We can solve the
harmonic oscillator problem using operator methods.
We write the Hamiltonian in terms of the operator

.

We compute the

If we apply the the commutator to the eigenfunction , we get which rearranges to the eigenvalue equation

This says that is an eigenfunction of with eigenvalue so it

This allows us to compute the

showing that the ground state energy is . Similarly,

A little more computation shows that

and that

These formulas are useful for all kinds of

Jim Branson 2013-04-22