# How to evaluate this nabla expression in spherical coordinates?

1. May 21, 2013

### Izzhov

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:

$(\vec{A}\cdot\nabla)\Psi$

where $\Psi$ is a scalar function of r, theta, and phi. How do you evaluate this expression in spherical coordinates in terms of the components of $\vec{A}$ and derivatives of $\Psi$?

2. May 21, 2013

### WannabeNewton

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 $A\cdot \nabla = A^{r}\partial_{r} + \frac{1}{r}A^{\theta}\partial_{\theta} + \frac{1}{r\sin\theta} A^{\phi}\partial_{\phi}$