Difficulty with Sakurai, Ch.1, Problem # 9

[quantumkiko]

Homework Statement

Determine the eigenvector for (S \cdot \hat{n}) |eigenvector> = (\hbar)/2 |eigenvector> where S = (\hbar)/2 \sigma. The sigmas are the Pauli spin matrices and \hat{n} = sin\beta cos\alpha \hat{i} + sin\beta\ sin\alpha \hat{j} + cos\beta \hat{k}

You have to solve for the coefficients a and b of

|eigenvector> = a|+> + b|->.

2. The attempt at a solution

It seems like when I try to solve for one coefficient, say a, is seems to vanish.

a(\cos\beta - 1) + b\sin\beta e^{-i\alpha} = 0
a\sin\beta e^{i\alpha} - b(\cos\beta + 1) = 0

For example, isolating b in the 1st equation,

b = - \frac{a(\cos\beta - 1)}{ \sin\beta e^{-i\alpha}}.

If you plug this in the 2nd equation you get a factor of a for all terms, therefore the a's cancel each other.

But Sakurai's solution is |eigenvector> = cos(\beta / 2)|+> + sin{\beta / 2} e^{i\alpha}|->. How can I get this?

 

[tiny-tim]

Hi quantumkiko!

(have an alpha: α and a beta: β and a sigma: σ )

Determine the eigenvector for (S \cdot \hat{n}) …
…
It seems like when I try to solve for one coefficient, say a, is seems to vanish.

a(\cos\beta - 1) + b\sin\beta e^{-i\alpha} = 0
a\sin\beta e^{i\alpha} - b(\cos\beta + 1) = 0

hmm … i can see σ₃ in there somewhere …

with eigenvectors which would be (a,b) = (1,0) or (0,1) if β = 0 or π …

but whatever happened to poor little σ₁ and σ₂? ​

(btw you have to put [noparse]before and after for the LaTeX to work in this forum[/noparse] )

 

[quantumkiko]

Thank you very much! ^^