Two-mode oscillator Casimir identity
= Two-mode oscillator Casimir identity
{title2=$H^2=4\omega^2J^2+\hbar^2\omega^2$}
For the <two-dimensional isotropic harmonic oscillator>, $H=\hbar\omega(N+1)$ and the <Schwinger boson representation> gives $J^2=\hbar^2N(N+2)/4$. Substitution yields the displayed relation, including its quantum constant term. The fixed-total-occupation sector is a spin-$j=N/2$ representation, so <energy> is $\hbar\omega(2j+1)$ with degeneracy $2j+1$.