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

The Theory of Scattering

Springer eBooksOct 8, 20080 citations

Potential Scattering Theory

WORLD SCIENTIFIC eBooksAug 1, 20090 citations

Study of One Dimensional Scattering by A Two-Step Potential

Francis Stephen Uralil

The purpose of this investigation is to contribute to the theory of one-dimensional scattering through a thorough study of the two-step potential. For this purpose we have studied the effect of changing a simple square well potential near its edge by adding a second step. The problem is treated using both perturbational and non-perturbational methods. The order of procedure is: (1) a general introduction, (2) multiple reflections of a wave packet by an attractive potential well, (3) discussion of the- two-step potential using perturbation theory, (4) calculation of depth of potential well for zero energy resonance, (5) conclusion.The purpose of this investigation is to contribute to the theory of one-dimensional scattering through a thorough study of the two-step potential. For this purpose we have studied the effect of changing a simple square well potential near its edge by adding a second step. The problem is treated using both perturbational and non-perturbational methods. The order of procedure is: (1) a general introduction, (2) multiple reflections of a wave packet by an attractive potential well, (3) discussion of the- two-step potential using perturbation theory, (4) calculation of depth of potential well for zero energy resonance, (5) conclusion.

Jan 1, 19720 citations

Multiple approaches to incorporating scattering states in non-degenerate perturbation theory

Don C. Colladay, John Eric Goff

The second-order perturbative Stark effect on the ground state of hydrogen is a typical example presented in many standard texts on quantum mechanics. Some texts miss the fact that the scattering states are significant contributors to the perturbative energy correction. The inclusion of scattering states has wider applicability than to just the Stark effect. An explicit calculation involving a finite-square well with a perturbation is used to illustrate the importance of including scattering states into the calculation. The second-order correction to the ground-state energy is obtained in three distinct ways. The first involves altering the problem by imposing additional boundary conditions at large distances to make the positive-energy spectrum discrete. The second makes use of the continuum scattering states directly. The third bypasses the use of scattering states by solving a differential equation for the first-order correction to the wave function.The second-order perturbative Stark effect on the ground state of hydrogen is a typical example presented in many standard texts on quantum mechanics. Some texts miss the fact that the scattering states are significant contributors to the perturbative energy correction. The inclusion of scattering states has wider applicability than to just the Stark effect. An explicit calculation involving a finite-square well with a perturbation is used to illustrate the importance of including scattering states into the calculation. The second-order correction to the ground-state energy is obtained in three distinct ways. The first involves altering the problem by imposing additional boundary conditions at large distances to make the positive-energy spectrum discrete. The second makes use of the continuum scattering states directly. The third bypasses the use of scattering states by solving a differential equation for the first-order correction to the wave function.

American Journal of PhysicsAug 19, 20201 citation

Quantum Scattering Theory

S. Zienau

NatureApr 6, 196833 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.