Further Exact Solutions of the Dirac Equation

We develop a non-commutative integration method for the Dirac equation in homogeneous spaces. The Dirac equation with an invariant metric is shown to be equivalent to a system of equations on a Lie group of transformations of a homogeneous space. This allows us to effectively apply the non-commutative integration method of linear partial differential equations on Lie groups. This method differs from the well-known method of separation of variables and to some extent can often supplement it. The general structure of the method developed is illustrated with an example of a homogeneous space which does not admit separation of variables in the Dirac equation. However, the basis of exact solutions to the Dirac equation is constructed explicitly by the non-commutative integration method. In addition, we construct a complete set of new exact solutions to the Dirac equation in the three-dimensional de Sitter space-time AdS3 using the method developed. The solutions obtained are found in terms of elementary functions, which is characteristic of the non-commutative integration method.

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Exact solutions of the nilpotent Dirac equation

Journal of Physics Conference Series ◽

10.1088/1742-6596/1557/1/012035 ◽

2020 ◽

Vol 1557 ◽

pp. 012035

Author(s):

P Rowlands ◽

S Rowlands

Keyword(s):

Dirac Equation ◽

Exact Solutions

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

Advances in High Energy Physics ◽

10.1155/2019/3423198 ◽

2019 ◽

Vol 2019 ◽

pp. 1-11 ◽

Cited By ~ 1

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.

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