The Dirac Equation

Introducing Spin

1. The Stern-Gerlach experiment
   The internal degree of freedom called 'spin' had to be introduced in order to interpret the result of the Stern-Gerlach experiment:
   A collimated beam of silver atoms is subjected to an inhomogeneous magnetic field with the main component (z-direction) being perpendicular onto the beam direction. If the silver atoms have a magnetic moment (which is essentially the magnetic moment of just one atomic electron) it experiences a force given (to a good approximation) by: The magnetic moment is generated by an angular momentum. As the directions of the magnetic moment vector will be randomly distributed one expects that the beam will be deflected in a continous way between if the angular momentum is classical.
   The result of the experiment however is that the beam gets split into two components. The interpretation is that μ_z and as a consequence S_z can only have two possible values. Numerically, it is found that As a consequence, the electron must carry an internal angular momentum, 'spin', which is quantized. In the following, the two quantum states are described by the kets |S_z;+>=|+> and |S_z;->=|->.
   By blocking the |-> component and performing then a Stern-Gerlach experiment in x-direction and blocking the |S_x;->=|-> component and again performing a Stern-Gerlach experiment in z-direction one finds that there are again |S_z;+> and |S_z;-> components in the atomic silver beam.
   This leads to the following conclusion: One can construct the operators S_x, S_y and S_z using the fact that the matrix representation of an observable A in terms of eigenkets of A is given by: One has for the identity operator: For S_z one has and for S_x and S_y when expressing |S_x;+-> and |S_y;+-> in terms of the eigenkets of S_z: |+> and |->.
2. Spin vs. orbital angular momentum
   
3. Connection to the Pauli-matrices
   
4. Fields with internal degrees of freedom: Isospin and Colour
   
5. Example of isospin conservation
   
6.