Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 80
Текст из файла (страница 80)
Instead of the time t we could have usedany parameter τ to describe x(τ ) and write the eikonal (4.62) asSE [x] =Zdτqgij (x)ẋi (τ )ẋj (τ ).(4.66)Einstein has certainly been inspired by this ancient description of classical trajectories when geometrizing the relativistic Kepler motion by attributing a dynamicalRiemannian geometry to spacetime.It is worth pointing out a subtlety in this variational principle, in view of aclosely related situation to be encountered later in Chapter 10. The variations aresupposed to be carried out at a fixed energyM 2ẋ + V (x).(4.67)2This is a nonholonomic constraint which destroys the independence of the variationδx(t) and δ ẋ. They are related byE=1∇V (x)δx.(4.68)MIt is, however, possible to regain the independence by allowing for a simultaneousvariation of the time argument in x(t) when varying x(t). As a consequence, wecan no longer employ the standard equality δ ẋ = dδx/dt which is necessary for thederivation of the Euler-Lagrange equation (4.63).
Instead, we calculateẋ δ ẋ = −δ ẋ =dx + dδxdd− ẋ = δx − ẋ δt,dt + dδtdtdt(4.69)which shows that variation and time derivatives no longer commute with each other.Combining this with the relation (4.68) we see that the variations of x and ẋ canbe made independent if we vary t along the orbit according to the relationẋ2dd1δt = ẋ δx + ∇V (x)δx.dtdtM(4.70)H. Kleinert, PATH INTEGRALS4.2 Saddle Point Approximation379With (4.69), the variations of the eikonal (4.64) areδSE [x] =Z!!Z∂LE∂LEd∂LE dδx +δx + dt LE −ẋδt ,∂ ẋ dt∂x∂ ẋdtdt(4.71)where we have kept the usual commutativity of variation and time derivative of thetime itself.
In the second integral, we may set−LE +∂LEẋ ≡ HE .∂ ẋ(4.72)The function HE arises from LE by the same combination of LE (x, ẋ) and∂LE /∂ ẋ(x, ẋ) as in a Legendre transformation which brings a Lagrangian to theassociated Hamiltonian [recall (1.13)]. But in contrast to the usual procedure we donot eliminate ẋ in favor of a canonical momentum variable ∂LE /∂ ẋ [recall (1.14)],i.e., the HE is a function HE (x, ẋ).
Note that it is not equal to the energy.The variation (4.71) shows that the extra variation δt of the time does notchangethe Euler-Lagrange equations for the above Lagrangian in Eq. (4.64), LE =q2M[E − V (x)]ẋ2 . Being linear in ẋ, the associated HE vanishes identically, sothat the second term disappears and we recover the ordinary equation of motion∂LEd ∂LE=.dt ∂ ẋ∂x(4.73)In general, however, we must keep the second term. Expressing dδt/dt via (4.70),we find#"ẋ d∂LE− HE 2δxδSE [x] =dt∂ ẋẋ dt#"Z1 1∂LE− HE 2 ∇V (x) δx,+dt∂xẋ MZ(4.74)and the general equation of motion becomes#"∂LEd ∂LEẋ1 1− HE 2 =− HE 2 ∇V (x),dt ∂ ẋ∂xẋẋ M(4.75)rather than (4.73).
Let us illustrate this by rewriting the eikonal as a functionalSE [x] =Zdt L0E (x, ẋ)=MZdt ẋ2 (t),(4.76)which is the same functional as (4.62) as long as the energy E is kept fixed. If weinsert the new Lagrangian L0E into (4.75), we obtain the correct equation of motionM ẍ = −∇V (x).(4.77)3804 Semiclassical Time Evolution AmplitudeIn this case, the equation of motion can actually be found more directly. We varythe eikonal (4.76) as follows:δSE [x] = MZδdt ẋ2 + MZdt ẋ δ ẋ + MZdt ẋδ ẋ.(4.78)In the last term we insert the relation (4.69) and writeδSE [x] = M"Z2δdt ẋ +Zdt ẋ δ ẋ +Zddt ẋ δx −dtZ#ddt ẋ δt .dt2(4.79)The two terms containing δt cancel each other, so that relation (4.70) is no longerneeded. Using now (4.68), we obtain directly the equation of motion (4.77).With the help of the eikonal (4.61), we write the classical action (4.59) asA(xb , xa ; tb − ta ) ≡ S(xb , xa ; E) − (tb − ta )E,(4.80)where E is given by (4.58).The action has the property that its derivatives with respect to the endpointsxb , xa at a fixed tb − ta yield the initial and final classical momenta:∂A(xb , xa ; tb − ta ) = ±p(xb,a ).∂xb,a(4.81)Indeed, the differentiation gives#"Zxb∂E∂p(x)∂A− (tb − ta )= p(xb ) +dx,∂xb∂E∂xbxa(4.82)∂p(x)M1== ,∂Ep(x)ẋ(4.83)and usingwe see thatZxbxa∂p(x) Z tbdt = tb − ta ,=dx∂Eta(4.84)so that the bracket in (4.82) vanishes, and (4.81) is indeed fulfilled [compare also(4.12)].
The relation (4.84) implies that the eikonal (4.61) has the energy derivative∂S(xb , xa ; E) = tb − ta .∂E(4.85)As a conjugate relation, the derivative of the action with respect to the time tb atfixed xb gives the energy with a minus sign [compare (4.10)]:∂A(xb , xa ; tb − ta ) = −E(xb , xa ; tb − ta ).∂tb(4.86)H.
Kleinert, PATH INTEGRALS4.2 Saddle Point Approximation381This is easily verified:∂A=∂tb"Zxbxa#∂p∂E− E = −E.− (tb − ta )dx∂E∂tb(4.87)Thus, the classical action function A(xb , xa ; tb − ta ) and the eikonal S(xb , xa ; E) areLegendre transforms of each other.The equation1(∂ A)2 + V (x) = ∂t A2M x(4.88)is, of course, the Hamilton-Jacobi equation (4.13) of classical mechanics.We have therefore found the leading term in the semiclassical approximation tothe amplitude [corresponding to the approximation (4.36)]:h̄→0(xb tb |xa ta ) −−−→ const × eiA(xb ,xa ;tb −ta )/h̄ .(4.89)In general, this leading term will be multiplied by a fluctuation factor(xb tb |xa ta ) = eiA(xb ,xa ;tb −ta )/h̄ F (xb , xa ; tb − ta ).(4.90)In contrast to the purely harmonic case in Eq.
(2.146) this will depend on the initialand final coordinates xa and xb .The calculation of the leading contribution to the fluctuation factor is the nextstep in the saddle point expansion of the path integral (4.33). For this we expand theaction (4.52) in the neighborhood of the classical orbit in powers of the fluctuationsδx(t) = x(t) − xcl (t).(4.91)This yields the fluctuation expansionδAδx(t)δx(t)taZδ2A1 tbdtdt0+δx(t)δx(t0 )(4.92)2 taδx(t)δx(t0 )Zδ3A1 tbδx(t)δx(t0 )δx(t00 ) + . . . ,dtdt0 dt00+3! taδx(t)δx(t0 )δx(t00 )A[x, ẋ] = A[xcl ] +Ztbdtwhere all functional derivatives on the right-hand side are evaluated along the classical orbit x(t) = xcl (t). The linear term in the quantum fluctuation δx(t) is absentsince A[x, ẋ] is extremal at xcl (t).
For a point particle, the quadratic term is12Ztbtaδ2Adtdtδx(t)δx(t0 ) =δx(t)δx(t0 )0ZtbtaM1dt(δ ẋ)2 + V 00 (xcl (t))(δx)2 .22(4.93)3824 Semiclassical Time Evolution AmplitudeThus the fluctuations behave like those of a harmonic oscillator with a timedependent frequencyΩ2 (t) =1 00V (xcl (t)).M(4.94)By definition, the fluctuations vanish at the endpoints:δx(ta ) = 0, δx(tb ) = 0.(4.95)If we include only the quadratic terms in the fluctuation expansion (4.92), we canintegrate out the fluctuations in the path integral (4.33).
Since x(t) and δx(t)differ only by a fixed additive function xcl (t), the measure of the path integral overx(t) transforms trivially into that over δx(t). Thus we conclude that the leadingsemiclassical limit of the amplitude is given by the product(xb tb |xa ta )sc = eiA(xb ,xa ;tb −ta )/h̄ Fsc (xb , xa ; tb − ta ),(4.96)with the semiclassical fluctuation factor [compare (2.193)]i Z ta MFsc (xb , xa ; tb − ta ) =Dδx(t) expdt [δ ẋ2 − Ω2 (t)δx2 ]h̄ tb21det (−∇∇ − Ω2 (t))−1/2= q2πih̄/MZ1= q2πih̄(tb − ta )/Mvuutdet (−∂t2 ).det (−∂t2 − Ω2 (t))(4.97)In principle, we would now have to solve the differential equation[−∂t2 − Ω2 (t)]yn (t) = [−∂t2 − V 00 (xcl (t))/M]yn (t) = λn yn (t),(4.98)and find the energies of the eigenmodes yn (t) of the fluctuations. The ratio offluctuation determinantsD0det (−∂t2 )=(4.99)Ddet (−∂t2 − Ω2 (t))in the second line of (4.97) would then be found from the product of ratios ofeigenvalues, λn /λ0n , where λ0n are the eigenvalues of the differential equation−∂t2 yn (t) = λ0n yn (t).(4.100)Fortunately, we can save ourselves all this work using the Gelfand-Yaglom methodof Section 2.4 which provides a much simpler and more direct way of calculatingfluctuation determinants with a time-dependent frequency without the knowledgeof the eigenvalues λn .H.
Kleinert, PATH INTEGRALS4.3 Van Vleck-Pauli-Morette Determinant4.3383Van Vleck-Pauli-Morette DeterminantAccording to the Gelfand-Yaglom method of Section 2.4, a functional determinantof the formdet (−∂t2 − Ω2 (t))is found by solving the differential equation (4.98) at zero eigenvalue[−∂t2 − Ω2 (t)]Da (t) = 0,(4.101)with the initial conditionsDa (ta ) = 0,Ḋa (ta ) = 1.(4.102)Then Da (tb ) is the desired fluctuation determinant. In Eq. (2.233), we have constructed the solution to these equations in terms of an arbitrary solution ξ(t) of thehomogenous equation[−∂t2 − Ω2 (t)]ξ(t) = 0asDren = ξ(t)ξ(ta)Ztbtadt0.ξ 2 (t0 )(4.103)(4.104)In general, it is difficult to find an analytic solution to Eq.
(4.103). In the presentfluctuation problem, however, the time-dependent frequency Ω(t) has a special formΩ2 (t) = V 00 (xcl (t))/M of (4.94). We shall now prove that, just as in the purelyharmonic action in Section 2.5, all information on the fluctuation determinant iscontained in the classical orbit xcl (t), and ultimately in the mixed spatial derivativesof the classical action A(xb , xa ; tb − ta ). In fact, the solution ξ(t) is simply equal tothe velocityξ(t) = ẋcl (t).(4.105)This is seen directly by differentiating the equation of motion (4.53) with respect tot, yielding∂t [M ẍcl + V 0 (xcl (t))] = [M∂t2 + V 00 (xcl (t))]ẋcl (t) = 0,(4.106)which is precisely the homogenous differential equation (4.103) for ẋcl (t).There is a simple symmetry argument to understand (4.105) as a completelygeneral consequence of the time translation invariance of the system.
The fluctuationδx(t) ∝ ẋcl (t) describes an infinitesimal translation of the classical solution xcl (t)in time, xcl (t) → xcl (t + ) = xcl + ẋcl + . . . . Interpreted as a translationalfluctuation of the solution xcl (t) along the time axis it cannot carry any energy λnand y0 (t) ∝ ẋcl (t) must therefore solve Eq. (4.98) with λ0 = 0.3844 Semiclassical Time Evolution AmplitudeWith the special solution (4.105), the functional determinant (4.104) becomesDren = ẋcl (tb )ẋcl (ta )Ztbtadt.ẋ2cl (t)(4.107)Note that also the Green-function of the quadratic fluctuations associated withEq. (4.103) can be given explicitly in terms of the classical solution xcl (t). For Dirichlet boundary conditions, it is equal to the combination (3.61) of the solutions Da (t)and Db (t) of the homogeneous differential equation (4.103) satisfying the boundaryconditions (2.221) and (2.222), whose d’Alembert construction (2.232) becomes hereDa (t) = ẋcl (t)ẋcl (ta )Zttadt,2ẋcl (t)Db (t) = ẋcl (tb )ẋcl (t)Zttbdt.ẋ2cl (t)(4.108)In Eqs.
(2.245) and (2.262) we have found two simple expressions for the fluctuation determinant in terms of the classical actionDren∂ ẋb=−∂xa!−1∂2= −MA∂xb ∂xa cl"#−1.(4.109)These were derived for purely quadratic actions with an arbitrary time-dependentfrequency Ω2 (t). But they hold for any action. First, the equality between thesecond and third expression is a consequence of the general relation (4.81).
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
