Stationary Perturbations in Quantum Mechanics

Abstract

We denote a quantum system with a time-independent Hamiltonian H ₀ as solvable (or sometimes also as exactly solvable) if we can calculate the energy eigenvalues and eigenstates of H ₀ analytically. The harmonic oscillator and the hydrogen atom provide two examples of solvable quantum systems. Exactly solvable systems provide very useful models for quantum behavior in physical systems. The harmonic oscillator describes systems near a stable equilibrium, while the Hamiltonian with a Coulomb potential is an important model system for atomic physics and for every quantum system which is dominated by Coulomb interactions. However, in many cases the Schrödinger equation will not be solvable, and we have to go beyond solvable model systems to calculate quantitative properties. In these cases we have to resort to the calculation of approximate solutions. The methods developed in the present chapter are applicable to perturbations of discrete energy levels by time-independent perturbations V of the Hamiltonian, $H_{0} \rightarrow H = H_{0} + V$.