Orbital angular momentum as sum of harmonic oscillators
On section 7.3 of <PERSON>'s "Quantum Mechanics: A Modern Development" there is a really nice argument on why the eigenvalues of the total angular momentum operator must be integer. By defining the operators $$q_1=\frac{Q_x+P_y}{\sqrt{2}},\quad q_2=\frac{Q_x-P_y}{\sqrt{2}}, \quad p_1=\frac{P_x-Q_y}{\sqrt{2}}, \quad p_2=\frac{P_x+Q_y}{\sqrt{2}},$$ the projection of the orbital angular momentum can be written as $$L_z=Q_xP_y-Q_yP_x=\frac{1}{2}(p_1^2+q_1^2)-\frac{1}{2}(p_2^2+q_2^2),$$ i.e., a difference of harmonic oscillators. The reason we can say these are indeed independent harmonic oscillators is that the following commutation relations are satisfied: $$[q_1,q_2]=[p_1,p_2]=0, \quad [q_a,p_b]=i\delta_{ab}.$$
Expressing this operator in terms of ladder operators, we can easily see that its eigenvalues would be of the form $(n_1+1/2)-(n_2+1/2)=n_1-n_2$, which is always an integer. 669
This means the quantum number $m$ is an integer, which in turn implies $l$ is an integer.
If we make the change $p_2,q_2 \rightarrow iq_2,ip_2$, then we have $$L_z=\frac{1}{2}(p_1^2+q_1^2)+\frac{1}{2}(p_2^2+q_2^2),$$ and the commutation relations are still satisfied (if we make instead $p_2,q_2 \rightarrow ip_2,iq_2$, then there is an unwanted minus sign in the expression for $[q_a,p_b]$). As a footnote, I think this is unphysical since $ip_2$ and $iq_2$ are not hermitian, but probably the following still makes mathematical sense:
We know that the eigenfunctions of $L_z$ are spherical harmonics $Y_l^m(\theta,\phi)$. On the other hand, the eigenfunctions of the harmonic oscillator are Hermite polynomials (with some factors).
My questions are: If we write $L_z$ in terms of the operators ${q_i,p_i}$, can we say the eigenfunctions are products of Hermite polynomials (2D harmonic oscillator)? And if so, would this give a relation between the product of Hermite polynomials and spherical harmonics by means of a change of variables?
I hope there are no trivial mistakes in my reasoning. 669