SPIN ANGULAR MOMENTUM

Introduction
What is spin? Why was it needed? \\
A point particle in classical theories is described by position $\vec{r}$
and momentum $\vec{p}$. Therefore the only quantity of dimension of an
angular momentum that can be constructed out of these two variables is
$\vec{r}\times\vec{p}$. This is just the orbital angular momentum $\vec{L}$.
Note that for a particle at rest, the orbital angular momentum will be
zero.

The spin was introduced as an additional property of an electron and is equal
to the angular momentum of electron at rest. The total angular momentum then
consists of spin angular momentum and orbital angular momentum.

The Uhlenbeck and Goudschmidt introduced spin to explain {\bf anomalous Zeeman
effect} and it also was needed to explain the {\bf fine structure of atomic
spectra}. At present a large number of elementary particles are known to have
non-zero spin.
%TODO Check the details of discovery of spin.
\paragraph*{You may want to ask Questions} on   anomalous Zeeman effect and
fine structure and also spin.

Goals

1.  According to the postulates of quantum mechanics there is a  hermitian operator corresponding to every dynamical variable, this must also be true for spin also.
2. Let us denote the operators assigned to spin as $\vec{S} = ( S_x, S_y, S_z)$. These are assumed to obey the commutation relations\
3. \begin{equation}
4. [S_x, S_y] = i\hbar S_z, \mbox{\rm etc}.
   \end{equation}
   
    The square of total  spin, $\vec{S}^2$ and the $z-$ component $S_z$  can be measured simultaneously.
5. 
   The eigenvalues, which are also the allowed values square of total spin $S^2$ and $S_z$, are
        $$ \begin{array}{|ccll|}
            \hline &&& \\
         S^2  & \to & s(s+1)\hbar^2 ; & s=0,\frac{1}{2}, 1, \frac{3}{2}. \cdots \\
         \hline &&&\\
         S_z  & \to & s, s-1, \ldots, -s+1, -s; & (2s+1)\text{values} \\
            && &\text{for a given total spin } s\\
              \hline
       \end{array} $$
   
   The spin operators for a spin half particle are given by $\frac{\hbar}{2} \vec{\sigma}$ where $\vec{\sigma}$ are the three Pauli matrices.
   
   Taking spin into account, the wave function of an electron has two components and is, in general, has the form
          \begin{equation}
               \Psi(x) = \begin{pmatrix}
                           \psi_1(x) \\
                           \psi_2(x)
                         \end{pmatrix}
          \end{equation}
6. In an important special case the two components are proportional. In such a case, $\psi_1(x)= c_1 \psi(x), \psi_2(x)=c_2 \psi(x)$ and the  wave function is usually written as a product of space part, $\psi(x)$, and a spin part, $\chi$, as follows

                 \begin{eqnarray}

1.                     \Psi(x) &=& \psi(x) \begin{pmatrix}
2.                                         c_1 \\ c_2
3.                                       \end{pmatrix} \equiv \psi(x) \chi, \\
4.                   \text{where~~~~} \chi &=& \begin{pmatrix}
5.                                         c_1 \\ c_2
6.                                       \end{pmatrix}.
7.                 \end{eqnarray}
8. In such a case $|c_1|^2$ and $|c_2|^2$ give the probabilities of $S_z$ value being $\pm \frac{1}{2}$.