Perturbation Theory for a Hamiltonian is a mathematical method used in quantum mechanics to approximate the energy levels and wavefunctions of a system when the system's Hamiltonian (the operator that describes its energy) is slightly perturbed by an additional term.
Perturbation Theory for a Hamiltonian is typically used when the system's Hamiltonian is too complex to solve exactly, or when the perturbation is small enough that the approximations made by the theory are accurate.
The main assumptions of Perturbation Theory for a Hamiltonian are that the perturbation is small and that the system has a discrete and non-degenerate energy spectrum. Additionally, the perturbation must not cause any degeneracies or singularities in the energy levels.
The basic process of Perturbation Theory for a Hamiltonian involves expanding the Hamiltonian into a series of terms, with the unperturbed Hamiltonian as the first term and the perturbation as subsequent terms. The energy and wavefunction of the system are then approximated by solving for the coefficients of this series.
One major limitation of Perturbation Theory for a Hamiltonian is that it only works for small perturbations. If the perturbation is large, the approximations made by the theory will not be accurate. Additionally, the theory can become very complicated and difficult to apply for systems with a large number of energy levels.
