We have two solutions for the 3D harmonic oscillator, one is in Cartesian coordinate, and the other is in spherical coordinate.

\Psi^C_{n_x n_y n_z}(x,y,z) = N_C H_{n_x}(x)H_{n_y}(y)H_{n_z}(z) \exp(-r^2/2)

\Psi^S_{n, l, m}(r,\theta,\phi) = N_S r^l \exp(-r^2/2) L_{\frac{n-l}{2}}^{l+\frac{1}{2}}(r^2)

We can project the \Psi^C onto \Psi^S or,

\displaystyle \Psi^C_{n_x n_y n_z} = \sum_{l,m} C_{lm} \Psi^S_{nlm}

Replace (x,y,z) \rightarrow (r, \theta, \phi) in the integration.Here are some results

n = 0 \rightarrow C_{00} = 1

(n_x, n_y, n_z) = (0,0,1) \rightarrow C_{1,0} = 1

(n_x, n_y, n_z) = (1,0,0) \rightarrow C_{1,-1} = 1/\sqrt{2} , C_{1,1}= -1/\sqrt{2}

(n_x, n_y, n_z) = (0,1,0) \rightarrow C_{1,-1} = -i/\sqrt{2} , C_{1,1}= i/\sqrt{2}

(n_x, n_y, n_z) = (0,0,2) \rightarrow C_{0,0} = -1/\sqrt{3} , C_{2,0}= \sqrt{2/3}

(n_x, n_y, n_z) = (1,0,1) \rightarrow C_{2,-1} = 1/\sqrt{2} , C_{2,1}= 1/\sqrt{2}

(n_x, n_y, n_z) = (1,1,0) \rightarrow C_{2,-2} = -i/\sqrt{2} , C_{2,2}= i/\sqrt{2}

(n_x, n_y, n_z) = (0,0,3) \rightarrow C_{1,0} = -\sqrt{3/5} , C_{3,0}= \sqrt{2/5}

(n_x, n_y, n_z) = (0,0,4) \rightarrow C_{0,0} = \sqrt{1/5} , C_{2,0}= -\sqrt{4/7} , C_{4,0}= \sqrt{8/35}

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