Self-consistent Approximations

Abstract

Even for a few particles the Schrödinger equation is prohibitively difficult to solve. Hence it is important to have approximations which work in various regimes. One such approximation, which has a nice unifying theme and connects to a large area of mathematics, is the one approximating solutions of n-particle Schrödinger equations by products of n one-particle functions (i.e. functions of 3 variables). This results in a single nonlinear equation in 3 variables, or several coupled such equations. The trade-off here is the number 9 of dimensions for the nonlinearity. This method, which goes under different names, e.g. the mean-field or self-consistent approximation, is especially effective when the number of particles, n, is sufficiently large.