In quantum physics, when you have the eigenstates of a system, you can determine the allowable states of the system and the relative probability that the system will be in any of those states.

The commutator of operators A, B is [A, B] = AB – BA, so note that the commutator of

is the following:

This is equal to the following:

This equation breaks down to

And putting together this equation with the Hamiltonian,

Okay, with the commutator relations, you’re ready to go. The first question is: if the energy of state | n > is En, what is the energy of the state a | n >? Well, to find this, rearrange the commutator

Then use this to write the action of

like this:

So a | n > is also an eigenstate of the harmonic oscillator, with energy

not En. That’s why a is called the annihilation or lowering operator: It lowers the energy level of a harmonic oscillator eigenstate by one level.

So what’s the energy level of

You can write that like this:

All this means that

is an eigenstate of the harmonic oscillator, with energy

not just En — that is, the

raises the energy level of an eigenstate of the harmonic oscillator by one level.

So now you know that

You can derive the following from these equations:

C and D are positive constants, but what do they equal? The states |n – 1> and |n + 1> have to be normalized, which means that <n – 1|n – 1> = <n + 1|n + 1> = 1. So take a look at the quantity using the C operator:

And because |n – 1> is normalized, <n – 1|n – 1> = 1:

But you also know that

the energy level operator, so you get the following equation:

< n | N | n > = C2

N | n > = n | n >, where n is the energy level, so

n < n | n > = C2

However, < n | n > = 1, so

This finally tells you, from a | n > = C | n – 1 >, that

That’s cool — now you know how to use the lowering operator, a, on eigenstates of the harmonic oscillator.