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Quantum Mechanics and Statistical physics is a bit hard , could we then study only the WKB approximation ?

In the form:

replace $ \sum_{n=0}^{\infty}exp(- \beta E_{n})=Z(\beta)\sim\iint dxdpexp(-p^{2}\beta +\beta V(x)) $

and for the Eigenvalue staircase $ \sum_{n} \delta (E-E_{n})\sim \iint dxdp \delta (E-p^{2}-V(x))$

so it is easier to evaluate quantities :)

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May I suggest, adding the "soft-question" tag. –  Argus Jun 8 '12 at 21:58
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For what purpose? The first you said is classical stat-mech, the second is a uniform density on the energy hypersurface, and it's not the most accurate semi-classical approximation, it has no leading $\hbar$ corrections. –  Ron Maimon Jun 9 '12 at 2:08

1 Answer 1

No, these semiclassical formulas are no good. The first formula just gives the free energy as the volume of classical phase space, and this is incorrect, since, for instance, it predicts the specific heat of a cold gas is 1.5k independent of temperature, and this vanishes for cold quantum gasses as the discrete energy levels of the box the gas is contained in become apparent.

The second approximation is not even leading order semiclassical, it's just the level density formula. It does not predict the spacing of levels, or how they are distributed, and misses (for example) random matrix statistics of chaotic billiards and large nuclei vs. integrable level spacings of Rydberg atoms. It is also completely wrong for describing atoms near the ground state where the energy levels are discrete.

Of course you can't do semiclassical approximation--- the world is quantum.

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