The Schro¨dinger Equation in Three Dimensions
In three dimensions (3-D), the ordinary differential equations of the previous chapter become partial differential equations. There are two general methods for solving partial differential equations: (1) use the separation of variables, or (2) use numerical methods. Although many realistic problems require numerical techniques, we will restrict our attention to those simpler problems that allow the separation of variables so that we obtain a set of three ordinary differential equations. The complications of the inseparable problems may often be managed through perturbation methods if the problems are nearly separable. Our first example demonstrates the technique for the separation of variables and leads us to an already solved one-dimension problem.