= Solution
Write $\alpha=m\omega/\hbar$. 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 $x_1e^{-\alpha(x_1^2+x_2^2)/2}$ and $x_2e^{-\alpha(x_1^2+x_2^2)/2}$. Thus in <Cartesian coordinates> the new <wavefunctions> are
$$
\boxed{\chi_0\propto e^{-\alpha(x_1^2+x_2^2)/2},\qquad
\chi_\pm\propto(x_1\pm ix_2)e^{-\alpha(x_1^2+x_2^2)/2}.}
$$
Using <polar coordinates> $x_1=r\cos\varphi$, $x_2=r\sin\varphi$ gives
$$
\boxed{\chi_0\propto e^{-\alpha r^2/2},\qquad
\chi_\pm\propto r e^{\pm i\varphi}e^{-\alpha r^2/2}.}
$$
The <orbital angular momentum> is $L_3=-i\hbar\partial_\varphi$, which immediately verifies its <eigenvalues> $0,\pm\hbar$. 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>.
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