Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 31
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With the matrix notation for the latticeoperator −∇∇ − ω 2 , we have to solve the multiple integral1FωN (tb , ta ) = q2πh̄i/MNYn,n0 =1Zdδxn∞−∞q2πh̄i/MNiM Xδxn [−∇∇ − ω 2]nn0 δxn0 .× exph̄ 2 n=1()(2.153)When going to the Fourier components of the paths, the integral factorizes in thesame way as for the free-particle expression (2.114). The only difference lies in theeigenvalues of the fluctuation operator which are nowΩm Ωm − ω 2 =122 [2 − 2 cos(νm )] − ω(2.154)instead of Ωm Ωm . For times tb , ta where all eigenvalues are positive (which willbe specified below) we obtain from the upper part of the Fresnel formula (1.334)directlyNY11qFωN (tb , ta ) = q.2πh̄i/M m=1 2 Ωm Ωm − 2 ω 2(2.155)H.
Kleinert, PATH INTEGRALS1152.3 Exact Solution for Harmonic OscillatorThe product of these eigenvalues is found by introducing an auxiliary frequency ω̃satisfyingωω̃≡.22sin(2.156)Then we decompose the product asNY222[ Ωm Ωm − ω ] ==N hYm=12i Ωm Ωm NYm=12 Ωm Ωmm=1N hYm=11 −Ni Ym=1sin2 ω̃2sin2 2(Nmπ+1)"2 Ωm Ωm − 2 ω 22 Ωm Ωm .#(2.157)The first factor is equal to (N + 1) by (2.118).
The second factor, the product ofthe ratios of the eigenvalues, is found from the standard formula7sin2 x 1 sin[2(N + 1)x]1 −=.2mπsin 2x(N + 1)sin 2(N +1)m=1NY(2.158)With x = ω̃/2, we arrive at the fluctuation determinantdetN (−2 ∇∇ − 2 ω 2 ) =NYm=1[2 Ωm Ωm − 2 ω 2] =sin ω̃(tb − ta ),sin ω̃(2.159)and the fluctuation factor is given byFωN (tb , ta )1=q2πih̄/Mssin ω̃, sin ω̃(tb − ta )where, as we have agreed earlier in Eq. (1.334),larger than zero.2.3.3√tb − ta < π/ω̃,(2.160)i means eiπ/4 , and tb − ta is alwaysThe iη -Prescription and Maslov-Morse IndexThe result (2.160) is initially valid only fortb − ta < π/ω̃.(2.161)In this time interval, all eigenvalues in the fluctuation determinant (2.159) are positive, and the upper version of the Fresnel formula (1.334) applies to each of theintegrals in (2.153) [this was assumed in deriving (2.155)].
If tb − ta grows largerthan π/ω̃, the smallest eigenvalue Ω1 Ω1 − ω 2 becomes negative and the integrationover the associated Fourier component has to be done according to the lower case ofthe Fresnel formula (1.334). The resulting amplitude carries an extra phase factor7I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 1.391.1.1162 Path Integrals — Elementary Properties and Simple Solutionse−iπ/2 and remains valid until tb − ta becomes larger than 2π/ω̃, where the secondeigenvalue becomes negative introducing a further phase factor e−iπ/2 .All phase factors emerge naturally if we associate with the oscillator frequency ωan infinitesimal negative imaginary part, replacing everywhere ω by ω − iη with aninfinitesimal η > 0. This is referred to as the iη-prescription.
Physically, it amountsto attaching an infinitesimal damping term to the oscillator, so that the amplitudebehaves like e−iωt−ηt and dies down to zero after a very long time (as opposed toan unphysical antidamping term which would make it diverge after a long time).Now, each time that tb − ta passes an integer multiple of π/ω̃, the square root ofsin ω̃(tb −ta ) in (2.160) passes a singularity in a specific way which ensures the properphase.8 With such an iη-prescription it will be superfluous to restrict tb − ta to therange (2.161).
Nevertheless it will sometimes be useful to exhibit the phase factorarising in this way in the fluctuation factor (2.160) for tb − ta > π/ω̃ by writingFωN (tb , ta )1=q2πih̄/Mssin ω̃e−iνπ/2 ,| sin ω̃(tb − ta )|(2.162)where ν is the number of zeros encountered in the denominator along the trajectory.This number is called the Maslov-Morse index of the trajectory9 .2.3.4Continuum LimitLet us now go to the continuum limit, → 0. Then the auxiliary frequency ω̃ tendsto ω and the fluctuation determinant becomes→0detN (−2 ∇∇ − 2 ω 2 ) −−−→sin ω(tb − ta ).ω(2.163)The fluctuation factor FωN (tb − ta ) goes over into1Fω (tb − ta ) = q2πih̄/Msω,sin ω(tb − ta )(2.164)with the phase for tb − ta > π/ω determined as above.In the limit ω → 0, both fluctuation factors agree, of course, with the free-particleresult (2.120).In the continuum limit, the ratios of eigenvalues in (2.157) can also be calculatedin the following simple way.
We perform the limit → 0 directly in each factor.This gives2 Ωm Ωm − 2 ω 22 Ωm Ωm=2 ω 21−2 − 2 cos(νm )8In the square root, we may equivalently assume tb − ta to carry a small negative imaginarypart. For a detailed discussion of the phases of the fluctuation factor in the literature, see Notesand References at the end of the chapter.9V.P. Maslov and M.V.
Fedoriuk, Semi-Classical Approximations in Quantum Mechanics, Reidel, Boston, 1981.H. Kleinert, PATH INTEGRALS1172.3 Exact Solution for Harmonic Oscillator→0−−−→ 1 −ω 2 (tb − ta )2.π 2 m2(2.165)As the number N goes to infinity we wind up with an infinite product of thesefactors. Using the well-known infinite-product formula for the sine function10sin x = xx21− 2 2 ,mπ∞Y!m=1(2.166)we find, with x = ω(tb − ta ),Ym2∞Y→0νmω(tb − ta )Ωm Ωm−−→,2 −22 =sin ω(tb − ta )Ωm Ωm − ωm=1 νm − ω(2.167)and obtain once more the fluctuation factor in the continuum (2.164).Multiplying the fluctuation factor with the classical amplitude, the time evolution amplitude of the linear oscillator in the continuum reads(xb tb |xa ta ) =ZDx(t) exp1= q2πih̄/Msih̄ZtbtadtM 2(ẋ − ω 2x2 )2ωsin ω(tb − ta )(2.168)()iMω× exp[(x2b + x2a ) cos ω(tb − ta ) − 2xb xa ] .2h̄ sin ω(tb − ta )The result can easily be extended to any number D of dimensions, where the actionisA=ZtbtadtM 2ẋ − ω 2 x2 .2(2.169)Being quadratic in x, the action is the sum of the actions of each component leadingto the factorized amplitude:(xb tb |xa ta ) =D Yi=1xib tb |xia ta(1=qD2πih̄/Msωsin ω(tb − ta )D)iMω[(x2 + x2a ) cos ω(tb − ta ) − 2xb xa ] ,× exp2h̄ sin ω(tb − ta ) b(2.170)where the phase of the second square root for tb − ta > π/ω is determined as in theone-dimensional case [see Eq.
(1.544)].10I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 1.431.1.1182.3.52 Path Integrals — Elementary Properties and Simple SolutionsUseful Fluctuation FormulasIt is worth realizing that when performing the continuum limit in the ratio of eigenvalues (2.167),we have actually calculated the ratio of the functional determinants of the differential operatorsdet(−∂t2 − ω 2 ).det(−∂t2 )(2.171)Indeed, the eigenvalues of −∂t2 in the space of real fluctuations vanishing at the endpoints aresimply2πm2,(2.172)νm =tb − taso that the ratio (2.171) is equal to the product∞2Yνm− ω2det(−∂t2 − ω 2 )=,2det(−∂t2 )νmm=1(2.173)which is the same as (2.167).
This observation should, however, not lead us to believe that theentire fluctuation factor Z tbZiM222Fω (tb − ta ) = Dδx expdt [(δ ẋ) − ω (δx) ](2.174)h̄ ta2could be calculated via the continuum determinant→011pFω (tb , ta ) −−−→ p2πh̄i/M det(−∂t2 − ω 2 )(false).(2.175)The product of eigenvalues in det(−∂t2 − ω 2 ) would be a strongly divergent expressiondet(−∂t2 − ω 2 ) ==∞Ym=12νm∞Y2(νm− ω2)(2.176)m=1∞ 2Yνmm=1∞ Yπ 2 m2sin ω(tb − ta )− ω2.=×22νm(t−t)ω(tb − ta )bam=1Only ratios of determinants −∇∇ − ω 2 with different ω’s can be replaced by their differentiallimits.
Then the common divergent factor in (2.176) cancels.Let us look at the origin of this strong divergence. The eigenvalues on the lattice and theircontinuum approximation start both out for small m as2Ωm Ωm ≈ νm≈π 2 m2.(tb − ta )2(2.177)2For large m ≤ N , the eigenvalues on the lattice saturate at Ωm Ωm → 2/2 , while the νm’s keepgrowing quadratically in m. This causes the divergence.The correct time-sliced formulas for the fluctuation factor of a harmonic oscillator is summarized by the following sequence of equations:"Z#NY1dδxniM TNpFω (tb − ta ) = pexpδx (−2 ∇∇ − 2 ω 2 )δxh̄ 22πh̄i/M n=12πh̄i/M=11pq,2πh̄i/M det (−2 ∇∇ − 2 ω 2 )N(2.178)H. Kleinert, PATH INTEGRALS1192.3 Exact Solution for Harmonic Oscillatorwhere in the first expression, the exponent is written in matrix notation with xT denoting the transposed vector x whose components are xn .
Taking out a free-particle determinant detN (−2 ∇∇),formula (2.132), leads to the ratio formulawhich yields1FωN (tb − ta ) = p2πh̄i(tb − ta )/MFωN (tb1− ta ) = p2πih̄/MdetN (−2 ∇∇ − 2 ω 2 )detN (−2 ∇∇)s−1/2,sin ω̃. sin ω̃(tb − ta )(2.179)(2.180)If we are only interested in the continuum limit, we may let go to zero on the right-hand side of(2.179) and evaluateFω (tb − ta )===−1/2det(−∂t2 − ω 2 )det(−∂t2 )−1/2∞ 2Y1νm − ω 2p2νm2πh̄i(tb − ta )/M m=1sω(tb − ta )1p.2πh̄i(tb − ta )/M sin ω(tb − ta )1p2πh̄i(tb − ta )/M(2.181)Let us calculate also here the time evolution amplitude in momentum space.
The Fouriertransform of initial and final positions of (2.170) [as in (2.133)] yieldsZZD−ipb xb /h̄(pb tb |pa ta ) = d xb edD xa eipa xa /h̄ (xb tb |xa ta )(2πh̄)D1= √D pD2πih̄M ω sin ω(tb − ta ) 21i(pb + p2a ) cos ω(tb − ta ) − 2pb pa .× exph̄ 2M ω sin ω(tb − ta )(2.182)The limit ω → 0 reduces to the free-particle expression (2.133), not quite as directly as in thex-space amplitude (2.170). Expanding the exponent 21(pb + pa ) cos ω(tb − ta ) − 2pb p2a2M ω sin ω(tb − ta )1 21222(p−p)−(p+p)[ω(t−t)]+...,=babaa2M ω 2 (tb − ta )2 band going to the limit ω → 0, the leading term in (2.182)i(2π)D12exp(pb − pa )pDh̄ 2M ω 2 (tb − ta )2πiω 2 (t − t )h̄Mb(2.183)(2.184)atends to (2πh̄)D δ (D) (pb − pa ) [recall (1.532)], while the second term in (2.183) yields a factor2e−ip (tb −ta )/2M , so that we recover indeed (2.133).2.3.6Oscillator Amplitude on Finite Time LatticeLet us calculate the exact time evolution amplitude for a finite number of time slices.
In contrast tothe free-particle case in Section 2.2.4, the oscillator amplitude is no longer equal to its continuum1202 Path Integrals — Elementary Properties and Simple Solutionslimit but -dependent. This will allow us to study some typical convergence properties of pathintegrals in the continuum limit.
Since the fluctuation factor was initially calculated at a finite in(2.162), we only need to find the classical action for finite . To maintain time reversal invarianceat any finite , we work with a slightly different sliced potential term in the action than before in(2.141), usingAN = N +1M X(∇xn )2 − ω 2 (x2n + x2n−1 )/2 ,2 n=1(2.185)or, written in another way,NANM X(∇xn )2 − ω 2 (x2n+1 + x2n )/2 .=2 n=0(2.186)This differs from the original time-sliced action (2.141) by having the potential ω 2 x2n replaced bythe more symmetric one ω 2 (x2n + x2n−1 )/2. The gradient term is the same in both cases and canbe rewritten, after a summation by parts, asN+1X(∇xn )2 =NXN X(∇xn )2 = xb ∇xb − xa ∇xa −xn ∇∇xn .n=0n=1(2.187)n=1This leads to a time-sliced actionAN =NMMM Xxb ∇xb − xa ∇xa − ω 2 (x2b + x2a ) − xn (∇∇ + ω 2 )xn .242 n=1(2.188)Since the variation of AN is performed at fixed endpoints xa and xb , the fluctuation factor is thesame as in (2.153).
The equation of motion on the sliced time axis is(∇∇ + ω 2 )xcl (t) = 0.(2.189)Here it is understood that the time variable takes only the discrete lattice values tn . The solutionof this difference equation with the initial and final values xa and xb , respectively, is given byxcl (t) =1[xb sin ω̃(t − ta ) + xa sin ω̃(tb − t)] ,sin ω̃(tb − ta )(2.190)where ω̃ is the auxiliary frequency introduced in (2.156).
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