Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 82
Текст из файла (страница 82)
This is found from the condition∂t= 0,∂xb(4.141)which after inserting (4.136), goes over into∂2S∂ 2 S ∂E∂ 2 S =+= 0.∂E∂xb t ∂E∂xb ∂E 2 ∂xbWe now use the relation(4.142)∂A ∂S ∂E ∂S∂S ∂E ∂E ∂S = − t=+ − t=∂xb t∂xb t ∂xb t∂xb ∂E ∂xb t ∂xb t∂xb E(4.143)3904 Semiclassical Time Evolution Amplitudeand find from it∂ 2 S ∂E∂2S∂ 2 A =+∂xb ∂xa t∂xb ∂xa ∂xb ∂E ∂xa(4.144)∂2S∂2S∂2S=−∂xb ∂xa ∂xa ∂E ∂xb ∂E,∂2S.∂E 2Thus the fixed-energy amplitude (4.133) takes the simple form1/2(xb |xa )E = DS eiS(xb ,xa ;E)/h̄ ,(4.145)with the 2 × 2-determinantDS =∂2S∂xb ∂xa∂2S∂E∂xa∂2S∂xb ∂E∂2S∂E 2.(4.146)The determinant can be simplified by the fact that a differentiation of the HamiltonJacobi equation!∂SH,x = E(4.147)∂xb bwith respect to xa leads to the equation∂2S∂H ∂ 2 S= ẋb= 0.∂pb ∂xb ∂xa∂xb ∂xa(4.148)It implies the vanishing of the upper left element in (4.146), reducing DS to∂2S ∂2SDS = −.∂xb ∂E ∂xa ∂E(4.149)Since ∂S/∂xb,a = ±pb,a and ∂p/∂E = 1/ẋ, one arrives atDS =1.ẋb ẋa(4.150)Let us calculate the semiclassical fixed-energy amplitude for a free particle.
Theclassical action function isA(xb , xa ; tb − ta ) =M (xb − xa )2,2 tb − ta(4.151)so that the function E(xb , xa ; tb − ta ) is given byE(xb , xa ; tb − ta ) = −∂ M (xb − xa )2M (xb − xa )2.=∂tb 2 tb − ta2 (tb − ta )2(4.152)H. Kleinert, PATH INTEGRALS4.6 Semiclassical Amplitude in Momentum Space391By a Legendre transformation, or directly from the defining equation (4.61), wecalculate√S(xb , xa ; E) = 2ME|xb − xa |.(4.153)From this we calculate the determinant (4.150) asDs =M,2E(4.154)and the fixed-energy amplitude (4.145) becomes(xb |xa )E =4.6sM i√2M E|xb −xa |/h̄.e2E(4.155)Semiclassical Amplitude in Momentum SpaceThe simple way of finding Fourier transforms in the semiclassical approximationcan be used to derive easily amplitudes in momentum space. Consider first thetime evolution amplitude (xb tb |xa ta )sc .
The momentum space version is given bythe two-dimensional Fourier integral [recall (2.37) and insert (4.96)](pb tb |pa ta )sc =Zdxb dxa e−i(pb xb −pa xa )/h̄ eiA(xb ,xa ;tb −ta )/h̄ F (xb , xa ; tb − ta ).(4.156)The semiclassical evaluation according to the general rule (4.48) yields√2πih̄(pb tb |pa ta )sc = √[−∂xb ∂xa A(xb , xa ; tb − ta )]1/2 ei[A(xb ,xa ;tb −ta )−pb xb +pa xa ]/h̄,(4.157)det Hwhere H is the matrixH=∂x2b A(xb , xa ; tb − ta ) ∂xb ∂xa A(xb , xa ; tb − ta )∂xa ∂xb A(xb , xa ; tb − ta )∂x2a A(xb , xa ; tb− ta ).(4.158)The exponent must be evaluated at the extremum with respect to xb and xa , whichlies atpb = ∂xb A(xb , xa ; tb − ta ),pa = −∂xb A(xb , xa ; tb − ta ).(4.159)The exponent contains then the Legendre transform of the action A(xb , xa ; tb − ta )which depends naturally on pb and pa :A(pb , pa ; tb − ta ) = A(xb , xa ; tb − ta ) − pb xb + pa xa .(4.160)The inverse Legendre transformation to (4.159) isxb = −∂pb A(xb , xa ; tb − ta ),xa = ∂xb A(xb , xa ; tb − ta ).(4.161)3924 Semiclassical Time Evolution AmplitudeThe important observation which greatly simplifies the result is that for a 2 × 2matrix Hab with (a, b = 1, 2), the matrix element −H12 /det H is equal to H12 .
Bywriting the matrix H and its inverse asH=∂pb∂pb∂xb∂xa∂p∂p− a − a∂xb∂xa,H−1=∂xb∂x− b∂pb∂pa∂x∂xa− a∂pb∂pawe see that, just as in the Eqs. (2.271) and (2.272):−1H12=,∂xa∂ 2 A(pb , pa ; tb − ta )=.∂pb∂pb ∂pa(4.162)(4.163)As a result, the semiclassical time evolution amplitude in momentum space (4.157)takes the simple form2πh̄(pb tb |pa ta )sc = √[−∂pb ∂pa A(pb , pa ; tb − ta )]1/2 eiA(pb ,pa ;tb −ta )/h̄.2πih̄(4.164)In D dimensions, this becomes(pb tb |pa ta ) = √1D2πih̄no1/2detD [−∂pi ∂pja A(pb , pa ; tb − ta )]beiA(pb ,pa ;tb −ta )/h̄ , (4.165)or(pb tb |pp ta ) = √1D2πih̄(detD"∂p− b∂xa#)1/2eiA(pb ,pa ;tb −ta )/h̄ ,(4.166)these results being completely analogous to the x-space expression (4.119) and(4.121), respectively.
As before, the subscripts a and b can be interchanged inthe determinant.If we apply these formulas to the harmonic oscillator with a time-dependentfrequency, we obtain precisely the amplitude (2.278). Thus in this case, the semiclassical time evolution amplitude (pb tb |pa ta )sc happens to coincide with the exactone.For a free particle with the action A(xb , xa ; tb − ta ) = M(xb − xa )2 /2(tb − ta ),the formula (4.157) cannot be applied since determinant of H vanishes, so that thesaddle point approximation is inapplicable. The formal infinity one obtains whentrying to apply Eq.
(4.157) is a reflection of the δ-function in the exact expression(2.133), which has no semiclassical approximation. The Legendre transform of theaction can, however, be calculated correctly and yields via the derivatives pa = pb ≡p = A(xb , xa ; tb − ta ) = M(xb − xa )/2(tb − ta ) the expressionA(pb , pa ; tb − ta ) = −p2(t − ta ),2 b(4.167)which agrees with the exponent of (2.133).H. Kleinert, PATH INTEGRALS4.7 Semiclassical Quantum-Mechanical Partition Function4.7393Semiclassical Quantum-Mechanical Partition FunctionFrom the result (4.96) we can easily derive the quantum-mechanical partition function in the semiclassical approximation:ZQM (tb − ta ) =Zdxa (xa tb |xa ta )sc =Zdxa F (xa , xa ; tb − ta )eiA(xa ,xa ;tb −ta )/h̄ .
(4.168)Within the semiclassical approximation the path integral, as the final trace integralmay be performed using the saddle point approximation. At the saddle point onehas [as in (4.127)]∂∂∂A(xa , xa ; tb − ta ) =A(xb , xa ; tb − ta ) +A(xb , xa ; tb − ta )∂xa∂xb∂xaxb =xaxb =xa= pb − pa = 0,(4.169)i.e., only classical orbits contribute whose momenta are equal at the coinciding endpoints. This restricts the orbits to periodic solutions of the equations of motion.The semiclassical limit selects, among all paths with xa = xb , the paths solving theequation of motion, ensuring the continuity of the internal momenta along thesepaths.
The integration in (4.168) enforces the equality of the initial and final momenta on these paths and permits a continuation of the equations of motion beyondthe final time tb in a periodic fashion, leading to periodic orbits. Along each of theseorbits, the energy E(xa , xa , tb − ta ) and the action A(xa , xa , tb − ta ) do not dependon the choice of xa . The phase factor eiA/h̄ in the integral (4.168) is therefore aconstant. The integral must be performed over a full period between the turningpoints of each orbit in the forward and backward direction.
It contains a nontrivialxa -dependence only in the fluctuation factor. Thus, (4.168) can be written asZQM (tb − ta ) =Zdxa F (xa , xa ; tb − ta ) eiA(xa ,xa ;tb −ta )/h̄ .(4.170)For the integration over the fluctuation factor we use the expression (4.117) and theequation∂ ∂1 ∂2AA(xb , xa ; tb − ta ) = −,∂xb ∂xaẋb ẋa ∂t2b(4.171)following from (4.113) and (4.86), and have11∂2AF (xb , xa ; tb − ta ) = √22πih̄ ẋ(tb )ẋ(ta ) ∂tb"#1/2.(4.172)Inserting xa = xb leads to1 ∂2A1F (xa , xa ; tb − ta ) = √2πih̄ ẋa ∂t2b"#1/2.(4.173)3944 Semiclassical Time Evolution AmplitudeThe action of a periodic path does not depend on xa ,so that the xb -integration in(4.168) requires only integrating 1/ẋa forward and back, which produces the totalperiod:tb − ta = 2Zx+x−dxa1=2ẋaZx+x−Hence we obtain from (4.168):dx qM2M[E − V (x)]1/2tb − ta ∂ 2 A ZQM (tb − ta ) = √2πih̄ ∂t2b .(4.174)eiA(tb −ta )/h̄−iπ .(4.175)There is a phase factor e−iπ associated with a Maslov-Morse index ν = 2, first introduced in the fluctuation factor (2.264).
In the present context, this phase factorarises from the fact that when doing the integral (4.170), the periodic orbit passesthrough the turning points x− and x+ where the integrand of (4.174) becomes singular, even though the integral remains finite. Near the turning points, the semiclassical approximation breaks down, as discussed in Section 4.1 in the context of theWKB approximation to the Schrödinger equation. This breakdown required specialattention in the derivation of the connection formulas relating the wave functions onone side of the turning points to those on the other side. There, the breakdown wascircumvented by escaping into the complex x-plane. When going around the singuq1/2larity in the clockwise sense, the prefactor 1/p(x) = 1/ 2M(E − V (x)) acquireda phase factor e−iπ/2 .
For a periodic orbit, both turning points had to be encircledproducing twice this phase factor, which is precisely the phase e−iπ given in (4.175).The result (4.175) takes an especially simple form after a Fourier transformaction:Z̃QM (E) =Z∞tadtb eiE(tb −ta )/h̄ ZQM (tb − ta )1= √2πih̄Z∞tadtb (tb 2 1/2∂ A− ta ) 2 ei[A(tb −ta )+(tb −ta )E]/h̄−iπ . ∂t (4.176)bIn the semiclassical approximation, the main contribution to the integral at a givenenergy E comes from the time where tb − ta is equal to the period of the particleorbit with this energy.
It is determined as in (4.133) by the extremum ofA(tb − ta ) + (tb − ta )E.Thus it satisfies−∂A(tb − ta ) = E.∂tb(4.177)(4.178)As in (4.134), the extremum determines the period tb − ta of the orbit with anenergy E. It will be denoted by t(E). The second derivative of the exponent is(i/h̄)∂ 2 A(tb − ta )/∂t2b . For this reason, the quadratic correction in the saddle pointH. Kleinert, PATH INTEGRALS4.7 Semiclassical Quantum-Mechanical Partition Function395approximation to the integral over tb cancels the corresponding prefactor in (4.176)and leads to the simple expressionZ̃QM (E) = t(E)ei[A(t)+t(E)E]/h̄−iπ .(4.179)The exponent contains again the eikonal S(E) = A(t) + t(E)E, the Legendre transform of the action A(t) defined byS(E) = A(t) − t∂A(t),∂t(4.180)where the variable t has to be replaced by E(t) = −∂A(t)/∂t.
Via the inverseLegendre transformation, the derivative ∂S(E)/∂E = t leads back toA(t) = S(E) −∂S(E)E.∂E(4.181)Explicitly, S(E) is given by the integral (4.61):S(E) = 2Zx+x−dx p(x) = 2Zx+x−qdx 2M[E − V (x)].(4.182)Finally, we have to take into account that the periodic orbit is repeatedly traversedfor an arbitrary number of times. Each period yields a phase factor eiS(E)/h̄−iπ . Thesum isZ̃QM (E) =∞Xn=1t(E)ein[S(E)/h̄−π] = −t(E)eiS(E)/h̄.1 + eiS(E)/h̄(4.183)This expression possesses poles in the complex energy plane at points where theeikonal satisfies the conditionS(En ) = 2πh̄(n + 1/2),n = 0, ±1, ±2, .
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
