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So, I have the problem of determining the spectrum of H and L, in terms of creation and annihilation operators of angular momentum... The problem goes along with what is happening on this page. However, my professor is asking us for the commutators of H and L with a quantity A, which is defined as $$A=\hat a_x^2+\hat a_y^2$$ When I do this out, writing H as $1+\hat a_x^{\dagger}\hat a_x+\hat a_y^{\dagger}\hat a_y$, I find that $$[H,A]=\hbar\omega(a_x^3+a_y^3)$$ I have no problem with this, except that when I want to construct an eigenket representation of $E_{nm}$in terms of the $\hat a_L^{\dagger}\text{ and } \hat a_R^{\dagger}$ and $A^{\dagger}$, I am not sure how to use this result to find this. I know that the result should be $E_{n,m}=\hbar\omega[2n+1+|m|]$ Thanks

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