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 E* _{n}*, 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 E* _{n}*. 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 E* _{n}* — 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** *> = C^{2}

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

*n** *< *n* | *n* > = C^{2}

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.

What about the raising operator,

First you rearrange the commutator

Then you follow the same course of reasoning you take with the *a* operator to show the following:

So at this point, you know what the energy eigenvalues are and how the raising and lowering operators affect the harmonic oscillator eigenstates.