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Quantum Mechanics Operators

  1. Mar 29, 2014 #1
    1. The problem statement, all variables and given/known data
    Given that [itex] \hat{p} = -i\hbar (\frac{\partial}{\partial r} + \frac{1}{r}) [/itex], show that [itex] \hat{p}^2 = -\frac{\hbar^2}{r^2} \frac{\partial}{\partial r}(r^2 \frac{\partial}{\partial r}) [/itex]


    2. Relevant equations

    Above

    3. The attempt at a solution
    I tried [itex]\hat{p}\hat{p} = -\hbar^2((\frac{\partial}{\partial r})^2 + \frac{1}{r} \frac{\partial}{\partial r} + \frac{\partial}{\partial r}\frac{1}{r} +\frac{1}{r^2}) [/itex].

    This gave me [itex] -\hbar^2((\frac{\partial}{\partial r})^2 + \frac{1}{r} \frac{\partial}{\partial r} )[/itex] instead of the 2 / r factor I needed.
     
    Last edited: Mar 29, 2014
  2. jcsd
  3. Mar 29, 2014 #2

    BvU

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    Nope. ##{\partial \over \partial r}{1\over r} ## gives ##{1\over r}{\partial \over \partial r} -{1\over r^2}##
    Remember p is an operator: you have to imagine there is something to the right of it to operate on.
     
  4. Mar 29, 2014 #3
    Thanks!

    I suppose it makes it easier if I had used a test function, and then taken it away.
     
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