A class $O$ of systems of noninteracting quantum-mechanical particles is defined, so that it is often justifiable to approximate a given system of particles as a system of class $O$. A "limit $L$" is defined as the limiting case which arises of the number of particles $N$ and the volume $V$ of the system are both allowed to tend to infinity so as to keep the ratio $\frac{N}{V}$ a finite and nonzero constant. It is shown that if one wishes to calculate the mean value of an extensive variable, as averaged over a canonical or over a grand canonical ensemble, for a system of class $O$ in the limit $L$, one can apply the continuous spectrum approximation directly to the particle quantum states (excepting, possibly, the states which belong to the lowest energy level) without using a limiting process.

• Received 14 December 1953

DOI:https://doi.org/10.1103/PhysRev.94.469