Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 48
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This freedom willbe removed below by imposing appropriate boundary conditions.In the fluctuation action (3.12), we now perform a quadratic completion by ashift of δx(t) to1δx̃(t) ≡ δx(t) +MZtbtadt0 Gω2 (t, t0 )j(t0 ).(3.17)Then the action becomes quadratic in both δx̃ and j:Afl =ZtbtadtZtbtaM1dtδx̃(t)Dω2 (t, t0 )δx̃(t0 ) −j(t)Gω2 (t, t0 )j(t0 ) .22M0(3.18)H.
Kleinert, PATH INTEGRALS2113.1 External SourcesThe Green function obeys the same boundary condition as the fluctuations δx(t):0Gω2 (t, t ) = 0 for(t = tb ,t0 arbitrary,t arbitrary, t0 = ta .(3.19)Thus, the shifted fluctuations δx̃(t) of (3.17) also vanish at the ends and run throughthe same functional space as the original δx(t). The measure of path integrationRDδx(t) is obviously unchanged by the simple shift (3.17).
Hence the path integralRDδx̃ over eiAfl /h̄ with the action (3.18) gives, via the first term in Afl , the harmonicfluctuation factor Fω (tb − ta ) calculated in (2.164):1Fω (tb − ta ) = q2πih̄/Msω.sin ω(tb − ta )(3.20)The source part in (3.18) contributes only a trivial exponential factorFj,fli= expA,h̄ j,fl(3.21)whose exponent is quadratic in j(t):Aj,fl = −1 Z tb Z tb 0dt j(t)Gω2 (t, t0 )j(t0 ).dt2M tata(3.22)The total time evolution amplitude in the presence of a source term can thereforebe written as the product(xb tb |xa ta )jω = (xb tb |xa ta )ω Fj,cl Fj,fl,(3.23)where (xb tb |xa t)ω is the source-free time evolution amplitude(i/h̄)Aω,cl(xb tb |xa ta )ω = e(1Fω (tb − ta ) = q2πih̄/Msωsin ω(tb − ta ))iMω× exp[(x2 + x2a ) cos ω(tb − ta ) − 2xb xa ] ,2h̄ sin ω(tb − ta ) b(3.24)and Fj,cl is an amplitude containing the classical action (3.11):Fj,cl = e(i/h̄)Aj,cl(i1= exph̄ sin ω(tb − ta )Ztbta)dt[xa sin ω(tb − t) + xb sin ω(t − ta )] j(t) .
(3.25)To complete the result we need to know the Green function Gω2 (t, t0 ) explicitly,which will be calculated in the next section.2123.23 External Sources, Correlations, and Perturbation TheoryGreen Function of Harmonic OscillatorAccording to Eq. (3.16), the Green function in Eq. (3.22) is obtained by invertingthe second-order differential operator −∂t2 − ω 2 :Gω2 (t, t0 ) = (−∂t2 − ω 2 )−1 δ(t − t0 ),t, t0 ∈ (ta , tb ).(3.26)As remarked above, this function is defined only up to solutions of the homogeneous differential equation associated with the operator −∂t2 − ω 2. The boundaryconditions removing this ambiguity are the same as for the fluctuations δx(t), i.e.,Gω2 (t, t0 ) vanishes if either t or t0 or both hit an endpoint ta or tb (Dirichlet boundarycondition).
The Green function is symmetric in t and t0 . For the sake of generality,we shall find the Green function also for the more general differential equation withtime-dependent frequency,[−∂t2 − Ω2 (t)]GΩ2 (t, t0 ) = δ(t − t0 ),(3.27)with the same boundary conditions.There are several ways of calculating this explicitly.3.2.1Wronski ConstructionThe simplest way proceeds via the so-called Wronski construction, which is basedon the following observation. For different time arguments, t > t0 or t < t0 , theGreen function GΩ2 (t, t0 ) has to solve the homogeneous differential equations(−∂t2 − ω 2 )GΩ2 (t, t0 ) = 0,(−∂t20 − ω 2 )GΩ2 (t, t0 ) = 0.(3.28)It must therefore be a linear combination of two independent solutions of the homogeneous differential equation in t as well as in t0 , and it must satisfy the Dirichletboundary condition of vanishing at the respective endpoints.Constant FrequencyIf Ω2 (t) ≡ ω 2, this implies that for t > t0 , Gω2 (t, t0 ) must be proportional to sin ω(tb −t) as well as to sin ω(t0 − ta ), leaving only the solutionGω2 (t, t0 ) = C sin ω(tb − t) sin ω(t0 − ta ),t > t0 .(3.29)t < t0 .(3.30)For t < t0 , we obtain similarlyGω2 (t, t0 ) = C sin ω(tb − t0 ) sin ω(t − ta ),The two cases can be written as a single expressionGω2 (t, t0 ) = C sin ω(tb − t> ) sin ω(t< − ta ),(3.31)H.
Kleinert, PATH INTEGRALS3.2 Green Function of Harmonic Oscillator213where the symbols t> and t< denote the larger and the smaller of the times t andt0 , respectively. The unknown constant C is fixed by considering coincident timest = t0 . There, the time derivative of Gω2 (t, t0 ) must have a discontinuity which givesrise to the δ-function in (3.15). For t > t0 , the derivative of (3.29) is∂t Gω2 (t, t0 ) = −Cω cos ω(tb − t) sin ω(t0 − ta ),(3.32)whereas for t < t0∂t Gω2 (t, t0 ) = Cω sin ω(tb − t0 ) cos ω(t − ta ).(3.33)At t = t0 we find the discontinuity∂t Gω2 (t, t0 )|t=t0 + − ∂t Gω2 (t, t0 )|t=t0 − = −Cω sin ω(tb − ta ).(3.34)Hence −∂t2 Gω2 (t, t0 ) is proportional to a δ-function:−∂t2 Gω2 (t, t0 ) = Cω sin ω(tb − ta )δ(t − t0 ).(3.35)By normalizing the prefactor to unity, we fix C and find the desired Green function:Gω2 (t, t0 ) =sin ω(tb − t> ) sin ω(t< − ta ).ω sin ω(tb − ta )(3.36)It exists only if tb − ta is not equal to an integer multiple of π/ω.
This restrictionwas encountered before in the amplitude without external sources; its meaning wasdiscussed in the two paragraphs following Eq. (2.161).The constant in the denominator of (3.36) is the Wronski determinant (or Wronskian) of the two solutions ξ(t) = sin ω(tb − t) and η(t) = sin ω(t − ta ) which wasintroduced in (2.215):˙W [ξ(t), η(t)] ≡ ξ(t)η̇(t) − ξ(t)η(t).(3.37)An alternative expression for (3.36) isGω2 (t, t0 ) =− cos ω(tb − ta − |t − t0 |) + cos ω(tb + ta − t − t0 ).2ω sin ω(tb − ta )(3.38)In the limit ω → 0 we obtain the free-particle Green function1(t − t> )(t< − ta )(tb − ta ) b111=−tt0 − (tb − ta )|t − t0 | + (ta + tb )(t + t0 ) − ta tb .
(3.39)tb − ta22G0 (t, t0 ) =2143 External Sources, Correlations, and Perturbation TheoryTime-Dependent FrequencyIt is just as easy to find the Green functions of the more general differential equation(3.27) with a time-dependent oscillator frequency Ω(t). We construct first a so-calledretarded Green function (compare page 38) as a product of a Heaviside function witha smooth functionGΩ2 (t, t0 ) = Θ(t − t0 )∆(t, t0 ).(3.40)Inserting this into the differential equation (3.27) we find[−∂t2 − Ω2 (t)]GΩ2 (t, t0 ) = Θ(t − t0 ) [−∂t2 − Ω2 (t)]∆(t, t0 )− δ̇(t − t0 ) − 2∂t ∆(t, t0 )δ(t − t0 ).(3.41)Expanding1∆(t, t0 ) = ∆(t, t) + [∂t ∆(t, t0 )]t=t0 (t − t0 ) + [∂t2 ∆(t, t0 )]t=t0 (t − t0 )2 + .
. . ,2(3.42)and using the fact that(t − t0 )δ̇(t − t0 ) = −δ(t − t0 ),(t − t0 )n δ̇(t − t0 ) = 0 for n > 1,(3.43)the second line in (3.41) can be rewritten as−δ̇(t − t0 )∆(t, t0 ) − δ(t − t0 )∂t ∆(t, t0 ).(3.44)By choosing the initial conditions∆(t, t) = 0,˙ t0 )| 0 = −1,∆(t,t =t(3.45)we satisfy the inhomogeneous differential equation (3.27) provided ∆(t, t0 ) obeys thehomogeneous differential equation[−∂t2 − Ω2 (t)]∆(t, t0 ) = 0,for t > t0 .(3.46)This equation is solved by a linear combination∆(t, t0 ) = α(t0 )ξ(t) + β(t0 )η(t)(3.47)of any two independent solutions η(t) and ξ(t) of the homogeneous equation[−∂t2 − Ω2 (t)]ξ(t) = 0,[−∂t2 − Ω2 (t)]η(t) = 0.(3.48)˙Their Wronski determinant W = ξ(t)η̇(t) − ξ(t)η(t)is nonzero and, of course, timeindependent, so that we can determine the coefficients in the linear combination(3.47) from (3.45) and find∆(t, t0 ) =i1 hξ(t)η(t0) − ξ(t0 )η(t) .W(3.49)H.
Kleinert, PATH INTEGRALS2153.2 Green Function of Harmonic OscillatorThe right-hand side contains the so-called Jacobi commutator of the two functionsξ(t) and η(t). Here we list a few useful algebraic properties of ∆(t, t0 ):∆(t, t0 ) =∆(tb , t)∆(t0 , ta ) − ∆(t, ta )∆(tb , t0 ),∆(tb , ta )∆(tb , t)∂tb ∆(tb , ta ) − ∆(t, ta ) = ∆(tb , ta )∂t ∆(tb , t),(3.50)(3.51)∆(t, ta )∂tb ∆(tb , ta ) − ∆(tb , t) = ∆(tb , ta )∂t ∆(t, ta ).(3.52)GΩ2 (t, t0 ) = Θ(t − t0 )∆(t, t0 ) + a(t0 )ξ(t) + b(t0 )η(t),(3.53)The retarded Green function (3.40) is so far not the unique solution of the differential equation (3.27), since one may always add a general solution of the homogeneous differential equation (3.48):with arbitrary coefficients a(t0 ) and b(t0 ).
This ambiguity is removed by the Dirichletboundary conditionsGΩ2 (tb , t) = 0,GΩ2 (t, ta ) = 0,tb 6= t,t 6= ta .(3.54)Imposing these upon (3.53) leads to a simple algebraic pair of equationsa(t)ξ(ta ) + b(t)η(ta ) = 0,a(t)ξ(tb ) + b(t)η(tb ) = ∆(t, tb ).(3.55)(3.56)Denoting the 2 × 2 -coefficient matrix byΛ=ξ(ta ) η(ta )ξ(tb ) η(tb )!,(3.57)we observe that under the conditiondet Λ = W ∆(ta , tb ) 6= 0,(3.58)the system (3.56) has a unique solution for the coefficients a(t) and b(t) in the Greenfunction (3.53).
Inserting this into (3.54) and using the identity (3.50), we obtainfrom this Wronski’s general formula corresponding to (3.36)GΩ2 (t, t0 ) =Θ(t − t0 )∆(tb , t)∆(t0 , ta ) + Θ(t0 − t)∆(t, ta )∆(tb , t0 ).∆(ta , tb )(3.59)At this point it is useful to realize that the functions in the numerator coincidewith the two specific linearly independent solutions Da (t) and Db (t) of the homogenous differential equations (3.48) which were introduced in Eqs.
(2.221) and (2.222).Comparing the initial conditions of Da (t) and Db (t) with that of the function ∆(t, t0 )in Eq. (3.45), we readily identifyDa (t) ≡ ∆(t, ta ),Db (t) ≡ ∆(tb , t),(3.60)2163 External Sources, Correlations, and Perturbation Theoryand formula (3.59) can be rewritten asΘ(t − t0 )Db (t)Da (t0 ) + Θ(t0 − t)Da (t)Db (t0 )GΩ2 (t, t ) =.Da (tb )0(3.61)It should be pointed out that this equation renders a unique and well-definedGreen function if the differential equation [−∂t2 − Ω2 (t)]y(t) = 0 has no solutionswith Dirichlet boundary conditions y(ta ) = y(tb ) = 0, generally called zero-modes.A zero mode would cause problems since it would certainly be one of the independent solutions of (3.49), say η(t).
Due to the property η(ta ) = η(tb ) = 0, however,the determinant of Λ would vanish, thus destroying the condition (3.58) which wasnecessary to find (3.59). Indeed, the function ∆(t, t0 ) in (3.49) would remain undetermined since the boundary condition η(ta ) = 0 together with (3.55) implies that˙also ξ(ta ) = 0, making W = ξ(t)η̇(t) − ξ(t)η(t)vanish at the initial time ta , and thusfor all times.3.2.2Spectral RepresentationA second way of specifying the Green function explicitly is via its spectral representation.Constant FrequencyFor constant frequency Ω(t) ≡ ω, the fluctuations δx(t) which satisfy the differentialequation(−∂t2 − ω 2 ) δx(t) = 0,(3.62)and vanish at the ends t = ta and t = tb , are expanded into a complete set oforthonormal functions:xn (t) =s2sin νn (t − ta ),tb − ta(3.63)with the frequencies [compare (2.105)]νn =πn.tb − ta(3.64)These functions satisfy the orthonormality relationsZtbtadt xn (t)xn0 (t) = δnn0 .(3.65)Since the operator −∂t2 − ω 2 is diagonal on xn (t), this is also true for the Greenfunction Gω2 (t, t0 ) = (−∂t2 − ω 2 )−1 δ(t − t0 ).
Let Gn be its eigenvalues defined byZtbtadt Gω2 (t, t0 )xn (t0 ) = Gn xn (t).(3.66)H. Kleinert, PATH INTEGRALS3.2 Green Function of Harmonic Oscillator217Then we expand Gω2 (t, t0 ) as follows:Gω2 (t, t0 ) =∞XGn xn (t)xn (t0 ).(3.67)n=1By definition, the eigenvalues of Gω2 (t, t0 ) are the inverse eigenvalues of the differential operator (−∂t2 − ω 2), which are νn2 − ω 2. ThusGn = (νn2 − ω 2 )−1 ,(3.68)and we arrive at the spectral representation of Gω2 (t, t0 ):Gω2 (t, t0 ) =∞sin νn (t − ta ) sin νn (t0 − ta )2 X.tb − ta n=1νn2 − ω 2(3.69)We may use the trigonometric relationsin νn (tb − t) = − sin νn [(t − ta ) − (tb − ta )] = −(−1)n sin νn (t − ta )to rewrite (3.69) asGω2 (t, t0 ) =∞2 Xsin νn (tb − t) sin νn (t0 − ta ).(−1)n+1tb − ta n=1νn2 − ω 2(3.70)These expressions make sense only if tb − ta is not equal to an integer multiple ofπ/ω, where one of the denominators in the sums vanishes.
This is the same rangeof tb − ta as in the Wronski expression (3.36).Time-Dependent FrequencyThe spectral representation can also be written down for the more general Greenfunction with a time-dependent frequency defined by the differential equation (3.27).If yn (t) are the eigenfunctions solving the differential equation with eigenvalue λnK(t)yn (t) = λn yn (t),(3.71)and if these eigenfunctions satisfy the orthogonality and completeness relationsZtbtadt yn (t)yn0 (t) = δnn0 ,Xnyn (t)yn (t0 ) = δ(t − t0 ),(3.72)(3.73)and if, moreover, there exists no zero-mode for which λn = 0, then GΩ2 (t, t0 ) has thespectral representationX yn (t)yn (t0 )0GΩ2 (t, t ) =.(3.74)λnnThis is easily verified by multiplication with K(t) using (3.71) and (3.73).It is instructive to prove the equality between the Wronskian construction andthe spectral representations (3.36) and (3.70).
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