- #1
mark.laidlaw19
- 21
- 0
Homework Statement
Hi all, I have a net spin operator that the problem has asked me to find:[tex]S=\frac{\hbar}{2} \left(\begin{array}{cc}\cos\alpha&\sin\alpha\\\sin\alpha&-\cos\alpha\end{array}\right)[/tex] and I need to write out the matrix representation with respect to the [itex]S_z[/itex] spinor basis.
Homework Equations
I know that to do this, I need to find the eigenvalues and eigenvectors (or eigenspinors) of this matrix, and then rewrite them in terms of the [itex]S_z[/itex] basis. From the characteristic equation of this matrix, I have found the eigenvalues to be [itex]\pm\frac{\hbar}{2}[/itex]
The Attempt at a Solution
I am pretty confident with most of this problem, however, when I try to find the eigenvectors of the spin operator above, I end up with this:
[tex]\left(\begin{array}{cc}\cos\alpha&\sin\alpha\\\sin\alpha&-\cos\alpha\end{array}\right) \left(\begin{array}{cc}a\\b\end{array}\right) = \left(\begin{array}{cc}a\\b\end{array}\right)[/tex]
I can't seem to solve these equations for a or b in order to find the eigenspinors, as once I substitute one equation into the other, the a and the b cancel out, and I'm wondering if there is a simple step that I am missing.
Many thanks in advance
Last edited: