Rotating harmonic oscillator: Violation of an equipartition theorem

The rotating harmonic oscillator in three dimensions (owing its relevance to the diatomic molecule) has been reinvestigated by Nieto and Gutschick. They have demonstrated that the ground-state energy, which is expected (on the basis of naive arguments involving the uncertainty principle) to acquire a contribution $\frac{1}{2}\hslash \omega$ from each degree of freedom, and hence should have a value $\frac{3}{2}\hslash \omega$, does, on the contrary, in the limit of large equilibrium separation (between the atoms), go to the asymptotic value of only $\frac{1}{2}\hslash \omega$, thereby violating what these authors called a "quantum folk theorem." The present Brief Report consists of showing that these conclusions can be elucidated analytically and further extended through the $\frac{1}{N}$ expansion ($N$ being the dimensionality of the space), leading, in particular, to the result that for the case $N=2\mathrm{}$ the approach to the asymptotic limit $\left(\frac{1}{2}\hslash \omega \right)$ is, surprisingly, from below, thereby providing an even stronger violation of this type of "equipartition theorem."