ABSTRACT
The Pauli spin matrices have many useful properties. The spin of a particle is a degree of freedom additional to the spatial degrees of freedom. It is independent of the spatial degrees of freedom in the sense that one can simultaneously specify the spin state and the spatial state of a particle. Non-relativistically one must use more indirect methods of observing the spin. Classically, a spinning distribution of charge will have a magnetic moment; quantum mechanical spinning particles also have magnetic moments associated with their spins, and one tries to observe these magnetic moments in experiments. Spins in magnetic fields have very interesting dynamical properties. The right side is just the torque exerted by a magnetic field on a magnetic moment. This equation says in operator language that the rate of change of the spin angular momentum is the applied torque.
