Work out the spin matrices for arbitrary spin , generalizing spin (Equations 4.145 and 4.147), spin 1 (Problem 4.31), and spin (Problem 4.52). Answer:

$$\begin{gathered} {\mathbf{S}}_{\mathbf{z}}\mathbf{=}\mathbf{\hslash }\mathbf{\left(}\begin{array}{ccccc}\mathbf{s}& \mathbf{0}& \mathbf{0}& \mathbf{\cdots }& \mathbf{0}\\ \mathbf{0}& \mathbf{s}\mathbf{-}\mathbf{1}& \mathbf{0}& \mathbf{\cdots }& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{s}\mathbf{-}\mathbf{2}& \mathbf{\cdots }& \mathbf{0}\\ \mathbf{⋮}& \mathbf{⋮}& \mathbf{⋮}& \mathbf{\cdots }& \mathbf{⋮}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{-}\mathbf{s}\end{array}\mathbf{\right)} \\ {\mathbf{S}}_{\mathbf{x}}\mathbf{=}\frac{\mathbf{\hslash }}{\mathbf{2}}\mathbf{\left(}\begin{array}{ccccccc}\mathbf{0}& {\mathbf{b}}_{\mathbf{s}}& \mathbf{0}& \mathbf{0}& \mathbf{\cdots }& \mathbf{0}& \mathbf{0}\\ {\mathbf{b}}_{\mathbf{s}}& \mathbf{0}& {\mathbf{b}}_{\mathbf{s}\mathbf{-}\mathbf{1}}& \mathbf{0}& \mathbf{\cdots }& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& {\mathbf{b}}_{\mathbf{s}\mathbf{-}\mathbf{1}}& \mathbf{0}& {\mathbf{b}}_{\mathbf{s}\mathbf{-}\mathbf{2}}& \mathbf{\cdots }& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{b}}_{\mathbf{s}\mathbf{-}\mathbf{2}}& \mathbf{0}& \mathbf{\cdots }& \mathbf{0}& \mathbf{0}\\ \mathbf{⋮}& \mathbf{⋮}& \mathbf{⋮}& \mathbf{⋮}& \mathbf{\cdots }& \mathbf{⋮}& \mathbf{⋮}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{\cdots }& \mathbf{0}& {\mathbf{b}}_{\mathbf{-}\mathbf{s}\mathbf{+}\mathbf{1}}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{\cdots }& {\mathbf{b}}_{\mathbf{-}\mathbf{s}\mathbf{+}\mathbf{1}}& \mathbf{0}\end{array}\mathbf{\right)} \\ {\mathbf{S}}_{\mathbf{y}}\mathbf{=}\frac{\mathbf{\hslash }}{\mathbf{2}}\mathbf{\left(}\begin{array}{ccccccc}\mathbf{0}& \mathbf{-}{\mathbf{ib}}_{\mathbf{s}}& \mathbf{0}& \mathbf{0}& \mathbf{\cdots }& \mathbf{0}& \mathbf{0}\\ {\mathbf{ib}}_{\mathbf{s}}& \mathbf{0}& \mathbf{-}{\mathbf{ib}}_{\mathbf{s}\mathbf{-}\mathbf{1}}& \mathbf{0}& \mathbf{\cdots }& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{-}{\mathbf{ib}}_{\mathbf{s}\mathbf{-}\mathbf{1}}& \mathbf{0}& \mathbf{-}{\mathbf{ib}}_{\mathbf{s}\mathbf{-}\mathbf{2}}& \mathbf{\cdots }& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{-}{\mathbf{ib}}_{\mathbf{s}\mathbf{-}\mathbf{2}}& \mathbf{0}& \mathbf{\cdots }& \mathbf{0}& \mathbf{0}\\ \mathbf{⋮}& \mathbf{⋮}& \mathbf{⋮}& \mathbf{⋮}& \mathbf{\cdots }& \mathbf{⋮}& \mathbf{⋮}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{\cdots }& \mathbf{0}& \mathbf{-}{\mathbf{ib}}_{\mathbf{s}\mathbf{+}\mathbf{1}}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{\cdots }& \mathbf{-}{\mathbf{ib}}_{\mathbf{s}\mathbf{+}\mathbf{1}}& \mathbf{0}\end{array}\mathbf{\right)}\mathbf{} \end{gathered}$$

where,${\mathbf{b}}_{\mathbf{j}}\mathbf{\equiv }\sqrt{\mathbf{(}\mathbf{s}\mathbf{+}\mathbf{j}\mathbf{)}\mathbf{(}\mathbf{s}\mathbf{+}\mathbf{1}\mathbf{-}\mathbf{j}\mathbf{)}}$

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.

$s_z|sm⟩=hm|sm⟩$

Write the matrix element of $s_z$.

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

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

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

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

Use the property of the $\delta$ function.

$({\mathrm{s}}_{*}{)}_{\mathrm{mw}}=\hslash {\mathrm{b}}_{\mathrm{n}}{\mathrm{\delta }}_{\mathrm{nm}+1}$

Write the matrix.

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

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" $s_x$.

${\mathrm{s}}_{\mathrm{x}}=\frac{\hslash }{2}0{\mathrm{b}}_{\mathrm{s}}00\cdots 0{\mathrm{b}}_{\mathrm{s}}0{\mathrm{b}}_{\mathrm{s}-1}0\dots 00{\mathrm{b}}_{\mathrm{s}-1}0{\mathrm{b}}_{\mathrm{s}-2}\dots \cdots 00{\mathrm{b}}_{\mathrm{s}-2}00{\mathrm{b}}_{-\mathrm{s}+1}000\dots {\mathrm{b}}_{-\mathrm{s}+1}0\dots {\mathrm{b}}_{-\mathrm{s}+1}0$

Write the value of$s_y$ .

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

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