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 \bgroup\color{black}$\pmatrix{1\cr 0}$\egroup, with z component of angular momentum \bgroup\color{black}$+{\hbar\over 2}$\egroup, and spin down \bgroup\color{black}$\pmatrix{0\cr 1}$\egroup, with \bgroup\color{black}$-{\hbar\over 2}$\egroup. The corresponding spin operators are

S_x ={\hbar\over 2} \left(\matrix{0 &1\cr 1 &0\cr}\right) \qqu...
S_z ={\hbar\over 2} \left(\matrix{1 &0\cr 0 &-1\cr}\right)

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

It is common to define the Pauli Matrices, \bgroup\color{black}$\sigma_i$\egroup, which have the following properties.

S_i &\equiv & {\hbar\over 2} \sigma_i. \\
...ma_y+\sigma_y\sigma_z=0 \\

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

Jim Branson 2013-04-22