Eigenstates of general spin matrix

[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

Homework Statement

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.

Homework 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$$\theta$$)

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

 

[Jhero]

Hi there,

I can understand your frustration with this problem. It can be quite tricky to understand how eigenvalues relate to matrix mechanics, especially when dealing with spin operators. However, I believe there are a few key things that you may be overlooking in your approach.

Firstly, when taking the dot product of n and S, you need to remember that S is a vector operator, meaning it operates on a vector (in this case, n) to produce another vector. So the resulting matrices from your dot product will also be vectors, not just numbers. This is where the entries in the upper right, middle, and lower left corners come into play.

Secondly, remember that the eigenvalues of a matrix are the values that satisfy the eigenvalue equation (Av = λv), where A is the matrix, v is the eigenvector, and λ is the eigenvalue. So in this case, you need to find the values of λ that satisfy the equation S{n} = λn, where n is the eigenvector. This will give you the eigenvalues for S{n}.

Finally, to find the corresponding eigenvectors, you can plug in the eigenvalues you found into the eigenvalue equation and solve for the eigenvector. This may result in complicated expressions, but as long as you have the correct eigenvalues, you should be on the right track.

I hope this helps and good luck with your problem! If you need any further clarification, please don't hesitate to ask.