ABSTRACT

We have already used the variational principle in Chapters 11 and 13, but here we will give a more careful exposition and describe some more general examples. The basic idea is simple: the expectation value of the Hamiltonian in any normalized state is given by ∑ | c k | 2   E k ≥ E 0   ∑ | c k | 2   =   E 0 , https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429438424/da053ade-e2d6-4199-a300-57784d0ed704/content/eq18_1.tif"/> where E 0 is the ground state energy. Thus, if we start from any state, and make changes that reduce the expectation value of the energy, we get a better approximation for the ground state. The art in employing the variational principle lies in using one’s intuition to find an ansatz for the ground state wave function, which captures important features of the physics of the system, while giving rise to calculations that are relatively simple. A lot of the words that are used to describe complicated physical systems are actually derived from clever variational approximations. The most famous example is the concept of single electron orbitals in complicated systems with interacting electrons.