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Finding the x, y, and z Equations for Three-Dimensional Free Particle Problems

At some point, your quantum physics instructor may want you to find the x, y, and z equations for three-dimensional free particle problems. Take a look at the x equation for the free particle,

image0.png

You can write its general solution as

image1.png

where Ay and Az are constants.

Because

image2.png

you get this for

image3.png

where A= Ax Ay Az.

The part in the parentheses in the exponent is the dot product of the vectors

image4.png

That is, if the vector a = (ax, ay, az) in terms of components and the vector b = (bx, by, bz), then the dot product of a and b is

image5.png

So here’s how you can rewrite the

image6.png
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