 Solution

Write $\alpha =m\omega /\hslash$. The first Hermite polynomials give ${\psi }_{0,0}\propto e^{-\alpha (x_1^2+x_2^2)/2}$ and ${\psi }_{1,0},{\psi }_{0,1}$ proportional, with the same constant, to $x_1{e}^{-\alpha (x_1^2+x_2^2)/2}$ and $x_2{e}^{-\alpha (x_1^2+x_2^2)/2}$. Thus in Cartesian coordinates the new wavefunctions areUsing polar coordinates $x_1=r\cos \phi$, $x_2=r\sin \phi$ givesThe orbital angular momentum is $L_3=-i\hslash {\partial }_{\phi }$, which immediately verifies its eigenvalues $0,\pm \hslash$. Multiplying these spatial wavefunctions by $e^{-i\omega t}$ for ${\chi }_0$ and $e^{-2i\omega t}$ for ${\chi }_{\pm }$ gives the corresponding full stationary states.