Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 59
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Kleinert, PATH INTEGRALS2713.17 Perturbation Expansion of Anharmonic Systems3.53ρ(ε) 2.521.510.50δ(ε − E0 )12345678δ(ε − E0 )3.53ρ(ε) 2.521.510.50ε/ω12345678ε/ωFigure 3.6 Density of states for weak and strong damping in natural units. On the left,the parameters are γ/ω = 0.2, ωD /ω = 10, on the right γ/ω = 5, ωD /ω = 10. For moredetails see Hanke and Zwerger in Notes and References.with γ = e2 /6c2 πM.
The power of ωm accompanying the friction constant is increased by two units. Adding a Drude correction for the high-frequency behaviorwe replace γ by ωm /(ωm + ωD ) and obtain instead of (3.445)Zωdamp =2∞ωm(ωm + ωD )(1 + γωD )kB T Y.32h̄ω m=1 ωm (1 + γωD ) + ωmωD + ωm ω 2 + ωD ω 2(3.469)The resulting partition function has again the form (3.450), except that w123 are thesolutions of the cubic equationw 3 (1 + γωD ) − w 2 ωD + wω 2 − ω 2 ωD = 0.(3.470)Since the electromagnetic coupling is small, we can solve this equation to lowestorder in γ.
If we also assume ωD to be large compared to ω, we find the rootseffw1 ≈ γpb/2 + iω,effw1 ≈ γpb/2 − iω,effw3 ≈ ωD /(1 + γpbωD /ω 2),(3.471)where we have introduced an effective friction constant of the photon batheffγpbe2ω2,= 26c πM(3.472)which has the dimension of a frequency, just as the usual friction constant γ in theprevious heat bath equations (3.451).3.17Perturbation Expansion of Anharmonic SystemsIf a harmonic system is disturbed by an additional anharmonic potential V (x), to becalled interaction, the path integral can be solved exactly only in exceptional cases.These will be treated in Chapters 8, 13, and 14. For sufficiently smooth and smallpotentials V (x), it is always possible to expand the full partition in powers of theinteraction strength. The result is the so-called perturbation series.
Unfortunately,it only renders reliable numerical results for very small V (x) since, as we shall prove2723 External Sources, Correlations, and Perturbation Theoryin Chapter 17, the expansion coefficients grow for large orders k like k!, making theseries strongly divergent. The can only be used for extremely small perturbations.Such expansions are called asymptotic (more in Subsection 17.10.1). For this reasonwe are forced to develop a more powerful technique of studying anharmonic systemsin Chapter 5. It combines the perturbation series with a variational approach andwill yield very accurate energy levels up to arbitrarily large interaction strengths.
Itis therefore worthwhile to find the formal expansion in spite of its divergence.Consider the quantum-mechanical amplitude(xb tb |xa ta ) =Zx(tb )=xbx(ta )=xa(iDx exph̄tbZtaM 2ω2dtẋ − M x2 − V (x)22"#),(3.473)and expand the integrand in powers of V (x), which leads to the seriesx(tb )=xbZ(xb tb |xa ta ) =x(ta )=xaiDx 1 −h̄ZtbtadtV (x(t))tbtb1dtV(x(t))dt1 V (x(t1 ))2ta2!h̄2 ta 2Z tbZ tbZ tbidt1 V (x(t1 )) + . . .dt2 V (x(t2 ))dt V (x(t3 ))+ 3tata3!h̄ ta 3 Z ti b M 2(3.474)× expẋ − ω 2 x2 .dth̄ ta2Z−ZIf we decompose the path integral in the nth term into a product (2.4), the expansioncan be rewritten asi tb(xb tb |xa ta ) = (xb tb |xa ta ) −dx1 (xb tb |x1 t1 )V (x1 )(xb tb |xa ta )(3.475)dth̄ ta 1Z tbZ tbZ1− 2dt2dt1 dx1 dx2 (xb tb |x2 t2 )V (x2 )(x2 t2 |x1 t1 )V (x1 )(x1 t1 |xa ta ) + .
. . .ta2!h̄ taZZA similar expansion can be given for the Euclidean path integral of a partitionfunctionZ=I(1Dx exp −h̄Zh̄β0M 2dτ(ẋ + ω 2 x2 ) + V (x)2),(3.476)where we obtainZ =Z1Dx 1 −h̄−Zh̄β0133!h̄(× exp −1dτ V (x(τ )) +2!h̄2Zh̄β01h̄Z0dτ3 V (x(τ3 ))h̄βdt"ZZ0h̄β0h̄βdτ2 V (x(τ2 ))dτ2 V (x(τ2 ))2M 2ωẋ + M x222#).ZZ0h̄β0h̄βdτ1 V (x(τ1 ))dτ1 V (x(τ1 )) + . .
.(3.477)H. Kleinert, PATH INTEGRALS2733.17 Perturbation Expansion of Anharmonic SystemsThe individual terms are obviously expectation values of powers of the EuclideaninteractionAint,e ≡Zh̄β0dτ V (x(τ )),(3.478)calculated within the harmonic-oscillator partition function Zω . The expectationvalues are defined byh. . .iω ≡Zω−1Z(1Dx . . . exp −h̄Zh̄βM 2(ẋ + ω 2x2 )dτ2).(3.479)11123Z = 1 − hAint,e iω +2 hAint,e iω −3 hAint,e iω + .
. . Zω .h̄2!h̄3!h̄(3.480)0With these, the perturbation series can be written in the formAs we shall see immediately, it is preferable to resum the prefactor into an exponential of a series111231 − hAint,eiω +2 hAint,e iω −3 hAint,e iω + . . .h̄ 2!h̄3!h̄11123hA i −hA i + . .
. .= exp − hAint,e iω +h̄2!h̄2 int,e ω,c 3!h̄3 int,e ω,c(3.481)The expectation values hA3int,e iω,c are called cumulants. They are related to theoriginal expectation values by the cumulant expansion:12hA2int,eiω,c ≡ hA2int,e iω − hAint,ei2ω(3.482)hA3int,eiω,c ≡ hA3int,e iω − 3hA2int,eiω hAint,e iω + 2hAint,e i3ω(3.483)= h[Aint,e − hAint,eiω ]2 iω ,3= h[Aint,e − hAint,eiω ] iω ,... .The cumulants contribute directly to the free energy F = −(1/β) log Z. From(3.481) and (3.480) we conclude that the anharmonic potential V (x) shifts the freeenergy of the harmonic oscillator Fω = (1/β) log[2 sinh(h̄βω/2)] by1∆F =β11123hAint,e iω −2 hAint,e iω,c +3 hAint,e iω,c + . . . .h̄2!h̄3!h̄D(3.484)EWhereas the original expectation values Anint,egrow for large β with the nthωpower of β, due to contributions of n disconnected diagrams of firstD orderE in gwhich are integrated independently over τ from 0 to h̄β, the cumulants Anint,,e areω12Note that the subtracted expressions in the second lines of these equations are particularlysimple only for the lowest two cumulants given here.2743 External Sources, Correlations, and Perturbation Theoryproportional to β, thus ensuring that the free energy F has a finite limit, the groundstate energy E0 .
In comparison with the ground state energy of the unperturbedharmonic system, the energy E0 is shifted by1∆E0 = limβ→∞ β11123hAint,eiω −2 hAint,e iω,c +3 hAint,e iω,c + . . . .h̄2!h̄3!h̄(3.485)There exists a simple functional formula for the perturbation expansion of thepartition function in terms of the generating functional Zω [j] of the unperturbedharmonic system.
Adding a source term into the action of the path integral (3.476),we define the generating functional of the interacting theory:Z[j] =I(1Dx exp −h̄Zh̄β0)M 2dτẋ + ω 2 x2 + V (x) − jx .2(3.486)The interaction can be brought outside the path integral in the form1Z[j] = e− h̄R h̄β0dτ V (δ/δj(τ ))Zω [j] .(3.487)The interacting partition function is obviouslyZ = Z[0].(3.488)Indeed, after inserting on the right-hand side the explicit path integral expressionfor Z[j] from (3.233):Zω [j] =Z(1Dx exp −h̄Zh̄β0)M 2(ẋ + ω 2 x2 ) − jxdτ2,(3.489)and expanding the exponential in the prefactor− h̄1eR h̄β0dτ V (δ/δj(τ ))1= 1−h̄Z0h̄βdτ V (δ/δj(τ ))h̄βh̄β1dτV(δ/δj(τ))dτ1 V (δ/δj(τ1 ))(3.490)2202!h̄2 0Z h̄βZ h̄β1 Z h̄β− 3dτ3 V (δ/δj(τ3 ))dτ2 V (δ/δ(τ2 ))dτ1 V (δ/δ(τ1 )) + .
. . ,003!h̄ 0+ZZthe functional derivatives of Z[j] with respect to the source j(τ ) generate insidethe path integral precisely the expansion (3.480), whose cumulants lead to formula(3.484) for the shift in the free energy.Before continuing, let us mention that the partition function (3.476) can, ofcourse, be viewed as a generating functional for the calculation of the expectationvalues of the action and its powers. We simply have to form the derivatives withrespect to h̄−1 :nn−1 ∂Z[j](3.491)hA i = Znω −1 .h̄ =0∂h̄−1H. Kleinert, PATH INTEGRALS3.18 Rayleigh-Schrödinger and Brillouin-Wigner Perturbation Expansion275For a harmonic oscillator where Z is given by (3.242), this yieldshAi = lim Zω−1 h̄2h̄→∞∂h̄ωβZω = lim h̄= 0.h̄→∞∂h̄2 sinh h̄ωβ/2(3.492)The same result is, incidentally, obtained by calculating the expectation value of theaction with analytic regularization:Dẋ2 (τ )EωD+ ω 2 x2 (τ )Eω=dω 0 ω 02+2π ω 02 + ω 2ZZdω 0 ω 2=2π ω 02 + ω 2Zdω 0= 0.
(3.493)2πThe integral vanishes by Veltman’s rule (2.500).3.18Rayleigh-Schrödinger and Brillouin-WignerPerturbation ExpansionThe expectation values in formula (3.484) can be evaluated by means of the socalled Rayleigh-Schrödinger perturbation expansion, also referred to as old-fashionedperturbation expansion. This expansion is particularly useful if the potential V (x)is not a polynomial in x. Examples are V (x) = δ(x) and V (x) = 1/x. In these twocases the perturbation expansions can be summed to all orders, as will be shownfor the first example in Section 9.5. For the second example the reader is referredto the literature.13 We shall explicitly demonstrate the procedure for the groundstate and the excited energies of an anharmonic oscillator.
Later we shall also giveexpansions for scattering amplitudes.To calculate the free-energy shift ∆F in Eq. (3.484) to first order in V (x), weneed the expectationhAint,e iω ≡Zω−1Z0h̄βdτ1Zdxdx1 (x h̄β|x1 τ1 )ω V (x1 )(x1 τ1 |x 0)ω .(3.494)The time evolution amplitude on the right describes the temporal development ofthe harmonic oscillator located initially at the point x, from the imaginary time 0up to τ1 .
At the time τ1 , the state is subject to the interaction depending on itsposition x1 = x(τ1 ) with the amplitude V (x1 ). After that, the state is carried to thefinal state at the point x by the other time evolution amplitude.To second order we have to calculate the expectation in V (x):Z h̄βZ h̄βZ1 2−1hA i ≡ Zωdτ2dτ1 dxdx2 dx1 (x h̄β|x2 τ2 )ω V (x2 )2 int,e ω00×(x2 τ2 |x1 τ1 )ω V (x1 )(x1 τ1 |x 0)ω .(3.495)The integration over τ1 is taken only up to τ2 since the contribution with τ1 > τ2would merely render a factor 2.13M.J. Goovaerts and J.T. Devreese, J. Math. Phys.
13, 1070 (1972).2763 External Sources, Correlations, and Perturbation TheoryThe explicit evaluation of the integrals is facilitated by the spectral expansion(2.293). The time evolution amplitude at imaginary times is given in terms of theeigenstates ψn (x) of the harmonic oscillator with the energy En = h̄ω(n + 1/2):(xb τb |xa τa )ω =∞Xψn (xb )ψn∗ (xa )e−En (τb −τa )/h̄ .(3.496)n=0The same type of expansion exists also for the real-time evolution amplitude.This leads to the Rayleigh-Schrödinger perturbation expansion for the energy shiftsof all excited states, as we now show.The amplitude can be projected onto the eigenstates of the harmonic oscillator.For this, the two sides are multiplied by the harmonic wave functions ψn∗ (xb ) andψn (xa ) of quantum number n and integrated over xb and xa , respectively, resultingin the expansionZdxb dxa ψn∗ (xb )(xb tb |xa ta )ψn (xa ) =Zdxb dxa ψn∗ (xb )(xb tb |xa ta )ω ψn (xa )1ii23× 1 + hn|Aint|niω −2 hn|Aint |niω −3 hn|Aint |niω + .
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