Cartesian number state of the three-dimensional isotropic harmonic oscillator
= Cartesian number state of the three-dimensional isotropic harmonic oscillator
{title2=$|n_1,n_2,n_3\rangle$}
From the normalized <ground state> annihilated by every $A_i$, the normalized Cartesian number states are
$$
|n_1,n_2,n_3\rangle
=\prod_{i=1}^3\frac{(A_i^\dagger)^{n_i}}{\sqrt{n_i!}}|0\rangle.
$$
They have energy $\hbar\omega(n_1+n_2+n_3+3/2)$.