Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 58
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In nature, there can bevarious different sources of dissipation. The most elementary of these is the deexcitation of atoms by radiation, which at zero temperature gives rise to the naturalline width of atoms. The photons may form a thermally equilibrated gas, the mostfamous example being the cosmic black-body radiation which is a gas of the photonsof 3 K left over from the big bang 15 billion years ago (and which create a sizablefraction of the blips on our television screens).The theoretical description is quite simple.
We decompose the vector potentialA(x, t) of electromagnetism into Fourier components of wave vector kA(x, t) =Xkck (x)Xk (t),ck =eikxq2Ωk V,Xk=Zd3 kV.(2π)3(3.430)The Fourier components Xk (t) can be considered as a sum of harmonic oscillatorsof frequency Ωk = c|k|, where c is the light velocity. A photon of wave vector k isa quantum of Xk (t).
A certain number N of photons with the same wave vectorcan be described as the Nth excited state of the oscillator Xk (t). The statisticalsum of these harmonic oscillators led Planck to his famous formula for the energyof black-body radiation for photons in an otherwise empty cavity whose walls havea temperature T . These will form the bath, and we shall now study its effecton the quantum mechanics of a charged point particle.
Its coupling to the vectorpotential is given by the interaction (2.610). Comparison with the coupling to the2663 External Sources, Correlations, and Perturbation Theoryheat bath in Eq. (3.396) shows that we simply have to replace −P− k ck Xk (τ )ẋ(τ ). The bath action (3.399) takes then the formAbath [x] = −Pi ci Xi (τ )x(τ )1 Z h̄β Z h̄β 0 idτ ẋ (τ )αij (x(τ ), τ ; x(τ 0 ), τ 0 )ẋj (τ 0 ),dτ2 00by(3.431)where αij (x, τ ; x0 , τ 0 ) is a 3 × 3 matrix generalization of the correlation function(3.400):αij (x, τ ; x0 , τ 0 ) =e2 Xic−k (x)ck (x0 )hX−k(τ )Xkj (τ 0 )i.2h̄c k(3.432)We now have to account for the fact that there are two polarization states for eachphoton, which are transverse to the momentum direction.
We therefore introduce atransverse Kronecker symbolT ijδk≡ (δ ij − k i k j /k2 )(3.433)iand write the correlation function of a single oscillator X−k(τ ) asjp0i0T ij0Gij−k0 k (τ − τ ) = hX̂−k0 (τ )X̂k0 (τ )i = h̄ δk δkk0 Gω 2 ,e k (τ − τ ),(3.434)withGpω2 ,e k (τ − τ 0 ) ≡1 cosh Ωk (|τ − τ 0 | − h̄β/2).2Ωksinh(Ωk h̄β/2)(3.435)Thus we finde2α (x, τ ; x , τ ) = 2cij000Zd3 k T ij eik(x−x ) cosh Ωk (|τ − τ 0 | − h̄β/2).δsinh(Ωk h̄β/2)(2π)3 k 2Ωk(3.436)At zero temperature, and expressing Ωk = c|k|, this simplifies toe2α (x, τ ; x , τ ) = 3cij000Z0d3 k T ij eik(x−x )−c|k||τ −τ |δ.2|k|(2π)3 k(3.437)Forgetting for a moment the transverse Kronecker symbol and the prefactor e2 /c2 ,the integral yields00GRe (x, τ ; x , τ ) =11,2 20 24π c (τ − τ ) + (x − x0 )2 /c2(3.438)which is the imaginary-time version of the well-known retarded Green function usedin electromagnetism.
If the system is small compared to the average wavelengthsin the bath we can neglect the retardation and omit the term (x − x0 )2 /c2 . In thefinite-temperature expression (3.437) this amounts to neglecting the x-dependence.H. Kleinert, PATH INTEGRALS2673.15 Harmonic Oscillator in Ohmic Heat BathThe transverse Kronecker symbol can then be averaged over all directions of thewave vector and yields simply 2δ ij /3, and we obtain the approximate functionαij (x, τ ; x0 , τ 0 ) =2e2 ij 1δ3c22πc2Zdω cosh ω(|τ − τ 0 | − h̄β/2)ω.2πsinh(ωh̄β/2)(3.439)This has the generic form (3.407) with the spectral function of the photon bathρpb (ω 0) =e2 0ω.3c2 π(3.440)This has precisely the Ohmic form (3.424), but there is now an important difference:the bath action (3.431) contains now the time derivatives of the paths x(τ ).
Thisgives rise to an extra factor ω 02 in (3.424), so that we may define a spectral densityfor the photon bath:e2.(3.441)ρpb (ω 0) ≈ 2Mγω 03 , γ = 26c πMIn contrast to the usual friction constant γ in the previous section, this has thedimension 1/frequency.3.15Harmonic Oscillator in Ohmic Heat BathFor a harmonic oscillator in an Ohmic heat bath, the partition function can becalculated as follows. SettingVren (x) =M 2 2ω x,2(3.442)the Fourier decomposition of the action (3.420) readsMh̄Ae =kB T(∞ hiω2 2 X2x0 +ωm+ ω 2 + ωm γm |xm |2 .2m=1)(3.443)The harmonic potential is the full renormalized potential (3.419).
Performing theGaussian integrals using the measure (2.439), we obtain the partition function forthe damped harmonic oscillator of frequency ω [compare (2.400)]Zωdampk T= Bh̄ω(∞Ym=1"2ωm+ ω 2 + ωm γ m2ωm#)−1.(3.444)For the Drude dissipation (3.426), this can be written asZωdamp =2∞kB T Yωm(ωm + ωD ).32h̄ω m=1 ωm + ωm ωD + ωm (ω 2 + γωD ) + ωD ω 2(3.445)Let w1 , w2 , w3 be the roots of the cubic equationw 3 − w 2 ωD + w(ω 2 + γωD ) − ω 2 ωD = 0.(3.446)2683 External Sources, Correlations, and Perturbation TheoryThen we can rewrite (3.445) asZωdamp =∞ωmωm ωm + ωDωmkB T Y.h̄ω m=1 ωm + w1 ωm + w2 ωm + w3 ωm(3.447)Using the product representation of the Gamma function8Γ(z) = n→∞limnnz Ymz m=1 m + z(3.448)and the fact thatw1 + w2 + w3 − ωD = 0,w1 w2 w3 = ω 2 ωD ,(3.449)the partition function (3.447) becomesZωdamp =1 ω Γ(w1 /ω1 )Γ(w2 /ω1 )Γ(w3 /ω1 ),2π ω1Γ(ωD /ω1 )(3.450)where ω1 = 2πkB T /h̄ is the first Matsubara frequency, such that wi /ω1 = wi β/2π.In the Ohmic limit ωD → ∞, the roots w1 , w2 , w3 reduce tow1 = γ/2 + iδ,w1 = γ/2 − iδ,withδ≡and (3.450) simplifies further toZωdamp =w3 = ωD − γ,qω 2 − γ 2 /4,(3.451)(3.452)1 ωΓ(w1 /ω1 )Γ(w2 /ω1 ).2π ω1(3.453)For vanishing friction, the roots w1 and w2 become simply w1 = iω, w2 = −iω, andthe formula9π(3.454)Γ(1 − z)Γ(z) =sin πzcan be used to calculateωππωΓ(iω/ω1 )Γ(−iω/ω1 ) = 1= 1,(3.455)ω sinh(πω/ω1)ω sinh(ωh̄/2kB T )showing that (3.450) goes properly over into the partition function (3.214) of theundamped harmonic oscillator.The free energy of the system isF (T ) = −kB T [log(ω/2πω1) − log Γ(ωD /ω1 )+ log Γ(w1 /ω1 ) + log Γ(w2 /ω1 ) + log Γ(w3 /ω1 )] .89(3.456)I.S.
Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.322.ibid., Formula 8.334.3.H. Kleinert, PATH INTEGRALS2693.15 Harmonic Oscillator in Ohmic Heat BathUsing the large-z behavior of log Γ(z)10log Γ(z) = z −1111log z − z + log 2π +−− O(1/z 5 ),2212z 360z 3(3.457)we find the free energy at low temperature11ω21++−F (T ) ∼ E0 −w1 w2 w1 w1 w2 w3!πγπ(kB T )2 = E0 − 2 (kB T )2 , (3.458)6h̄6ω h̄whereE0 = −h̄[w log(w1 /ωD ) + w2 log(w2 /ωD ) + w3 log(w3 /ωD )]2π 1(3.459)is the ground state energy.For small friction, this reduces toh̄ωγωDγ24ωE0 =+log−1+22πω16ωπωD!+ O(γ 3 ).(3.460)The T 2 -behavior of F (T ) in Eq.
(3.458) is typical for Ohmic dissipation.At zero temperature, the Matsubara frequencies ωm = 2πmkB T /h̄ move arbitrarily close together, so that Matsubara sums become integrals according to theruleZ ∞dωm1 X.(3.461)−−−→T →0h̄β m2π0Applying this limiting procedure to the logarithm of the product formula (3.445),the ground state energy can also be written as an integralh̄E0 =2πZ0∞2ω 3 + ωmωD + ωm (ω 2 + γωD ) + ωD ω 2dωm log m,2ωm(ωm + ωD )"#(3.462)which shows that the energy E0 increases with the friction coefficient γ.It is instructive to calculate the density of states defined in (1.579). Invertingthe Laplace transform (1.578), we have to evaluate1 Z η+i∞dβ eiεβ Zωdamp (β),ρ(ε) =2πi η−i∞(3.463)where η is an infinitesimally small positive number.
In the absence of friction, theP−βh̄ω(n+1/2)yieldsintegral over Zω (β) = ∞n=0 eρ(ε) =∞Xn=0δ(ε − (n + 1/2)h̄ω).(3.464)In the presence of friction, we expect the sharp δ-function spikes to be broadened.The calculation is done as follows: The vertical line of integration in the complex2703 External Sources, Correlations, and Perturbation TheoryFigure 3.5 Poles in complex β-plane of Fourier integral (3.463) coming from the Gammafunctions of (3.450)β-plane in (3.463) is moved all the way to the left, thereby picking up the polesof the Gamma functions which lie at negative integer values of wiβ/2π. From therepresentation of the Gamma function11Γ(z) =Z∞1dt tz−1e−t +(−1)nn=0 n!(z + n)∞X(3.465)we see the size of the residues. Thus we obtain the sumRn,1 =∞ X31 Xρ(ε) =Rn,i e−2πnε/wi ,ω n=1 i=1(3.466)ω (−1)n−1 Γ(−nw2 /w1 )Γ(−nw3 /w1 ),Γ(−nωD /w1 )w12 (n − 1)!(3.467)with analogous expressions for Rn,2 and Rn,3 .
The sum can be done numericallyand yields the curves shown in Fig. 3.6 for typical underdamped and overdampedsituations. There is an isolated δ-function at the ground state energy E0 of (3.459)which is not widened by the friction. Right above E0 , the curve continues from afinite value ρ(E0 + 0) = γπ/6ω 2 determined by the first expansion term in (3.458).3.16Harmonic Oscillator in Photon Heat BathIt is straightforward to extend this result to a photon bath where the spectral densityis given by (3.441) and (3.468) becomesZωdamp1011k T= Bh̄ω(∞Ym=1"23ωm+ ω 2 + ωmγ2ωm#)−1,(3.468)ibid., Formula 8.327.ibid., Formula 8.314.H.
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