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Total Angular Momentum Measurements

  1. Oct 23, 2014 #1
    1. The problem statement, all variables and given/known data
    Consider a particle with orbital momentum ##l=1## and spin ##s = 1/2## to be in the state described by
    $$\Psi = \frac{1}{\sqrt{5}}| 1,1\rangle|\downarrow\rangle+\frac{2}{\sqrt{5}}|1,0\rangle|\uparrow\rangle$$

    If the total angular momentum is measured what would be the possible outcomes? What are the corresponding probabilities?

    2. Relevant equations
    ##\mathbf{J} = \mathbf{L}+\mathbf{S}##

    3. The attempt at a solution
    Okay, so on the surface this seems pretty simple, but I want to make sure that I am not thinking about this wrong.

    For the first state: ##J=1-1/2 =1/2## with probability ##1/5## and the second state ##J=1+1/2=3/2## with probability ##4/5##. Is this correct?
     
  2. jcsd
  3. Oct 23, 2014 #2

    vela

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    No, it's not correct. You need to express the state in terms of eigenstates of ##J^2##.
     
  4. Oct 23, 2014 #3
    On second looks, it's not as easy as I thought. For some reason I thought it was ##|j,m\rangle## and instead it is ##|l,m\rangle|s_z\rangle##

    What would be a good way to approach this problem? In thinking about it again, I need to determine ##j##, but I am not sure how to go about doing so.
     
  5. Oct 23, 2014 #4

    vela

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    Look up the appropriate Clebsch-Gordon coefficients.
     
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