§ Why commutator is important for QM
- Suppose we have an operator with eigenvector , eigenvalue . So .
- Now suppose we have another operator such that for some constant .
- Compute , which implies:
- So is an eigenvector of with eigenvalue .
- This is how we get "ladder operators" which raise and lower the state. If we have a state with some eigenvalue , the operator like gives us an "excited state" from which eigenvalue .