= Two-dimensional oscillator level ordering
{title2=$1\leq\omega_2/\omega_1<3/2$}
For a two-dimensional <anisotropic quantum harmonic oscillator> in the indicated range, the six lowest <number states>, counted with multiplicity, have occupations $(0,0),(1,0),(0,1),(2,0),(1,1),(0,2)$. Their excitation <energies>, divided by $\hbar$, are $0,\omega_1,\omega_2,2\omega_1,\omega_1+\omega_2,2\omega_2$. These are ordered because $\omega_2<2\omega_1$ and $2\omega_2<3\omega_1$; every omitted state has at least three quanta and costs at least $3\hbar\omega_1$. At equal <angular frequencies>, the <two-dimensional isotropic harmonic oscillator> has <energy eigenvalue> $\hbar\omega(N+1)$ and <quantum degeneracy> $N+1$, since there are $N+1$ ordered nonnegative pairs with sum $N$.
Back to article page
