BrillouinWigner

Brillouin-Wigner perturbation theory is a method used in quantum mechanics to approximate the energy levels of a system when the Hamiltonian cannot be solved exactly. It is particularly useful for systems where the interaction between particles is weak, allowing for a perturbative approach. The theory was developed by Léon Brillouin and Eugene Wigner in the 1930s.

The Brillouin-Wigner method starts with a known unperturbed Hamiltonian, which can be solved exactly, and a small perturbation term. The energy levels of the unperturbed system serve as the zeroth-order approximation. The perturbation is then treated as a small correction to these energy levels. The first-order correction to the energy is given by the expectation value of the perturbation with respect to the unperturbed eigenstates. Higher-order corrections can be calculated iteratively, although they become increasingly complex.

One of the key advantages of the Brillouin-Wigner method is its ability to handle systems with degenerate energy levels. In such cases, the standard Rayleigh-Schrödinger perturbation theory may fail, as it cannot distinguish between degenerate states. The Brillouin-Wigner method, however, can handle degeneracies by using a more sophisticated approach to calculate the energy shifts.

The Brillouin-Wigner perturbation theory has been widely applied in various fields of physics, including atomic and molecular physics, condensed matter physics, and nuclear physics. It provides a powerful tool for understanding the effects of weak interactions on the energy levels of quantum systems. Despite its limitations, such as the requirement for a small perturbation and the potential for slow convergence in higher orders, the Brillouin-Wigner method remains an essential technique in quantum mechanics.