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

Gordon Baym

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

Stationary Perturbation Theory with Spatially Well-separated Potentials

Kim, Seok, Choonkyu Lee

We present a new perturbation theory for quantum mechanical energy eigenstates when the potential equals the sum of two localized, but not necessarily weak potentials $V_{1}(\vec{r})$ and $V_{2}(\vec{r})$ , with the distance $L$ between the respective centers of the two taken to be quite large. It is assumed that complete eigenfunctions of the local Hamiltonians (i.e., in the presence of $V_{1}(\vec{r})$ or $V_{2}(\vec{r})$ only) are available as inputs to our perturbation theory. If the two local Hamiltonians have degenerate bound-state energy levels, a systematic extension of the molecular orbital theory (or the tight-binding approximation) follows from our formalism. Our approach can be viewed as a systematic adaptation of the multiple scattering theory to the problem of bound states.We present a new perturbation theory for quantum mechanical energy eigenstates when the potential equals the sum of two localized, but not necessarily weak potentials $V_{1}(\vec{r})$ and $V_{2}(\vec{r})$, with the distance $L$ between the respective centers of the two taken to be quite large. It is assumed that complete eigenfunctions of the local Hamiltonians (i.e., in the presence of $V_{1}(\vec{r})$ or $V_{2}(\vec{r})$ only) are available as inputs to our perturbation theory. If the two local Hamiltonians have degenerate bound-state energy levels, a systematic extension of the molecular orbital theory (or the tight-binding approximation) follows from our formalism. Our approach can be viewed as a systematic adaptation of the multiple scattering theory to the problem of bound states.

arXiv:quant-ph/0209046Sep 6, 20020 citations

Time-independent perturbation for bound states

Bipin R. Desai

Cambridge University Press eBooksDec 3, 20090 citations

We present a new perturbation theory for quantum mechanical energy eigenstates when the potential equals the sum of two localized, but not necessarily weak potentials

V_{1}(\vec{r})

and

V_{2}(\vec{r})

, with the distance

L

between the respective centers of the two taken to be quite large. It is assumed that complete eigenfunctions of the local Hamiltonians (i.e., in the presence of

V_{1}(\vec{r})

V_{2}(\vec{r})

only) are available as inputs to our perturbation theory. If the two local Hamiltonians have degenerate bound-state energy levels, a systematic extension of the molecular orbital theory (or the tight-binding approximation) follows from our formalism. Our approach can be viewed as a systematic adaptation of the multiple scattering theory to the problem of bound states.