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How to evaluate this nabla expression in spherical coordinates?

  1. May 21, 2013 #1
    I'm currently working out the Schrödinger equation for a proton in a constant magnetic field for a research project, and while computing the Hamiltonian I came across this expression:

    [itex](\vec{A}\cdot\nabla)\Psi[/itex]

    where [itex]\Psi[/itex] is a scalar function of r, theta, and phi. How do you evaluate this expression in spherical coordinates in terms of the components of [itex]\vec{A}[/itex] and derivatives of [itex]\Psi[/itex]?
     
  2. jcsd
  3. May 21, 2013 #2

    WannabeNewton

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    We have in spherical coordinates, ##\nabla = \hat{r}\partial_{r} + \frac{1}{r}\hat{\theta}\partial_{\theta} + \frac{1}{r\sin\theta} \hat{\phi}\partial_{\phi}## and ##A = A^{r}\hat{r} + A^{\theta}\hat{\theta} + A^{\phi}\hat{\phi}## so the operator is just ##A\cdot \nabla = A^{r}\partial_{r} + \frac{1}{r}A^{\theta}\partial_{\theta} + \frac{1}{r\sin\theta} A^{\phi}\partial_{\phi}##
     
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