- #1

gfd43tg

Gold Member

- 953

- 49

Hello,

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.

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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