# Del operator and wave function

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1. Aug 28, 2015

### apenn121

I've been given the question "What is ∇exp(ipr/ħ) ?"

I recognise that this is the del operator acting on a wave function but using the dot product of momentum and position in the wave function is new to me. The dot product is always scalar so I was wondering if it would be correct in writing that it is equal to prcosθ and then using the spherical coordinate del operator on the wave function to find the result. £

2. Aug 28, 2015

### blue_leaf77

If you do that, you will be assuming the momentum is directed toward z axis, the result will not be so general then. Just expand the dot product in terms of Cartesian components.
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3. Aug 28, 2015

### RUber

In Cartesian coordinates, you will often see something like:
$\vec p = \hat x p_x+ \hat y p_y+\hat z p_z$ and $\vec r = \hat x f_x(x,y,z)+ \hat y f_y(x,y,z)+\hat z f_z(x,y,z)$
Then $i\vec p \cdot \vec r = i (p_x f_x(x,y,z)+p_y f_y(x,y,z ) + p_z f_z(x,y,z))$
$\nabla W = \hat x (i \frac{\partial}{\partial x} (p_x f_x(x,y,z)+p_y f_y(x,y,z ) + p_z f_z(x,y,z)))W + \hat y (i \frac{\partial}{\partial y} (p_x f_x(x,y,z)+p_y f_y(x,y,z ) + p_z f_z(x,y,z)))W + \hat z (i \frac{\partial}{\partial z} (p_x f_x(x,y,z)+p_y f_y(x,y,z ) + p_z f_z(x,y,z)))W$