How to Determine Harmonic Oscillator Eigenstates of a System

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

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is the following:

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This is equal to the following:

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This equation breaks down to

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And putting together this equation with the Hamiltonian,

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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

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Then use this to write the action of

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like this:

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

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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

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You can write that like this:

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All this means that

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is an eigenstate of the harmonic oscillator, with energy

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not just En — that is, the

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

So now you know that

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You can derive the following from these equations:

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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:

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

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But you also know that

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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

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

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

What about the raising operator,

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First you rearrange the commutator

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

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

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