Notes 27 : The Variational Method 3 3 . Families of Trial Wave Functions

Very few realistic problems in quantum mechanics are exactly solvable, so approximation methods are a virtual necessity for understanding the physics of real systems. In Notes 22 we considered bound state perturbation theory, which allows us to find the discrete energy eigenvalues and eigenstates of a system that is close to a solvable system. But what do we do when there is no exactly solvable system close to a given system, as often happens in practice? One answer is the variational method, which we discuss in these notes. The variational method is particularly useful for finding approximations to the ground state energy and eigenfunction of a system. It is harder to use for excited states, with some exceptions that we will mention in this set of notes. Nevertheless, the variational method is a vital one in the repertoire of techniques useful in quantum mechanics.