91影视

The spin related matrices are known as spin matrices. These are number of matrices. These matrices are complex that include involutory, unitary, and Hermitian.

Write equation 4.135.

$s_z|sm鉄�=hm|sm鉄�$

Write the matrix element of $s_z$.

Write the matrix (diagonal matrix) with values of m ranging from s to -s along the diagonal.

${\mathrm{S}}_{\mathrm{z}}=鈩�\left(\begin{array}{ccccc}\mathrm{s}& 0& 0& 鈰�& 0\\ 0& \mathrm{s}-1& 0& 鈰�& 0\\ 0& 0& \mathrm{s}-2& 0& 鈰�\\ 鈰�& 鈰�& 鈰�& 鈰�& 鈰�\\ 0& 0& 0& 0& -\mathrm{s}\end{array}\right)$

Determine the value of ${\left(s_{+}\right)}_{nm}$.

Here,$b_{m+1}=\sqrt{(s-m)(s+m+1)}$ .

Use the property of the $未$ function.

$({\mathrm{s}}_{*}{)}_{\mathrm{mw}}=鈩�{\mathrm{b}}_{\mathrm{n}}{\mathrm{未}}_{\mathrm{nm}+1}$

Write the matrix.

${\mathrm{s}}_{+}=鈩�\left[\begin{array}{cccccc}0& {\mathrm{b}}_{\mathrm{s}}& 0& 0& 鈰�& 0\\ 0& 0& {\mathrm{b}}_{\mathrm{s}-1}& 0& 鈰�& 0\\ 0& 0& 0& {\mathrm{b}}_{\mathrm{s}-2}& 鈰�& 0\\ 鈰�& 鈰�& 鈰�& 鈰�& 鈰�& {\mathrm{b}}_{-\mathrm{s}+1}\\ 0& 0& 0& 0& 鈰�& 0\end{array}\right]$

Write the value of ${\left(s\_\right)}_{nm}$.

role="math" localid="1658146052379"

Write the value of s_ .

$$\begin{gathered} \mathrm{s\_}=\sout{\mathrm{h}}000.....0{\mathrm{b}}_{\mathrm{s}}000...00{\mathrm{b}}_{\mathrm{s}-1}.......00{\mathrm{b}}_{\mathrm{s}-2}.......0000{\mathrm{s}}_{\mathrm{x}} \\ =\frac{1}{2}\left[{\mathrm{s}}_{+}+{\mathrm{s}}_{-}\right] \end{gathered}$$

Write the value ofrole="math" localid="1658143674691" $s_x$.

${\mathrm{s}}_{\mathrm{x}}=\frac{鈩�}{2}0{\mathrm{b}}_{\mathrm{s}}00鈰�0{\mathrm{b}}_{\mathrm{s}}0{\mathrm{b}}_{\mathrm{s}-1}0鈥�00{\mathrm{b}}_{\mathrm{s}-1}0{\mathrm{b}}_{\mathrm{s}-2}鈥�鈰�00{\mathrm{b}}_{\mathrm{s}-2}00{\mathrm{b}}_{-\mathrm{s}+1}000鈥�{\mathrm{b}}_{-\mathrm{s}+1}0鈥�{\mathrm{b}}_{-\mathrm{s}+1}0$

Write the value of$s_y$ .

$$\begin{gathered} {\mathrm{s}}_{\mathrm{y}}=\frac{1}{2\mathrm{i}}[{\mathrm{s}}_{+}-{\mathrm{s}}_{-}] \\ {\mathrm{s}}_{\mathrm{y}}=\frac{鈩�}{2\mathrm{i}}\left[\begin{array}{cccccc}0& {\mathrm{b}}_{\mathrm{s}}& 0& 0& 鈰�& 0\\ -{\mathrm{b}}_{\mathrm{s}}& 0& -{\mathrm{b}}_{\mathrm{s}-1}& 0& 鈰�& 0\\ 鈰�& -{\mathrm{b}}_{\mathrm{s}-1}& 0& -{\mathrm{b}}_{\mathrm{s}-2}& 鈰�& 0\\ 0& 0& 0& 0& 鈰�& -{\mathrm{b}}_{-\mathrm{s}-1}\\ 0& 0& 0& 0& -{\mathrm{b}}_{-\mathrm{s}-1}& 0\end{array}\right] \end{gathered}$$

Thus, the spin matrices are$\frac{鈩�}{2\mathrm{i}}\left[\begin{array}{cccccc}0& {\mathrm{b}}_{\mathrm{s}}& 0& 0& 鈰�& 0\\ -{\mathrm{b}}_{\mathrm{s}}& 0& -{\mathrm{b}}_{\mathrm{s}-1}& 0& 鈰�& 0\\ 鈰�& -{\mathrm{b}}_{\mathrm{s}-1}& 0& -{\mathrm{b}}_{\mathrm{s}-2}& 鈰�& 0\\ 0& 0& 0& 0& 鈰�& -{\mathrm{b}}_{-\mathrm{s}-1}\\ 0& 0& 0& 0& -{\mathrm{b}}_{-\mathrm{s}-1}& 0\end{array}\right]$