Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 94
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Then the independent variation of W1Ω (x0 ) with respect to these two parameters yields both (5.24) and theminimization condition (5.37) for Ω2 (x0 ).From the extremal W1Ω (x0 ) we obtain the approximation for the partition function and the free energy−F1 /kB TZ1 = e=Z∞−∞dx0 −W1 (x0 )/kB Te≤ Z.le (h̄β)(5.39)We leave it to the reader to calculate the second derivative of W1Ω (x0 ) with respectto Ω2 (x0 ) and to prove that it is nonnegative, implying that the above extremalsolution is a local minimum.H.
Kleinert, PATH INTEGRALS4615.4 Accuracy of Variational Approximation5.4Accuracy of Variational ApproximationThe accuracy of the approximate effective classical potential W1 (x0 ) can be estimated by the following observation: In the limit of high temperatures, the approximation is perfect by construction, due the shrinking width (5.25) of the nonzerofrequency fluctuations. This makes W1 (x0 ) in (5.31) converge against V (x0 ), just asthe exact effective classical potential in Eq.
(3.809).In the opposite limit of low temperatures, the integral over x0 in the generalexpression (3.808) is dominated by the minimum of the effective classical potential.If its position is denoted by xm , we have the saddle point approximation (see Section4.2)Zdx0 −[V eff cl (xm )]00 (x0 −xm )2 /2kB T−V eff cl (xm )/kB Te.(5.40)Z−−−→ eT →0le (h̄β)The exponential of the prefactor yields the leading low-temperature behavior of thefree energy:F−−−→ V eff cl (xm ).(5.41)T →0The Gaussian integral over x0 contributes a termh̄∆F = kB T logk TBs[V eff cl (xm )]00 ,M(5.42)which accounts for the entropy of x0 fluctuations around xm [recall Eq.
(1.564)].Moreover, at zero temperature, the free energy F converges against the groundstate energy E (0) of the system, so thatE (0) = V eff cl (xm ).(5.43)The minimum of the approximate effective classical potential, W1Ω (x0 ) with respectto Ω(x0 ) supplies us with a variational approximation to the free energy F1 , whichin the limit T → 0 yields a variational approximation to the ground state energy(0)E1 = F1 |T =0 ≡ W1 (xm )|T =0 .(5.44)By taking the T → 0 limit in (5.32) we see thatlim W1Ω (x0 ) =T →0i1hh̄Ω(x0 ) − MΩ2 (x0 )a2 (x0 ) + Va2 (x0 ).2(5.45)In the same limit, Eq. (5.24) giveslim a2 (x0 ) =T →0so thath̄,2MΩ(x0 )1h̄21lim W1Ω (x0 ) = h̄Ω(x0 ) + Va2 (x0 ) =+ Va2 (x0 ) .T →048 Ma2 (x0 )(5.46)(5.47)4625 Variational Perturbation TheoryThe right-hand side is recognized as the expectation value of the Hamiltonian operatorp̂2Ĥ =+ V (x)(5.48)2Min a normalized Gaussian wave packet of width a centered at x0 :ψ(x) =11exp − 2 (x − x0 )2 .2 1/44a(2πa )Indeed,DĤEψ≡Z∞−∞dxψ ∗ (x)Ĥψ(x) =(5.49)1 h̄2+ Va2 (x0 ).8 Ma2(5.50)Let E1 be the minimum of this expectation under the variation of x0 and a2 :DE1 = minx0 ,a2 ĤEψ.(5.51)This is the variational approximation to the ground state energy provided by theRayleigh-Ritz method .In the low temperature limit, the approximation F1 to the free energy convergestoward E1 :lim F1 = E1 .(5.52)T →0The approximate effective classical potential W1 (x0 ) is for all temperatures andx0 more accurate than the estimate of the ground state energy E0 by the minimalexpectation value (5.51) of the Hamiltonian operator in a Gaussian wave packet.
Forpotentials with a pronounced unique minimum of quadratic shape, this estimate isknown to be excellent.In Table 5.1 we list the energies E1 = W1 (0) for a particle in an anharmonicoscillator potential. Its action will be specified in Section 5.7, where the approximation W1 (x0 ) will be calculated and discussed in detail. The table shows that thisapproximation promises to be quite good [2].With the effective classical potential having good high- and low-temperaturelimits, it is no surprise that the approximation is quite reliable at all temperatures.Even if the potential minimum is not smooth, the low-temperature limit can beof acceptable accuracy.
An example is the three-dimensional Coulomb system forwhich the limit (5.51) becomes (with the obvious optimal choice x0 = 0)E1 = mina3 h̄22 e2√√−8 Ma2π 2a2!3 h̄2.=−8 Ma2min(5.53)qThe minimal value of a is amin = 9π/32aH where aH = h̄2 /Me2 is the Bohrradius (4.336) of the hydrogen atom. In terms of it, the minimal energy has thevalue E1 = −(4/3π)e2 /aH . This is only 15% percent different from the true ground(0)state energy of the Coulomb system Eex= −(1/2)e2 /aH .
Such a high degree ofH. Kleinert, PATH INTEGRALS4635.5 Weakly Bound Ground State Energy in Finite-Range Potential WellTable 5.1 Comparison of variational energy E1 = limT →0 F1 , obtained from Gaussian(0)trial wave function, with exact ground state energy Eex . The energies of the first two(1)(2)(0)(1)(0)excited states Eex and Eex are listed as well. The level splitting ∆Eex = Eex − Eex tothe first excited state is shown in column 6. We see that it is well approximated by thevalue of Ω(0), as it should (see the discussion after Eq. (5.21).g/40.10.20.30.40.50.60.70.80.91.010501005001000E10.56030.60490.64160.67340.70170.72730.75090.77210.79320.81251.53132.54763.19245.42586.8279(0)Eex0.5591460.6024050.6379920.6687730.6961760.7210390.7439040.7651440.7850320.8037711.504972.499713.131385.319896.69422(1)Eex1.769501.950542.094642.216932.324412.421022.509232.590702.666632.737895.321618.9151011.187319.043423.9722(2)Eex3.138623.536303.844784.102844.327524.528124.710334.877935.033605.1792910.347117.437021.906937.340747.0173(0)∆Eex1.210351.348101.456651.548161.628231.699981.765331.825561.862861.934123.816946.413398.0559013.723517.2780Ω(0)1.2221.3701.4871.5851.6271.7491.8191.8841.9442.0004.0006.7448.47414.44618.190a2 (0)0.40940.36500.33630.31540.29910.28590.27490.26540.25720.25000.12500.07410.05900.03460.0275accuracy may seem somewhat surprising since the exact Coulomb wave functionψ(x) = (πa3H )−1/2 exp(−r/aH ) is far from being a Gaussian.The partition function of the Coulomb system can be calculated only after subtracting the free-particle partition function and screening the 1/r -behavior downto a finite range.
The effective classical free energy F1 of the Coulomb potentialobtained by this method is, at any temperature, more accurate than the difference(0)between E1 and Eex. More details will be given in Section 5.10.5.5Weakly Bound Ground State Energyin Finite-Range Potential WellThe variational approach allows us to derive a simple approximation for the boundstate energy in an arbitrarily shaped potential of finite range, for which the bindingenergy is very weak. Precisely speaking, the falloff of the ground state wave functionhas to lie outside the range of the potential.
A typical example for this situation isthe binding of electrons to Cooper pairs in a superconductor. The attractive forcecomes from the electron-phonon interaction which is weakened by the Coulombrepulsion. The potential has a complicated shape, but the binding energy is soweak that the wave function of a Cooper pair reaches out to several thousand latticespacings, which is much larger than the range of the potential, which extends onlyover maximally a hundred lattice spacings. In this case one may practically replacethe potential by an equivalent δ-function potential.4645 Variational Perturbation TheoryThe present considerations apply to this situation. Let us assume the absoluteminimum of the potential to lie at the origin.
The first-order variational energy atthe origin is given byΩ(5.54)W1 (0) = + Va2 (0),2where by (5.30)Z ∞022dx0√ 0 e−x0 /2a V (x00 ).Va2 (0) =(5.55)2−∞2πaBy assumption, the binding energy is so small that the ground state wave functiondoes not fall off within the range of V (x0 ). Hence we can approximateVa2 (0) ≈sΩπZ∞−∞dx00 V (x00 ),(5.56)where we have inserted a2 = 1/2Ω. Extremizing this in Ω yields the approximateground state energy Z ∞21(0)−E1 ≈ −dx00 V (x00 ) .(5.57)2π−∞By applying this result to a simple δ-function potential at the origin,V (x) = −gδ(x),g > 0,(5.58)we find an approximate ground state energy(0)E1 = −1 2g .2π(5.59)The exact value is1E (0) = − g 2 .(5.60)2The failure of the variational approximation is due to the fact that outside qthe rangeof the potential, the wave function is a simple exponential e−k|x| with k = −2E (0) ,and not a Gaussian.
In fact, if we consider the expectation value of the Hamiltonianoperator1 d2H=−− gδ(x)(5.61)2 dx2for a normalized trial wave function√ψ(x) = Ke−K|x|,(5.62)we obtain a variational energyW1 =K2− gK,2(5.63)H. Kleinert, PATH INTEGRALS4655.6 Possible Direct Generalizationswhose minimum gives the exact ground state energy (5.60). Thus, problems of thepresent type call for the development of a variational perturbation theory for whichEq. (5.54) and (5.55) readK2W (0) =+ VK (0),(5.64)2whereZ ∞dx00 −K|x0|VK (0) = KeV (x0 ).(5.65)−∞ aFor an arbitrary attractive potential whose range is much shorter than a, this leadsto the correct energy for a weakly bound ground state(0)E15.61≈− −2Z∞−∞dx00V (x0 )2.(5.66)Possible Direct GeneralizationsLet us remark that there is a possible immediate generalization of the above variational procedure.One may treat higher components xm with m > 0 accurately, say up to m = m̄ − 1, where m̄ issome integer > 1, using the ansatz(ZZ1 h̄/kB T M ẋ2 (τ )+ Ω2 (x0 , .
. . , xm̄ )Zm̄ ≡ Dx(τ ) exp −h̄ 022!2 m̄−1X× x(τ ) − x0 −(xm e−iωm τ + c.c.) e−(1/kB T )Lm̄ (x0 ,...,xm̄ )m=1=Z∞−∞dx0le (h̄β)m̄−1Y Zn=1im2dxreωm+ Ω2 (x0 )m dxm22πkB T /M ωmωmh̄Ω(x0 )/2kB Te−Lm̄ (x0 ,...,xm̄ )/kB T ,sinh(h̄Ω(x0 )/2kB T )×(5.67)with the trial function Lm̄ :Lm̄ (x0 , . .
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