# Hermitian operators in spherical coordinates

1. Nov 3, 2007

### noospace

Hi,

I have a general question. How do I show that an operator expressed in spherical coordinates is Hermitian? e.g. suppose i have the operator $i \partial /\partial \phi$. If the operator was a function of x I know exactly what to do, just check

$\int_\mathbb{R} \psi_l^\ast \hat{A} \psi_m dx = \int_\mathbb{R} (\hat{A} \psi_l)^\ast \psi_m dx$.

Do i need to check that

$\int\int\int Y_{lm}^\ast \hat{A} Y_{lm} r^2\sin\theta d\theta\d\phi dr= \int\int\int (\hat{A}Y_{lm})^\ast Y_{lm} r^2\sin\theta d\theta\d\phi dr$

or is there a simpler way?

[EDIT: I figured it out. The operator I have has no r or theta dependence so the integral is trivial.]

Last edited: Nov 3, 2007