PerturbationsHamiltonian

PerturbationsHamiltonian refers to the study of how small changes, or perturbations, to a system's Hamiltonian affect its behavior. The Hamiltonian, in classical mechanics and quantum mechanics, describes the total energy of a system. When a system is exactly solvable, its Hamiltonian is known precisely. However, many real-world systems are too complex to be solved exactly. In such cases, a perturbation approach is employed.

The core idea is to split the Hamiltonian into two parts: a solvable part, H0, and a small, unsolvable part, H1. The total Hamiltonian is then H = H0 + H1. The solutions to H0 are known, and the effect of H1 is then treated as a small correction to these known solutions. This correction can be calculated using perturbation theory, which provides a systematic way to approximate the energy levels and wavefunctions (or trajectories in classical mechanics) of the perturbed system.

Perturbation theory can be applied in various forms, such as non-degenerate or degenerate perturbation theory, depending on whether the energy levels of the unperturbed system are distinct or repeated. The accuracy of the approximation depends on the size of the perturbation H1 relative to H0. If H1 is sufficiently small, the perturbation series converges, yielding accurate results. This method is fundamental in many areas of physics, including atomic physics, molecular spectroscopy, celestial mechanics, and quantum field theory, allowing for the analysis of complex systems that would otherwise be intractable.