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arXiv:1401.8070

Special rectangular (double-well and hole) potentials

Zafar Ahmed, Tanayveer Singh Bhatia, Shashin Pavaskar, Achint Kumar

Published Jan 31, 2014• 1 citation
arXivHTMLPDF

Abstract

We revisit a rectangular barrier as well as a rectangular well (pit) between two rigid walls. The former is the well known double-well potential and the latter is a hole potential. Let $|V_0|$ be the height (depth) of the barrier (well) then for a fixed geometry of the potential, we show that in the double-well, $E=V_0(>0)$, and in the hole potential ($V_0 <0$), $E=0$, can be energy eigenvalues provided $V_0$ admits some special discrete values. These states have been missed out earlier which emerge only when one seeks the special zero-energy solution of one-dimensional Schr{ö}dinger equation as $ψ(x)=Bx+C$.

DOI
10.48550/arxiv.1401.8070

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We revisit a rectangular barrier as well as a rectangular well (pit) between two rigid walls. The former is the well known double-well potential and the latter is a hole potential. Let ∣V0∣|V_0|∣V0​∣ be the height (depth) of the barrier (well) then for a fixed geometry of the potential, we show that in the double-well, E=V0(>0)E=V_0(>0)E=V0​(>0), and in the hole potential (V0<0V_0 <0V0​<0), E=0E=0E=0, can be energy eigenvalues provided V0V_0V0​ admits some special discrete values. These states have been missed out earlier which emerge only when one seeks the special zero-energy solution of one-dimensional Schr{ö}dinger equation as ψ(x)=Bx+Cψ(x)=Bx+Cψ(x)=Bx+C.
Abstract We study a general double Dirac delta potential to show that this is the simplest yet still versatile solvable potential to introduce double wells, avoided crossings, resonances and perfect transmission ( T = 1). Perfect transmission energies turn out to be the critical property of symmetric and anti-symmetric cases wherein these discrete energies are found to correspond to the eigenvalues of a Dirac delta potential placed symmetrically between two rigid walls. For well(s) or barrier(s), perfect transmission (or zero reflectivity, R ( E ) ) at energy E = 0 is non-intuitive. However, this has been found earlier and called the ‘threshold anomaly’. Here we show that it is a critical phenomenon and we can have 0 ≤ R ( 0 ) < 1 when the parameters of the double delta potential satisfy an interesting condition. We also invoke a zero-energy and zero curvature eigenstate ( ψ ( x ) = Ax + B ) of the delta well between two symmetric rigid walls for R ( 0 ) = 0 . We resolve that the resonant energies and the perfect transmission energies are different and they arise differently.
Two rectangular models described by the one-dimensional Schr\"odinger equation with sharply localized potentials are suggested. The potentials have a multilayer thin structure being composed from adjacent barriers and wells. Their peculiar tunneling properties are studied in considerable detail. Particularly, in the zero-range limit when the potentials are squeezed to a single point, sharp peaks with total transmission are observed at certain (positive and negative) quantized values of the potential strength constant forming infinite discrete sets. Beyond these sets, the barrier-well structures behave as a perfectly reflecting wall. The transcendental equations with respect to potential strengths, the solutions of which determine transmission (resonance) sets, are derived. In this regard, both the models are exactly solvable. The energy dependence of an incident particle is shown to reveal a resonance behavior, being completely different from that observed in a typical double-barrier structure.
In some molecules such as ammonia, double vibrations are observed at small energies. These small vibrations are thought to arise from double well potentials. To explain these small energy gaps, double potential wells are investigated, and complex calculations are made. Until recently, since the exact solution of the Schrödinger equation was not known, approximate solutions were always obtained. The exact solution of the Schrödinger equation is now known. In previous studies, a simple procedure for the general solution of the radial Schrödinger Equation has been found for spherical symmetric potentials without making any approximation. In this article, the Schrodinger equation was solved with this new solution method. It has been applied to the oscillations through a barrier tunneling potential for a particle in a central quadratic deformed potential well (double potential well). It is seen that there is no need to look for complex potentials to explain the double wells observed in small vibrations in ammonia and similar molecules, they were compared with experimentally measured energies. The results were found to be fully compatible. It has been seen that there can always be double well potentials at small energies for all molecules, atomic nuclei, and similar particles with angular momentum other than zero.
The solution to a problem in quantum mechanics is generally a linear superposition of states. The solutions for double well potentials epitomize this property, and go even further than this: they can often be described by an effective model whose low energy features can be described by two states—one in which the particle is on one side of the barrier, and a second where the particle is on the other side. Then the ground state remains a linear superposition of these two macroscopic-like states. In this paper, we illustrate that this property is achieved similarly with an attractive potential that separates two regions of space, as opposed to the traditionally repulsive one. In explaining how this comes about we revisit the concept of “orthogonalized plane waves,” first discussed in 1940 to understand electronic band structure in solids, along with the accompanying concept of a pseudopotential. We show how these ideas manifest themselves in a simple double well potential, whose “barrier” consists of a moat instead of the conventional wall.