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How to Find the Energy Eigenstate of a Harmonic Oscillator in Position Space

In quantum physics, you can use operators to determine the energy eigenstate of a harmonic oscillator in position space. The charm of using the operators a and

image0.png

is that given the ground state, | 0 >, those operators let you find all successive energy states. If you want to find an excited state of a harmonic oscillator, you can start with the ground state, | 0 >, and apply the raising operator,

image1.png

For example, you can do this:

image2.png

And so on. In general, you have this relation:

image3.png

Can’t you get a spatial eigenstate of this eigenvector? Something like

image4.png

not just | 0 >? Yes, you can. In other words, you want to find

image5.png

So you need the representations of

image6.png

in position space.

The p operator is defined as

image7.png

Because

image8.png

you can write

image9.png

And writing

image10.png

this becomes

image11.png

Okay, what about the a operator? You know that

image12.png

And that

image13.png

Therefore,

image14.png

You can also write this equation as

image15.png

Okay, so that’s a in the position representation. What’s

image16.png

That turns out to be this:

image17.png

Now’s the time to be clever. You want to solve for | 0 > in the position space, or < x | 0 >. Here’s the clever part — when you use the lowering operator, a, on | 0 >, you have to get 0 because there’s no lower state than the ground state, so a | 0 > = 0. And applying the < x | bra gives you < x | a | 0 > = 0.

That’s clever because it’s going to give you a homogeneous differential equation (that is, one that equals zero). First, you substitute for a:

image18.png

Multiplying both sides by

image19.png

gives you the following

image20.png

The solution to this compact differential equation is

image21.png

That’s a gaussian function, so the ground state of a quantum mechanical harmonic oscillator is a gaussian curve, as you see in the figure.

The ground state of a quantum mechanical harmonic oscillator.
The ground state of a quantum mechanical harmonic oscillator.
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