phys. Lett. 137 , 162 (1987) (http://www.physik.fu-berlin.de/~kleinert/154)]. We have picked the worst case, g = 0.4. The solidlines represent the higher approximation W3 (x0 ), to be calculated in Section 5.13.kB T 1 ∂sinh(h̄ΩN (x0 )/2kB T )log(5.86)M Ω ∂Ωh̄Ω(x0 )/2kB T"#kB Th̄Ω(x0 )1h̄ΩN (x0 )=coth−1 ,2kB T cosh(ΩN (x0 )/2)MΩ2 (x0 ) 2kB T=where mmax = N/2 for even and (N − 1)/2 for odd N [recall (2.383)], and ΩN (x0 )is defined bysinh[ΩN (x0 )/2] ≡ Ω(x0 )/2(5.87)[see Eq.
(2.391)]. The trial potential W1 (x0 ) now readsW1 (x0 ) ≡ kB T logMsinh h̄ΩN (x0 )/2kB T+ Va2 (x0 ) (x0 ) − Ω2 (x0 )a2 (x0 ),h̄Ω(x0 )/2kB T2(5.88)rather than (5.32). Minimizing this in a2 (x0 ) gives again (5.37) and (5.38) for Ω2 (x0 ).In Fig. 5.6 we have plotted the resulting approximate effective classical potentialW1 (x0 ) of the double-well potential (5.78) with g = 0.4 at a fixed large value β = 20for various numbers of lattice points N + 1. It is interesting to compare these plots4725 Variational Perturbation Theory1.0g =0.4, β =200.80.6W10.40.20.0-2-10x012Figure 5.6 Effective classical potential W1 (x0 ) for double-well potential (5.78) withg = 0.4 at fixed low temperature T = 1/β = 1/20, for various numbers of time slicesN + 1 = 2 (×t), 4 (4), 8 (5), 16 (3), 32 (+), 64 ( ).
The dashed line represents the originalpotential V (x0 ). For the source of the data points, see the previous figure caption.with the exact curves, obtained again from Monte Carlo simulations. For N = 1,the agreement is exact. For small N, the agreement is good near and outside thepotential minima. For larger N, the exact effective classical potential has oscillationswhich are not reproduced by the approximation.5.8Particle DensitiesIt is possible to find approximate particle densities from the optimal effective classicalpotential W1 (x0 ) [5, 6]. Certainly, the results cannot be as accurate as those for thefree energies. In Schrödinger quantum mechanics, it is well known that variationalmethods can give quite accurate energies even if the trial wave functions are onlyof moderate quality.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
