## Use Commutators to Derive HO Energies

We have computed the commutators

Apply to the energy eigenfunction .

This equation shows that is an eigenfunction of with eigenvalue . Therefore, lowers the energy by .

Now, apply to the energy eigenfunction .

is an eigenfunction of with eigenvalue . raises the energy by .

We cannot keep lowering the energy because the HO energy cannot go below zero.

The only way to stop the lowering operator from taking the energy negative, is for the lowering to give zero for the wave function. Because this will be at the lowest energy, this must happen for the ground state. When we lower the ground state, we must get zero.

Since the Hamiltonian contains in a convenient place, we can deduce the ground state energy.

The ground state energy is and the states in general have energies

since we have shown raising and lowering in steps of . Only a state with energy can stop the lowering so the only energies allowed are

It is interesting to note that we have a number operator for

Subsections
Jim Branson 2013-04-22