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 Ay and Az are constants.

Because

you get this for

where A= Ax Ay Az.

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

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

So here’s how you can rewrite the