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Energies of a two particle system

  1. Jan 18, 2013 #1
    Hey,

    This question is on determining the energies of a two particle system given the Hamiltonian, I believe it to be simple enough but would like you guys to check it and fill in any gaps in my reasoning

    Hamiltonian.png

    So I believe the eigenvalues of J^2 and J^2(z) are given by:

    [tex]\hat{J}^{2}:j(j+1)\: ,\: \hat{J}^{2}_{z}:m(m+1)\Leftrightarrow \hbar=1[/tex]

    ('z' subscript same as '3')

    and so the energy of state 1,1 is :

    [tex]2(\alpha+\beta)[/tex]

    The second part state that j=3, therefore m=3,2,1,0,-1,-2,-3
    and so we just pop these into our eigenvalue equations above to attain the energies :

    [tex]|3,3> : 12\alpha+12\beta\: ,\: |3,2>:12\alpha+6\beta[/tex]

    etc.

    Is this right?

    Thanks for any comment/help!
    SK
     
    Last edited: Jan 18, 2013
  2. jcsd
  3. Jan 18, 2013 #2

    G01

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    There is a mistake in your work. What is the eigenvalue of the Jz operator?
     
  4. Jan 18, 2013 #3
    The eigenvalue of the Jz operator is 'm', so does that mean the eigenvalue of the Jz^2 operator is m(m+1)?

    Oh and 'z' is the same as '3' for the subscripts!
     
  5. Jan 18, 2013 #4

    G01

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    No. If [itex]J_z|j,m>=m|j,m>[/itex] then what is

    [tex]J_z^2|j,m>=J_z(J_z|j,m>)=?[/tex]
     
  6. Jan 18, 2013 #5
    Oh, m^2?
     
  7. Jan 18, 2013 #6

    G01

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    Yep. :smile:
     
  8. Jan 18, 2013 #7
    Oh ok, well I just assumed it was the same as the J^2 eigenvalue j(j+1), does the J^2 operator raise the state by 1 then?
     
  9. Jan 18, 2013 #8

    G01

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    You are confused about the J^2 operator. It is a total angular momentum operator (squared), not a ladder operator. The eigenvalues for the J^2 operator are correct as you have them: [itex] J^2 |j,m>=j(j+1)|j,m>[/itex]

    If you don't understand why the J^2 operator has a different eigenvalues than the J_z operator, you should review the derivation of these eigenvalue equations in your textbook or with your instructor.
     
  10. Jan 18, 2013 #9
    It's not that I don't understand why they're different but more why the J^2 is equal to j(j+1), I'm sure we've 'shown' it somewhat before but these things are easily forgotten by myself.

    Thanks anyway G01 for being patient with me and helping!
    SK
     
  11. Jan 18, 2013 #10

    G01

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    You're welcome!
     
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