# Eigenstates of general spin matrix

1. Oct 16, 2007

### dshave

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

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

2. Relevant equations
3.19: (3x3 matrix)

R($$\theta$$j) =

.5(1 + cos$$\theta$$) -(1/Sqrt(2))sin$$\theta$$ .5(1-cos$$\theta$$)
(1/Sqrt(2))(sin$$\theta$$) cos$$\theta$$ -(1/Sqrt(2))sin$$\theta$$
.5(1 - cos$$\theta$$) (1/Sqrt(2))sin$$\theta$$ .5(1+cos$$\theta$$)

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