Problems 571∗14.10. For the discrete correlation functionGAB,P(t) defined in eqn. (14.6.8), we
could analyze the importance of the phase factor Φ(x 1 ,...,x 2 P) by calculating
its fluctuation
(δΦ)^2 ≡〈Φ^2 〉−〈Φ〉^2 =〈(Φ−〈Φ〉)^2 〉
with respect to an equilibrium discrete path integral consisting of 2Pimag-
inary time points. Using the path-integral virial theorem in eqn. (12.6.33),
derive a virial estimator for the above average.14.11. Derive analytical expressions for the imaginary-time mean-square displace-
ment of a free particle in 1, 2, and 3 dimensions. In particular, show that
inddimensions,R^2 (τ) is an inverted parabola, symmetric about the point
τ=β ̄h/2. For each number of dimensions, sketch the graph ofR^2 (τ) as a func-
tion ofτ. Finally, determine the numerical value ofR^2 (β ̄h/2) in angstroms
atT= 300 K for an electron and for a proton.