## 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 , with z component of angular momentum , and spin down , with . The corresponding spin operators are

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

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

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

Jim Branson 2013-04-22