Stationary States of One Particle in 3 Dimensions

The three-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 \(U_{x}(x), U_{y}(y)\) and \(U_{z}(z)\) are arbitrary functions.