I have to prove that [itex]\left\langle lm | \vec{\hat{r}} \times \vec{\hat{p}} | lm\right \rangle = \left\langle lm | \vec{\hat{p}} \times \vec{\hat{r}} | lm \right \rangle[/itex], where [itex] | lm \rangle [/itex] are eigenkets of angular momentum operator [itex] \hat{L}^2 [/itex](adsbygoogle = window.adsbygoogle || []).push({});

And I can't figure out a way to do this correctly. I wrote the angular momentum operator-vector in terms of its components, [itex] \hat{L_x} [/itex], [itex] \hat{L_y} [/itex] and [itex] \hat{L_z} [/itex], and only the [itex] \hat{k} [/itex] component survives the bra-ket operation, because I can write [itex] \vec{L_x} [/itex] y [itex] \vec{L_y} [/itex] in terms of ladder operators, and after lowering and rising the eigenstates, the corresponding eigenkets (except for the [itex] \hat{k} [/itex] component) cancel with [itex] \langle lm | [/itex] because of the orthogonality property of these eigenkets. Probably there is a wrong sign in the problem statement. Looks trivial in that case, but I want to know your opinion.

Thanks in advance for any advice.

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# Angular momentum theory problem probably wrong sign

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