scholarly journals Exactly solvable quantum few-body systems associated with the symmetries of the three-dimensional and four-dimensional icosahedra

2016 ◽  
Vol 1 (1) ◽  
Author(s):  
Thibault Scoquart ◽  
Joseph Seaward ◽  
Steven Glenn Jackson ◽  
Maxim Olshanii
Keyword(s):  
Quantum Particle ◽  
Three Dimensional ◽  
Particle Systems ◽  
Parametric Family ◽  
Half Line ◽  
Reflection Groups ◽  
Exactly Solvable ◽  
Solvable Quantum ◽  
Regular Icosahedron

The purpose of this article is to demonstrate that non-crystallographic reflection groups can be used to build new solvable quantum particle systems. We explicitly construct a one-parametric family of solvable four-body systems on a line, related to the symmetry of a regular icosahedron: in two distinct limiting cases the system is constrained to a half-line. We repeat the program for a 600600-cell, a four-dimensional generalization of the regular three-dimensional icosahedron.

scholarly journals HIDDEN SCALE IN QUANTUM MECHANICS

2009 ◽  
Vol 24 (27) ◽  
pp. 2203-2211 ◽  
Author(s):  
PULAK RANJAN GIRI
Keyword(s):  
Wave Packet ◽  
Quantum Particle ◽  
Dimensional Space ◽  
Free Particle ◽  
Three Dimensional ◽  
Particle Systems ◽  
Quantum Mechanical ◽  
Scale Invariant ◽  
Mechanical Models ◽  
Three Dimensional Space

We show that the intriguing localization of a free particle wave-packet is possible due to a hidden scale present in the system. Self-adjoint extensions (SAE) is responsible for introducing this scale in quantum mechanical models through the nontrivial boundary conditions. We discuss a couple of classically scale invariant free particle systems to illustrate the issue. In this context it has been shown that a free quantum particle moving on a full line may have localized wave-packet around the origin. As a generalization, it has also been shown that particles moving on a portion of a plane or on a portion of a three-dimensional space can have unusual localized wave-packet.

scholarly journals Accelerated and Airy–Bloch oscillations

2016 ◽  
Vol 30 (26) ◽  
pp. 1650189 ◽  
Author(s):  
Stefano Longhi
Keyword(s):  
Quantum Particle ◽  
Dynamical Behavior ◽  
Periodic Breathing ◽  
Quantum Diffusion ◽  
Bloch Oscillations ◽  
Classical Motion ◽  
Accelerated Motion ◽  
Exactly Solvable ◽  
Solvable Quantum ◽  
Analytical Expressions

A quantum particle subjected to a constant force undergoes an accelerated motion following a parabolic path, which differs from the classical motion just because of wave packet spreading (quantum diffusion). However, when a periodic potential is added (such as in a crystal) the particle undergoes Bragg scattering and an oscillatory (rather than accelerated) motion is found, corresponding to the famous Bloch oscillations (BOs). Here, we introduce an exactly-solvable quantum Hamiltonian model, corresponding to a generalized Wannier–Stark Hamiltonian [Formula: see text], in which a quantum particle shows an intermediate dynamical behavior, namely an oscillatory motion superimposed to an accelerated one. Such a novel dynamical behavior is referred to as accelerated BOs. Analytical expressions of the spectrum, improper eigenfunctions and propagator of the generalized Wannier–Stark Hamiltonian [Formula: see text] are derived. Finally, it is shown that acceleration and quantum diffusion in the generalized Wannier–Stark Hamiltonian are prevented for Airy wave packets, which undergo a periodic breathing dynamics that can be referred to as Airy–Bloch oscillations.

scholarly journals SUSUSY Quantum Mechanics

1997 ◽  
Vol 12 (01) ◽  
pp. 171-176 ◽  
Author(s):  
David J. Fernández C.
Keyword(s):  
Quantum Mechanics ◽  
Shift Operator ◽  
Second Order ◽  
Parametric Family ◽  
First Order ◽  
Shift Operators ◽  
Exactly Solvable ◽  
Exactly Solvable Hamiltonians

The exactly solvable eigenproblems in Schrödinger quantum mechanics typically involve the differential "shift operators". In the standard supersymmetric (SUSY) case, the shift operator turns out to be of first order. In this work, I discuss a technique to generate exactly solvable eigenproblems by using second order shift operators. The links between this method and SUSY are analysed. As an example, we show the existence of a two-parametric family of exactly solvable Hamiltonians, which contains the Abraham–Moses potentials as a particular case.

scholarly journals RESONANCES ON HEDGEHOG MANIFOLDS

2013 ◽  
Vol 53 (5) ◽  
pp. 416-426 ◽  
Author(s):  
Pavel Exner ◽  
Jiří Lipovský
Keyword(s):  
Real Axis ◽  
Quantum Particle ◽  
Single Point ◽  
Three Dimensional ◽  
High Energy ◽  
Quantum Graphs ◽  
The Real ◽  
Momentum Plane ◽  
Embedded Eigenvalues ◽  
Aharonov Bohm

We discuss resonances for a nonrelativistic and spinless quantum particle confined to a two- or three-dimensional Riemannian manifold to which a finite number of semiinfinite leads is attached. Resolvent and scattering resonances are shown to coincide in this situation. Next we consider the resonances together with embedded eigenvalues and ask about the high-energy asymptotics of such a family. For the case when all the halflines are attached at a single point we prove that all resonances are in the momentum plane confined to a strip parallel to the real axis, in contrast to the analogous asymptotics in some metric quantum graphs; we illustrate this on several simple examples. On the other hand, the resonance behaviour can be influenced by a magnetic field. We provide an example of such a ‘hedgehog’ manifold at which a suitable Aharonov-Bohm flux leads to absence of any true resonance, i.e. that corresponding to a pole outside the real axis.

scholarly journals QUASI-EXACTLY-SOLVABLE QUANTUM LATTICE SOLITONS

2006 ◽  
Vol 21 (06) ◽  
pp. 1221-1238
Author(s):  
YVES BRIHAYE ◽  
NATHALIE DEBERGH ◽  
ANCILLA NININAHAZWE
Keyword(s):  
Coupling Constant ◽  
Quantum Lattice ◽  
Exactly Solvable ◽  
Solvable Quantum

We extend the exactly solvable Hamiltonian describing f quantum oscillators considered recently by J. Dorignac et al. We introduce a new interaction which we choose to be quasi-exactly solvable. The properties of the spectrum of this new Hamiltonian are studied as functions of the new coupling constant. We point out that both the original and the by us modified Hamiltonians are related to adequate Lie structures.

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