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.

 

[denian]

I cannot give a definitive answer without further context and information about the specific system and approximation being considered. However, in general, using an integral to approximate a sum can be a useful tool in certain cases. It is commonly used in physics and other sciences to simplify complex systems and make calculations more manageable. However, whether this specific approximation holds in general will depend on the specific system and the range of values for 'a' and 'V(x)' being considered. More rigorous analysis and testing would be necessary to determine the validity and usefulness of this approximation.