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In many books I read about semiclassical approximation applied to the field of Bose-Einstein condensation. But I don't understand what it really means. For example I read that an expression like this $f_\textbf p (\textbf r) \frac{d^3\textbf rd^3\textbf p}{(2\pi \hbar)^3}$ for the state density. Have anybody a good explanation applied to this case? Or a general definition of semiclassical approximation?

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I recently updated the wikipedia article on statistical ensembles which might be relevant. Basically, in classical physics the probability distribution for the state of a system is written as an integral over position and momentum as in your equation. It turns out to be necessary to choose an arbitrary unit of action (energy times time) in order to define "one state" and make the units work out. It also turns out that if you make this action unit equal to Planck's constant, then the number of classical states contained in the ensemble is roughly the same as in quantum mechanics, at least in the limit when quantum mechanics is behaving classically. That is the semiclassical limit.

For a full explanation, note that your distribution function only has 6 coordinates, whereas a statistical ensemble would have 6N coordinates (3 momenta and 3 coordinates for each particle). In other words, even though they have a potentially complex multi-particle system, they are only tracking the distribution of single-particle parameters. This means they are not interested in all the correlations between different particles. Such an approach is useful if the particles are non-interacting, and also if they are weakly interacting in which case a sort of "molecular chaos" sets in, like the chaotic motions of particles in a gas.

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