Spherical Symmetric Potentials in 3 Dimensions

The stationary Schrödinger equation in three dimensions

$$\left[-\frac{\hbar^2\Delta}{2m} + V({\bf x}) \right]\Psi({\bf x}) =E \Psi({\bf x})$$ (278)

contains the Laplace operator

$$\Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial z^2}+\frac{\partial^2}{\partial z^2}.$$ (279)

For general $V({\bf x})$, it is usually very difficult to solve this equation. There are, however, a few cases which are physically very important and can be solved exactly.