(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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?

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# Homework Help: Difficulty with Sakurai, Ch.1, Problem # 9

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