1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Addition of angular momentum - Finding the second tower states

  1. Nov 10, 2014 #1

    DataGG

    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data

    I'm supposed to calculate all the states for a system with ##l=1## and ##s=1/2##. Let's say ##\vec{J} = \vec{L} + \vec{S}##. I want to find the Klebsch-Gordon coefficients.

    I know that said system has 2 towers, one with ##j=3/2## and the other with ##j=1/2##. I've calculated all the states for ##j=3/2## but now I'm having some problems with ##j=1/2##.

    So, for the second tower, we've two states: ##|j,j_z>=|1/2, 1/2>## and ##|j,j_z>=|1/2, -1/2>##

    How am I supposed to find ##|j,j_z>=|1/2, 1/2>##? If I do that, finding ##|j,j_z>=|1/2, -1/2>## should be easily done by applying the ##J _## operator.

    2. Relevant equations

    $$J _ |j, j_z>=\hbar \sqrt{j(j+1)-j_z(j_z-1)}|j,j_z-1>$$
    $$S _ |s, s_z>=\hbar \sqrt{s(s+1)-s_z(s_z-1)}|s,s_z-1>$$
    $$L _ |j, j_z>=\hbar \sqrt{l(l+1)-l_z(l_z-1)}|l,l_z-1>$$
    3. The attempt at a solution

    Well, I've done well for the tower with ##j=3/2##. Now with this second tower, I don't know where to begin from. I think this is because for ##j = j_z = 3/2##, we know that ##j_z = l_z + s_z## which means ##l_z =1 ## and ##s_z = 1/2##. There's no other way.

    For the case with ##j=1/2##, we've two options. ## l_z=0, s_z=1/2##, which is to say ##|l_z,s_z>=|0, 1/2> ## and ##l_z=1, s_z=-1/2##, which is to say ##|l_z,s_z>=|1, -1/2>##. Should I sum those states somehow?
     
    Last edited: Nov 10, 2014
  2. jcsd
  3. Nov 10, 2014 #2

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    You already know that one of the jz = 1/2 states is part of the j=3/2 representation. Since there are only two of those, which is the remaining one?
     
  4. Nov 10, 2014 #3

    DataGG

    User Avatar
    Gold Member

    I'm not sure I understand what you're saying.. I know that, for ##j_z = 1/2## there's two states. One being for ##j=3/2## and the other for ##j=1/2##. That is:

    ##|j, j_z> = |3/2, 1/2>## and ##|j, j_z> = |1/2, 1/2>##. Now I need to write this last state using ##l_z## and ##s_z##, in order to find the Klebsch-Gordon coefficients.
     
  5. Nov 10, 2014 #4

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Yes, I am fully aware of that. What I am saying is that you know what the state with ##j_z = 1/2## and ##j = 3/2## is, since you have already computed the states with ##j = 3/2##. The state you are searching for must be orthogonal to this.
     
  6. Nov 10, 2014 #5

    DataGG

    User Avatar
    Gold Member

    Oh!! I forgot that. I don't know why they need to be orthogonal though, but I guess that's a discussion for another thread. I'll see if I can solve it having that in mind!

    Thank you Orodruin!
     
  7. Nov 10, 2014 #6

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    What can you say about the operator ##\hat J = \hat L + \hat S##?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Addition of angular momentum - Finding the second tower states
Loading...