Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 71
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It is then easy to calculate the effective classicalBoltzmann factor for the Hamiltonian [recall (2.665)]H(p, x) =M1 2 M 2 2p + ω⊥ x⊥ (τ ) + ωk2 z 2 (τ ) + ωB lz (p(τ ), x(τ )),2M22(3.825)where lz (p, x) is the z-component of the angular momentum defined in Eq. (2.623).We have shifted the center of momentum integration to p0 , for later convenience(see Subsection 5.11.2). The vector x⊥ = (x, y) denotes the orthogonal part of x. Asin the generalized magnetic field action (2.665), we have chosen different frequenciesin front of the harmonic oscillator potential and of the term proportional to lz , forgenerality.
The effective classical Boltzmann factor follows immediately from (2.679)by “undoing” the momentum integrations in px , py , and using (3.824) for the motionin the z-direction:h̄βωk /2h̄βω− /2h̄βω+ /2e−βH(p0 ,x0 ),sinh h̄βω+ /2 sinh h̄βω− /2 sinh h̄βωk /2(3.826)where ω± ≡ ωB ± ω⊥ , as in (2.675). As in Eq. (3.820), the restrictions of the pathintegrals over x and p to the fixed averages x0 = x̄ and p0 = p̄ give rise to the extranumerators in comparison to (2.679).B(p0 , x0 )= e−βH3.25.3eff cl (p0 ,x0 )= le3 (h̄β)High- and Low-Temperature BehaviorWe have remarked before in Eq.
(3.809) that in the limit T → ∞, the effectiveclassical potential V eff cl (x0 ) converges by construction against the initial potentialV (x0 ). There exists, in fact, a well-defined power series in h̄ω/kB T which describesH. Kleinert, PATH INTEGRALS3313.25 Effective Classical Potentialthis approach. Let us study this limit explicitly for the effective classical potentialof the harmonic oscillator calculated in (3.820), after rewriting it assinh(h̄ω/2kB T ) M 2 2+ ω x0(3.827)h̄ω/2kB T2"#M 2 2 h̄ωh̄ω−h̄ω/kB T=ω x0 ++ kB T log(1 − e) − log.22kB TVωeff cl (x0 ) = kB T logDue to the subtracted logarithm of ω in the brackets, the effective classical potentialhas a power seriesVωeff cl (x0 )11 h̄ωM 2 2ω x0 + h̄ω −=224 kB T2880h̄ωkB T!3+ ... .(3.828)This pleasant high-temperature behavior is in contrast to that of the effective potential which reads for the harmonic oscillatorVωeff (x0 ) = kB T log [2 sinh(h̄ω/2kB T )] +=M 2 2ω x02M 2 2 h̄ωω x0 ++ kB T log(1 − e−h̄ω/kB T ),22(3.829)as we can see from (3.793).
The logarithm of ω prevents this from having a powerseries expansion in h̄ω/kB T , reflecting the increasing width of the unsubtractedfluctuations.Consider now the opposite limit T → 0, where the final integral over the Boltzmann factor B(x0 ) can be calculated exactly by the saddle-point method. In thislimit, the effective classical potential V eff cl (x0 ) coincides with the Euclidean versionof the effective potential:V eff cl (x0 ) → V eff (x0 ) ≡ Γe [X]/β X=x0T →0,(3.830)whose real-time definition was given in Eq. (3.663).Let us study this limit again explicitly for the harmonic oscillator, where itbecomesT →0 h̄ωMh̄ω+ ω 2 x20 − kB T log,(3.831)Vωeff cl (x0 ) −−−→22kB Ti.e., the additional constant tends to h̄ω/2.
This is just the quantum-mechanicalzero-point energy which guarantees the correct low-temperature limith̄ω Z ∞ dx0 −M ω2 x20 /2kB TekB T −∞ le (h̄β)T →0−h̄ω/2kB T=e−h̄ω/2kB T .Zω −−−→ e(3.832)The limiting partition function is equal to the Boltzmann factor with the zero-pointenergy h̄ω/2.3323.25.43 External Sources, Correlations, and Perturbation TheoryAlternative Candidate for Effective Classical PotentialIt is instructive to compare this potential with a related expression which can bedefined in terms of the partition function density defined in Eq. (2.325):Ṽωeff cl (x) ≡ kB T log [le (h̄β) z(x)] .(3.833)This quantity shares with Vωeff cl (x0 ) the property that it also yields the partitionfunction by forming the integral [compare (2.324)]:∞ZZ=−∞dx0 −Ṽ eff cl (x0 )/kB Te.le (h̄β)(3.834)It may therefore be considered as an alternative candidate for an effective classicalpotential.For the harmonic oscillator, we find from Eq.
(2.326) the explicit form"# Mω2h̄ω h̄ωh̄ω 2log++kB T log 1−e−2h̄ω/kB T +tanhx .(3.835)2kB T2h̄kB Tk TṼωeff cl (x) = − BThis shares with the effective potential V eff (X) in Eq. (3.829) the unpleasant property of possessing no power series representation in the high-temperature limit.The low-temperature limit of Ṽωeff cl (x) looks at first sight quite similar to (3.831):T →0Ṽ eff cl (x0 ) −−−→Mω 2 kB T2h̄ωh̄ω+ kB Tx −log,2h̄2kB T(3.836)and the integration leads to the same result (3.832) in only a slightly different way:T →0−h̄ω/2kB TZω −−−→ es−h̄ω/2kB T=e2h̄ωkB TZ∞−∞dx −M ωx2 /h̄ele (h̄β).(3.837)There is, however, an important difference of (3.836) with respect to (3.831). Thewidth of a local Boltzmann factor formed from the partition function density (2.325):B̃(x) ≡ le (h̄β) z(x) = e−Ṽeff cl (x)/kBT(3.838)is much wider than that of the effective classical Boltzmann factor B(x0 ) =eff cle−V (x0 )/kB T .
Whereas B(x0 ) has a finite width for T → 0, the Boltzmann factor B̃(x) has a width growing to infinity in this limit. Thus the integral over xin (3.837) converges much more slowly than that over x0 in (3.832). This is theprincipal reason for introducing V eff cl (x0 ) as an effective classical potential ratherthan Ṽ eff cl (x0 ).H. Kleinert, PATH INTEGRALS3333.25 Effective Classical Potential3.25.5Harmonic Correlation Function without Zero ModeBy construction, the correlation functions of η(τ ) have the desired subtracted form(3.802):h̄ p 0h̄ cosh ω(|τ − τ 0 |−h̄β/2)10hη(τ )η(τ )iω = Gω2 ,e (τ −τ ) =−,M2Mωsinh(βh̄ω/2)h̄βω 20(3.839)with the square width as in (3.803):hη 2 (τ )iω ≡ a2ω = Gpω02 ,e (0) =1βh̄ω1coth−,2ω2h̄βω 2(3.840)which decreases withincreasing temperature. This can be seen explicitly by addingRa current term − dτ j(τ )η(τ ) to the action (3.814) which winds up in the exponentof (3.816), replacing λ/β by j(τ ) + λ/β and multiplies the exponential in (3.817) bya factor12Mh̄β 2(Zh̄β0dτ(Z0h̄β0h20dτ λ + λβj(τ ) + λβj(τ )iGpω2 ,e (τ0−τ ))1 Z h̄β Z h̄β 0× expdτdτ j(τ )Gpω2 ,e (τ − τ 0 )j(τ 0 ) .2Mh̄ 00)(3.841)In the first exponent, one of the τ -integrals over Gpω2 ,e (τ − τ 0 ), say τ 0 , produces afactor 1/ω 2 as in (3.818), so that the first exponent becomes)(1λβ Z h̄β2 h̄βdτ j(τ ) .λ 2 +2 22Mh̄β 2ωω 0(3.842)If we now perform the integral over λ, the linear term in λ yields, after a quadraticcompletion, a factor)(Z h̄βZ h̄β1dτ 0 j(τ )j(τ 0 ) .exp −dτ02Mβh̄2 ω 2 0(3.843)Combined with the second exponential in (3.841) this leads to a generating functionalfor the subtracted correlation functions (3.839):Zωx0 [j]()βh̄ω/22 21 Z h̄β Z h̄β 0=dτdτ j(τ )Gpω02 ,e (τ −τ 0 )j(τ 0 ) .e−βM ω x0 /2 expsin(βh̄ω/2)2Mh̄ 00(3.844)For j(τ ) ≡ 0, this reduces to the local Boltzmann factor (3.820).3343 External Sources, Correlations, and Perturbation Theory3.25.6Perturbation ExpansionWe can now apply the perturbation expansion (3.480) to the path integral over η(τ )in Eq.
(3.813) for the effective classical Boltzmann factor B(x0 ). We take the actionAe [x] =Zh̄βdτ0M 2ẋ + V (x) ,2(3.845)and rewrite it asAe = h̄βV (x0 ) + A(0)e [η] + Aint,e [x0 ; η],(3.846)with an unperturbed actionA(0)e [η]=Zh̄β0M 2Mdτη̇ (τ ) + Ω2 (x0 )η 2 (τ ) , Ω2 (x0 ) ≡ V 00 (x0 )/M,22and an interactionAint,e [x0 ; η] =Zh̄β0dτ V int (x0 ; η(τ )),(3.847)(3.848)containing the subtracted potential1V int (x0 ; η(τ )) = V (x0 + η(τ )) − V (x0 ) − V 0 (x0 )η(τ ) − V 00 (x0 )η 2 (τ ).2(3.849)This has a Taylor expansion starting with the cubic termV int (x0 ; η) =1 0001V (x0 )η 3 + V (4) (x0 )η 4 + . .
. .3!4!(3.850)Since η(τ ) has a zero temporal average, the linear term 0h̄β dτ V 0 (x0 )η(τ ) is absentin (3.847). The effective classical Boltzmann factor B(x0 ) in (3.813) has then theperturbation expansion [compare (3.480)]RB(x0 ) =1−Ex01 D 2 Ex01 D 3 Ex01D−+...BΩ (x0 ). (3.851)Aint,e +AAint,e Ωint,e ΩΩh̄2!h̄23!h̄3The harmonic expectation values are defined with respect to the harmonic pathintegralBΩ (x0 ) =Z(0)Dη δ̃(η̄) e−Ae[η]//h̄.(3.852)For an arbitrary functional F [x] one has to calculatehF [x]ixΩ0=BΩ−1 (x0 )Z(0)Dη δ̃(η̄) F [x] e−Ae[η]/h̄.(3.853)Some calculations of local expectation values are conveniently done with theexplicit Fourier components of the path integral. Recalling (3.806) and expandingH.
Kleinert, PATH INTEGRALS3353.25 Effective Classical Potentialthe action (3.814) in its Fourier components using (3.804), they are given by theproduct of integralshF [x]ixΩ0=[ZΩx0 ]−1∞Ym=1"Zimdxre− M Σ∞ [ω 2 +Ω2 (x0 )]|xm |2m dxmF [x].e kB T m=1 m2πkB T /Mωm#(3.854)This implies the correlation functions for the Fourier componentsDxm x∗m0Ex0Ω= δmm0kB T1.2M ωm + Ω2 (x0 )(3.855)From these we can calculate once more the correlation functions of the fluctuationsη(τ ) as follows:D0η(τ )η(τ )Ex0Ω=*∞Xxm x∗m0 e−i(ωm −ωm0 )τm,m0 6=0+x 0Ω=211 X.
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