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For the spin angular momentum operator, the eigenvalue problem can be formed into matrix form. I will use ##S_{z}## as my example

$$S_{z} | \uparrow \rangle = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \frac {\hbar}{2} \begin{pmatrix} 1 \\ 0 \end{pmatrix}$$

$$S_{z} | \downarrow \rangle = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix}= \frac {-\hbar}{2} \begin{pmatrix} 0 \\ 1 \end{pmatrix}$$

Why is it that the matrix on the left hand side is assumed to be a 2x2 matrix? Also, why is it when solving for a,b,c, and d the second equation is used. I don't understand how I can just say

$$\begin{pmatrix} a \\ c \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \frac {\hbar}{2} \begin{pmatrix} 1 \\ 0 \end{pmatrix}$$

$$\begin{pmatrix} b \\ d \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix}= \frac {-\hbar}{2} \begin{pmatrix} 0 \\ 1 \end{pmatrix}$$

I don't understand how I can just isolate the second column of the 2x2 matrix for the spin down case, and isolate the first column for spin up.

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# Spin angular momentum operator queries

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