Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

European Journal of Physics

Quadrupolar, triple δ-function potential in one dimension

S. H. Patil

Published Apr 9, 2009• 7 citations
Publisher

Abstract

The energy and parity eigenstates for quadrupolar, triple δ-function potential are analysed. Using the analytical solutions in specific domains, simple expressions are obtained for even- and odd-parity bound-state energies. The Heisenberg uncertainty product is observed to have a minimum for a specific strength of the potential. The scattering states have a transmission resonance for some energies, and the poles of the transmission coefficients correspond to bound-state energies. These features are of considerable pedagogical interest in the understanding of quantum-mechanical properties of bound and scattering states, for graduate and general physicists.

DOI
10.1088/0143-0807/30/3/020

Related papers

Re-discussion on the Bound Energy and Scattering Amplitude Pole by the Schrdinger and the Heisenberg Representation

Wang Zhi-liang

One-dimensional δ-potential well bound state and one-dimensional δ-scattering potential were discussed by Schrdinger and Heisenberg representation in the wave function in the time-dependent state.It is proved more generally that one-dimensional δ-potential well restraint level is the δ-barrier scattering amplitude pole.

Journal of Jiamusi University•Jan 1, 2009•0 citations

Resonance scattering from a double δ-function potential

I. Richard Lapidus

The transmission coefficient and total cross section as functions of energy are obtained for scattering of particles by a double δ-function potential in one dimension. This example may be useful for the discussion of resonance scattering in an introductory quantum mechanics course. (AIP)

American Journal of Physics•Jul 1, 1982•13 citations

Completeness of the energy eigenfunctions for the one-dimensional δ-function potential

S. H. Patil

We develop energy eigenfunctions for the one-dimensional δ-function potential, with well-defined parity and appropriate orthogonalization. We use these elementary functions to present an explicit demonstration of the completeness of the energy eigenfunctions. It provides a simple illustration of an essential quantum mechanical property, which is pedagogically important.

American Journal of Physics•Aug 1, 2000•16 citations

Quantum solutions for the delta ring and delta shell

Luis F. Castillo-Sánchez, Julio C. Gutiérrez-Vega

The quantum description of a free particle subject to a delta circular ring and a delta spherical shell is studied. We determine the bound eigenstates and energy spectra for different angular momentum states. The normalization constants of the bound states and the transmission and reflection coefficients of the scattering states are calculated in closed form. The ground state of the delta ring is always present regardless of the potential strength. In contrast, the delta shell requires a threshold of the delta strength for the first eigenstate to appear. For the ring and the shell, the wave functions tend to be degenerate at high energies, and the effect of angular momentum on the wave functions becomes negligible. This work generalizes the standard problem of the one-dimensional delta potential studied in textbooks.

American Journal of Physics•Jun 20, 2025•2 citations

The poles of one-dimensional quantum scattering and the energy of bound state

Lü Xiao

The analytic properties of one-dimensional Schrodinger equation are discussed. The relations between the pole of transmission coefficients and the energy of bound states are obtained. Calculations are performed for an example.

College Physics•Jan 1, 2002•0 citations

Transmission Coefficients of One-dimensional Symmetric Double Rectangular Potential andδPotential

Luo Zhi-Quan

The scattering of a free particle by one - dimensional symmetric double rectangular potential(ODSDRP) andδpotential barriers is analyzed.Bound - states energy levels of the free particle in the ODSDRP andδpotential wells are discussed.Transmission coefficients and equation of bound -states energy levels are strictly derived.

Journal of China West Normal University•Jan 1, 2011•0 citations
The energy and parity eigenstates for quadrupolar, triple δ-function potential are analysed. Using the analytical solutions in specific domains, simple expressions are obtained for even- and odd-parity bound-state energies. The Heisenberg uncertainty product is observed to have a minimum for a specific strength of the potential. The scattering states have a transmission resonance for some energies, and the poles of the transmission coefficients correspond to bound-state energies. These features are of considerable pedagogical interest in the understanding of quantum-mechanical properties of bound and scattering states, for graduate and general physicists.
The quantum description of a free particle subject to a delta circular ring and a delta spherical shell is studied. We determine the bound eigenstates and energy spectra for different angular momentum states. The normalization constants of the bound states and the transmission and reflection coefficients of the scattering states are calculated in closed form. The ground state of the delta ring is always present regardless of the potential strength. In contrast, the delta shell requires a threshold of the delta strength for the first eigenstate to appear. For the ring and the shell, the wave functions tend to be degenerate at high energies, and the effect of angular momentum on the wave functions becomes negligible. This work generalizes the standard problem of the one-dimensional delta potential studied in textbooks.