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Homework Help: Linear combination of states

  1. Apr 23, 2003 #1
    The following linear combination of states is considered in almost all quantum mechanics textbooks when they try to explain the addition of spin 1/2 and orbital angular momentum. The thing I don't understand is how the left hand side is equal to the right. Please, if you can, explain how.

    |j,m + 1/2> = a|lm,+> + b|lm,-> where |+> is spin up and |-> is spin down.


  2. jcsd
  3. Apr 23, 2003 #2

    Tom Mattson

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    Write the spin and orbital states separately, and the total as a direct product of the two, as follows:


    Then use the ladder operators to work your way down.


    L=1, S=1/2


    Apply the ladder operator J-=L-+S-, noting that on the RHS the operator only acts on its respective ket.


    and keep working down.

    Try to work out the combination for |J=3/2,MJ=1/2>. If you get stuck, post what you came up with and I'll help you through it.
  4. Apr 23, 2003 #3

    Tom Mattson

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    What the hell, I'm feeling ambitious!




    Putting them together and solving for |3/2,1/2> yields:


    Try the rest, and let me know if you need help.

    Remember: Once you get to J=1/2, you will have a parity change. See me if you need help on that, too.
  5. Apr 23, 2003 #4
    Thanks Tom for your time and help.

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