Matrix representation of the z-component of Spin Angular Momentum | Quantum Science Philippines

## Matrix representation of the z-component of Spin Angular Momentum

By: Maria Christine L. Lugo, MS Physics I, MSU-IIT

Show that

$S_{z} =\hbar\left( \begin{array}{cccc} 1& 0&0&0 \\ 0 &0&0&0 \\0&0&0&0\\0&0&0&-1 \end{array} \right)$

Solution:

$S_{z}$ can be written in the form

$S_{z} = S_{1z} + S_{2Z}$

and the eigenstate is,

$S_{z}|sm> = m\hbar |sm>$

Now, solving for the eigenvalues of the orthonormal basis we have,

$\begin{array}{lll} S_{z}|++>& = &(S_{1z} + S_{2z})|++>\\ \\&=& (m_{1}\hbar + m_{2}\hbar) |++>\\ \\&= &(\frac{1}{2}\hbar + \frac{1}{2}\hbar)|++>\\ \\&=&\hbar|++> \end{array}$

$\begin{array}{lll} S_{z}|-+>& = &(S_{1z} + S_{2z})|-+>\\ \\&=&(\frac{-1}{2}+\frac{1}{2})|-+>\\ \\&=&0\end{array}$

$\begin{array}{lll} S_{z}|+-> &=& (S_{1z} + S_{2z})|+->\\ \\&=&(\frac{-1}{2}+\frac{1}{2})|+->\\ \\&=&0\end{array}$

$\begin{array}{lll}S_{z}|- -> &=& (S_{1z} + S_{2z})|- ->\\ \\&=&(\frac{-1}{2}+\frac{-1}{2}|+->)\\ \\&=&-\hbar|- ->\end{array}$

Therefore using the results we can form a matrix representation of;

$S_{z} =\hbar\left( \begin{array}{cccc} 1& 0&0&0 \\ 0 &0&0&0 \\0&0&0&0\\0&0&0&-1 \end{array} \right)$