How to Estimate a Particle's Location by Applying Schrödinger's Equation to a Wave Packet

If you have a number of solutions to the Schrödinger equation, any linear combination of those solutions is also a solution. So that’s the key to getting a physical particle: You add various wave functions together so that you get a wave packet, which is a collection of wave functions of the form

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such that the wave functions interfere constructively at one location and interfere destructively (go to zero) at all other locations:

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This is usually written as a continuous integral:

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What is

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It’s the amplitude of each component wave function, and you can find

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from the Fourier transform of the equation:

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Because

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you can also write the wave packet equations like this, in terms of p, not k:

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Well, you may be asking yourself just what’s going on here. It looks like

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That looks pretty circular.

The answer is that the two previous equations aren’t definitions of

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they’re just equations relating the two. You’re free to choose your own wave packet shape yourself — for example, you may specify the shape

image10.png

Here’s an example in which you get concrete, selecting an actual wave packet shape. Choose a so-called Gaussian wave packet, which you can see in the figure — localized in one place, close to zero in the others.

A Gaussian wave packet.
A Gaussian wave packet.

The amplitude

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you may choose for this wave packet is

image13.png

You start by normalizing

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to determine what A is. Here’s how that works:

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Substituting in

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gives you this equation:

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Doing the integral (that means looking it up in math tables) gives you the
following:

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So here’s your wave function:

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This little gem of an integral can be evaluated to give you the following:

image20.png

So that’s the wave function for this Gaussian wave packet (Note: The exp[–x2/a2] is the Gaussian part that gives the wave packet the distinctive shape that you see in the figure) — and it’s already normalized.

Now you can use this wave packet function to determine the probability that the particle will be in, say, the region

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The probability is

image22.png

In this case, the integral is

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And this works out to be

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So the probability that the particle will be in the region

image25.png

Cool!

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