# Semiclassical approximation for the exponential sum (partition function)

For any real and positive $s$ and in the sense of semiclassical approximation, is this valid?

$\sum_{n}exp(-sE_{n})\sim \iint_{C}dxdpe^{-s(p^{2}+V(x))}$ valid for every $s$

Here simply both the sum and integral are convergent, so in this case the approximation can be seen as the replacement of a sum by an integral.

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It looks like you assume $\hbar=1$ and $m=1$. If not then what is $p$ here? If $s$ has dimension of inverse energy then $p^2$ must have dimension of energy. But since the sum is dimensionless $p$ must have dimension of inverse length to compensate $dx$. And what is $C$? Is it the contour on the $(x,p)$ plain that correspond to quasiclassical trajectory of the upper state? – Maksim Zholudev Dec 20 '11 at 20:45
I mean, the two sides are clearly semiclassically equal. What more than yes can you say? – Ron Maimon Dec 21 '11 at 3:48
thanks to your comments, yes i forgot Maksim , i am using units where $2m=1= \hbar$ – Jose Javier Garcia Dec 21 '11 at 11:12
The answer is "yes". If you want someone to derive the formula here it is a do-my-homework question. If you tried to do it and have encountered some difficulties, please describe them in the question. – Maksim Zholudev Dec 21 '11 at 13:08