Symmetry groups and degeneracy

Professor Susskind presents an example of rotational symmetry and derives the angular momentum operator as the generator of this symmetry. He then presents the concept of degenerate states, and shows that any two symmetries that do not commute imply degeneracy. Symmetries that do not commute can form a symmetry group, and the generators of these symmetries form a Lie algebra.

The angular momentum generators in three dimensions are an example of a symmetry group.  Professor Susskind then derives the raising and lowering operators from the angular momentum generators, and shows how they are used to raise or lower the magnetic quantum number of a system between degenerate energy states.  Due to reflection symmetry, these states must have whole- or half-integer values for the magnetic quantum number.