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Question on the Hydrogen atom

  1. Jul 26, 2011 #1
    1. The problem statement, all variables and given/known data
    Use
    [tex][H_{0},r_{j}]=\frac{i\hbar}{\mu}p_{j}[/tex]
    for the Hydrogen atom (where the j's denote the jth components in Cartesian coordinates) to prove that
    [tex]<n_{f},l_{f},m_{l,f}|p_{j}|n_{i},l_{i},m_{l,i}>=-i\mu\omega<n_{f},l_{f},m_{l,f}|r_{j}|n_{i},l_{i},m_{l,i}>[/tex]


    2. Relevant equations
    [tex][H_{0},r_{j}]=\frac{i\hbar}{\mu}p_{j}[/tex]

    3. The attempt at a solution
    I'm really at a loss on how to begin here. I don't see how I can use the commutator to prove this.
     
  2. jcsd
  3. Jul 26, 2011 #2
    Just substitute for [itex]p_{j}[/itex] the communtator [itex][H_{0},r_{j}][/itex]
    [tex]<n_{f},l_{f},m_{l,f}|p_{j}|n_{i},l_{i},m_{l,i}> = \frac{\mu}{i\hbar}<n_{f},l_{f},m_{l,f}|[H, r_j]|n_{i},l_{i},m_{l,i}>[/tex]
    and take it from there. Remember that [itex]H[/itex] is Hermitian and what its eigenstates and eigenvalues are.
     
  4. Jul 27, 2011 #3
    Thanks a lot mathfeel. It took me a while when you mentioned the hermicity of the Hamiltonian, but after staring it down for a straight 5 minutes I felt dumb since it's so obvious, hahaha.
     
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