The infinite well and Dirac delta function potentials as pedagogical, mathematical and physical models in quantum mechanics

Belloni, Mario; Robinett, R. W.

doi:10.1016/j.physrep.2014.02.005

Cited by 116 publications

(127 citation statements)

References 216 publications

Supporting

Mentioning

121

Contrasting

Unclassified

Order By: Relevance

“…As a consequence, the first order perturbed state and second order perturbed energy require renormalization [39,58,59]. Note that a similar difficulty would also exist when using perturbation theory to extrapolate from the solutions of H ∞ γ (which is solvable for certain traps shapes) to H τ γ .…”

Section: Split Well Hamiltonianmentioning

confidence: 99%

Infinite barriers and symmetries for a few trapped particles in one dimension

Harshman

2017

Phys. Rev. A

View full text Add to dashboard Cite

This article investigates the properties of a few interacting particles trapped in a few wells and how these properties change under adiabatic tuning of interaction strength and inter-well tunneling. While some system properties are dependent on the specific shapes of the traps and the interactions, this article applies symmetry analysis to identify generic features in the spectrum of stationary states of few-particle, few-well systems. Extended attention is given to a simple but flexible threeparameter model of two particles in two wells in one dimension. A key insight is that two limiting cases, hard-core repulsion and no inter-well tunneling, can both be treated as emergent symmetries of the few-particle Hamiltonian. These symmetries are the mathematical consequences of infinite barriers in configuration space. They are necessary to explain the pattern of degeneracies in the energy spectrum, to understand how degeneracies are broken for models away from limiting cases, and to explain separability and integrability. These symmetry methods are extendable to more complicated models and the results have practical consequences for stable state control in fewparticle, few-well systems with ultracold atoms in optical traps.

show abstract

Section: Split Well Hamiltonianmentioning

confidence: 99%

Infinite barriers and symmetries for a few trapped particles in one dimension

Harshman

2017

Phys. Rev. A

View full text Add to dashboard Cite

show abstract

“…Therefore, it is reasonable to propose a Gaussian-like ansatz for the wave function in the static case. In order to meet the hard-wall condition, however, we modify the Gaussian function such that it has the form of a so-called mirror solution [55,56],…”

Section: Modelmentioning

confidence: 99%

“…In order to extend this approximate solution to the case where V 0 < µ, we must work out the corresponding hard-wall TF solution. With the help of the mirror analogy [55,56], we obtain the approximate TF wave function…”

Section: Hard-wall Thomas-fermi Solutionmentioning

confidence: 99%

Quasi one-dimensional Bose–Einstein condensate in a gravito-optical surface trap

Akram¹,

Girodias²,

Pelster³

2016

J. Phys. B: At. Mol. Opt. Phys.

View full text Add to dashboard Cite

Abstract. We study both static and dynamic properties of a weakly interacting BoseEinstein condensate (BEC) in a quasi one-dimensional gravito-optical surface trap, where the downward pull of gravity is compensated by the exponentially decaying potential of an evanescent wave. First, we work out approximate solutions of the Gross-Pitaevskii equation for both a small number of atoms using a Gaussian ansatz and a larger number of atoms using the Thomas-Fermi limit. Then we confirm the accuracy of these analytical solutions by comparing them to numerical results. From there, we numerically analyze how the BEC cloud expands non-ballistically, when the confining evanescent laser beam is shut off, showing agreement between our theoretical and previous experimental results. Furthermore, we analyze how the BEC cloud expands non-ballistically due to gravity after switching off the evanescent laser field in the presence of a hard-wall mirror which we model by using a blue-detuned far-off-resonant sheet of light. There we find that the BEC shows significant self-interference patterns for a large number of atoms, whereas for a small number of atoms, a revival of the BEC wave packet with few matter-wave interference patterns is observed.

show abstract

“…The aim of this paper is to study the RSs of simple one-dimensional (1D) quantum-mechanical systems, such as double and triple quantum wells, for better understanding of their properties, as well as for generating an analytic basis of RSs for its further use in the RSE treating more complicated potentials. In this work, we take a well-known simplification of a multiple-quantum well/barrier potential, approximating it with a sequence of Dirac delta functions, a model which is widely used in physics [12]. Bound states in such potentials are known in the literature [13], as well as periodic solutions of the famous Kronig-Penney potential [14] modeling the electronic band structure of a 1D crystal lattice.…”

Section: Introductionmentioning

confidence: 99%

Resonant states in double and triple quantum wells

Tanimu

Muljarov

2018

J. Phys. Commun.

View full text Add to dashboard Cite

The full set of resonant states in double and triple quantum well/barrier structures is investigated. This includes bound, anti-bound and normal resonant states which are all eigensolutions of Schrödinger's equation with generalized outgoing wave boundary conditions. The transformation of resonant states and their transitions between different subgroups as well as the role of each subgroup in observables, such as the quantum transmission, is analyzed. The quantum well potentials are modeled by Dirac delta functions; therefore, as part of this study, the well known problem of bound states in delta-like potentials is also revisited.

show abstract

The infinite well and Dirac delta function potentials as pedagogical, mathematical and physical models in quantum mechanics

Cited by 116 publications

References 216 publications

Infinite barriers and symmetries for a few trapped particles in one dimension

Infinite barriers and symmetries for a few trapped particles in one dimension

Quasi one-dimensional Bose–Einstein condensate in a gravito-optical surface trap

Resonant states in double and triple quantum wells

Contact Info

Product

Resources

About