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Show that it is a energy eigenstate and find the corresponding energy

  1. Apr 29, 2013 #1
    1. The problem statement, all variables and given/known data
    Hi.

    I'm looking at a hydrogen atom, which normalized stationary states is defined as |nlm>

    The hydrogen atom is described by the normalized wavefunction:
    [tex]\left| \psi \right\rangle =\frac{1}{\sqrt{2}}\left( \left| 210 \right\rangle +\left| 211 \right\rangle \right)[/tex]
    Now, show that [itex]\left| \psi \right\rangle[/itex] is an energy eigenstate, and find the corresponding energy.


    2. Relevant equations

    I'm told that:

    [tex]{{L}^{2}}\left| nlm \right\rangle =l\left( l+1 \right){{\hbar }^{2}}\left| nlm \right\rangle[/tex]
    [tex]{{L}_{z}}\left| nlm \right\rangle =m\hbar \left| nlm \right\rangle[/tex]
    [tex]{{L}_{+}}={{L}_{x}}+i{{L}_{y}}[/tex]
    [tex]{{L}_{-}}={{L}_{x}}-i{{L}_{y}}[/tex]


    3. The attempt at a solution

    In my mind, it seems so easy, but I don't have my book at my side, so I can't even check how it is done. Does it has something to do with:
    [tex]H\left| \psi \right\rangle =E\left| \psi \right\rangle[/tex]

    If so, what hamiltonian am I suppose to use ?

    Well, I'm kinda lost right now, so I was hoping to get a push in the right direction.

    Thanks in advance.


    Regards
     
  2. jcsd
  3. Apr 30, 2013 #2

    Simon Bridge

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    Yes - it is an energy eigenstate if it is an eigenvector of the Hamiltonian.
    You will need the Hamiltonian for the hydrogen potential.

    Note: out of the quantum numbers shown in the state vectors, which refer to the energy eigenstates?
     
  4. Apr 30, 2013 #3
    Ahh yes, thank you very much :D
     
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