SUPERSYMMETRY APPROACHES TO THE BOUND STATES OF THE GENERALIZED WOODS–SAXON POTENTIAL

2004 ◽  
Vol 19 (08) ◽  
pp. 615-625 ◽  
Author(s):  
H. FAKHRI ◽  
J. SADEGHI
Keyword(s):  
Differential Equation ◽  
Quantum Mechanics ◽  
Bound States ◽  
Differential Operators ◽  
Energy Levels ◽  
Negative Energy ◽  
Saxon Potential ◽  
First Order ◽  
Jacobi Differential Equation ◽  
Algebraic Solutions

Using the associated Jacobi differential equation, we obtain exactly bound states of the generalization of Woods–Saxon potential with the negative energy levels based on the analytic approach. According to the supersymmetry approaches in quantum mechanics, we show that these bound states by four pairs of the first-order differential operators, represent four types of the laddering equations. Two types of these supersymmetry structures, suggest the derivation of algebraic solutions by two different approaches for the bound states.

scholarly journals On first-order differential operators with Bohr-Neugebauer type property

1989 ◽  
Vol 12 (3) ◽  
pp. 473-476 ◽  
Author(s):  
Aribindi Satyanarayan Rao
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Linear Operator ◽  
Real Line ◽  
Differential Operators ◽  
Bounded Solution ◽  
Almost Periodic ◽  
Bounded Linear ◽  
First Order ◽  
Type Property

We consider a differential equationddtu(t)-Bu(t)=f(t), where the functions u and f map the real line into a Banach space X and B: X→X is a bounded linear operator. Assuming that any Stepanov-bounded solution u is Stepanov almost-periodic when f is Bochner almost-periodic, we establish that any Stepanov-bounded solution u is Bochner almost-periodic when f is Stepanov almost-periodic. Some examples are given in which the operatorddt-B is shown to satisfy our assumption.

Nuclear Physics Methods for Problems in Relativistic Quantum Mechanics

1998 ◽  
Vol 07 (05) ◽  
pp. 559-571
Author(s):  
Marcos Moshinsky ◽  
Verónica Riquer
Keyword(s):  
Quantum Mechanics ◽  
Nuclear Physics ◽  
Bound States ◽  
Relativistic Quantum ◽  
Negative Energy ◽  
Relativistic Quantum Mechanics ◽  
Variational Parameter ◽  
Approximate Methods ◽  
Rest Energy ◽  
Quark Models

Atomic and molecular physicists have developed extensive and detailed approximate methods for dealing with the relativistic versions of the Hamiltonians appearing in their fields. Nuclear physicists were originally more concerned with non-relativistic problems as the energies they were dealing with were normally small compared with the rest energy of the nucleon. This situation has changed with the appearance of the quark models of nucleons and thus the objective of this paper is to use the standard variational procedures of nuclear physics for problems in relativistic quantum mechanics. The 4 × 4α and β matrices in the Dirac equation are replaced by 2 × 2 matrices, one associated with ordinary spin and the other, which we call sign spin, is mathematically identical to the isospin of nuclear physics. The states on which our Hamiltonians will act will be the usual harmonic oscillator ones with ordinary and sign spin and the frequency ω of the oscillator will be our only variational parameter. The example discussed as an illustration will still be the Coulomb problem as the exact energies of the relativistic bound states are available for comparison. A gap of the order of 2mc2 is observed between states of positive and negative energy, that permits the former to be compared with the exact results.

SUPERSYMMETRIC QUANTUM MECHANICS APPLIED TO NONRELATIVISTIC QUARK MODELS

1993 ◽  
Vol 08 (08) ◽  
pp. 1437-1455 ◽  
Author(s):  
E.J.O. GAVIN ◽  
H. FIEDELDEY ◽  
H. LEEB ◽  
S.A. SOFIANOS
Keyword(s):  
Quantum Mechanics ◽  
Bound States ◽  
Approximation Scheme ◽  
Energy Levels ◽  
Supersymmetric Quantum Mechanics ◽  
S Wave ◽  
Nucleon Spectrum ◽  
Quark Potential ◽  
Quark Models ◽  
Original Potential

We examine the effect of changing the energy levels and normalization constants of bound states corresponding to baryons and mesons in nonrelativistic quark models. We do this by applying the transformations of supersymmetric quantum mechanics (SUSYQM) to the potentials used in these models. In particular, we fit the spectra and leptonic decay widths of [Formula: see text] and [Formula: see text] mesons by modifying several existing [Formula: see text] potentials by means of supersymmetric transformations. It is found that the potentials are unchanged beyond 2 fm, and that fitting the widths induces greater oscillations in the potentials than those generated by adjusting the energy levels only. Transformations of SUSYQM are applied to the hypercentral potential in order to accommodate the Roper resonance in the s-wave nucleon spectrum. The quark-quark potential found by inverting the transformed hypercentral potential via a new exact Abel transform differs significantly from the original potential up to 5 fm from the origin and violates the concavity requirement. The [Formula: see text] potential related to this potential by Lipkin’s rule does not reproduce the meson spectrum. As the Hall-Post lower bound is also accurate for baryons, the results of the application of supersymmetric transformations in this approximation scheme are also considered and compared to the upper bound of the hypercentral approximation.

BOUND STATES FOR SCHRÖDINGER HAMILTONIANS: PHASE SPACE METHODS AND APPLICATIONS

1996 ◽  
Vol 08 (04) ◽  
pp. 503-547 ◽  
Author(s):  
PH. BLANCHARD ◽  
J. STUBBE
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Dynamical Systems ◽  
Bound States ◽  
Energy Levels ◽  
Schrödinger Operators ◽  
Particle Systems ◽  
Existence And Nonexistence ◽  
Phase Space Methods ◽  
Special Case

Properties of bound states for Schrödinger operators are reviewed. These include: bounds on the number of bound states and on the moments of the energy levels, existence and nonexistence of bound states, phase space bounds and semi-classical results, the special case of central potentials, and applications of these bounds in quantum mechanics of many particle systems and dynamical systems. For the phase space bounds relevant to these applications we improve the explicit constants.

UPPER AND LOWER LIMITS ON THE EIGENVALUES OF THE SPINLESS SALPETER EQUATION

2005 ◽  
Vol 20 (30) ◽  
pp. 7277-7284 ◽  
Author(s):  
FABIAN BRAU
Keyword(s):  
Differential Equation ◽  
Relativistic Quantum ◽  
Energy Levels ◽  
Relativistic Quantum Mechanics ◽  
Coupling Constant ◽  
Logarithmic Potentials ◽  
First Order ◽  
Cornell Potential ◽  
Meson Spectroscopy ◽  
Lower Limits

In the context of relativistic quantum mechanics, we obtain a nonlinear first order differential equation for the energy as a function of the coupling constant of a central potential. This differential equation is only exact for power law and logarithmic potentials in the massless limit. For other potentials, we discuss under which conditions the differential equation yields rigorous upper and lower limits on the value of energy levels. These results are applied to the Cornell potential used in meson spectroscopy. We also show that the method applies to noncentral potentials.

scholarly journals The critical transition of Coulomb impurities in gapped graphene

2020 ◽  
Vol 2020 (8) ◽  
Author(s):  
M. Asorey ◽  
A. Santagata
Keyword(s):  
Boundary Conditions ◽  
Bound States ◽  
Energy Levels ◽  
Bound State ◽  
Negative Energy ◽  
Spectral Flow ◽  
Supercritical Regime ◽  
Gapped Graphene ◽  
Critical Charge ◽  
Common Framework

AbstractThe effect of supercritical charge impurities in graphene is very similar to the supercritical atomic collapses in QED for Z > 137, but with a much lower critical charge. In this sense graphene can be considered as a natural testing ground for the analysis of quantum field theory vacuum instabilities. We analyze the quantum transition from subcritical to supercritical charge regimes in gapped graphene in a common framework that preserves unitarity for any value of charge impurities. In the supercritical regime it is possible to introduce boundary conditions which control the singular behavior at the impurity. We show that for subcritical charges there are also non-trivial boundary conditions which are similar to those that appear in QED for nuclei in the intermediate regime 118 < Z < 137. We analyze the behavior of the energy levels associated to the different boundary conditions. In particular, we point out the existence of new bound states in the subcritical regime which include a negative energy bound state in the attractive Coulomb regime. A remarkable property is the continuity of the energy spectral flow under variation of the impurity charge even when jumping across the critical charge transition. We also remark that the energy levels of hydrogenoid bound states at critical values of charge impurities act as focal points of the spectral flow.

Representations of the n-dimensional rotation group

1961 ◽  
Vol 57 (3) ◽  
pp. 469-475 ◽  
Author(s):  
A. P. Stone
Keyword(s):  
Bound States ◽  
Differential Operators ◽  
Energy Levels ◽  
Radial Wave Function ◽  
Rotation Group ◽  
Wave Mechanics ◽  
Radial Wave ◽  
Shift Operators ◽  
Branching Rule ◽  
Infinitesimal Operators

ABSTRACTThe commutators of the infinitesimal operators of the n-dimensional rotation group Rn with vector operators under Rn are expressed in a vectorial notation. The infinitesimal operators for the representations (l 0…0) are treated in detail. Shift operators for l are constructed and are used to derive the branching rule for these representations. The energy levels and degeneracy of bound states of a particle under an inverse square force in n dimensions are found by wave mechanics and by expressing the Hamiltonian in terms of Casimir's operator for Rn+1. Differential operators which transform one radial wave function into another are obtained.

scholarly journals Fast Vacuum Fluctuations and the Emergence of Quantum Mechanics

2021 ◽  
Vol 51 (3) ◽  
Author(s):  
Gerard ’t Hooft
Keyword(s):  
Quantum Mechanics ◽  
Ground State ◽  
Quantum Interference ◽  
Direct Detection ◽  
Energy Levels ◽  
Vacuum Fluctuations ◽  
Heavy Particles ◽  
Evolving Systems ◽  
Gedanken Experiment ◽  
Classical Computer

AbstractFast moving classical variables can generate quantum mechanical behavior. We demonstrate how this can happen in a model. The key point is that in classically (ontologically) evolving systems one can still define a conserved quantum energy. For the fast variables, the energy levels are far separated, such that one may assume these variables to stay in their ground state. This forces them to be entangled, so that, consequently, the slow variables are entangled as well. The fast variables could be the vacuum fluctuations caused by unknown super heavy particles. The emerging quantum effects in the light particles are expressed by a Hamiltonian that can have almost any form. The entire system is ontological, and yet allows one to generate interference effects in computer models. This seemed to lead to an inexplicable paradox, which is now resolved: exactly what happens in our models if we run a quantum interference experiment in a classical computer is explained. The restriction that very fast variables stay predominantly in their ground state appears to be due to smearing of the physical states in the time direction, preventing their direct detection. Discussions are added of the emergence of quantum mechanics, and the ontology of an EPR/Bell Gedanken experiment.

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