= Cartesian and spherical bases of the first oscillator shell
{title2=$\phi_{+1}=-(\phi_x+i\phi_y)/\sqrt2,\quad\phi_0=\phi_z,\quad\phi_{-1}=(\phi_x-i\phi_y)/\sqrt2$}
In the isotropic three-dimensional <quantum harmonic oscillator>, the first excited Cartesian <energy eigenstates> are proportional to $x,y,z$ times the common Gaussian $e^{-r^2/(2a_0^2)}$. If $\phi_x,\phi_y,\phi_z$ are normalized with the same real phase convention, their combinations in the title are normalized <angular momentum eigenstates> with $l=1$ and $m=1,0,-1$.
Indeed, $z=r\cos\theta$ and $x\pm iy=r\sin\theta e^{\pm i\varphi}$ identify a common radial factor times the <spherical harmonics> $Y_{10},Y_{1,\pm1}$. The transformation is unitary, so it changes the basis rather than the threefold orbital degeneracy.
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