Spin 1/2 and other 2 State Systems

The angular momentum algebra defined by the commutation relations between the operators requires that the total angular momentum quantum number must either be an integer or a half integer. The half integer possibility was not useful for orbital angular momentum because there was no corresponding (single valued) spherical harmonic function to represent the amplitude for a particle to be at some position.

The half integer possibility is used to represent the internal angular momentum of some particles. The simplest and most important case is spin one-half. There are just two possible states with different z components of spin: spin up $\pmatrix{1\cr 0}$, with z component of angular momentum $+{\hbar\over 2}$, and spin down $\pmatrix{0\cr 1}$, with $-{\hbar\over 2}$. The corresponding spin operators are

$$\begin{eqnarray*} S_x ={\hbar\over 2} \left(\matrix{0 &1\cr 1 &0\cr}\right) \qqu... ...qquad S_z ={\hbar\over 2} \left(\matrix{1 &0\cr 0 &-1\cr}\right) \end{eqnarray*}$$

These satisfy the usual commutation relations from which we derived the properties of angular momentum operators.

It is common to define the Pauli Matrices, $\sigma_i$, which have the following properties.

$$\begin{eqnarray*} S_i &\equiv & {\hbar\over 2} \sigma_i. \\ \vec{S}&=&{\hbar\ov... ...ma_y+\sigma_y\sigma_z=0 \\ \{\sigma_i,\sigma_j\}&=&2\delta_{ij} \end{eqnarray*}$$

The last two lines state that the Pauli matrices anti-commute. The $\sigma$ matrices are the Hermitian, Traceless matrices of dimension 2. Any 2 by 2 matrix can be written as a linear combination of the $\sigma$ matrices and the identity.