# 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,

You can write its general solution as

where A* _{y}* and A

*are constants.*

_{z}Because

you get this for

where A= A* _{x}* A

*A*

_{y}*.*

_{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*

*) in terms of components and the vector*

_{z}**= (**

*b**b*

*,*

_{x}*b*

*,*

_{y}*b*

*), then the dot product of*

_{z}**and**

*a***is**

*b*So here’s how you can rewrite the