On the Notion of Pressure in a Canonical Ensemble

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Abstract
Recently a controversy has arisen between J. De Boer and H. S. Green concerning the notion of pressure in a canonical ensemble. According to Green, only classically is the pressure as derived from the partition function equal to that obtained from the virial theorem, while at low temperatures, at which quantum effects become important, there will be considerable deviations between the two. De Boer attempts to prove that the two pressures are actually identical. We have come to the same conclusion, and shall show this in several ways; first, by considering a simple example (Section 2), and then in general using the energy representation (Section 3). We believe that the discrepancy between the two pressures which Green has found is in fact due to improper handling of the effect of the wall of the vessel in which the particles are contained. Finally, we are of the opinion that Green's criticism of De Boer's calculation is not justified, and in the last section arguments are given to show that the traces of all commutators of interest in quantum statistical mechanics are zero.

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