I've been trying to calculate the mean squared displacement of a particle confined to a one-dimensional box, and I managed to get an answer in terms of an infinite series of the basic form
∞∑n=1(−1)nn2exp(−an2).
I can't figure out if this series has a simple solution; I can't find it in any table of series, nor can I seem to expand the exponential and collect like terms without running into ∑(−1)2 terms. Does this have a solution or is this as far as I can go? Thanks in advance.
Answer
By using the Jacobian Theta function the resulting series can be placed into the form
f(a)=∞∑n=1(−1)nn2e−an2=a2−π212−12∫a0θ(12,iuπ)du.
This is developed by the following. Differentiate f(a) to obtain
f′(a)=∞∑n=1(−1)n+1e−an2.
Now,
θ(x,it)=1+2∞∑n=1e−πtn2cos(2nπx)
which yields
f′(a)=12(1−θ(12,iaπ)).
Integrating with respect to a yields
f(a)=a2−12∫a0θ(12,iuπ)du+c0
Since f(0)=−12ζ(2) then the presented result is obtained.
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