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Find states for the angular momentum j = 2 from two independent subsystems with j = 1

  1. Jan 3, 2012 #1
    1. The problem statement, all variables and given/known data
    There are two independent subsystems with angular momenta j1 = j2 = 1. States have to be found for the whole system with angular momentum j = 2.

    2. Relevant equations

    Basic procedure for addition of Angular Momentum in Quantum Mechanics

    3. The attempt at a solution

    Basically j = j1 + j2

    I go through the states starting with 2 using the equations in the guide.
    So basically we are limited to states
    |2, 2>
    |2, 1>, |1, 1>
    |2, 0>, |1, 0>, |0, 0>

    But I'm not sure if it's the right way to go through the states. Also, maybe there are better guides since I'm not even sure if this one applies to the problem. I'm looking into Clebsch–Gordan coefficients at the moment.
     
    Last edited by a moderator: Mar 4, 2015
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  3. Jan 4, 2012 #2

    vela

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    Re: Find states for the angular momentum j = 2 from two independent subsystems with j

    What do you mean "the right way to go through the states"?
     
  4. Jan 4, 2012 #3
    Re: Find states for the angular momentum j = 2 from two independent subsystems with j

    Well I have found that there are 2*2+1 = 9 states which are |2, 2>, |2, 1>, |2, 0>, |2, -1>, |2, -2>, |1, 1>, |1, 0>, |1, -1>, |0, 0>.
    In 'big O' notation,
    1X1 = 2+1+0+|-1|+|-2|
    3*3 = 9
    So these seem to be all the possible states. I'm not sure if it means I have found all the states. I'm not sure how they have come up with
    \ket{j, j-1} = \sqrt{j_1 \over j} \ket{j_1, j_1-1, j_2, j_2} + \sqrt{j_2 \over j} \ket{j_1, j_1, j_2, j_2 - 1}
     
  5. Jan 4, 2012 #4

    vela

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    Re: Find states for the angular momentum j = 2 from two independent subsystems with j

    The states with total angular momentum j=2 are |2, 2>, |2, 1>, |2, 0>, |2, -1>, and |2, -2>. You're probably expected to find them in terms of the eigenstates of J1 and J2.
     
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