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arXiv:1511.08885 · Journal of Physics A Mathematical and General

Quantum exotic: a repulsive and bottomless confining potential

Miloslav Znojil

Published Apr 10, 1998• 9 citations
arXivHTMLPDF

Abstract

On a simple model we demonstrate that even in a classically repulsive regime (i.e. at couplings which make the potential decreasing to in some directions) quantum mechanics may still support the purely discrete spectrum of bound states. In our example, there exists a critical boundary of this domain of stability where a further increase of repulsion causes an explosive escape of particles in infinity.

DOI
10.1088/0305-4470/31/14/019

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On a simple model we demonstrate that even in a classically repulsive regime (i.e. at couplings which make the potential decreasing to in some directions) quantum mechanics may still support the purely discrete spectrum of bound states. In our example, there exists a critical boundary of this domain of stability where a further increase of repulsion causes an explosive escape of particles in infinity.
It is well known that quantum mechanics allows the penetration into classically forbidden regions (tunneling). Less well known seems to be the fact that in some sense the converse is true also. Potentials with classically allowed regions where a particle can move freely to infinity can nevertheless lead to bound states in quantum mechanics due to the stringent requirements of the boundary conditions, thus forbidding an escape to infinity. This effect is demonstrated by using an obvious generalization of the well-known one-dimensional (D=1) square well potential to arbitrary space dimensions.
It is known that the attractive singular inverse-square potential gives rise to the critical quantum collapse in the framework of the three-dimensional (3D) linear Schroedinger equation. This article summarizes theoretical results which demonstrate suppression of the collapse, caused by this singular potential, and the creation of the otherwise missing ground state (GS) in a 3D gas of bosonic particles, carrying an electric dipole moment, which are pulled to the central electric charge, with repulsive contact interactions between the particles. In the mean-field approximation, the repulsive interactions are represented by the cubic term in the respective Gross-Pitaevskii (GP) equation. In addition to the GS, excited states with angular momentum are briefly considered too. Another topic considered in the article is 1D and 2D bound states in the linear Schroedinger and GP equations with the repulsive potential, which demonstrates a singularity at r --> infinity. A very recent result is that such a potential, growing faster than the negative harmonic-oscillator potential, produces a full spectrum of counter-intuitive normalizable (localized) bound states. The article puts forward perspectives for further studies of linear and nonlinear bound states existing under the action of the potentials with the singularity at r --> 0 or r --> infinity.
In quantum mechanics a localized attractive potential typically supports a (possibly infinite) set of bound states, characterized by a discrete spectrum of allowed energies, together with a continuum of scattering states, characterized (in one dimension) by an energy-dependent phase shift. The 1∕x2 potential on 0<x<∞ confounds all of our intuitions and expectations. Resolving its paradoxes requires sophisticated theoretical machinery: regularization, renormalization, anomalous symmetry-breaking, and self-adjoint extensions. Our goal is to introduce the essential ideas at a level accessible to advanced undergraduates.
The existence of surface states of nonrelativistic particles at the interface between a vacuum and a layer of a material on a reflecting substrate, which are caused by the absorption in the layer rather than the presence of discrete levels, is established. The wave functions of such states (surface matter waves) are found and the domains of their existence in the complex plane of the “optical” potential (both attractive and repulsive) of the layer are determined. Threshold effects of the count of microparticles at the intersection of the boundaries of these domains are revealed. The spatial characteristics of the surface matter waves are calculated. An example of the layer composition providing the existence of such waves is given for ultracold neutrons. The increase in the loss of trapped ultracold neutrons owing to the excitation of the surface matter waves of these particles in the trap walls is discussed.
Exchange statistics endows identical particles with an effective "statistical potential", whose familiar exact form is two-body and purely repulsive for fermions. Here we construct an exact collective many-body form: the thermodynamics of NNN trapped ideal fermions maps onto classical distinguishable particles governed by a single potential -- exactly for harmonic confinement at all temperatures, and to leading semiclassical order for arbitrary potentials. The associated force separates canonically into pairwise contributions, which for N≥3N\geq 3N≥3 can turn attractive, governed by a simple geometric criterion. Classical minimization reproduces observed few-body Pauli-crystal symmetries and agrees with the N=55N=55N=55 ground-state probability maximum at sub-percent shell accuracy. Heating drives discrete structural transitions accompanied by a crossover of the strongest force from attractive to repulsive. Both the potential and its forces are directly computable from existing single-shot imaging data, turning quantum exchange into measurable classical mechanics.