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

  1. Dec 3, 2014 #1
    1. The problem statement, all variables and given/known data
    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.

    2. Relevant equations
    ##u(r)=r*R(r)##

    3. 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.
     
  2. jcsd
  3. Dec 3, 2014 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Where did you follow that suggestion?
     
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