Basic Concept

  Although many approaches exist, during our study of Many-Body Perturbation Theory (MBPT) it will become evident that Rayleigh-Schrödinger Perturbation Theory (RSPT) is an excellent starting point. Chapter 9 of Ref. [2]\ contains an excellent introduction to RSPT.

Let be the Hamiltonian for the quantum mechanical system of interest. As always, we are interested in the eigenvalues and eigenfunctions which are solutions to the Schrödinger equation.

  Assume a similar system exists for which non-degenerate solutions (eigenvalues and eigenfunctions) are already known or easily obtained.

  Let the difference between and be denoted as .

Introduce a physically motivated perturbation parameter .

The Schrödinger equation for the system becomes
 
Since depends on , the eigenfunctions and eigenvalues of must also depend on . Thus and can be expanded in a power series with respect to .

The following abbreviations were employed in the expansion.

To obtain the energy or wavefunction of a system you merely sum the corrections. Obviously the convergence of these power series becomes a key issue since the summations must be truncated in practice.

Using these expansions of and in the Schrödinger equation (Eqn 10) for the system of interest we have
 
Expanding this we obtain terms with various powers of on both the l.h.s. and r.h.s. If equality is to hold for all , all terms on the l.h.s. of Eqn. 15 must equal to those on the r.h.s. for a given power of lambda. This gives rise to the so called ``kth order Schrödinger equations.''