## Harmonic Oscillator Solution with Operators

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

.

We compute the commutators

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 lowers the energy by . Since the energy must be positive for this Hamiltonian, the lowering must stop somewhere, at the ground state, where we will have

This allows us to compute the ground state energy like this

showing that the ground state energy is . Similarly, raises the energy by . We can travel up and down the energy ladder using and , always in steps of . The energy eigenvalues are therefore

A little more computation shows that

and that

These formulas are useful for all kinds of computations within the important harmonic oscillator system. Both and can be written in terms of and .

Jim Branson 2013-04-22