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Eigenstates of general spin matrix

  1. Oct 16, 2007 #1
    Hello,
    I'm having a terrible difficulty solving problem 3.18 in A Modern Approach to Quantum Mechanics by John Townsend. I have done literally hours of work and am beginning to think I don't understand how eigenvalues relate to matrix mechanics as well as I thought. Please excuse my TeX it is my first time :redface:

    1. The problem statement, all variables and given/known data
    Determine the eigenstates of [tex]S{n}[/tex] = S [tex]\bullet[/tex] n for a spin-1 particle, where the spin operator S = [tex]S{x}[/tex]i + [tex]S{y}[/tex]j + [tex]S{z}[/tex]k and n = sin[tex]\theta[/tex]cos[tex]\phi[/tex]i + sin[tex]\theta[/tex]sin[tex]\phi[/tex]j + cos[tex]\theta[/tex]k. Use the matrix representation of the rotation operator in 3.19 to check your result when [tex]\phi[/tex] = 0.

    2. Relevant equations
    3.19: (3x3 matrix)

    R([tex]\theta[/tex]j) =

    .5(1 + cos[tex]\theta[/tex]) -(1/Sqrt(2))sin[tex]\theta[/tex] .5(1-cos[tex]\theta[/tex])
    (1/Sqrt(2))(sin[tex]\theta[/tex]) cos[tex]\theta[/tex] -(1/Sqrt(2))sin[tex]\theta[/tex]
    .5(1 - cos[tex]\theta[/tex]) (1/Sqrt(2))sin[tex]\theta[/tex] .5(1+cos[tex]\theta[/tex])


    3. The attempt at a solution

    I thought I could solve this by simple taking the dot product of n and S as described in the problem and adding the three resulting matrices. However it is impossible for this to reduce to 3.19 when phi = 0 because Sx Sy and Sz for spin-1 particles do not have entries in the upper right, middle, and lower left corners (among other problems). Even so, using this matrix I followed through and was able to get eigenvalues -1,0 and 1 (despite some very ridiculous algebra) which seemed reasonable to me. I attempted to plug eigenvalue 1 into the eigenvalue equation but the resulting eigenvector is very complicated (which makes me think it is wrong). Anyway I can't reduce my matrix so I think I'm doing something wrong at the very beginning but I can't figure out what.

    Any help would be much appreciated :smile:
     
  2. jcsd
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