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Eigenstates in a 3-state spin 1 system

  1. May 6, 2015 #1

    dyn

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    I have been looking at the solution to a question and I don't understand how the eigenstates are calculated. The question concerns a 3-state spin-1-system with angular momentum l=1. The 3 eigenstates of L3 are given as ## \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} ## , ## \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} ## , ## \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} ## which leads to the z-component of angular momentum as L3 = ## \hbar ## ## \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix} ##. When I try to calculate the eigenvalues and eigenvectors of the L3 matrix using determinants I get no answer. Can anybody tell me how to get the eigenvalues and eigenvectors ? Thanks
     
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  3. May 6, 2015 #2

    DrClaude

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    Staff: Mentor

    Could you post the original question? It will help to know what is the assumed starting point.
     
  4. May 6, 2015 #3

    dyn

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    The original question is - A quantum particle is known to have total angular momentum one ie. l = 1. Use the eigenstates of L2 and L3 , denoted by | l , m > = | 1 , m > as a basis and find the matrix representation of operators L1 , L2 and L2 in this 3-D subspace. Use | 1 , 1> = ## \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} ## = | 1 > , | 1 , 0 > = ## \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} ## = | 2 > , | 1 , -1 > = ## \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} ## as our basis. You may find it useful to compute the matrix representation of operators L+ , L- and L3 .
     
  5. May 6, 2015 #4

    DrClaude

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    Staff: Mentor

    Ok, so the idea is to use your knowledge of how operators like ##L_3## work on the basis states to build the matrix. For instance, you (should) know that
    $$
    L_3 | 1, -1 \rangle = -\hbar | 1, -1 \rangle
    $$
    so you build the matrix accordingly. Since you are using the eigenstates of ##L_3## to build the matrix, you know that the resulting matrix is diagonal, with the eigenvalues as the diagonal elements. You do the same for ##L^2##. As for ##L_1## and ##L_2##, you will indeed find it useful to use the operators ##L_+## and ##L_-##.
     
  6. May 6, 2015 #5

    dyn

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    The bit that I don't understand is that if I was given the L3 matrix I can't work out the eigenvalues and eigenvectors from it
     
  7. May 6, 2015 #6

    dyn

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    I've got it now !
     
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