Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 37
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As in Section 2.3.6, we consider the amplitudeZZRDp − ττb dτ [−ipẋ+p2 /2M+MΩ2 (τ )x2 /2]/h̄ae(xb τb |xa τa ) =D0 x2πh̄H. Kleinert, PATH INTEGRALS1492.11 Time-Dependent Harmonic Potential=ZDxe−R τbτadτ [M ẋ2 +Ω2 (τ )x2 ]/2h̄.(2.406)The time-sliced fluctuation factor is [compare (2.195)]F N (τa − τb ) = detN +1 [−2 ∇∇ + Ω2 (τ )]−1/2 ,(2.407)with the continuum limitF (τa − τb ) =kB Th̄det(−∂τ2 + Ω2 (τ ))det0 (−∂τ2 )−1/2.(2.408)Actually, in the thermal case it is preferable to use the oscillator result for normalizing thefluctuation factor, rather than the free-particle result, and to work with the formula−1/21det(−∂τ2 + Ω2 (τ ))F (τb , τa ) =.(2.409)2 sinh(βh̄ω/2)det(−∂τ2 + ω 2 )This has the advantage that the determinant in the denominator contains no zero eigenvalue whichwould require a special treatment as in (2.402); the operator −∂τ2 + ω 2 is positive.As in the quantum-mechanical case, the spectrum of eigenvalues is not known for general Ω(τ ).It is, however, possible to find a differential equation for the entire determinant, analogous to theGelfand-Yaglom formula (2.202), with the initial condition (2.207), although the derivation is nowmuch more tedious.
The origin of the additional difficulties lies in the periodic boundary conditionwhich introduces additional nonvanishing elements −1 in the upper right and lower left corners ofthe matrix −2 ∇∇ [compare (2.97)]:2 −1 0... 00 −1 −1 2 −1... 000 .... .2− ∇∇ = .(2.410). 000 . . . −1 2 −1 −1 00. . . 0 −1 2To better understand the relation with the previous result we shall replace the corner elements−1 by −α which can be set equal to zero at the end, for a comparison. Adding to −2 ∇∇ atime-dependent frequency matrix we then consider the fluctuation matrix2 + 2 Ω2N +1−10 ...
0−α−12 + 2 Ω2N −1 . . . 00−2 ∇∇ + 2 Ω2 = .......−α0. . . −1 2 + 2 Ω210(2.411)Let us denote the determinant of this (N + 1) × (N + 1) matrix by D̃N +1 . Expanding it along thefirst column, it is found to satisfy the equationD̃N +1 = (2 + 2 Ω2N +1 )2 + 2 Ω2N..×detN .0+detN (2.412)−1 000...00.... .
. −1 2 + 2 Ω21−1−10...02 + 2 Ω2N −1−10−12 + 2 Ω2N −20000 ...0 ...−1 . . .0000−α00.... . . −1 2 + 2 Ω211502 Path Integrals — Elementary Properties and Simple SolutionsN +1+(−1)αdetN −12 + 2 Ω2N−1...0−12 + 2 Ω2N −1000 ...0 ...−1 . . .0−α00...000. . . 2 + 2 Ω22−1.The first determinant was encountered before in Eq. (2.197) (except that there it appeared with−2 Ω2 instead of 2 Ω2 ). There it was denoted by DN , satisfying the difference equation(2.413)−2 ∇∇ + 2 Ω2N +1 DN = 0,with the initial conditionsD1= 2 + 2 Ω21 ,D2= (2 + 2 Ω21 )(2 + 2 Ω22 ) − 1.(2.414)The second determinant in (2.412) can be expanded with respect to its first column yielding−DN −1 − α.(2.415)The third determinant is more involved.
When expanded along the first column it gives(−1)N 1 + (2 + 2 Ω2N )HN −1 − HN −2 ,(2.416)with the (N − 1) × (N − 1) determinantHN −1 ≡ (−1)N −1(2.417)0 2 + 2 Ω2N −1−1×detN −1 ...00−12 + 2 Ω2N −200 ...0 ...−1 . . .0000−α00...000. . . −1 2 + 2 Ω22−1.By expanding this along the first column, we find that HN satisfies the same difference equationas DN :(−2 ∇∇ + 2 Ω2N +1 )HN = 0.(2.418)However, the initial conditions for HN are different:0−α H2 = = α(2 + 2 Ω22 ),2 + 2 Ω22 −1 00−αH3 = − 2 22+Ω−103−12 + 2 Ω22 −1 = α (2 + 2 Ω22 )(2 + 2 Ω23 ) − 1 .(2.419)(2.420)They show that HN is in fact equal to αDN −1 , provided we shift Ω2N by one lattice unit upwardsto Ω2N +1 .
Let us indicate this by a superscript +, i.e., we write+HN = αDN−1 .(2.421)Thus we arrive at the equationD̃N +1= (2 + 2 Ω2N )DN − DN −1 − α++−α[1 + (2 + 2 Ω2N )αDN−2 − αDN −3 ].(2.422)H. Kleinert, PATH INTEGRALS1512.11 Time-Dependent Harmonic Potential+Using the difference equations for DN and DN, this can be brought to the convenient form+D̃N +1 = DN +1 − α2 DN−1 − 2α.(2.423)For quantum-mechanical fluctuations with α = 0, this reduces to the earlier result in Section 2.3.6.For periodic fluctuations with α = 1, the result is+D̃N +1 = DN +1 − DN−1 − 2.(2.424)+In the continuum limit, DN +1 − DN−1 tends towards 2Ḋren , where Dren (τ ) = Da (t) is theimaginary-time version of the Gelfand-Yaglom function in Section 2.4 solving the homogenousdifferential equation (2.208), with the initial conditions (2.206) and (2.207), or Eqs. (2.221).
Thecorresponding properties are now: 2−∂τ + Ω2 (τ ) Dren (τ ) = 0, Dren (τ ) = 0,Ḋren (τ ) = 1.(2.425)In terms of Dren (τ ), the determinant is given by the Gelfand-Yaglom-like formula→0−−→ 2[Ḋren (h̄β) − 1],det(−2 ∇∇ + Ω2 )T −(2.426)and the partition function reads1ZΩ = r hi.2 Ḋren (h̄β) − 1(2.427)The result may be checked by going back to the amplitude (xb tb |xa ta ) of Eq. (2.255), continuingit to imaginary times t = iτ , setting xb = xa = x, and integrating over all x. The result is1ZΩ = q,2 Ḋa (tb ) − 1tb = ih̄β,(2.428)in agreement with (2.427).As an example, take the harmonic oscillator for which the solution of (2.425) isDren (τ ) =1sinh ωτω(2.429)[the analytically continued (2.209)]. Then2[Ḋren (τ ) − 1] = 2(cosh βh̄ω − 1) = 4 sinh2 (βh̄ω/2),(2.430)and we find the correct partition function:Zω==no−1/2 2[Ḋren (τ ) − 1]1.2 sinh(βh̄ω/2)τ =h̄β(2.431)On a sliced imaginary-time axis, the case of a constant frequency Ω2 ≡ ω 2 is solved as follows.From Eq.
(2.203) we take the ordinary Gelfand-Yaglom function DN , and continue it to Euclideanω̃e , yielding the imaginary-time versionDN =sinh(N + 1)ω˜e .sinh ω˜e (2.432)1522 Path Integrals — Elementary Properties and Simple Solutions+Then we use formula (2.424), which simplifies for a constant Ω2 ≡ ω 2 for which DN−1 = DN −1 ,and calculateD̃N +11[sinh(N + 2)ω˜e − sinh N ω˜e ] − 2sinh ω˜e = 2 [cosh(N + 1)ω˜e − 1] = 4 sinh2 [(N + 1)ω˜e /2].=(2.433)Inserting this into Eq. (2.378) yields the partition function11=Zω = q,2 sinh(h̄ω˜e β/2)D̃N +1(2.434)in agreement with (2.395).2.12Functional Measure in Fourier SpaceThere exists an alternative definition for the quantum-statistical path integral whichis useful for some applications (for example in Section 2.13 and in Chapter 5).
Thelimiting product formula (2.402) suggests that instead of summing over all zigzagconfigurations of paths on a sliced time axis, a path integral may be defined withthe help of the Fourier components of the paths on a continuous time axis. As in(2.372), but with a slightly different normalization of the coefficients, we expandthese paths here asx(τ ) = x0 + η(τ ) ≡ x0 +∞ Xxm eiωm τ + cc ,m=1x0 = real,x−m ≡ x∗m .(2.435)Note that the temporal integral over the time-dependent fluctuations η(τ ) is zero,dτ η(τ ) = 0, so that the zero-frequency component x0 is the temporal average0of the fluctuating paths:R h̄/kB Tx0 = x̄ ≡kB T Z h̄/kB Tdτ x(τ ).h̄ 0(2.436)In contrast to (2.372) which was valid on a sliced time axis and was thereforesubject to a restriction on the range of the m-sum, the present sum is unrestrictedand runs over all Matsubara frequencies ωm = 2πmkB T /h̄ = 2πm/h̄β.
In terms ofxm , the Euclidean action of the linear oscillator isAeM h̄/kB Tdτ (ẋ2 + ω 2 x2 )=2 0"#∞Mh̄ ω 2 2 X222=x +(ω + ω )|xm | .kB T 2 0 m=1 mZ(2.437)The integration variables of the time-sliced path integral were transformedto theQ R∞Fourier components xm in Eq. (2.380). The product of integrals n −∞dx(τn )H. Kleinert, PATH INTEGRALS1532.12 Functional Measure in Fourier Spaceturned into the product (2.382) of integrals over real and imaginary parts of xm . Inthe continuum limit, the result is∞Z−∞dx0∞ ZY∞m=1 −∞d Re xmZ∞−∞d Im xm .(2.438)Placing the exponential e−Ae /h̄ with the frequency sum (2.437) into the integrand,2the product of Gaussian integrals renders a product of inverse eigenvalues (ωm+ω 2 )−1for m = 1, .
. . , ∞, with some infinite factor. This may be determined by comparisonwith the known continuous result (2.402) for the harmonic partition function. Theinfinity is of the type encountered in Eq. (2.176), and must be divided out of themeasure (2.438). The correct result (2.400) is obtained from the following measureof integration in Fourier spaceIDx ≡Z"Z#∞∞ Z ∞ d Re x d Im xdx0 Ymm.2le (h̄β) m=1 −∞ −∞ πkB T /Mωm∞−∞(2.439)2The divergences in the product over the factors (ωm+ ω 2 )−1 discussed after2Eq.
(2.176) are canceled by the factors ωmin the measure. It will be convenient tointroduce a short-hand notation for the measure on the right-hand side, writing itasIDx ≡Z∞−∞dx0 I 0D x.le (h̄β)(2.440)The denominator of the x0 -integral is the length scale le (h̄β) associated with βdefined in Eq. (2.345).Then we calculateZωx0 ≡I0−Ae /h̄D xe="∞ ZYm=1∞−∞#d Re xm d Im xm −M h̄[ω2 x20 /2+P∞ (ωm2 +ω 2 )|x |2 /k Tm ] Bm=1e2−∞ πkB T /MωmZ∞−M ω 2 x20 /2kB T=e∞Ym=1"2ωm+ ω22ωm#−1.(2.441)The final integral over the zero-frequency component x0 yields the partition functionZω =I−Ae /h̄Dx e=Z∞−∞2∞ωm+ ω2k T Ydx0 x0Zω = B2le (h̄β)h̄ω m=1ωm"#−1,(2.442)as in (2.402).The same measure can be used for the more general amplitude (2.406), as isobvious from (2.408).
With the predominance of the kinetic term in the measureof path integrals [the divergencies discussed after (2.176) stem only from it], it caneasily be shown that the same measure is applicable to any system with the standardkinetic term.1542 Path Integrals — Elementary Properties and Simple SolutionsIt is also possible to find a Fourier decomposition of the paths and an associatedintegration measure for the open-end partition function in Eq.
(2.405). We begin byconsidering the slightly reduced set of all paths satisfying the Neumann boundaryconditionsẋ(τa ) = va = 0,ẋ(τb ) = vb = 0.(2.443)They have the Fourier expansionx(τ ) = x0 + η(τ ) = x0 +∞Xn=1xn cos νn (τ − τa ),νn = nπ/β.(2.444)The frequencies νn are the Euclidean version of the frequencies (3.64) for Dirichletboundary conditions. Let us calculate the partition function for such paths byanalogy with the above periodic case by a Fourier decomposition of the actionMAe =2Zh̄/kB T0∞Mh̄ ω 2 2 1 Xdτ (ẋ + ω x ) =(ν 2 + ω 2 )x2n ,x0 +kB T 22 n=1 n2"2 2#(2.445)and of the measureIDx ≡Z∞≡Z∞−∞−∞∞ Z ∞ Z ∞d xndx0 Yle (h̄β) n=1 −∞ −∞ πkB T /2Mνn2Idx0D 0 x.le (h̄β)"#(2.446)We now perform the path integral over all fluctuations at fixed x0 as in (2.441):ZωN,x0 ≡I0−Ae /h̄D xe="∞ ZYn=1∞−∞Z∞−∞−M ω 2 x20 /2kB T=e#P∞ 2 2d xn−M h̄[ω 2 x20 /2+ n=1 (νn+ω )|xn |2 ]/kB Te2πkB T /2Mνn∞Yn=1"νn2 + ω 2ωn2#−1.(2.447)Using the product formula (2.176), this becomesZωN,x0=sωh̄βMexp −β ω 2x20 .sinh ωh̄β2(2.448)The final integral over the zero-frequency component x0 yields the partition functionZωN1=le (h̄β)s2πh̄1√.Mω sinh ωh̄β(2.449)We have replaced the denominator in the prefactor 1/le (h̄β) by the length scale1/le (h̄β) of Eq.
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