554 Quantum time-dependent statistical mechanics
to make contact with the treatment of Section 14.3, we sett 0 =−∞in eqn. (14.5.9)
to obtain
〈Aˆ〉t=〈Aˆ〉+∫t−∞dsΦAV(t−s)Fe(s). (14.5.11)According to eqn. (14.5.11), whenVˆ is the operator we choose to measure in the
non-equilibrium ensemble, we find
〈Vˆ〉t=〈Vˆ〉+∫t−∞dsΦVV(t−s)Fe(s), (14.5.12)which involves a quantum autocorrelation function ofVˆ.
We now consider the special case of a monochromatic field of frequencyω, for
whichFe(t) =F(ω) exp(−iωt). Substituting this field form into eqn. (14.5.12) yields
〈Vˆ〉t=〈Vˆ〉+F(ω)∫t−∞dsΦVV(t−s)e−iωs. (14.5.13)Because the lower limit of the time integral in eqn. (14.5.13) is−∞, it is necessary to
ensure that the potentially oscillatory integrand yields a convergent result. Thus, we
multiply the integrand by aconvergence factorexp(ǫt), which decays to 0 ast→−∞.
After the integral is performed, the limitǫ→ 0 +(the limit thatǫapproaches 0 from the
positive side) is taken. The use of convergence factors is a formaldevice, the necessity
of which depends on the behavior of the autocorrelation function.For nearly perfect
solids and glassy systems, one would expect the decay of the correlation function to
be very slow, requiring the use of the convergence factor. For ordinary liquids, the
correlation function should decay rapidly to zero, obviating the need for this factor.
For generality, we retain it in the present discussion. Introducing aconvergence factor
into eqn. (14.5.13) gives
〈Vˆ〉t=〈Vˆ〉+F(ω) lim
ǫ→ 0 +∫t−∞dsΦVV(t−s)e−iωseǫs. (14.5.14)We now change the integration variables in eqn. (14.5.14) fromttoτ=t−s, which
yields
〈Vˆ〉t=〈Vˆ〉+F(ω) lim
ǫ→ 0 +e(iω+ǫ)t∫∞
0dτΦVV(τ)e−(iω+ǫ)τ. (14.5.15)Eqn. (14.5.15) involves a Fourier–Laplace transform of the after-effect function at a
complex variablez=ω−iǫ. Let the functionχVV(z) denote this Laplace transform
(see Appendix D)
χVV(z) =∫∞
0dτΦVV(τ)e−izτ, (14.5.16)which is referred to as thesusceptibility. Eqn. (14.5.15) can now be expressed as
〈Vˆ〉t=〈Vˆ〉+F(ω) lim
ǫ→ 0 +ei(ω−iǫ)tχVV(ω−iǫ). (14.5.17)