The Stationary Schrödinger Equation

Much of what we will be concerned with in this lecture are the solutions of the Schrödinger equation for a particle of mass $m$ in a potential $V({\bf x})$,

$$i\hbar\frac{\partial}{\partial t} \Psi({\bf x},t) =\left[-\frac{\... ...^2\Delta}{2m} + V({\bf x}) \right]\Psi({\bf x},t)\equiv \hat{H}\Psi({\bf x},t).$$ (78)

Like Newton's laws in classical mechanics, the Schrödinger equation is so important that generation of physicists have worked out how to solve it for physically interesting cases. Unfortunately, in general (i.e. for a general form of the potential $V({\bf x})$) the Schrödinger equation is not exactly soluble, and one has to retreat to approximate methods. There are, however, important classes of solutions that can be obtained exactly, most of which were `milestones' in the development of the theory. Among them are the hydrogen atom, the harmonic oscillator, or one-dimensional problems which we will discuss in this chapter.