Girvin, Yang Modern Condensed Matter Physics 2019 SSP · Problem 3.15

Lattices and Symmetries

Physics — Condensed Matter · gaussian-distribution-moments-cumulants

Problem

Consider a variable

x

with range

(-\infty, +\infty)

and a Gaussian distribution function

p(x) = (1/\sqrt{2\pi}\sigma)e^{-x^2/(2\sigma^2)}

. (i) Check that

p(x)

is properly normalized. (ii) Calculate

\langle x^{2n} \rangle

for all positive integer

n

s. (iii) Show that

\langle e^{-i\alpha x} \rangle = e^{-\alpha^2 \langle x^2 \rangle /2}

. (iv) Expand

e^{-i\alpha x}

as a Taylor series in

x

, calculate the average for every single term, resum the series, and show that you recover the result obtained in (iii).