N-dimensional harmonic oscillator yields monotonic series for the mathematical constant π

摘要

Recently, Mavromatis (1990) has shown that the usual quantum mechanical ideas of matrix elements, expansion in complete sets, and the like, coupled to the three-dimensional harmonic oscillator problem, lead to an infinite set of series expansions for the mathematical constant π. In this paper his results are extended by considering a harmonic oscillator in a general space of N dimensions. It is found that in each odd dimension one obtains series for π, while in each even dimension, series for 1/π.