##

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