Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 35
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(2.311)–(2.313). matrix equations.2.8Path Integrals and Quantum StatisticsThe path integral approach is useful to also understand the thermal equilibriumproperties of a system. We assume the system to have a time-independent Hamiltonian and to be in contact with a reservoir of temperature T . As explained in Section 1.7, the bulk thermodynamic quantities can be determined from the quantumstatistical partition functionZ = Tr e−Ĥ/kB T =Xe−En /kB T .(2.321)nH. Kleinert, PATH INTEGRALS1372.8 Path Integrals and Quantum StatisticsThis, in turn, may be viewed as an analytic continuation of the quantum-mechanicalpartition function(2.322)ZQM = Tr e−i(tb −ta )Ĥ/h̄to the imaginary timetb − ta = −ih̄≡ −ih̄β.kB T(2.323)In the local particle basis |xi, the quantum-mechanical trace corresponds to anintegral over all positions so that the quantum-statistical partition function can beobtained by integrating the time evolution amplitude over xb = xa and evaluatingit at the analytically continued time:Z≡Z∞−∞dx z(x) =Z∞−β Ĥ−∞dx hx|e|xi =Z∞−∞dx (x tb |x ta )|tb −ta =−ih̄β .(2.324)The diagonal elementsz(x) ≡ hx|e−β Ĥ |xi = (x tb |x ta )|tb −ta =−ih̄β(2.325)play the role of a partition function density.
For a harmonic oscillator, this quantityhas the explicit form [recall (2.168)]1zω (x) = q2πh̄/Ms()ωMωh̄βω 2exp −tanhx .sinh h̄βωh̄2(2.326)By splitting the Boltzmann factor e−β Ĥ into a product of N + 1 factors e−Ĥ/h̄with = h̄/kB T (N + 1), we can derive for Z a similar path integral representation just as for the corresponding quantum-mechanical partition function in (2.40),(2.46):Z≡NY+1 Z ∞n=1−∞dxn(2.327)× hxN +1 |e−Ĥ/h̄ |xN ihxN |e−Ĥ/h̄ |xN −1 i × . . . × hx2 |e−Ĥ/h̄ |x1 ihx1 |e−Ĥ/h̄ |xN +1 i.As in the quantum-mechanical case, the matrix elements hxn |e−Ĥ/h̄ |xn−1 i are reexpressed in the form−Ĥ/h̄hxn |e|xn−1 i ≈Z∞−∞dpn ipn (xn −xn−1 )/h̄−H(pn ,xn )/h̄e,2πh̄(2.328)with the only difference that there is now no imaginary factor i in front of theHamiltonian. The product (2.327) can thus be written asZ≈NY+1n=1"Z∞−∞dxnZ∞−∞#1dpnexp − AN,2πh̄h̄ e(2.329)1382 Path Integrals — Elementary Properties and Simple Solutionswhere ANe denotes the sumANe=N+1 hXn=1i−ipn (xn − xn−1 ) + H(pn , xn ) .(2.330)In the continuum limit → 0, the sum goes over into the integralAe [p, x] =Z0h̄βdτ [−ip(τ )ẋ(τ ) + H(p(τ ), x(τ ))],(2.331)and the partition function is given by the path integralZZDp −Ae [p,x]/h̄e.(2.332)Z = Dx2πh̄In this expression, p(τ ), x(τ ) may be considered as paths running along an “imaginary time axis” τ = it.
The expression Ae [p, x] is very similar to the mechanicalcanonical action (2.27). Since it governs the quantum-statistical path integrals itis called quantum-statistical action or Euclidean action, indicated by the subscripte. The name alludes to the fact that a D-dimensional Euclidean space extendedby an imaginary-time axis τ = it has the same geometric properties as a D + 1dimensional Euclidean space. For instance, a four-vector in a Minkowski spacetimehas a square length dx2 = −(cdt)2 + (dx)2 . Continued to an imaginary time, thisbecomes dx2 = (cdτ )2 + (dx)2 which is the square distance in a Euclidean fourdimensional space with four-vectors (cτ, x).Just as in the path integralH for thequantum-mechanical partition function (2.46),Rthe measure of integration Dx Dp/2πh̄ in the quantum-statistical expression(2.332) is automatically symmetric in all p’s and x’s:IDxZINY+1 ZZ ∞DpDp Zdxn dpn=.Dx =2πh̄2πh̄−∞ 2πh̄n=1(2.333)The symmetry is of course due to the trace integration over all initial ≡ final positions.Most remarks made in connection with Eq.
(2.46) carry over to the presentcase. The above path integral (2.332) is a natural extension of the rules of classical statistical mechanics. According to these, each cell in phase space dxdp/h isoccupied with equal statistical weight, with the probability factor e−E/kB T . In quantum statistics, the paths of all particles fluctuate evenly over the cells in path phaseQspace n dx(τn )dp(τn )/h (τn ≡ n), each path carrying a probability factor e−Ae /h̄involving the Euclidean action of the system.2.9Density MatrixThe partition function does not determine any local thermodynamic quantities. Important local information resides in the thermal analog of the time evolution amplitude hxb |e−Ĥ/kB T |xa i. Consider, for instance, the diagonal elements of this amplituderenormalized by a factor Z −1 :ρ(xa ) ≡ Z −1 hxa |e−Ĥ/kB T |xa i.(2.334)H.
Kleinert, PATH INTEGRALS1392.9 Density MatrixIt determines the thermal average of the particle density of a quantum-statisticalsystem. Due to (2.327), the factor Z −1 makes the spatial integral over ρ equal tounity:Z ∞dx ρ(x) = 1.(2.335)−∞By inserting into (2.334) a complete set of eigenfunctions ψn (x) of the Hamiltonianoperator Ĥ, we find the spectral decompositionρ(xa ) =Xn|ψn (xa )|2 e−βEn.Xe−βEn .(2.336)nSince |ψn (xa )|2 is the probability distribution of the system in the eigenstate |ni,Pwhile the ratio e−βEn / n e−βEn is the normalized probability to encounter the system in the state |ni, the quantity ρ(xa ) represents the normalized average particledensity in space as a function of temperature.Note the limiting properties of ρ(xa ).
In the limit T → 0, only the lowest energystate survives and ρ(xa ) tends towards the particle distribution in the ground stateT →0ρ(xa ) −−−→ |ψ0 (xa )|2 .(2.337)In the opposite limit of high temperatures, quantum effects are expected to becomeirrelevant and the partition function should converge to the classical expression(1.536) which is the integral over the phase space of the Boltzmann distributionT →∞Z−−−→ Zcl =Z∞−∞dxZ∞−∞dp −H(p,x)/kB Te.2πh̄(2.338)We therefore expect the large-T limit of ρ(x) to be equal to the classical particledistributionT →∞Zcl−1ρ(x) −−−→ ρcl (x) =Z∞−∞dp −H(p,x)/kB Te.2πh̄(2.339)Within the path integral approach, this limit will be discussed in more detail inSection 2.13. At this place we roughly argue as follows: When going in the originaltime-sliced path integral (2.327) to large T , i.e., small τb − τa = h̄/kB T , we maykeep only a single time slice and writeZ≈Z∞−∞dx hx|e−Ĥ/h̄ |xi,(2.340)withhx|e−Ĥ |xi ≈Z∞−∞dpn −H(pn ,x)/h̄e.2πh̄(2.341)After substituting = τb − τa this gives directly (2.339).
Physically speaking, thepath has at high temperatures “no (imaginary) time” to fluctuate, and only oneterm in the product of integrals needs to be considered.1402 Path Integrals — Elementary Properties and Simple SolutionsIf H(p, x) has the standard formH(p, x) =p2+ V (x),2M(2.342)the momentum integral is Gaussian in p and can be done using the formulaZ∞−∞1dp −ap2 /2h̄e=√.2πh̄2πh̄a(2.343)This leads to the pure x-integral for the classical partition functionZcl =Zdx∞−∞q2πh̄2 /MkB Te−V (x)/kB T =Z∞−∞dx −βV (x)e.le (h̄β)(2.344)In the second expression we have introduced the lengthq2πh̄2 β/M.le (h̄β) ≡(2.345)It is the thermal (or Euclidean) analog of the characteristic length l(tb − ta ) introduced before in (2.121). It is called the de Broglie wavelength associated with thetemperature T = 1/kB β, or short thermal de Broglie wavelength.Omitting the x-integration in (2.344) renders the large-T limit ρ(x), the classicalparticle distributionT →∞ρ(x) −−−→ ρcl (x) = Zcl−11e−V̄ (x) .le (h̄β)(2.346)For a free particle, the integral over x in (2.344) diverges.
If we imagine the lengthof the x-axis to be very large but finite, say equal to L, the partition function isequal toLZcl =.(2.347)le (h̄β)In D dimensions, this becomesZcl =VD,Dle (h̄β)(2.348)where VD is the volume of the D-dimensional system. For a harmonic oscillatorwith potential Mω 2 x2 /2, the integral over x in (2.344) is finite and yields, in theD-dimensional generalizationlDZcl = D ω ,(2.349)l (h̄β)wherelω ≡s2πβMω 2(2.350)H.
Kleinert, PATH INTEGRALS1412.9 Density Matrixis the classical length scale defined by the frequency of the harmonic oscillator. Itis related to the quantum-mechanical one λω of Eq. (2.296) bylω le (h̄β) = 2π λ2ω .(2.351)Thus we obtain the mnemonic rule for going over from the partition function of aharmonic oscillator to that of a free particle: we must simply replacelω −−−→ L,(2.352)ω→0ors1βM−−−→L.ω ω→02πThe real-time version of this is, of course,1−−−→ω ω→0s(2.353)(tb − ta )ML.2πh̄(2.354)Let us write down a path integral representation for ρ(x).
Omitting in (2.332)the final trace integration over xb ≡ xa and normalizing the expression by a factorZ −1 , we obtainρ(xa ) = Z −1Z= Z −1Zx(h̄β)=xbx(0)=xax(h̄β)=xbx(0)=xaD0xZDp −Ae [p,x]/h̄e2πh̄Dxe−Ae [x]/h̄ .(2.355)The thermal equilibrium expectation of an arbitrary Hermitian operator Ô isgiven byX −βEnhÔiT ≡ Z −1ehn|Ô|ni.(2.356)nIn the local basis |xi, this becomeshÔiT = Z−1ZZ∞−∞dxb dxa hxb |e−β Ĥ |xa ihxa |Ô|xb i.(2.357)An arbitrary function of the position operator x̂ has the expectationhf (x̂)iT = Z −1ZZ∞−∞dxb dxa hxb |e−β Ĥ |xa iδ(xb − xa )f (xa ) =Zdxρ(x)f (x).
(2.358)The particle density ρ(xa ) determines the thermal averages of local observables.If f depends also on the momentum operator p̂, then the off-diagonal matrixelements hxb |e−β Ĥ |xa i are also needed. They are contained in the density matrixintroduced for pure quantum systems in Eq. (1.221), and reads now in a thermalensemble of temperature T :ρ(xb , xa ) ≡ Z −1 hxb |e−β Ĥ |xa i,(2.359)1422 Path Integrals — Elementary Properties and Simple Solutionswhose diagonal values coincide with the above particle density ρ(xa ).It is useful to keep the analogy between quantum mechanics and quantum statistics as close as possible and to introduce the time translation operator along theimaginary time axisÛe (τb , τa ) ≡ e−(τb −τa )Ĥ/h̄ ,τb > τa ,(2.360)defining its local matrix elements as imaginary or Euclidean time evolution amplitudes(xb τb |xa τa ) ≡ hxb |Ûe (τb , τa )|xa i,τb > τa .(2.361)As in the real-time case, we shall only consider the causal time-ordering τb > τa .Otherwise the partition function and the density matrix do not exist in systems withenergies up to infinity.
Given the imaginary-time amplitudes, the partition functionis found by integrating over the diagonal elementsZ=Z∞−∞dx(x h̄β|x 0),(2.362)and the density matrixρ(xb , xa ) = Z −1 (xb h̄β|xa 0).(2.363)For the sake of generality we may sometimes also consider the imaginary-timeevolution operators for time-dependent Hamiltonians and the associated amplitudes.They are obtained by time-slicing the local matrix elements of the operator1 Z τbdτ Ĥ(−iτ ) .Û(τb , τa ) = Tτ exp −h̄ τa(2.364)Here Tτ is an ordering operator along the imaginary-time axis.It must be emphasized that the usefulness of the operator (2.364) in describingthermodynamic phenomena is restricted to the Hamiltonian operator Ĥ(t) depending very weakly on the physical time t.
The system has to remain close to equilibriumat all times. This is the range of validity of the so-called linear response theory (seeChapter 18 for more details).The imaginary-time evolution amplitude (2.361) has a path integral representation which is obtained by dropping the final integration in (2.329) and relaxing thecondition xb = xa :(xb τb |xa τa ) ≈N ZYn=1∞−∞dxn" NY+1 Zn=1∞−∞#dpnexp −AN/h̄.e2πh̄(2.365)The time-sliced Euclidean action isANe =N+1 hXn=1−ipn (xn − xn−1 ) + H(pn , xn , τn )i(2.366)H.
Kleinert, PATH INTEGRALS1432.9 Density Matrix(we have omitted the factor −i in the τ -argument of H). In the continuum limitthis is written as a path integralZ(xb τb |xa τa ) =0Dx1Dpexp − Ae [p, x]2πh̄h̄Z(2.367)[by analogy with (2.332)]. For a Hamiltonian of the standard form (2.7),p2+ V (x, τ ),2MH(p, x, τ ) =with a smooth potential V (x, τ ), the momenta can be integrated out, just as in(2.51), and the Euclidean version of the pure x-space path integral (2.52) leads to(2.53):(xb τb |xa τa ) =Z(1 Z h̄βMDx exp −(∂τ x)2 + V (x, τ )dτh̄ 02NY1≈ q2πh̄/Mn=1Z−∞+11 NXM× exp − h̄ n=1 2(dxn∞"q2πβ/M)xn − xn−1(2.368)2+ V (xn , τn )#).From this we calculate the quantum-statistical partition functionZ =Z=Z∞−∞dx (x h̄β|x 0)dxZx(h̄β)=xx(0)=x−Ae [x]/h̄Dx e=IDx e−Ae [x]/h̄ ,(2.369)where Ae [x] is the Euclidean version of the Lagrangian actionAe [x] =ZτbτaM 02dτx + V (x, τ ) .2(2.370)The prime denotes differentiation with respect to the imaginary Htime.
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