How Do You Apply the Del Operator to a Momentum-Dependent Wave Function?

[apenn121]

I've been given the question "What is ∇exp(ip⋅r/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. £

 

[blue_leaf77]

so I was wondering if it would be correct in writing that it is equal to prcosθ

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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[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 $i\vec p \cdot \vec r = i (p_x f_x(x,y,z)+p_y f_y(x,y,z ) + p_z f_z(x,y,z))$
So you gradient would be
$\nabla W = \hat x (i \frac{\partial}{\partial x} (p_x f_x(x,y,z)+p_y f_y(x,y,z ) + p_z f_z(x,y,z)))W + \hat y (i \frac{\partial}{\partial y} (p_x f_x(x,y,z)+p_y f_y(x,y,z ) + p_z f_z(x,y,z)))W + \hat z (i \frac{\partial}{\partial z} (p_x f_x(x,y,z)+p_y f_y(x,y,z ) + p_z f_z(x,y,z)))W$
In many cases, the functions of position might only depend on one of the spatial variables which can simplify the derivatives.