[NOTES/QM-20002] Spin Wave Function and Spin Operators

                              Contents 

1. Representation of Spin Wave Function
2. Representation of Spin Operators
3. Spin 1 matrices
4. Spin 1/2 matrices 

Representation of Spin Wave Function

In order to describe the spin degrees of freedom, it is convenient to introduce a representation. For this we need to select a complete commuting set of hermitian operators and construct an orthonormal basis from their simultaneous eigenvectors. A suitable set consists of $\vec{S}^2$ and $S_z$. In order to proceed further, we want to work with an explicit representation of the spin. We arrange the eigenvectors $\ket{s,m}$ in {\em descending order} in $m$ to get a basis $\big\{\ket{s,m}\big| m=s,s-1,\cdots,-s+1,-s \big\}$. An arbitrary state vector $\ket{x}$ is then a linear combination of the basis elements \begin{equation} \ket{x} = \sum_{m=-s}^{s} \alpha_m \ket{sm} \label{E1} \end{equation}
The interpretation of the numbers $\alpha_m$ is that square of its modulus, $|\alpha_k|^2,$ gives the probability that $S_z$ will have the corresponding value $m\hbar$. Following the convention of arranging the basis vectors in the order of decreasing values for the spin projection $S_z$, the $\ket{x}$ will be represented by a column vector \begin{equation} \chi=\begin{pmatrix}\alpha_s\\ \alpha_{s-1}\\\vdots \\ \alpha_{-s} \label{E2} \end{pmatrix} \end{equation} with $(2s+1)$ components.

Representation of Spin Operators

The spin operators $\vec{S}$ will be represented by matrices with $(2s+1)$ rows and $(2s+1)$ columns. First of all, the matrix for $S_z$ will be diagonal matrix with eigenvalues of $S_z$ appearing along the main diagonal. \begin{equation} S_z= \hbar\begin{pmatrix} s & 0& 0&\cdots&\cdots&0\\0&s-1&0&\cdots& \cdots&0\\0&0&s-2&\cdots&\cdots&0\\0&0&0&\cdots&\cdots\\\cdots&\cdots&\cdots& \cdots&\cdots&0\\0&0&0&\cdots&\cdots&-s\end{pmatrix} \label{E3} \end{equation}
The matrices for $S_x$ and $S_y$ are found by first obtaining the matrices for $S_\pm$ and using $S_x=\frac{1}{2}(S_+ + S_-)$ and $\frac{-i}{2}(S_+-S_-)$. To construct these matrices one needs to know the matrix elements $\matrixelement{s,m^{'}}{S_\pm}{s,m}$ which can be computed by making use of the result \begin{equation} \boxed{ S_{\pm}\ket{s,m}= \sqrt{s(s+1)-m(m\pm1)}\,\hbar\,\ket{s,m\pm1}} \label{E4} \end{equation}

Spin 1 matrices 

The corresponding result for the spin one matrices is
$S_x=\frac{\hbar}{\surd 2} \begin{pmatrix}0&1&0\\1&0&1\\ 0&1&0\end{pmatrix}$; & $S_y=\frac{\hbar}{\surd 2}\begin{pmatrix}0&-i&0\\i&0&-i\\0&i&0\end{pmatrix}$; &$S_z=\hbar \begin{pmatrix}1&0&0\\0&0&0\\0&0&-1 \end{pmatrix}$.
and is left as an exercise for the reader.

Spin 1/2 matrices

We shall give the answer for spin $\frac{1}{2}$ and spin $1$ matrices. The spin half matrices are related to the Pauli matrices $\sigma_x,\sigma_y,\sigma_z$ and are given by \begin{equation} \vec{S}=\frac{\hbar}{2}\,\vec{\sigma} \label{E5} \end{equation} This result is derived in the solved problem below. \end{frame}