Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 101
Текст из файла (страница 101)
The corresponding analyticexpression for W3Ω (x0 ) isg2 2 g4 4a + a28#" 2g32 6g2 4 g2 g4 4 2 g42 4 4 g42 81a +a a + a2 a + a2 + + a2−2!h̄Ω 2 22 28246W3Ω (x0 ) = V (x0 ) + FΩx0 +1g2gg2g+ 2 2 g23a63 + 3 2 4 (a42 )2 + 2 4 a63 a2(5.197)423!h̄ Ω!g2 g42 4 2 2 g2 g42 6 4 g2 g42 10 g2 g32 8+3(a2 ) a +aa +a +a44 36 32 3"!3g43 4 2 2 g43 6 6 g43 10 2 g43 12 3g32g4 10 3g32 g4 8 2+(a a ) + a3 a + a3 a + a3 +a0 +aa16 28484 34 3!#.5005 Variational Perturbation TheoryThe quantities a2LV are ordered in the same way as the associated diagrams in (5.196).As before, we have omitted the variable x0 in all but the first three terms, for brevity.The optimal trial frequency Ω(x0 ) is found numerically by searching, at eachvalue of x0 , for the real roots of the first derivate of W3Ω (x0 ) with respect to Ω(x0 ).Just as for zero temperature, the solution happens to be unique.By calculating the integralZ3 =Z∞−∞dx0 −W3 (x0 )/kB Te,le (h̄β)(5.198)we obtain the approximate free energy1F3 = − ln Z3 .β(5.199)The results are listed in Table 5.7 for various coupling constants g and temperatures.They are compared with the exact free energyX1exp(−βEn ),Fex = − logβn(5.200)whose energies En were obtained by numerically solving the Schrödinger equation.g0.0020.42.04.020200200080000β2.01.05.01.05.010.01.05.01.05.010.05.00.11.010.00.13.0F10.4279370.2260840.5591550.4926850.6994310.7009340.6573960.8098351.181021.241581.243532.545872.69975.408275.452518.151718.501F30.4279370.2260750.5586780.4925780.6961800.6962850.65710510.8039111.178641.225161.225152.501172.698345.323195.322518.047018.146Fex0.4277410.2260740.5586750.4925790.6961180.6961760.65710490.8037581.178631.224591.224592.499712.698345.319895.319918.045118.137Table 5.7 Free energy of anharmonic oscillator with potential V (x) = x2 /2 + gx4 /4 forvarious coupling strengths g and β = 1/kT .We see that to third order, the new approximation yields energies which arebetter than those of F1 by a factor of 30 to 50.
The remaining difference withrespect to the exact energies lies in the fourth digit.H. Kleinert, PATH INTEGRALS5015.15 Convergence of Variational Perturbation ExpansionIn the high-temperature limit, all approximations WN (x0 ) tend to the classicalresult V (x0 ), as they should. Thus, for small β, the approximations W3 (x0 ) andW1 (x0 ) are practically indistinguishable.The accuracy is worst at zero temperature. Using the T → 0 -limits of theΩFeynman integrals a2LV given in (3.550) and (3D.14), the approximation W3 takesthe simple formh̄Ω(x0 ) g2 2 g4 4+ a + a28" 22#1 g2 4 g2 g4 6 g32 2 4 g42 7 8a +a +a +a−(5.201)2h̄Ω 226 324 2124 8282 13 102 35 103 37 1233 6g a + g2 g3 a + g2 g4 3a + g3 g4 a + g2 g4 a + g4 a ,+3316646h̄Ω2 2 2W3Ω (x0 ) = V (x0 ) +where a2 = h̄/2MΩ.
As in (5.197), we have omitted the arguments x0 in Ω andgi , for brevity. At zero temperature, the remaining integral over x0 in the partitionfunction (5.198) receives its only contribution from the point x0 = 0, where W3 (0)is minimal. There it reduces to the energy W3 of Eq. (5.192).5.15Convergence of Variational Perturbation ExpansionFor a single interaction xp , the approximation WN at zero temperature can easilybe carried to high orders [13, 14].
The perturbation coefficients are available exactlyfrom recursion relations, which were derived for the anharmonic oscillator with p = 4in Appendix 3C. The starting point is the ordinary perturbation expansion for theenergy levels of the anharmonic oscillatorE(n)=ω∞Xk=0(n)Ekg4ω 3k.(5.202)It was remarked in (3C.27) and will be proved in Section 17.10 [see Eq. (17.323)](n)that the coefficients Ek grow for large k like(n)Ek1−−−→ −πs6 12n(−3)k Γ(k + n + 1/2).π n!(5.203)Using Stirling’s formula3n! ≈ (2π)1/2 nn−1/2 e−n ,this amounts to(n)Ek√"#n+k2 3 (−4)n −3(k + n)−−−→ −.πn!e(n)(5.204)(5.205)Thus, Ek grows faster than any power in k. Such a strong growth implies thatthe expansion has a zero radius of convergence. It is a manifestation of the fact that3M.
Abramowitz and I. Stegun, op. cit., Formulas 6.1.37 and 6.1.38.5025 Variational Perturbation Theory0.80.75N =1N =30.7WN =50.650.6N =11121.532.53.54Ω0.590.58N =2N =4N =60.57N =10W0.560.5511.522.5Ω33.54Figure 5.15 Typical Ω-dependence of N th approximations WN at T = 0 for increasingorders N . The coupling constant has the value g/4 = 0.1.
The dashed horizontal lineindicates the exact energy.the energy possesses an essential singularity in the complex g-plane at the expansionpoint g = 0. The series is a so-called asymptotic series. The precise form of thesingularity will be calculated in Section 17.10 with the help of the semiclassicalapproximation.If we want to extract meaningful numbers from a divergent perturbation seriessuch as (5.202), it is necessary to find a convergent resummation procedure.
Such aprocedure is supplied by the variational perturbation expansion, as we now demonstrate for the ground state energy of the anharmonic oscillator.Truncating the infinite sum (5.202) after the Nth term, the replacement (5.188)followed by a re-expansion in powers of g up to order N leads to the approximationWN at zero temperature:WNΩ=ΩNXl=0(0)εlg4Ω3l,(5.206)H. Kleinert, PATH INTEGRALS5035.15 Convergence of Variational Perturbation ExpansionW0.55914635N =21N =150.55914625N =111.61.421.82.2ΩFigure 5.16 New plateaus in WN developing for higher orders N ≥ 15 in addition to theminimum which now gives worse results.
For N = 11 the new plateau is not yet extremal,but it is the proper region of least Ω-dependence yielding the best approximation to theexact energy indicated by the dashed horizontal line. The minimum has fallen far belowthis value and is no longer useful. The figure looks similar for all couplings (in the plot,g = 0.4). The reason is the scaling property (5.215) proved in Appendix 5B.with the re-expansion coefficients(0)εl=lXj=0(0)Ej(1 − 3j)/2l−j!(−4σ)l−j .(5.207)Here σ denotes the dimensionless function of Ω1 2 Ω(Ω2 − 1)σ ≡ − Ωr =.2g(5.208)In Fig. 5.15 we have plotted the Ω-dependence of WN for increasing N at the couplingconstant g/4 = 0.1.
For odd and even N, an increasingly flat plateau develops atthe optimal energy.At larger orders N ≥ 15, the initially flat plateau is deformed into a minimumwith a larger curvature and is no longer a good approximation. However, a newplateau has developed yielding the best energy. This is seen on the high-resolutionplot in Fig. 5.16. At N = 11, the new plateau is not yet extremal but close to thecorrect energy.The worsening extrema in Fig. 5.16 correspond here to points leaving the optimaldashed into the upward direction. The newly forming plateaus lie always on thedashed curve.The set of all extremal ΩN -values for odd N up to N = 91 is shown in Fig. 5.17.The optimal frequencies with smallest curvature are marked by a fat dot.
In Subsection 17.10.5. we shall derive thatΩN (Ω2N − 1)σN =g(5.209)5045 Variational Perturbation Theory10g/4 =186ΩN420020406080100NFigure 5.17 Trial frequencies ΩN extremizing the variational approximation WN atT = 0 for odd N ≤ 91. The coupling is g/4 = 1. The dashed curve corresponds to theapproximation (5.211) [related to ΩN via (5.208)]. The frequencies on this curve producethe fastest convergence. The worsening extrema in Fig. 5.16 correspond here to pointsleaving the optimal dashed into the upward direction.
The newly forming plateaus liealways on the dashed curve.6g/4 =1ΩN42010N2030Figure 5.18 Extremal and turning point frequencies ΩN in variational approximationWN at T = 0 for even and odd N ≤ 30. The coupling is g/4 = 1. The dashed curvecorresponds to the approximation (5.211) [related to ΩN via (5.208)].grows for large N likeσN ≈ cN ,c = 0.186047 . . . ,(5.210)H.
Kleinert, PATH INTEGRALS5055.15 Convergence of Variational Perturbation Expansion100g/4 =110-5 |E-Eex|10-1010-1510-2002040N6080100Figure 5.19 Difference between approximate ground state energies E = WN and exactenergies Eex for odd N corresponding to the ΩN -values shown in Fig. 5.17. The coupling isg/4 = 1. The lower curve follows roughly the error estimate to be derived in Eq.
(17.409).The extrema in Fig. (5.17) which move away from the dashed curve lie here on horizontalcurves whose accuracy does not increase.so that ΩN grows like ΩN ≈ (cNg)1/3 . For smaller N, the best ΩN -values in Fig. 5.17can be fitted with the help of the corrected formula (5.210):σN ≈ cN 1 +6.85.N 2/3(5.211)The associated ΩN -curve is shown as a dashed line. It is the lower envelope of theextremal frequencies.The set of extremal and turning point frequencies ΩN is shown in Fig. 5.18 foreven and odd N up to N = 30. The optimal extrema with smallest curvatureare again marked by a fat dot.
The theoretical curve for an optimal convergencecalculated from (5.211) and (5.209) is again plotted as a dashed line.In Table 5.8, we illustrate the precision reached for large orders N at variouscoupling constants g by a comparison with accurate energies derived from numericalsolutions of the Schrödinger equation.The approach to the exact energy values is illustrated in Fig. 5.19 which showsthat a good convergence is achieved by using the lowest of all extremal frequencies,which lie roughly on the dashed theoretical curve in Fig. 5.17 and specify the positionof the plateaus. The frequencies ΩN on the higher branches leaving the dashedcurve in that figure, on the other hand, do not yield converging energy values.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
