scholarly journals DIRAC OSCILLATORS AND QUASI-EXACTLY SOLVABLE OPERATORS

2005 ◽  
Vol 20 (25) ◽  
pp. 1875-1885 ◽  
Author(s):  
Y. BRIHAYE ◽  
A. NININAHAZWE
Keyword(s):  
Dirac Equation ◽  
Dirac Operator ◽  
Spherically Symmetric ◽  
Dirac Oscillator ◽  
Generic Polynomial ◽  
Exactly Solvable ◽  
Spectral Equation ◽  
Radial Variable ◽  
Radial Dirac Equation

The Dirac equation is formulated in the background of three types of physically relevant potentials: scalar, vector and "Dirac-oscillator" potentials. Assuming these potentials to be spherically-symmetric and with generic polynomial forms in the radial variable, we construct the corresponding radial Dirac equation. Cases where this linear spectral equation is exactly solvable or quasi-exactly solvable are worked out in details. When available, relations between the radial Dirac operator and some super-algebra are pointed out.

scholarly journals A Novel Exact Solution of the 2+1-Dimensional Radial Dirac Equation for the Generalized Dirac Oscillator with the Inverse Potentials

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
ZiLong Zhao ◽  
ZhengWen Long ◽  
MengYao Zhang
Keyword(s):  
Exact Solution ◽  
Quantum Mechanics ◽  
Dirac Equation ◽  
Exact Solutions ◽  
Bethe Ansatz ◽  
Dirac Oscillator ◽  
Solvable Models ◽  
Ansatz Method ◽  
Radial Dirac Equation

The generalized Dirac oscillator as one of the exact solvable models in quantum mechanics was introduced in 2+1-dimensional world in this paper. What is more, the general expressions of the exact solutions for these models with the inverse cubic, quartic, quintic, and sixth power potentials in radial Dirac equation were further given by means of the Bethe ansatz method. And finally, the corresponding exact solutions in this paper were further discussed.

scholarly journals SOLUTION OF THE DIRAC EQUATION FOR POTENTIAL INTERACTION

2003 ◽  
Vol 18 (27) ◽  
pp. 4955-4973 ◽  
Author(s):  
A. D. ALHAIDARI
Keyword(s):  
Dirac Equation ◽  
Three Dimensional ◽  
Nonrelativistic Limit ◽  
Potential Interaction ◽  
Spherically Symmetric ◽  
Local Interactions ◽  
Shape Invariant ◽  
Exactly Solvable ◽  
Relativistic Extension

An effective approach for solving the three-dimensional Dirac equation for spherically symmetric local interactions, which we have introduced recently, is reviewed and consolidated. The merit of the approach is in producing Schrödinger-like equations for the spinor components that could simply be solved by correspondence with well-known exactly solvable nonrelativistic problems. Taking the nonrelativistic limit will reproduce the nonrelativistic problem. The approach has been used successfully in establishing the relativistic extension of all classes of shape-invariant potentials as well as other exactly solvable nonrelativistic problems. These include the Coulomb, Oscillator, Scarf, Pöschl–Teller, Woods–Saxon, etc.

scholarly journals Exotic fermionic fields and minimal length

2020 ◽  
Vol 80 (8) ◽  
Author(s):  
J. M. Hoff da Silva ◽  
D. Beghetto ◽  
R. T. Cavalcanti ◽  
R. da Rocha
Keyword(s):  
Dirac Equation ◽  
Quantum Gravity ◽  
Dirac Operator ◽  
Uncertainty Principle ◽  
Generalized Uncertainty Principle ◽  
Minimal Length ◽  
Dynamical Equations ◽  
Spacetime Manifold ◽  
Fermionic Fields

Abstract We investigate the effective Dirac equation, corrected by merging two scenarios that are expected to emerge towards the quantum gravity scale. Namely, the existence of a minimal length, implemented by the generalized uncertainty principle, and exotic spinors, associated with any non-trivial topology equipping the spacetime manifold. We show that the free fermionic dynamical equations, within the context of a minimal length, just allow for trivial solutions, a feature that is not shared by dynamical equations for exotic spinors. In fact, in this coalescing setup, the exoticity is shown to prevent the Dirac operator to be injective, allowing the existence of non-trivial solutions.

scholarly journals SPECTRUM OF THE RELATIVISTIC PARTICLES IN VARIOUS POTENTIALS

2005 ◽  
Vol 20 (12) ◽  
pp. 911-921 ◽  
Author(s):  
RAMAZAN KOÇ ◽  
MEHMET KOCA
Keyword(s):  
Quantum Mechanics ◽  
Hypergeometric Functions ◽  
Two Dimensions ◽  
Dirac Oscillator ◽  
Confluent Hypergeometric Functions ◽  
Solvable Potentials ◽  
Exactly Solvable Potentials ◽  
Exactly Solvable ◽  
Nonrelativistic Quantum Mechanics ◽  
Relativistic Particles

We extend the notion of Dirac oscillator in two dimensions, to construct a set of potentials. These potentials become exactly and quasi-exactly solvable potentials of nonrelativistic quantum mechanics when they are transformed into a Schrödinger-like equation. For the exactly solvable potentials, eigenvalues are calculated and eigenfunctions are given by confluent hypergeometric functions. It is shown that, our formulation also leads to the study of those potentials in the framework of the supersymmetric quantum mechanics.

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