Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 57
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(3.346)–(3.349) in an abbreviated notation. Inserting (3.370) into(0)(3.368) and performing the Gaussian momentum integration, over the exponentials in Zω [0, 0](1)and Zω [k, j], the result is( Z)1 h̄β(xb h̄β|xa 0)[k, j] = (xb h̄β|xa 0)[0, 0] × expdτ [xcl (τ )j(τ ) + pcl (τ )k(τ )]h̄ 0(!#)Z h̄β Z h̄β(D)(D)1Gxx (τ1 , τ2 ) Gxp (τ1 , τ2 )j(τ2 )× expdτ2 [(j(τ1 ), k(τ2 ))dτ1,(D)(D)k(τ2 )2h̄2 0Gpx (τ1 , τ2 ) Gpp (τ1 , τ2 )0(3.374)(D)where the Green functions Gab (τ1 , τ2 ) have now Dirichlet boundary conditions. In particular, the(D)Green function Gab (τ1 , τ2 ) is equal to (3.36) continued to imaginary time.
The Green functions(D)(D)Gxp (τ1 , τ2 ) and Gpp (τ1 , τ2 ) are Dirichlet versions of Eqs. (3.346)–(3.349) which arise from theabove Gaussian momentum integrals.After performing the integrals, the first factor without currents issZω2πh̄2(xb h̄β|xa 0)[0, 0] = lim lim limpτb ↑h̄β τa ↓0 τa0 ↓0 2πh̄Gxx (τa0 , τa0 ))()(Gpxp 2 (τa0 , τb )Gpxp 2 (τa0 , τa )1p2p2− Gpp (τa , τa ) + xb− Gpp (τb , τb )× expxaGpxx (τa0 , τa0 )Gpxx (τa0 , τa0 )2h̄2 p 0 Gxp (τa , τa )Gpxp (τa0 , τb )p−2xa xb−G(τ,τ).(3.375)pp a bGpxx (τa0 , τa0 )Performing the limits usingh̄lim lim Gpxp (τa0 , τa ) = −i ,2τa ↓0 τa0 ↓0(3.376)where the order of the respective limits turns out to be important, we obtain the amplitude (2.403):sMω(xb h̄β|xa 0)[0, 0] =2πh̄ sinh h̄βω 2Mω(xa + x2b ) cosh h̄βω − 2xa xb× exp −.(3.377)2h̄ sinh h̄βωThe first exponential in (3.374) contains a complicated representation of the classical path p 0Gxp (τa , τa )Gpxx (τ, τa0 )ipxcl (τ ) = lim lim limxa+ Gxp (τa , τ )τb ↑h̄β τa ↓0 τa0 ↓0 h̄Gpxx (τa0 , τa0 )2603 External Sources, Correlations, and Perturbation Theory−xbGpxp (τa0 , τb )Gpxx (τ, τa0 )p+G(τ,τ),xp bGpxx (τa0 , τa0 )(3.378)and of the classical momentumpcl (τ )i= lim lim limτb ↑h̄β τa ↓0 τa0 ↓0 h̄Gpxp (τa0 , τa )Gpxp (τa0 , τ )pxa− Gpp (τa , τ )Gpxx (τa0 , τa0 ) p 0Gxp (τa , τb )Gpxp (τa0 , τ )p−xb− Gpp (τb , τ ).Gpxx (τa0 , τa0 )(3.379)Indeed, inserting the explicit periodic Green functions (3.346)–(3.349) and going to the limits weobtainxcl (τ )=xa sinh ω(h̄β − τ ) + xb sinh ωτsinh h̄βω(3.380)−xa cosh ω(h̄β − τ ) + xb cosh ωτ,sinh h̄βω(3.381)andpcl (τ )= iM ωthe first being the imaginary-time version of the classical path (3.6), the second being related toit by the classical relation pcl (τ ) = iM dxcl (τ )/dτ .The second exponential in (3.374) quadratic in the currents contains the Green functions withDirichlet boundary conditionsGpxx (τ1 , 0)Gpxx (τ2 , 0),Gpxx (τ1 , τ1 )Gpxx (τ1 , 0)Gpxp (τ2 , 0)p,G(D)xp (τ1 , τ2 ) = Gxp (τ1 , τ2 ) +Gpxx (τ1 , τ1 )Gpxp (τ1 , 0)Gpxx (τ2 , 0)pG(D),px (τ1 , τ2 ) = Gpx (τ1 , τ2 ) +Gpxx (τ1 , τ1 )Gpxp (τ1 , 0)Gpxp (τ2 , 0)pG(D)(τ,τ)=G(τ,τ)−.1212ppppGpxx (τ1 , τ1 )pG(D)xx (τ1 , τ2 ) = Gxx (τ1 , τ2 ) −(3.382)(3.383)(3.384)(3.385)After applying some trigonometric identities, these take the formh̄[cosh ω(h̄β −|τ1 −τ2 |)−cosh ω(h̄β −τ1 −τ2 )],2M ω sinh h̄βωih̄G(D){θ(τ1 −τ2 ) sinh ω(h̄β −|τ1 −τ2 |)xp (τ1 , τ2 ) =2 sinh h̄βω−θ(τ2 −τ1 ) sinh ω(h̄β − |τ2 −τ1 |)+sinh ω(h̄β − τ1 −τ2 )},ih̄{θ(τ1 −τ2 ) sinh ω(h̄β − |τ1 −τ2 |)G(D)px (τ1 , τ2 ) = −2 sinh h̄βω−θ(τ2 −τ1 ) sinh ω(h̄β − |τ2 −τ1 |)−sinh ω(h̄β − τ1 −τ2 )},M h̄ωG(D)[cosh ω(h̄β − |τ1 −τ2 |) + cosh ω(h̄β −τ1 −τ2 )].pp (τ1 , τ2 ) =2 sinh h̄βωG(D)xx (τ1 , τ2 ) =(3.386)(3.387)(3.388)(3.389)The first correlation function is, of course, the imaginary-time version of the Green function (3.206).Observe the symmetry properties under interchange of the time arguments:(D)G(D)xx (τ1 , τ2 ) = Gxx (τ2 , τ1 ) ,G(D)px (τ1 , τ2 )=−G(D)px (τ2 , τ1 ) ,(D)G(D)xp (τ1 , τ2 ) = −Gxp (τ2 , τ1 ) ,G(D)pp (τ1 , τ2 )=G(D)pp (τ2 , τ1 ) ,(3.390)(3.391)H.
Kleinert, PATH INTEGRALS2613.13 Particle in Heat Bathand the identity(D)G(D)xp (τ1 , τ2 ) = Gpx (τ2 , τ1 ).(3.392)In addition, there are the following derivative relations between the Green functions with Dirichletboundary conditions:G(D)xp (τ1 , τ2 ) =G(D)px (τ1 , τ2 ) =G(D)pp (τ1 , τ2 ) =∂ (D)∂ (D)Gxx (τ1 , τ2 ) = iMG (τ1 , τ2 ) ,∂τ1∂τ2 xx∂ (D)∂ (D)G (τ1 , τ2 ) = −iMG (τ1 , τ2 ) ,iM∂τ1 xx∂τ2 xx∂2h̄M δ(τ1 −τ2 ) − M 2G(D) (τ1 −τ2 ) .∂τ1 ∂τ2 xx−iM(3.393)(3.394)(3.395)Note that Eq. (3.382) is a nonlinear alternative to the additive decomposition (3.142) of aGreen function with Dirichlet boundary conditions: into Green functions with periodic boundaryconditions.3.13Particle in Heat BathThe results of Section 3.8 are the key to understanding the behavior of a quantummechanical particle moving through a dissipative medium at a fixed temperature T .We imagine the coordinate x(t) a particle of mass M to be coupled linearly to aheat bath consisting of a great number of harmonic oscillators Xi (τ ) (i = 1, 2, 3, .
. .)with various masses Mi and frequencies Ωi . The imaginary-time path integral inthis heat bath is given byYI(xbh̄β|xa 0) =iDXi (τ )Zx(h̄β)=xbx(0)=xa(Dx(τ )1 Z h̄β X Mi2(Ẋi + Ω2i Xi2 )dτ× exp −h̄ 02i(1× exp −h̄Z0h̄β)(3.396)"XM 2dτẋ + V (x(τ )) −ci Xi (τ )x(τ )2i#)1,i Zi×Qwhere we have allowed for an arbitrary potential V (x). The partition functions ofthe individual bath oscillators()I1 Z h̄βMi22 2(Ẋi + Ωi Xi )Zi ≡DXi (τ ) exp −dτh̄ 021=(3.397)2 sinh(h̄βΩi /2)have been divided out, since their thermal behavior is trivial and will be of no interestin the sequel.
The path integrals over Xi (τ ) can be performed as in Section 3.1leading for each oscillator label i to a source expression like (3.243), in which ci x(τ )plays the role of a current j(τ ). The result can be written as(xb h̄β|xa 0) =Zx(h̄β)=xbx(0)=xa(1Dx(τ ) exp −h̄Z0h̄β)M 21ẋ + V (x(τ )) − Abath [x] ,dτ2h̄(3.398)2623 External Sources, Correlations, and Perturbation Theorywhere Abath [x] is a nonlocal action for the particle motion generated by the bathAbath [x] = −12h̄βZdτ0Zh̄β0dτ 0 x(τ )α(τ − τ 0 )x(τ 0 ).(3.399)The function α(τ − τ 0 ) is the weighted periodic correlation function (3.248):α(τ − τ 0 ) =X=Xc2iii1 pG 2 (τ − τ 0 )Mi Ωi ,ec2i cosh Ωi (|τ − τ 0 | − h̄β/2).2Mi Ωisinh(Ωih̄β/2)(3.400)Its Fourier expansion has the Matsubara frequencies ωm = 2πkB T /h̄∞1 X0αm e−iωm (τ −τ ) ,h̄β m=−∞α(τ − τ 0 ) =(3.401)with the coefficientsαm =1c2i.2Mi ωm + ωi2Xi(3.402)Alternatively, we can write the bath action in the form corresponding to (3.276)asAbath [x] = −12Zh̄β0dτZ∞−∞dτ 0 x(τ )α0 (τ − τ 0 )x(τ 0 ),(3.403)with the weighted nonperiodic correlation function [recall (3.277)]α0 (τ − τ 0 ) =Xi0c2ie−Ωi |τ −τ | .2Mi Ωi(3.404)The bath properties are conveniently summarized by the spectral density of thebathX c2iρb (ω 0 ) ≡ 2πδ(ω 0 − Ωi ).(3.405)i 2Mi ΩiThe frequencies Ωi are by definition positive numbers.
The spectral density allowsus to express α0 (τ − τ 0 ) as the spectral integral0α0 (τ − τ ) =Z0∞00dω 0ρb (ω 0 )e−ω |τ −τ | ,2π(3.406)and similarlyα(τ − τ 0 ) =Z∞0dω 0cosh ω 0 (|τ − τ 0 | − h̄β/2)ρb (ω 0 ).2πsinh(ω 0h̄β/2)(3.407)H. Kleinert, PATH INTEGRALS2633.13 Particle in Heat BathFor the Fourier coefficients (3.402), the spectral integral readsαm =Z∞0dω 02ω 00ρ (ω ) 2.2π bωm + ω 02(3.408)It is useful to subtract from these coefficients the first term α0 , and to invert thesign of the remainder making it positive definite. Thus we splitαm = 2Z∞02ωmdω 0 ρb (ω 0)1−22π ω 0ωm+ ω 02!= α0 − gm .(3.409)Then the Fourier expansion (3.401) separates asα(τ − τ 0 ) = α0 δ p (τ − τ 0 ) − g(τ − τ 0 ),(3.410)where δ p (τ − τ 0 ) is the periodic δ-function (3.279):δ p (τ − τ 0 ) =∞∞X01 Xδ(τ − τ 0 − nh̄β),e−iωm (τ −τ ) =h̄β m=−∞n=−∞(3.411)the right-hand sum following from Poisson’s summation formula (1.197). The subtracted correlation functiong(τ − τ 0 ) =∞1 X0g(ωm )e−iωm (τ −τ ) ,h̄β m=−∞(3.412)has the coefficientsgm =Xi2c2iωm=2Mi ωm+ Ω2iZ∞02dω 0 ρb (ω 0 ) 2ωm.22π ω 0 ωm+ ω 02(3.413)The corresponding decomposition of the bath action (3.399) isAbath [x] = Aloc + A0bath [x],(3.414)whereA0bath [x]1=2Zh̄β0dτh̄βZ0anddτ 0 x(τ )g(τ − τ 0 )x(τ 0 ),(3.415)α0 h̄βdτ x2 (τ ),(3.416)2 0is a local action which can be added to the original action in Eq.
(3.398), changingmerely the curvature of the potential V (x). Because of this effect, it is useful tointroduce a frequency shift ∆ω 2 via the equationZAloc = −2M∆ω ≡ −α0 = −2Z0∞X c2idω 0 ρb (ω 0 )=−2.2π ω 0i Mi Ωi(3.417)2643 External Sources, Correlations, and Perturbation TheoryThen the local action (3.416) becomesM∆ω 22Aloc =Zh̄β0dτ x2 (τ ).(3.418)This can be absorbed into the potential of the path integral (3.398), yielding arenormalized potentialM(3.419)Vren (x) = V (x) + ∆ω 2 x2 .2With the decomposition (3.414), the path integral (3.398) acquires the form(xb h̄β|xa 0) =Zx(h̄β)=xbx(0)=xa()M 211 Z h̄βẋ + Vren (x(τ )) − A0bath [x] .dτDx(τ ) exp −h̄ 02h̄(3.420)The subtracted correlation function (3.412) has the propertyZh̄β0dτ g(τ − τ 0 ) = 0.(3.421)Thus, if we rewrite in (3.415)1x(τ )x(τ 0 ) = {x2 (τ ) + x2 (τ 0 ) − [x(τ ) − x(τ 0 )]2 },2(3.422)the first two terms do not contribute, and we remain withA0bath [x] = −14Z0h̄βdτZ0h̄βdτ 0 g(τ − τ 0 )[x(τ ) − x(τ 0 )]2 .(3.423)If the oscillator frequencies Ωi are densely distributed, the function ρb (ω 0 ) iscontinuous.
As will be shown later in Eqs. (18.208) and (18.311), an oscillator bathintroduces in general a friction force into classical equations of motion. If this isto have the usual form −Mγ ẋ(t), the spectral density of the bath must have theapproximationρb (ω 0 ) ≈ 2Mγω 0(3.424)[see Eqs. (18.208), (18.311)]. This approximation is characteristic for Ohmic dissipation. In general, a typical friction force increases with ω only for small frequencies;for larger ω, it decreases again.
An often applicable phenomenological approximation is the so-called Drude form2ωDρb (ω ) ≈ 2Mγω 2,ωD + ω 0200(3.425)where 1/ωD ≡ τD is Drude’s relaxation time. For times much shorter than theDrude time τD , there is no dissipation. In the limit of large ωD , the Drude formdescribes again Ohmic dissipation.H. Kleinert, PATH INTEGRALS2653.14 Heat Bath of PhotonsInserting (3.425) into (3.413), we obtain the Fourier coefficients for Drude dissipation2gm = 2MγωDZ0∞2dω2ωmωD1.22 22 = M|ωm |γ2π ωD + ω ωm + ω|ωm | + ωD(3.426)It is customary, to factorizegm ≡ M|ωm |γm,(3.427)so that Drude dissipation corresponds toγm = γωD,|ωm | + ωD(3.428)and Ohmic dissipation to γm ≡ γ.The Drude form of the spectral density gives rise to a frequency shift (3.417)∆ω 2 = −γωD ,(3.429)which goes to infinity in the Ohmic limit ωD → ∞.3.14Heat Bath of PhotonsThe heat bath in the last section was a convenient phenomenological tool to reproduce the Ohmic friction observed in many physical systems.
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