The bound states of a weak attractive potential never come out of doing perturbation theory on free particle states. This is a quite general rule: if the perturbation commutes with a constant of the motion, A, of the unperturbed Hamiltonian, then matrix elements of the perturbation between eigenstates of A with different eigenvalues must vanish. This rule, an example of a selection rule, often simplifies the problem of selecting the correct linear combination of degenerate states with which to do the perturbation expansion. The Rayleigh-Schrodinger perturbation theory – yields rather complicated expressions for the change in the states in second order and beyond. There is a somewhat different form of the perturbation theory, due to Brillouin and Wigner that enables to see more clearly the structure of the perturbed states.The bound states of a weak attractive potential never come out of doing perturbation theory on free particle states. This is a quite general rule: if the perturbation commutes with a constant of the motion, A, of the unperturbed Hamiltonian, then matrix elements of the perturbation between eigenstates of A with different eigenvalues must vanish. This rule, an example of a selection rule, often simplifies the problem of selecting the correct linear combination of degenerate states with which to do the perturbation expansion. The Rayleigh-Schrodinger perturbation theory – yields rather complicated expressions for the change in the states in second order and beyond. There is a somewhat different form of the perturbation theory, due to Brillouin and Wigner that enables to see more clearly the structure of the perturbed states.

CRC Press eBooksMar 5, 20180 citations

The role of continuum states in perturbation theory

Viktor Urumov, Gjorgji Ivanovski

An example of a three-dimensional Schrödinger equation is solved exactly and it is shown that the potential can sustain an arbitrary number of bound states. Nevertheless, in the second-order perturbation theory all the contributions for all bound states come from the continuum states if the perturbation is spherically symmetric. This is also true for any central potential that does not show accidental degeneracy and has both bound and unbound states, when the perturbation is another central potential.An example of a three-dimensional Schrödinger equation is solved exactly and it is shown that the potential can sustain an arbitrary number of bound states. Nevertheless, in the second-order perturbation theory all the contributions for all bound states come from the continuum states if the perturbation is spherically symmetric. This is also true for any central potential that does not show accidental degeneracy and has both bound and unbound states, when the perturbation is another central potential.

American Journal of PhysicsOct 1, 19832 citations

Stationary-State Perturbation Theory

Stephen P. Martin

Graduate texts in physicsJan 1, 20260 citations

Approximate Perturbation Theory for High-Order Electromagnetic Transitions

Howard R. Reiss

An approximation method is developed which yields very simple and general analytical results for electromagnetic bound-bound or bound-free transitions for any arbitrarily high order of interaction with the electromagnetic field. The approximation is perturbative in the sense that transition probabilities are proportional to the field intensity raised to the power $N$ , where $N$ is the order of the interaction. Limiting values of electromagnetic field strength are given within which the approximation is valid. These limits depend upon the binding potential which produces the bound states. A feature of the results is that they show the onset of high-order dominance effects which can occur at high-field intensities.An approximation method is developed which yields very simple and general analytical results for electromagnetic bound-bound or bound-free transitions for any arbitrarily high order of interaction with the electromagnetic field. The approximation is perturbative in the sense that transition probabilities are proportional to the field intensity raised to the power $N$, where $N$ is the order of the interaction. Limiting values of electromagnetic field strength are given within which the approximation is valid. These limits depend upon the binding potential which produces the bound states. A feature of the results is that they show the onset of high-order dominance effects which can occur at high-field intensities.

Physical Review AAug 1, 197228 citations

N

, where

N

is the order of the interaction. Limiting values of electromagnetic field strength are given within which the approximation is valid. These limits depend upon the binding potential which produces the bound states. A feature of the results is that they show the onset of high-order dominance effects which can occur at high-field intensities.