Stationary States of One Particle in 3 Dimensions

Abstract

The 3-dimensional plane wave is the product of one-dimensional plane waves and the kinetic energy is the sum of the contributions of motions along x, y, z. More generally, the problem is separable into Cartesian coordinates if the potential energy is of the form

$$\begin{aligned} V(x,y, z)=U_{x}(x)+U_{y}(y)+U_{z}(z) \end{aligned}$$

where $U_{x}(x), U_{y}(y)$ e $U_{z}(z)$ are arbitrary functions.

The partial differential equations are quite a bit harder to solve than the ordinary ones, unless the symmetry allows us to separate the variables. Fortunately, among the most interesting stationary problems, there are some that can be solved analytically.