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Quantum, Spin, Orbital Angular momentum, operators

  1. Nov 25, 2013 #1
    1. The problem statement, all variables and given/known data

    If a particle has spin 1/2 and is in a state with orbital angular momentum L, there are two basis states with total z-component of angular momentum m*hbar l L,s,Lz,sz > which can be expressed in terms of the individual states ( l L,s,Lz,sz > = l L,Lz > l s,sz > ) as

    l a > = l L, 1/2 , m-1/2 , 1/2 > = l L, m-1/2 > l 1/2, 1/2 > = l L, m-1/2 > l alpha >
    l b > = l L, 1/2 , m+1/2 , -1/2 > = l L, m+1/2 > l 1/2, -1/2 > = l L, m-1/2 > l beta >

    Using these two states (l a > and l b >) as a basis, show that the matrix representation of the operator L.S is: where L.S = 1/2 ( L+S- + L-S+) + LzSz note all operators here with hats

    L.S = Hbar ^2 /2 { (m-1/2) [ (L +1/2)^2 - m^2 ]^1/2

    [ (L +1/2)^2 - m^2 ]^1/2 (m-1/2) }
    2. Relevant equations
    trying to find <a|L.S|a>, <a|L.S|b>, <b|L.S|a>, <b|L.S|b> (those are the matrix elements), by expanding L.S as in the equation, and then using the actions of the given operators on the basis vectors .





    3. The attempt at a solution

    <alpha|beta> = <beta|alpha> = 0

    LzSz |a> = Lz |l,m-> Sz |alpha> = hbar m- |l,m-> hbar/2 |alpha> ...

    am confused on how to contract with the other matrix terms,

    any help much appreciated
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 25, 2013 #2

    vela

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    You have the correct approach. What do the raising and lowering operators do to a state? For instance, what does ##\hat{L}_-## do to ##\lvert l, m \rangle##? You should be able to find this in your textbook if you don't already know.
     
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