Smooth Curve Approximation to the Energy-Level Distribution for Quantum Harmonic Oscillators

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Abstract
It is pointed out that the distribution of energy levels for a set of quantum oscillators may be expressed exactly as the inverse Laplace transform of a function closely related to the quantum partition function. This observation leads quite easily to a smooth curve approximation for the energy‐level distribution which can be expressed in terms of a generalized Bernoulli polynomial. A connection between this approximation and those given by Schlag and Sandsmark and by Whitten and Rabinovitch is established. A detailed comparison with Whitten and Rabinovitch's results is made for the special case of degenerate oscillators.