Elementary Lectures in Statistical Mechanics pp 208-214 | Cite as

# Kirkwood-Wigner Theorem

Chapter

## Abstract

This Aside treats quantum corrections to the partition function of a nearly classical system. The argument given here follows very closely the derivation of Kirkwood [1], who showed that the procedure described below may be applied recursively to obtain corrections of arbitrarily high order in Planck’s constant

*h*.From our point of view, the work has two major consequences for the constant*C*of the canonical-ensemble statistical weight, namely that:- (1)
the expectation of classical statistical mechanics agrees with the quantum expectation at low order in

*h*; and - (2)
*C*is shown from quantum mechanics to contain an extra factor*h*^{ − }^{3N }, 3*N*being the number of canonical coordinates (or canonically conjugate momenta) in the system.

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## References

- [1]J. G. Kirkwood,
*Phys. Rev.***44**, 31 (1933);**45**, 116 (1934). Note also F. H. Stillinger’sIntroduction to*John Gamble Kirkwood Collected Works*, Volume 1.*Quantum Statistics and Collective Phenomena*, Gordon and Breach, New York (1965).ADSCrossRefGoogle Scholar

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