3.8. External Source in Quantum-Statistical Path Integral

Transcription

1 3.8. External Source in Quantu-Statistical Path Integral This section studies the quantu statistical evolution alitude ( x b, x a ) ω x ex - ħ d M x + ω x - () x (3.) which will be evaluated in two different ways Continuation of Real-Tie Result The easiest way to obtain the result is by alying the analytic continuation t -i t b - t a -i ( b - a ) -i (3.a) to the real-tie results. Using the subscrit "e" to denote the Euclidean (iaginary-tie) version of a quantity, we get fro (3.68) & (3.7) ( x b, x a ) ω e - ħ cl, e F ω ( b, a ) e - ħ e ext [] F ω ( b, a ) where e () -i (t) t -i M π ħ ω sinh ω ex - ħ e ext [] (3.) (3.a) with F ω ( b, a ) M π ħ ext e [] cl,e +, fl,e ω sinh ω [(3.7) used.] (3.b) ω,cl,e +,cl,e +, fl, e [(3.69) used.] e +,e +,e e + e (3.) M ω ω,cl,e e sinh ω x b + x a cosh ω - x b x a (3.3) b,cl,e,e sinh ω d xb sinh ω( - a ) + x a sinh ω ( b -) () a - sinh ω d x b sinh ω + x a sinh ω ( - ) () (3.4), fl, e,e - b M d d ' () G ω, e(, ') (') [(3.67) used.] a a - M d d ' () G ω, e(, ') (') (3.5) G ω, e(, ') ω sinh ω sinh ω( b -) sinh ω(' - a ) > ' Shifting the origin to a, we have the equivalent for G ω, e(, ') sinh ω( - ) sinh ω ' ω sinh ω > ' (3.6a)

2 3.8._ExternalSourceInQuantuStatisticalPathIntegral.nb Alternatively, analytic continuation of the differential eq. [see (3.6)] gives t +ω G ω (t, t') -δ(t - t') - +ω G ω (-i, -i ') -δ -i ( - ') i δ( - ') Defining we get G ω, e(, ') ig ω (-i, -i ') (3.8) - +ω G ω, e(, ') δ( - ') (3.7) For the Dirichlet B.C., the solution is the Euclidean version of (3.36), naely G ω, e(, ') ω sinh ω sinh ω ( b - > ) sinh ω ( < - a ) Shifting the origin to a, we have G ω, e(, ') ω sinh ω sinh ω ( - >) sinh ω < ω sinh ω cosh ω ( - > + < ) -cosh ω ( - > - < ) cosh ω ( - -' ) -cosh ω ( - - ') (.6) ω sinh ω The Euclidean versions of (3.73-4) are A e (ω) i A(ω) t b -t a -i b M ω d e -ω ( - a ) () a M ω d e -ω () (3.) B e (ω) i B(ω) t b -t a -i M ω b M ω d e -ω ( b - ) () a d e -ω ( - ) () -e - ω A e (-ω) (3.) Either by using (3.-) on (3.4) or, ore easily, by analytic continuing (3.75), we get,cl,e,e M ω sinh ω x b e ω B e (ω) -A e (ω) + x a e ω A e (ω) -B e (ω) (3.9) Siilarly, (3.5) or (3.76) gives, fl, e,e M ω 8 sinh ω e ω A e (ω) + B e (ω) - B e (ω) A e (ω) - 4 M ω d d ' () e - ω - ' (') (3.) The quantu statistical artition function is ust the trace of the quantu statistical evolution alitude: Z ω - d x ( x, x ) ω

3 3.8._ExternalSourceInQuantuStatisticalPathIntegral.nb 3 F ω ( ) - For a source-free oscillator, d x ex - ħ e ext [] x b x a x (3.3a) e ext [] ω,cl,e e so that, with the hel of (3.b) & (3.3), we have Z ω F ω ( ) - d x ex - ħ e x b x a x M π ħ ω sinh ω d x ex - - ħ M ω x sinh ω cosh ω - M π ħ ω sinh ω π ħ sinh ω M ω (cosh ω - ) (cosh ω - ) (3.4) sinh ω In the resence of a source, there is another x-deendent ter given by [see (3.9)] M ω x,cl,e,e sinh ω e ω B e (ω) -A e (ω) + e ω A e (ω) -B e (ω) M ω x sinh ω e ω - B e (ω) +A e (ω) The x-integral thus becoes d x ex - M ω - ħ sinh ω cosh ω - x + e ω - ( B e + A e ) x - d x ex - ħ M ω cosh ω - sinh ω x + 4 e ω - cosh ω - ( B e + A e ) (3.4a) ex - ħ r,e where M ω cosh ω - r,e - sinh ω 4 e ω - cosh ω - ( B e + A e ) (3.4b) Using we have e ω - cosh ω - e ω - sinh ( ω / ) e ω - e ω / - e - ω / e ω r,e - M ω e ω 8 sinh ω ( B e + A e ) (3.6) Since the x-integral in (3.4a) gives the sae result as the source-free case (3.4), (3.3a) becoes

4 4 3.8._ExternalSourceInQuantuStatisticalPathIntegral.nb Z ω Z ω ex- ħ r, e +, fl, e Z ω ex- ħ r, e +,e (3.6a) Z ω ex - ħ e (3.) where [ see (3.) & (3.6) ] Using e r, e +, fl, e r, e +,e - M ω e ω 8 sinh ω ( B e + A e ) + M ω 8 sinh ω e ω A e + B e - A e B e - 4 M ω d d ' () e - ω - ' (') - M ω e ω 8 sinh ω ( B e + A e ) M ω + 8 sinh ω e ω A e + B e - A e B e M ω - 4 sinh ω e ω + A e B e - M ω e ω / cosh ( ω / ) sinh ω ħ ω / M ω eβ - 4 sinh ( ω /) A e B e (3.5) becoes e A e B e ħ ω / M ω eβ - 4 sinh ( ω /) A e B e - 4 M ω Using (3.-), we have M ω e ω / 4 sinh ( ω /) A e B e e ω / 4 M ω sinh ( ω /) - ω / e 4 M ω sinh ( ω /) M ω e ω - 4 M ω e ω - d d ' () e - ω - ' (') (3.7) d e -ω () d ' e -ω ( - ) (') d d ' e -ω ( - ' ) () (') d d ' e -ω ( - ' ) () (') d d ' e -ω ( - ' ) + e ω ( - ' ) () (') where we ve used ' to obtain the nd ter in the square bracket. (3.7) thus becoes e - 4 M ω d d ' () (') e ω - e-ω ( - ' ) + e ω ( - ' ) + e - ω - e () (') ω -ω ( - ' ) + e ω ( - ' ) > ' 4 M ω d d ' e ω - e -ω ( - ' ) + e ω + ω ( - ' ) < ' - ' (3.5)

5 3.8._ExternalSourceInQuantuStatisticalPathIntegral.nb 5 where - 4 M ω d d ' () (') - 4 M ω d d ' () (') - M d cosh ω - G ω,e() ω sinh( ω / ) e ω ( - - ' ) + e ω - ' e ω - cosh ω - - ' sinh( ω / ) (3.8) d ' () (') G ω,e( - ') (3.8a) ϵ [, ) (3.9) i G ω (-i ) tb -t a -i is ust the Euclidean version of the eriodic Green function in (3.99). For coleteness, we shall also calculate the artition function for oen aths, Z oen ω []. To begin, the x-integral now involves [ see (3.3-4) & (3.9) ] I - - d x b d x b - - d x a ex- ħ ( ω,cl,e +,cl,e ) M ω d x a ex - ħ sinh ω x b + x a cosh ω - x b x a (3.a) + x b e ω B e - A e + x a e ω A e - B e Using Matheatica [ see 3.8._Code.nb ] to do the integral, we get I π ħ M ω ex M ω 4 ħ ( + coth ω) A e + B e (3.b) Thus, (3.3a) becoes where Z ω oen [] π ħ M ω sinh ω ex - ħ oen (3.) oen - M ω 4 ( + coth ω) A e + B e +,e - M ω e ω 4 sinh ω A e + B e +,e Using (3.-) & oen M ω - 8 sinh ω e ω A e + B e + A e B e - 4 M ω d d ' () e - ω - ' (') - 8 M ω sinh ω d d ' () (') e ω e -ω e -ω ' + e -ω ( - ) e -ω ( - ' ) + e -ω e -ω ( - ' ) + e ω - e - ω e - ω - ' (3.a)

6 6 3.8._ExternalSourceInQuantuStatisticalPathIntegral.nb - 8 M ω sinh ω d d ' () (') e -ω ( + ' - ) + e ω ( + ' - ) (3.b) For any syetric function f(', ) f(, '), + e -ω ( - '+ ) + e -ω ( - ' - ) - e -ω ( - ' + ) d d ' f(, ') d d d d ' + d ' + d ' f(, ') d ' f(, ') ' d ' d f(, ') [c.f. (3.67-a)] (3.c) While the st two ters inside the square bracket in (3.b) can be re-written using (3.c), the last 3 ters require secial treatent as follows d d ' () (') e -ω ( - '+ ) + e -ω ( - ' - ) - e -ω ( - ' + ) d + d ' e -ω ( - '+ ) + e -ω ( - ' - ) d ' e -ω ( - '+ ) + e ω ( - ' + ) - e ω ( - ' - ) () (') d d ' e -ω ( - '+ ) + e -ω ( - ' - ) () (') + d 4 d d ' d e -ω ( - '+ ) + e ω ( - ' + ) - e ω ( - ' - ) () (') d ' e -ω ( - '+ ) -ω ( - ' - ) + e d + e ω ( - '- ) + e -ω ( - ' - ) - e -ω ( - ' + ) () (') d ' e ω ( - '- ) + e -ω ( - ' - ) () (') d ' () (') cosh[ ω ( - ' - ) ] (3.b) can therefore be written as oen - M ω sinh ω d d ' () (') cosh[ ω ( + ' - ) ] + cosh[ ω ( - ' - ) ] where - M ω sinh ω - M d d ' () (') cosh ω ( - ) cosh ω ' d d ' () (') G oen ω, e(, ') (3.3) G oen cosh ω ( - ) cosh ω ' ω,e (, ') ω sinh ω cosh ω ( - >) cosh ω < ω sinh ω for > ' (3.3)

7 3.8._ExternalSourceInQuantuStatisticalPathIntegral.nb 7 For sall ω, we have [ see 3.8._Code.nb ], G oen ω,e (, ') ω > + > + < + Oω (3.3a) Using > ' > ' < ' - ' + + ' (3.3a) becoes G oen ω,e (, ') ω which is the iaginary-tie version of (3.57) Calculation at Iaginary Tie - ' - ( + ' ) + + ' + Oω (3.3) We now calculate Z ω directly fro Z ω x b x a x() ex - ħ e[] (3.33) with the Euclidean action e [] d M x + ω x - () x (3.34) Using the eriodic B.C., the surface ters fro a artial integration cancel out & we have Define e [] d M x - +ω x - () x (3.35) D ω, e(, ') - +ω δ( - ') - ' ϵ [, ) (3.36) G ω,e(, ') - +ω - δ( - ') G ω,e( - ') - D ω, e(, ') (3.37) Setting we have x() x' () + M x() - +ω x() x' () - +ω x' () + M + M + M d ' G ω,e(, ') (') (3.38) d ' G ω,e(, ') (') - +ω x' () d ' x' () - +ω G ω,e(, ') (') d ' G ω,e(, ') (') - +ω x' () - +ω x' () + M () x' () + M d '' G ω,e(, '') ('') d ' () G ω,e(, ') (') and () x() () x' () + M d ' () G ω,e(, ') (')

8 8 3.8._ExternalSourceInQuantuStatisticalPathIntegral.nb (3.35) thus becoes e [] d M x' - +ω x' - M For, we have [c.f. (.48a)] d Z ω x' ex - x' b x' a ħ d M x' - +ω x' d ' () G ω,e(, ') (') (3.39) x' ex - x' b x' a ħ d d ' M x' () D ω, e(, ') x' (') det D ω, e For the eriodic B.C., det D ω, e - ex ω + ω - ω π (3.4) lnω + ω (3.4) ex ln ω + [ (.57) used. ] ω ex ln sinh ω [See.5.3 ] 4 sinh ω (3.4a) (3.4) thus becoes Z ω sinh ω (3.4) As with (3.), Z ω Z ω ex - ħ e[] (3.43) where e [] - M d d ' () G ω,e(, ') (') (3.44) is the -deendent art of e [] in (3.39). Using the set of orthonoral, colete, and eriodic eigenfunctions ϕ () e -i ω as basis, (3.37) becoes G ω,e ( - ') - +ω - δ( - ') - +ω e -i ω ( - ' ) ω + ω e-i ω ( - ' ) For low teeratures, β >> so that ω <<, we have f(ω ) Δω π d ω f(ω ) π f (ω ) ω π (3.44a) (3.45)

9 3.8._ExternalSourceInQuantuStatisticalPathIntegral.nb 9 so that d ω G ω,e( - ') - π d ω - π ω ω + ω e-i ω ( - ' ) i ω e -ω ( - ' ) e ω ( - ' ) β ω - i ω - ω + i ω e-i ω ( - ' ) > ' < ' C.W. lower contour closed in C.C.W. uer lane ω e-ω - ' (3.46) For finite teertures, we use the Poisson suation forula to write d μ G ω,e( - ') π ei μ n μ + ω e-i μ ( - ' ) Since G ω,e n - ω e -ω - ' + n (3.46a) n - is eriodic, we need consider only the riary interval ϵ [, ) for which G ω,e() - ω e -ω ( + n ) + e ω ( + n ) n n - ω e -ω ( + n ) + e ω ( - n ) n n ω e-ω + ( e -ω + e ω ) e -n ω ω e-ω + ( e -ω + e ω ) ω ω e -ω + e ω ( - ) - e - ω coshω - n e - ω - e - ω sinh ω ϵ [, ) (3.48) in agreeent with (3.9). Since (3.48) is not exlicitly eriodic, values outside [, ) ust be calculated using the eriodicity. For sall ω, we have [ see 3.8._Code.nb ], G ω,e() ω - + Noting that the ter in (3.45) is G ω,e + + Oω (3.49), we define ω () G ω,e() - ω (3.5) ω + ω e-i ω (3.5a) which has a finite ω liit G,e () ϵ [, ) (3.5)

10 3.8._ExternalSourceInQuantuStatisticalPathIntegral.nb The following grah lots G ω,e() for 3 values of ω [ see 3.8._Code.nb ]. G ω,e()βħ / βħ βħω βħω 5 βħω (3.5) can also be derived directly fro (3.5a) [c.f..5.6]. Setting ω, we have G,e () ω e-i ω (-) ω e-i ω - ω π (3.5) (-) n π n! -i π - n (3.5a) Using n (-) (-) -n + (-)- +-n -n -n (-) n even (3.5a) becoes G,e () - -n n odd π π n,, 4,... n,, 4,... (-) where [ see Abraowitz & Stegun, Forula ] η(z) (-) - -n -n n! -i π - η( - n) n! -i π - n is the Rieann eta function related to the Rieann zeta function by n (3.53) (3.53a) (3.54) η(z) - -z ζ(z) (3.55) Since [ see Abraowitz & Stegun, Forulae 3..(4, & 4) ] we have ζ(- n) n,, 3,... (3.55a) ζ() - ζ() π 6

11 3.8._ExternalSourceInQuantuStatisticalPathIntegral.nb η( - n) η () ζ() π n η() -ζ() n n 4, 6, 8,... so that (3.53a) reduces to G,e () - π η() + η() -i π! π π π + π in agreeent with (3.5).! -i π - π - (3.55b) (3.55) The Bernoulli olynoials are defined as [see Gradshteyn & Ryzhik, Forula 9.6 ] n B n (x) k n k B k x n-k (3.59) where B n are the Bernoulli nubers given by [ G&R, 9.6 ] n B n k n k B k (3.59a) They can also be obtained fro the generating function [ G&R, 9.6 ] e x t e t - B n (x) t n- n n! and have the exansion [ G&R, 9.6 ] (3.6) B n (x) (-)n- ( n)! cos k π x x, n,, 3,... (3.6) ( π) n k k n Soe secial values are [ G&R, 9.67 ] B (x) x -, B (x) x - x + 6 (3.6) (3.55) can therefore be written as G,e () B This relation is not accidental but arises fro the relation [ A&S 3..6 ] n ( π) ζ( n) ( n)! Anti-eriodic B.C. (3.58) B n n,, 3,... (3.6a) The anti-eriodic Green function is related to the eriodic one by the substitution

12 3.8._ExternalSourceInQuantuStatisticalPathIntegral.nb ω π ω f π + Using the Poisson suation forula for ferions [see (3.5) ] - f + - d μ n - (-) n e π i μ n f(μ) (3.63b) (3.46a) becoes a G ω,e( - ') ω (-) n e -ω - ' + n (3.63c) n - a Since G ω,e is anti-eriodic, we need consider only the riary interval ϵ [, ) for which a G ω,e() - ω (-) n e -ω ( + n ) + (-) n e ω ( + n ) n n - ω (-) n e -ω ( + n ) + (-) n e ω ( - n ) n n ω e-ω + ( e -ω + e ω ) (-) n e -n ω ω e-ω + ( e -ω + e ω ) ω ω e -ω - e ω ( - ) + e - ω sinhω - n -e - ω + e - ω (3.63a) cosh ω ϵ [, ) (3.63) Caution: (3.63) differs fro Kleinert s version by a negative sign. Using (3.8), the analytic continuation of (3.3) gives G ω, e() i G ω (-i ) which agrees with (3.63). sin ω -i - -i ω cos -i ω sinh ω - - ω cosh ω Since (3.63) is not exlicitly anti-eriodic, values outside [, ) ust be calculated using the antieriodicity. a The following grah lots G ω,e() for 3 values of ω [ see 3.8._Code.nb ].

13 3.8._ExternalSourceInQuantuStatisticalPathIntegral.nb 3 a G ω,e()βħ / βħ βħω βħω 5 βħω Caution: This grah differs fro Kleinert s Fig.3.4 by a negative sign. For ω <<, (3.63) becoes [ see 3.8._Code.nb ], a G ω,e() Oω ϵ [, ) (3.64) Unlike the eriodic version (3.49), there is no singularity at ω. Siilar to the eriodic case, we now try to obtain (3.64) directly fro the ferionic version of (3.45). Setting ω, we have G a,e () Using f f - ω e-i ω f ω - π π ( - ) + -ω f - (3.65) becoes G a,e () f ω e-i ω f + f ω e-i ω f + f ω e-i ω f + ω cosω f f Caution: the su (3.66) for obvious reasons. Using ω f π + f ω e-i ω - - f e i ω f - f ω - f ω ei ω f in (3.66a) cannot be converted to the for - (3.65) (3.66a) as given in Kleinert s sinω f - sinω f cos π + + cosω f sin π + (3.66a) becoes (-) cosω f

14 4 3.8._ExternalSourceInQuantuStatisticalPathIntegral.nb G a,e () (-) ω sinω f f (-) f ω π n, 3, 5,... n, 3, 5,... - (3.66) (-) (n-) / f ω n! (-) + -n - n (-) (n-) / Consider [ see Abraowitz & Stegun, Forulae 3.. ] (-) β(n) n,, 3,... ( + ) n n! π - n (3.67) (-) (3.68) n + n which is related to a generalization of the Reiann zeta function called the Hurwitz zeta function ζ(s, q) Re s > & Re q > (3.69) ( + q) s Indeed, (-) ( + q ) s k ( k + q) - s ( k + + q) s - s k k + qs k + ( + q) s (3.68) thus becoes β(n) ζ s, q s - ζ s, ( + q) (3.7) n ζ n, 4 - ζ n, 3 4 (3.7) Near s, we have [ see G&R, ] ζ(s, q) - ψ(q) + O[s - ] (3.7) s - where the si function is defined as [G&R 8.36] ψ(x) d ln Γ(x) d x Hence, for n, (3.7) gives β() ζ, 4 - ζ, 3 4 Using [G&R ] -ψ 4 + ψ 3 4 (3.73a) ψ 4 -γ - π - 3 ln (3.74) ψ 3 4 -γ + π - 3 ln where γ Euler s constant

15 3.8._ExternalSourceInQuantuStatisticalPathIntegral.nb 5 (3.73a) becoes β() π 4 (3.73) in agreeent with [ A&S 3..3 ]. Caution: G&R use C to denote Euler s constant and set γ e C [see G&R, age xxviii ]. In the for of (3.68), β(-n) sees to be divergent. However, the for (3.7) gives finite results. Unfortunately, the clai that β(-n) for n, 3, 5,... does not hold nuerically [ see 3.8._Code.nb ]. Thus, (3.67) cannot be evaluated as described in Kleinert s text. Periodic Source () Using [see (3.46a)], G ω,e(, ') ω e -ω - ' + n (3.77) n - the source ter (3.44) can be written as e [] - M d - 4 M ω n M ω - For a eriodic source, n - 4 M ω n - (' - n ) () (3.76a) becoes e [] - 4 M ω - 4 M ω d ' () G ω,e(, ') (') d d ' () (') e -ω - ' + n (-n) d d '' () ('' + n ) e -ω -n - '' (+n) d d '' () ('' - n ) e -ω - '' (3.76a) n d n - d - n (n+) d ' () (') e -ω n Periodic or Anti-eriodic Potential Ω() - ' d ' () (') e -ω - ' (3.76) The following is sily the Euclidean version of the Wronski construction discussed in 3.5. The eq. for the Green function is [ c.f. (3.58a) ] -Ω, () G a Ω, e(, ') δ, a ( - ') (3.78) where [ c.f. (3.59) ] δ,a ( - ') n - (±) n δ( - ' - n ) (3.79) Note that excet for the defining eq, there is no exlicit aearance of t n in any eq. in 3.5. Furtherore, derivatives of t in the aear hoogeneously. Therefore, the Euclidean versions of the

16 6 3.8._ExternalSourceInQuantuStatisticalPathIntegral.nb results can be obtained by sily relacing t with without worrying about factors of -i. Fro (3.66), we get Δ(, a ) ± Δ( b, )Δ( b, ') ± Δ(', a ), G a Ω, e(, ') G Ω,e(, ') + Δ( b, a ) Δ,a ( a, b ) (3.8) where [see (3.59)] G Ω,e(, ') - Δ( b, a ) (Θ( - ') Δ( b, ) Δ(', a ) + Θ(' - ) Δ(, a ) Δ( b, ')) (3.8) and [see (3.64c-d) & (3.65)] W ξ η ξ η ξ η - ξ η (3.8a) Δ(, ') ξ() η(') - ξ(') η() (3.8) W Δ,a ( a, b ) ± [ b Δ( b, a ) + a Δ( a, b )] (3.83) ω For a tie-indeendent otential ω, the defining eq. is, (- -ω ) G a ω, e (, ') δ, a ( - ') (3.84a) Siilar to (3.45-6), we have G ω,e Since G ω,e G ω,e ( - ') ( - -ω ) - δ( - ') n - ( - -ω ) - n e -i ω ( - ' ) i ω - ω e-i ω ( - ' ) d μ π ei μ n i μ - ω e-i μ ( - ' ) e -ω ( - ' -n ) - ' - n > - ' - n < n - Θ ( - ' - n ) e -ω - ' -n C.W. lower contour closed in C.C.W. uer lane is eriodic, we need consider only the riary interval ϵ [, ) for which [c.f. (3.48) ] () n -ω ( + n ) e e -ω - e - ω e-ω + n b ω (3.84) e -ω e ω / e ω / - e - ω / e-ω ( - / ) Siilarly, following (3.63), we have a G ω,e () f i ω - ω e-i ωf - n - (-) n -ω -n Θ( - n ) e sinh( ω / ) (3.84a)

17 3.8._ExternalSourceInQuantuStatisticalPathIntegral.nb 7 (-) n e -ω ( + n ) ϵ [, ) n e -ω + e - ω e-ω f - n ω (3.85) e -ω e ω / e ω / + e - ω / e-ω ( - / ) (3.85a) cosh( ω / ) Ω() For a tie-deendent otential Ω(), the defining eq. is, [- -Ω() ] G a ω, e (, ') δ, a ( - ') (3.87) The analytic continuation of (3.) is G Ω (, ') Θ( - ') ex - ' d '' Ω('') (3.88) The eriodic & anti-eriodic versions are obtained by suerosition:, a (, ') G ω, e n - n (±) n G Ω ( - ' + n ) - ' ϵ [, ) (±) n ex - ' + n which reduces to (3.84-5) for Ω() ω. d '' Ω('') (3.89)