Does this hold in general ? (as an approximation only)

[tpm]

Does this hold in general ?? (as an approximation only)

for every real or pure complex number 'a' can we use as an approximation:

$$\sum _{n} exp(-aE_{n}) \sim \int_{-\infty}^{\infty} dx \int_{-\infty}^{\infty} dp exp(-ap^{2}-aV(x))$$

So for every x V(x) > 0 in case of real and positive a ... then i would like to know if this approximation could be useful to describe the 'Semi-classical behaviour' of the sum over energies (trace) replaced by an integral.

 

[StatMechGuy]

What you've written is more or less the classical partition function. It has all manner of problems at low temperatures (i.e. a goes to infinity), but as long as a is very small, this approximation works okay.