 Cartesian and spherical bases of the first oscillator shell

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 ${\varphi }_x,{\varphi }_y,{\varphi }_z$ 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 $x\pm iy=r\sin \theta e^{\pm i\phi }$ identify a common radial factor times the spherical harmonics $Y_{10},Y_{1,\pm 1}$. The transformation is unitary, so it changes the basis rather than the threefold orbital degeneracy.