U(0)=0 for real expectation values of momentum

[rtellez700]

Homework Statement

The position-space representation of the radial component of the momentum operator is given by
$p_r \rightarrow \frac{\hbar}{i}\left ( \frac{\partial }{\partial r} + \frac{1}{r}\right )$

Show that for its expectation value to be real:$\left \langle \psi|p_r|\psi \right \rangle = \left \langle \psi|p_r|\psi \right \rangle ^{*}$, the radial wave function must satisfy the condition $u(0)=0$. Suggestion: Express the expectation value in position space in spherical coordinates and integrate by parts.

Homework Equations

$u(r)=r*R(r)$

The Attempt at a Solution

I think this can be solved for a general \psi but I'm having a hard time figuring out where the integration by parts would even come into play. Any insight on how to approach this problem would be appreciated.

 

[mfb]

Suggestion: Express the expectation value in position space in spherical coordinates

Where did you follow that suggestion?