Ladder operator

A ladder operator maps weight vectors to weight vectors whose weights differ by one root. Positive-root vectors raise weights and negative-root vectors lower them.

www.physics.udel.edu/~jim/PHYS424_17F/Class%20Notes/Class_5.pdf by James MacDonald shows it well.

The operators are a natural guess on the lines of "if p and x were integers".

And then we can prove the ladder properties easily.

The commutator appear in the middle of this analysis.