How to Add Time Dependence and Get a Physical Equation for Three-Dimensional Free Particle Problems

At some point, your quantum physics instructor may want you to add time dependence and get a physical equation for a three-dimensional free particle problem. You can add time dependence to the solution for

image0.png

if you remember that, for a free particle,

image1.png

That equation gives you this form for

image2.png

Because

image3.png

the equation turns into

image4.png

In fact, now that the right side of the equation is in terms of the radius vector r, you can make the left side match:

image5.png

That’s the solution to the Schrödinger equation, but it’s unphysical. Why? Trying to normalize this equation in three dimensions, for example, gives you the following, where A is a constant:

image6.png

(Remember that the asterisk symbol [*] means the complex conjugate. A complex conjugate flips the sign connecting the real and imaginary parts of a complex number. The limits on the integral just mean to integrate over all of space, like this:

image7.png

Thus, the integral diverges and you can’t normalize

image8.png

as written here. So what do you do here to get a physical particle?

The key to solving this problem is realizing that if you have a number of solutions to the Schrödinger equation, then any linear combination of those solutions is also a solution. In other words, you add various wave functions together so that you get a wave packet, which is a collection of wave functions of the form

image9.png

such that

  • The wave functions interfere constructively at one location.

  • They interfere destructively (go to zero) at all other locations.

Look at the time-independent version:

image10.png

However, for a free particle, the energy states are not separated into distinct bands; the possible energies are continuous, so people write this summation as an integral:

image11.png

So what is

image12.png

It’s the three-dimensional analog of

image13.png

That is, it’s the amplitude of each component wave function. You can find

image14.png

from the Fourier transform of

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like this:

image16.png

In practice, you choose

image17.png

yourself. Look at an example, using the following form for

image18.png

which is for a Gaussian wave packet (Note: The exponential part is what makes this a Gaussian wave form):

image19.png

where a and A are constants. You can begin by normalizing

image20.png

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

image21.png

Okay. Performing the integral gives you

image22.png

which means that the wave function is

image23.png

You can evaluate this equation to give you the following, which is what the time-independent wave function for a Gaussian wave packet looks like in 3D:

image24.png
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