The variational principle in quantum mechanics, lecture 5

1. Lecture 5: Mean-field theory and the Gross-Pitaevskii equation

In the previous lecture we saw two applications of the variational principle to the class of product states and fermionic gaussian states, respectively. In both cases we obtained an effective equation for the ground-state properties involving only a single effective particle (a single spin in the first case and a single majorana fermion in the second case).

In this lecture we continue our study of mean-field theory in the bosonic setting, in particular to the description of Bose-Einstein condensates. Here we find a similar result: we’ll obtain a nonlinear effective equation for the condensate in terms of a single effective particle degree of freedom. Before we do this we need to review some of the formalism for the description of quantum fields and coherent states.

The pdf version of the notes can be found here.