Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 92
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Note thatZ τbdτ hηi (τ )i = 0,(4B.12)(4B.13)τaandτbτbτbτbh̄dτ 0 Gij (τ, τ 0 ) =∆τ (∆τ 2 − τa2 )δij ,12MτaτaτaτaZ τbZ τb∆τh̄ ∆τ 22− τa δij +∆xi ∆xj .dτ hηi (τ )ηj (τ )i =dτ Gij (τ, τ ) =M612τaτaZZdτ00dτ hηi (τ )ηj (τ )i =ZdτZ(4B.14)(4B.15)Thus we obtain the semiclassical imaginary-time amplitudeD/2M∆τM∆x2 −V (x)(4B.16)exp −2πh̄∆τ2h̄∆τh̄∆τ 2 2∆τ∆τ 322× 1−∇ V (x) −(∆x∇) V (x) +[∇V (x)] + . . . .12M24h̄24M h̄(xb τb |xa τa ) =This agrees precisely with the real-time amplitude (4.259).For the partition function at inverse temperature β = (τb − τa )/h̄, this implies the semiclassicalapproximationZZ =dD x (x h̄β|x 0)≈M2πh̄2 βD/2 ZdD x̄1−h̄2 β 2 2h̄2 β 3∇ V (x) +[∇V (x)]2 e−βV (x) .12M24M(4B.17)A partial integration simplifies this toZD/2 ZMh̄2 β 2 2D∇ V (x) e−βV (x)≈d x̄ 1 −24M2πh̄2 βD/2Zh̄2 β 2 2M≈dD x̄−.exp−βV(x)∇V(x)24M2πh̄2 β(4B.18)Actually, it is easy to calculate all terms in (4B.16) proportional to V (x) and its derivatives.Instead of the expansion (4B.6), we evaluateZ1 τbdτ hV (x + (τ )) − V (x)i .(xb τb |xa τa ) = e−βV (x) (∆x/2 τb |−∆x/2 τa ) 1 −h̄ τa(4B.19)By rewriting V (x + ) as a Fourier integralZdD kV (x + ) =Ṽ (k) exp [ik (x + )] ,(4B.20)(2π)DH.
Kleinert, PATH INTEGRALSAppendix 4B Derivation of Semiclassical Time Evolution Amplitude451we obtain(xb τb |xa τa ) = (∆x/2 τb |−∆x/2 τa )ZZD E1 τbdD kik (τ )−βV (x)ikx1−e−1.× eṼ (k)edτh̄ τa(2π)D(4B.21)The expectation value can be calculated using Wick’s theorem (3.307) asZ τbZ τbD E Z τb22dτ eik (τ ) =dτ e−ki kj hηi (τ )ηj (τ )i/2 =dτ e−ki kj [A (τ,τ )δij +B (τ,τ )∆xi ∆xj ]/2 .τaτaτa(4B.22)where A2 (τ, τ ), B 2 (τ, τ ) are from (4B.12):A2 (τ, τ ) =h̄(∆τ − τ )τ,M ∆τ2B 2 (τ, τ ) =(τ − τ̄ ).∆τ 2Inserting the inverse of the Fourier decomposition (4B.20),ZṼ (k) = dD η V (x + ) exp [−ik (x + )] ,whereis now a time-independent variable of integration, we findZ τb ZZ τb DZ Dd k −(1/2)ki Gij (τ,τ )kj −iki ηi (τ ) Edτ dDη V (x + )dτ eik (τ ) =e.(2π)Dτaτa(4B.23)(4B.24)(4B.25)After a quadratic completion of the exponent, the momentum integral can be performed and yieldsZ τbZ τbZD E−1/2 −(1/2)ηi G−1(τ,τ )ηjik (τ )Dijdτ [det G(τ, τ )]e.
(4B.26)dτ e=d η V (x + )τaτaUsing the transverse and longitudinal projection matricesPijT = δij −2∆xi ∆xj,(∆x)2PijL =∆xi ∆xj,(∆x)2(4B.27)2satisfying P T = P T , P L = P L , we can decompose Gij (τ, τ 0 ) asGij ≡ A2 (τ, τ 0 )PijT + A2 (τ, τ 0 ) + B 2 (τ, τ 0 )(∆x)2 PijL .(4B.28)It is then easy to find the determinantdet G(τ, τ 0 ) = [A2 (τ, τ 0 )]D−1 [A2 (τ, τ 0 ) + B 2 (τ, τ 0 )(∆x)2 ],(4B.29)and the inverse matrixG−1ij (τ )1∆xi ∆xj1∆xi ∆xj.= 2δij −+ 2A (τ, τ 0 )(∆x)2A (τ, τ 0 ) + B 2 (τ, τ 0 )(∆x)2 (∆x)2(4B.30)Inserting (4B.26) back into (4B.21), and taking the correction into the exponent, we arrive atR τb−(1/h̄)dτ Vsm (x,τ )τa(xb τb |xa τa ) = (∆x/2 τb |−∆x/2 τa ) e,(4B.31)where Vsm (x, τ ) is the harmonically smeared potentialZ−1−1/2Vsm (x, τ ) ≡ [det G(τ, τ )]dD η V (x + )e−(1/2)ηi Gij (τ )ηj .(4B.32)4524 Semiclassical Time Evolution AmplitudeBy expanding V (x + ) to second order in , the exponent in (4B.31) becomesZ τb1−βV (x) −Vij (x)dτ Gij (τ, τ ) + .
. . .2h̄τa(4B.33)According to Eq. (4B.15), we haveZ τbh̄ ∆τ 2∆τdτ Gij (τ, τ ) =δij +∆xi ∆xj ,M 612τa(4B.34)so that we reobtain the first two correction terms in the curly brackets of (4B.16)The calculation of the higher-ordercorrections becomes quite tedious. One rewrites the exηi (τ )∂iV (x) and (4B.19) aspansion (4B.2) as V x + (τ ) = e∞X(−1)n(xb τb |xa τa ) = (∆x/2 τb |−∆x/2 τa ) ×n!n=0*∞ ZYn=0τbdτn eηi (τn )∂iτa+V (x) .(4B.35)Now we apply Wick’s rule (3.307) for harmonically fluctuating variables, to re-expressEDeη(τ )∂i = ehηi (τ )ηj (τ )i/2 = eGij (τ,τ )∂i ∂j /2 ,DE0000eηi (τ )∂i eηi (τ )∂i = e[hηi (τ )ηj (τ )i∂i ∂j +hηi (τ )ηj (τ )i∂i ∂j +2hηi (τ )ηj (τ )i∂i ∂j ]/20 00= e[Gij (τ,τ )∂i ∂j +Gij (τ ,τ )∂i ∂j +2Gij (τ,τ )∂i ∂j ]/2....(4B.36)Expanding the exponentials and performing the τ -integrals in (4B.35) yields all desired higherorder corrections to (4B.16).For ∆x = 0, the expansion has been driven to high orders in Ref.
[1] (including a minimalinteraction with a vector potential).Notes and ReferencesFor the eikonal expansion, see the original works byG. Wentzel, Z. Physik 38, 518 (1926);H.A. Kramers, Z. Physik 39, 828 (1926);L. Brillouin, C. R. Acad. Sci. Paris 183, 24 (1926);V.P. Maslov and M.V. Fedoriuk, Semiclassical Approximation in Quantum Mechanics, Reidel,Dordrecht, 1982;J.B. Delos, Semiclassical Calculation of Quantum Mechanical Wave Functions, Adv.
Chem. Phys.65, 161 (1986);M.V. Berry and K.E. Mount, Semiclassical Wave Mechanics, Rep. Prog. Phys. 35, 315 (1972);and the references quoted in the footnotes.For the semiclassical expansion of path integrals seeR. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D 10, 4114, 4130 (1974),R. Rajaraman, Phys. Rep. 21C, 227 (1975);S. Coleman, Phys. Rev. D 15, 2929 (1977); and in The Whys of Subnuclear Physics, Erice Lectures1977, Plenum Press, 1979, ed. by A. Zichichi.Recent semiclassical treatments of atomic systems are given inR.S. Manning and G.S.
Ezra, Phys. Rev. 50, 954 (1994).Chaos 2, 19 (1992).H. Kleinert, PATH INTEGRALSNotes and References453Semiclassical scattering is treated inJ.M. Rost and E.J. Heller, J. Phys. B 27, 1387 (1994).For the semiclassical approach to chaotic systems see the textbookM.C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer, Berlin, 1990.Section 12.4 of that book also calculates the action (4.499) and the eikonal (4.517) of the Coulombsystem. Sections 6.3 and 6.4 of that book discuss the properties of the stability matrix.Applications to complex highly excited atomic spectra are described byH.
Friedrich and D. Wintgen, Phys. Rep. 183, 37 (1989);P. Cvitanović and B. Eckhardt, Phys. Rev. Lett. 63, 823 (1991);G. Tanner, P. Scherer, E.B. Bogomonly, B. Eckhardt, and D. Wintgen, Phys. Rev. Lett. 67, 2410(1991);G.S. Ezra, K. Richter, G. Tanner, and D. Wintgen, J. Phys. B 24, L413 (1991);B. Eckhardt and D. Wintgen, J. Phys. A 24, 4335 (1991);D. Wintgen, K. Richter, and G. Tanner, Chaos 2, 19 (1992).The individual citations refer to the following works:[1] D.
Fliegner, M.G. Schmidt, and C. Schubert, Z. Phys. C64, 111 (1994) (hep-ph/9401221);D. Fliegner, P. Haberl, M.G. Schmidt, and C. Schubert, Ann. Phys. (N.Y.) 264, 51 (1998)(hep-th/9707189).[2] J. Schwinger, Phys. Phys. A 22, 1827 (1980), A24, 2353 (1981).[3] The form (4.539) of the scattering amplitude was first derived byP. Pechukas, Phys. Rev. 181, 166 (1969).See alsoJ.M. Rost and E.J. Heller, J. Phys. B 27, 1387 (1994).For rainbow and glory scattering see the paper by Pechukas and byK.W.
Ford and J.A. Wheeler, Ann. Phys. 7, 529 (1959).For the semiclassical treatment of the Coulomb problem seeA. Northcliffe and I.C. Percival, J. Phys. B 1, 774, 784 (1968); A. Northcliffe, I.C. Percival,and M.J. Roberts, J. Phys. B 2, 590, 578 (1968).For an alternative path integral formula for the scattering matrix seeW.B. Campbell, P. Finkler, C.E. Jones, and M.N. Misheloff, Phys. Rev.
D 12, 2363 (1975).[4] C.M. Bender, K. Olaussen, and P.S. Wang, Phys. Rev. D 16, 1740 (1977).H. Kleinert, PATH INTEGRALSNovember 20, 2006 ( /home/kleinert/kleinert/books/pathis3/pthic5.tex)Who can believe what varies every day,Nor ever was, nor will be at a stay?John Dryden (1631-1700), Hind and the Panther (1687)5Variational Perturbation TheoryMost path integrals cannot be performed exactly.
It is therefore necessary to developapproximation procedures which allow us to approach the exact result with anydesired accuracy, at least in principle. The perturbation expansion of Chapter 3does not serve this purpose since it diverges for any coupling strength. Similardivergencies appear in the semiclassical expansion of Chapter 4.The present chapter develops a convergent approximation procedure to calculateEuclidean path integrals at a finite temperature. The basis for this procedure is avariational approach due to Feynman and Kleinert, which was recently extended toa systematic and uniformly convergent variational perturbation expansion [1].5.1Variational Approach to Effective ClassicalPartition FunctionStarting point for the variational approach will be the path integral representation(3.813) for the effective classical potential introduced in Section 3.25.
Explicitly, theeffective classical Boltzmann factor B(x0 ) for a quantum system with an actionAe =Zh̄β0dτM 2ẋ (τ ) + V (x(τ ))2(5.1)has the path integral representation [recall (3.813) and (3.806)]−V eff cl (x0 )/kB TB(x0 ) ≡ e=I0−Ae /h̄D xe=∞Ym=1∞M X1 Z h̄/kB T22× exp −ω |x | −dτ VkB T m=1 m mh̄ 0""Zimdxrem dxm×2πkB T /Mωmx0 +#∞Xm=1(xm e−iωm τ + c.c.)(5.2)!#,imwith the notation xrem = Re xm , xm = Im xm . To make room for later subscripts,we shall in this chapter write A instead of Ae . In Section 3.25 we have derived aeff clperturbation expansion for B(x0 ) = e−βV (x0 ) .
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