Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 96
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This has also been seen in the Eq. (5.53) estimate to theground state energy of the Coulomb system by a Gaussian wave packet. The energyis a rather global property of the system. For physical quantities such as particledensities which contain local information on the wave functions, the approximationis expected to be much worse.
Let us nevertheless calculate particle densities ofa quantum-mechanical system. For this we tie down the periodic particle orbit inthe trial partition function Z1 for an arbitrary time at a particular position, say xa .Mathematically, this is enforced with the help of a δ-function:δ(xa − x(τ )) = δ(xa − x0 −∞X(xm e−iωm τ + c.c.))m=1H. Kleinert, PATH INTEGRALS4735.8 Particle Densities=Z∞−∞∞Xdkexp ik xa − x0 −(xm e−iωm τ + c.c.)2πm=1("#).(5.89)With this, we write the path integral for the particle density [compare (2.345)]ρ(xa ) = Z−1IDx δ(xa − x(τ ))e−A/h̄Z∞dx0le (h̄β)(5.90)and decomposeρ(xa ) = Z −1×Z−∞Zdk ik(xa −x0 )e2π∞Dxδ̃(x̄ − x0 )e−ikΣm=1 (xm +c.c.) e−A/h̄ .(5.91)The approximation W1 (x0 ) is based on a quasiharmonic treatment of the xm -Figure 5.7 Approximate particle density (5.94) of anharmonic oscillator for g = 40, asPcompared with the exact density ρ(x) = Z −1 n |ψn (x)|2 e−βEn , obtained by integratingthe Schrödinger equation numerically.
The curves are labeled by their β values with thesubscripts 1, ex, cl indicating the approximation.fluctuations for m > 0. For harmonic fluctuations we use Wick’s rule of Section 3.10to evaluateD∞e−ikΣm=1 (xm e−iωm τ +c.c.)Ex0Ω≈ e−k2 Σ∞m=1xh|xm |2 iΩ0 ≡ e−k2 a2 /2 ,(5.92)4745 Variational Perturbation TheoryFigure 5.8 Particle density (5.94) in double-well potential (5.78) for the worst choiceof the coupling constant, g = 0.4. Comparison is made with the exact density ρ(x) =PZ −1 n |ψn (x)|2 e−βEn obtained by integrating the Schrödinger equation numerically. Thecurves are labeled by their β values with the subscripts 1, ex, cl indicatingthe approxima2 /2a2 √−x2tion.
For β → ∞, the distribution tends to the Gaussian e/ 2πa with a2 = 1.030(see Table 5.1).which is true for any τ . Thus we could have chosen any τ in the δ-function (5.89)to find the distribution function. Inserting (5.92) into (5.91) we can integrate out kand find the approximation to the particle densityρ(xa ) ≈ Z−1Z∞−∞22dx0 e−(xa −x0 ) /2a (x0 ) −V eff cl (x0 )/kB Tqe.le (h̄β)2πa2 (x )(5.93)0By inserting for V eff cl (x0 ) the approximation W1 (x0 ), which for Z yields the approximation Z1 , we arrive at the corresponding approximation for the particle distributionfunction:22Z ∞dx0 e−(xa −x0 ) /2a (x0 ) −W1 (x0 )/kB T−1qρ1 (xa ) = Z1e.(5.94)−∞ le (h̄β)2πa2 (x )0R∞This has obviously the correct normalization −∞ dxa ρ1 (xa ) = 1.
Figure 5.7 showsa comparison of the approximate particle distribution functions of the anharmonicoscillator with the exact ones. Both agree reasonably well with each other. InFig. 5.8, the same plot is given for the double-well potential at a coupling g = 0.4.Here the agreement at very low temperature is not as good as in Fig. 5.7. Compare,for example, the zero-temperature curve ∞1 with the exact curve ∞ex . The firsthas only a single central peak, the second a double peak.
The reason for thisdiscrepancy is the correspondence of the approximate distribution to an optimalH. Kleinert, PATH INTEGRALS5.9 Extension to D Dimensions475Gaussian wave function which happens to be centered at the origin, in spite of thedouble-well shape of the potential. In Fig. 5.3 we see the reason for this: Theapproximate effective classical potential W1 (x0 ) has, at small temperatures up toT ∼ 1/10, only one minimum at the origin, and this becomes the center of theoptimal Gaussian wave function.
For larger temperatures, there are two minima andthe approximate distribution function ρ1 (x) corresponds roughly to two Gaussianwave packets centered around these minima. Then, the agreement with the exactdistribution becomes better. We have intentionally chosen the coupling g = 0.4,where the result would be about the worst. For g 0.4, both the true and theapproximate distributions have a single central peak. For g 0.4, both have twopeaks at small temperatures.
In both limits, the approximation is acceptable.5.9Extension to D DimensionsThe method can easily be extended to approximate the path integral of a particlemoving in a D-dimensional x-space. Let xi be the D components of x. Then thetrial frequency Ω2ij (x0 ) in (5.7) must be taken as a D × D -matrix.
In the special√case of V (x) being rotationally symmetric and depending only on r = x2 , wemay introduce, as in the discussion of the effective action in Eqs. (3.681)–(3.686),longitudinal and transverse parts of Ω2ij (x0 ) via the decompositionΩ2ij (x0 ) = Ω2L (r0 )PLij (x0 ) + Ω2T (r0 )PT ij (x0 ),(5.95)wherePLij (x̂0 ) = x0i x0j /r02 ,PT ij (x̂0 ) = δij − x0i x0j /r02 ,(5.96)are projection matrices into the longitudinal and transverse directions of x0 . Analogous projections of a vector are defined by xL ≡ PL (x0 ) x, xT ≡ PT (x0 ) x. Thenthe anisotropic generalization of W1Ω (x0 ) becomes"sinh h̄ΩT /2kB Tsinh h̄ΩL /2kB T+ (D − 1) logW1 (r0 ) = kB T logh̄ΩL /2kB Th̄ΩT /2kB ThiM 2 2−ΩL aL + (D − 1)Ω2T a2T + VaL ,aT ,2#(5.97)with all functions on the right-hand side depending only on r0 . The bold-face superscript indicates the presence of two variational frequencies≡ (ΩL , ΩT ).
Thesmeared potential is now given byD Z ∞Y11dδxVaL ,aT (r0 ) = qqiD−1−∞2πa2L 2πa2Ti=112−22× exp − a−2δx+aδxV (x0 + δx),TT2 L L(5.98)4765 Variational Perturbation Theorywhich can also be written as1ZZVaL ,aT (r0 ) = qdδxL d(2π)D a2L aT2D−2D−1δxL2 δxT2δxT exp − 2 − 2 V (x0 +δx). (5.99)2aL 2aT#"For higher temperatures where the smearing widths a2L , a2T are small, we set V (x) ≡v(r 2 ), so thatV (x) = v(r02 + 2r0 δxL + δx2L + δx2T ) ==v(r02λ(2r0 δxL +δx2L +δx2T )+ ∂λ )e∞Xn1 (n) 2 v (r0 ) 2r0 δxL + δx2L + δx2Tn=0 n!.(5.100)λ=0Inserting this into the right-hand side of (5.98), we find112r02 λ2 a2L /(1−2a2L λ)2VaL ,aT (r0 ) = v(r0 + ∂λ ) qqD−1 e1 − 2a2L λ 1 − 2a2T λ,(5.101)λ=0which has the expansionVaL ,aT (r0 ) = v(r02 ) + v 0 (r02 )[a2L + (D − 1)a2T ]1+ v 00 (r02 )[3a4L + 2(D − 1)a2L a2T + (D 2 − 1)a4T + 4r02 a2L ] + .
. . .2(5.102)The prime abbreviates the derivative with respect to r02 .In general it is useful to insert into (5.98) the Fourier representation for thepotentialZdD k ikxV (x) =e V (k),(5.103)(2π)Dwhich makes the x-integration Gaussian, so that (5.99) becomesVa2 ,a2 (r0 ) =LTZdD ka2L 2 a2T 2V (k) exp − kL − kT − ir0 kL .22(2π)D!(5.104)Exploiting the rotational symmetry of the potential by writing V (k) ≡ v(k 2 ), wedecompose the measure of integration asZZ ∞SD−1 Z ∞dD k=dkdkT kTD−2 ,LDD0(2π)(2π) −∞whereSD =2π D/2Γ(D/2)(5.105)(5.106)H.
Kleinert, PATH INTEGRALS4775.10 Application to Coulomb and Yukawa Potentialsis the surface of a sphere in D dimensions, and further with kL = k cos φ, kT =k sin φ:ZZ ∞SD−1 Z 1dD kd cos φdk k D−1 .(5.107)D =D0(2π)(2π) −1This brings (5.104) to the formZ ∞SD−1 Z 1dudk k D−1 v(k 2 )Va2 ,a2 (r0 ) =DL T−1(2π) −1hi12 2222aL u + aT (1 − u ) k − ir0 u .× exp −2(5.108)The final effective classical potential is found by minimizing W1 (r0 ) at each r0 inaL , aT , ΩL , ΩT .
To gain a rough idea about the solution, it is usually of advantage tostudy first the isotropic approximation obtained by assuming a2T (r0 ) = a2L (r0 ), andto proceed later to the anisotropic approximation.5.10Application to Coulomb and Yukawa PotentialsThe effective classical potential can be useful also for singular potentials as long asthe smearing procedure makes sense. An example is the Yukawa potentialV (r) = −(e2 /r)e−mr ,(5.109)which reduces to the Coulomb potential for m ≡ 0. Using the Fourier representationV (r) = 4πZd3 keikx= 4π(2π)3 k 2 + m2∞Z0dτZd3 k −τ (k2 +m2 )e,(2π)3(5.110)we easily calculate the isotropically smeared potentiald3 x12V (r)√32 (x0 − x)22a2πaZ ∞12222m2 a2 /2= −e 2πeda0 qexp −r02 /2a0 − m2 a0 /23a222πa0Va2 (r0 ) = −e2Zexp −√2 222 22em a /2 1 Z r0 / 2a22√= −edte−(t +m r0 /4t ) .π r0 02(5.111)In the Coulomb limit m → 0, the smeared potential becomes equal to the Coulombpotential multiplied by an error function,e2Va2 (r0 ) = − erf(r0 / 2a2 ),r0q(5.112)where the error function is defined by2erf(z) ≡ √πZ0z2dxe−x .(5.113)4785 Variational Perturbation TheoryFigure 5.9 Approximate effective classical potential W1 (r0 ) of Coulomb system at various temperatures (in multiples of 104 K).
It is calculated once in the isotropic (dashedcurves) and once in the anisotropic approximation. The improvement is visible in theinsert which shows W1 /V . The inverse temperature values of the different curves areβ = 31.58, 15.78, 7.89, 3.945, 1.9725, 0.9863, 0 atomic units, respectively.The smeared potential is no longer singular at the origin,e2Va2 (0) = −as2 m2 a2 /2e.π(5.114)The singularity has been removed by quantum fluctuations.
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