Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 102
Текст из файла (страница 102)
Thesharper minima in Fig. 5.16 correspond precisely to those branches which no longerdetermine the region of weakest Ω-dependence.The anharmonic oscillator has the remarkable property that a plot of the ΩN values in the N, σN -plane is universal in the coupling strengths g; the plots do not5065 Variational Perturbation TheoryN1234510152025exactN1234510152025exactg/4 =0.10.56030737110.55915213930.55915421880.55914574080.55914615960.55914632660.55914632720.55914632720.55914632720.5591463272g/4 =502.54758039962.50312132532.50069962792.49959801252.49962132272.49970719602.49970894032.49970890792.49970877312.4997087726g/4 = 0.30.64162986210.63808873470.63803575980.63798787130.63798990840.63799176770.63799178380.63799178360.63799178320.6379917832g/4 =2004.00846088123.93655860483.93255382033.93074881273.93078578923.93092867433.93093162833.93093157323.93093133963.9309313391g/4 =0.50.70166164290.69637694990.69625363260.69616849780.69617174750.69617577820.69617582310.69617430590.69617582080.6961758208g/4 =10006.82795331366.70400326066.69703286386.69390361786.69396679716.69421616806.69422136316.69422126596.69422085226.6942208505g/4 =1.00.81250000000.80419009460.80391405280.80375634570.80376152320.80377053290.80377065960.80377065750.80377065130.8037706514g/4 =800013.63528259313.38659848613.37256118913.36626903813.36639534713.36689807913.36690858313.36690838713.36690755113.366907544g/4 =2.00.96440355980.95229362980.95179976940.95154441980.95155174500.95156822490.95156849330.95156848870.95155841210.9515684727g/4 =2000018.50165871218.16397996718.14490838918.13636164218.13653306018.13721620018.13723048118.13723021418.13723002218.137229073Table 5.8 Comparison of the variational approximations WN at T = 0 for increasing Nwith the exact ground state energy at various coupling constants g.depend on g.
To see the reason for this, we reinsert explicitly the frequency ω (whichwas earlier set equal to unity). Then the re-expanded energy WN in Eq. (5.206) hasthe general scaling formWNΩ = ΩwN (ĝ, ω̂ 2 ),(5.212)where wN is a dimensionless function of the reduced coupling constant and frequencyĝ ≡g,Ω3ω̂ ≡ω,Ω(5.213)respectively. When differentiating (5.212),"#dddWN = 1 − 3ĝ − 2ω̂ 2 2 wN (ĝ, ω̂ 2 ),dΩdĝdω̂(5.214)we discover that the right-hand side can be written as a product of ĝ N and a dimensionless polynomial of order N depending only on σ = Ω(Ω2 − ω 2 )/g:dW Ω = ĝ N pN (σ).dΩ N(5.215)A proof of this will be given in Appendix 5B for any interaction xp . The universaloptimal σN -values are obtained from the zeros of pN (σ).It is possible to achieve the same universality for the optimal frequencies of theeven approximations WN by determining them from the extrema of pN (σ) ratherthan from the turning points of WN as a function of Ω.H.
Kleinert, PATH INTEGRALS5075.15 Convergence of Variational Perturbation ExpansionThe universal functions pN (σ) are found most easily by replacing the variable σ(0)in the coefficients εl of the re-expansion (5.206) by its ω = 0 -limit σ|ω=0 = Ω3 /g =1/ĝ. This yields the simpler expressionWNΩ= ΩwN (ĝ, 0) = ΩNXĝ4(0)εll=0with(0)εl=lX!(1 − 3j)/2l−j(0)Ejj=0!l,(5.216)(−4/ĝ)l−j .(5.217)The derivative of WN with respect to Ω yieldspN (σ) = ĝ−N"#d1 − 3ĝwN (ĝ, 0).dĝĝ=1/σ(5.218)(0)In Section 17.10, we show the re-expansion coefficients εk in (5.207) to be for(0)large k proportional to Ek :(0)(0)εk ≈ e−2σN Ek ,σN =ΩN (Ω2N − 1)g(5.219)[see Eq. (17.396)]. Thus, at any fixed Ω, the re-expanded series has the same asymptotic growth as the original series with the same vanishing radius of convergence.The behavior (5.219) can be seen in Fig.
5.20(a) where we have plotted the logarithmlog Sklog Sk3010N= 120010N= 5-1030k40N =10-1010020N= 910203040N =2050-20kN =30-30a)b)Figure 5.20 Logarithmic plot of kth terms in re-expanded perturbation series at acoupling constant g/4 = 1:(a) Frequencies ΩN extremizing the approximation WN . The dashed curves indicate thetheoretical asymptotic behavior (5.219).(b) Frequencies ΩN corresponding to the dashed curve in Fig. 5.17. The minima lie foreach N precisely at k = N , producing the fastest convergence. The curves labeled Ω = ωindicate the kth term in the original perturbation series.of the absolute value of the kth termSk =(0)εkĝ4!k(5.220)5085 Variational Perturbation Theoryof the re-expanded perturbation series (5.202) for various optimal values ΩN andg = 40.
All curves show a growth ∝ k k . The terms in the original series start growing immediately (precocious growth). Those in the re-expanded series, on the otherhand, decrease initially and go through a minimum before they start growing (retarded growth). The dashed curves indicate the analytically calculated asymptoticbehavior (5.219).The increasingly retarded growth is the reason why energies obtained from thevariational expansion converge towards the exact result.
Consider the terms Skof the resummed series with frequencies ΩN taken from the theoretical curve ofoptimal convergence in Fig. 5.17 (or 5.18). In Fig. 5.20(b) we see that the terms Skare minimal at k = N, i.e., at the last term contained in the approximation WN . Ingeneral, a divergent series yields an optimal result if it is truncated after the smallestterm Sk . The size of the last term gives the order of magnitude of the error in thetruncated evaluation. The re-expansion makes it possible to find, for every N, afrequency ΩN which makes the truncation optimal in this sense.5.16Variational Perturbation Theory for Strong-CouplingExpansionFrom the ω → 0 -limit of (5.206), we obtain directly the strong-coupling behaviorof WN . Since Ω = (g/ĝ)1/3 , we can writeWN = (g/ĝ)1/3 wN (ĝ, 0),(5.221)and evaluate this at the optimal value ĝ = 1/σN .
The large-g behavior of WN isthereforeWN −−−→(g/4)1/3 b0 ,(5.222)b0 = (4/ĝ)1/3 wN (ĝ, 0)|ĝ=1/σN .(5.223)g→∞with the coefficientThe higher corrections to the leading behavior (5.222) are found just as easily.By expanding wN (ĝ, ω̂ 2 ) in powers of ω̂ 2 ,WN =and insertinggĝ!1/3 1 + ω̂ 2dω̂ 4+2!dω̂ 2ω̂ 2 =ddω̂ 2!2+ . . . wN (ĝ, ω̂ 2),(5.224)ω̂ 2 =0ĝ 2/3,(g/ω 3)2/3(5.225)we obtain the expansion 1/3 "gWN =4b0 + b1g4ω 3−2/3+ b2g4ω 3−4/3#+ ... ,(5.226)H.
Kleinert, PATH INTEGRALS5095.16 Variational Perturbation Theory for Strong-Coupling Expansionlog |α0 − αex0 |81/3Figure 5.21b0 and b1 .log |α1 − αex1 |271/3641/3N 1/31251/381/32161/3271/3641/3N 1/31251/32161/3Logarithmic plot of N -behavior of strong-coupling expansion coefficientsFigure 5.22 Oscillations of approximate strong-coupling expansion coefficient b0 asa function of N when approaching exponentially fast the exact limit. The exponentialbehavior has been factored out. The upper and lower points show the odd-N and even-Napproximations, respectively.with the coefficients1bn =n!ĝ4!(2n−1)/3ddω̂ 2!n2 wN (ĝ, ω̂ ).ĝ=1/σN(5.227),ω̂ 2 =0The derivatives on the right-hand side have the expansionsddω̂ 2!nwN (ĝ, ω̂ 2 ) =NXl=0(0)εn lĝ4!l,(5.228)5105 Variational Perturbation TheoryTable 5.9 Coefficients bn of strong-coupling expansion of ground state energy of anharmonic oscillator obtained from a perturbation expansion of order 251.
An extremelyprecise value for b0 was given by F. Vinette and J.<b>Текст обрезан, так как является слишком большим</b>.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
