Finding 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,

You can write its general solution as

where A_y and A_z are constants.

Because

you get this for

where A= A_x A_y A_z.

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

That is, if the vector a = (a_x, a_y, a_z) in terms of components and the vector b = (b_x, b_y, b_z), then the dot product of a and b is

So here’s how you can rewrite the