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Hermitian operators in spherical coordinates

  1. Nov 3, 2007 #1
    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 [itex]i \partial /\partial \phi[/itex]. If the operator was a function of x I know exactly what to do, just check

    [itex]\int_\mathbb{R} \psi_l^\ast \hat{A} \psi_m dx = \int_\mathbb{R} (\hat{A} \psi_l)^\ast \psi_m dx[/itex].

    Do i need to check that

    [itex]\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[/itex]

    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
  2. jcsd
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