Dirac’s reduced radial equations and the problem of additional solutions

Abstract

We show that additional solutions must be ignored (in differences of the Schrodinger and Klein-Gordon equations) in the Dirac equation, where usually passed the second order radial equation, called the reduced equation, instead of a system. Analogously to the Schrodinger equation, in this process the Dirac's delta function appears, which was unnoted during the full history of quantum mechanics. This unphysical term we remove by a boundary condition at the origin. However, the distribution theory imposes on the radial function strong restriction and by this reason practically for all potentials, even regular, use of these reduced equations is not permissible. At the end we include consideration in the framework of two-dimensional Dirac equation. We show that even here the additional solution does not survives as a result of usual physical requirements.

Hypervirial and Ehrenfest Theorems in Spherical Coordinates: Systematic Approach

“…This condition corresponds to the Dirichlet boundary condition for reduced u rR  wave function (for details see [11][12][13][14], also the Appendix below). Some authors consider the condition (3.1) as too restrictive and recommends other boundary conditions, which also guarantee the self-adjointness of the reduced radial Hamiltonian.…”

“…Here the (+) sign corresponds to repulsion, while the (-) sign -to attraction. For such potential the wave function has the following behavior [11][12][13][14]:…”

Analytical Bound State Solutions of the Klein-Fock-Gordon Equation for the Sum of Hulthén and Yukawa Potential within SUSY Quantum Mechanics

“…In principle, the radial function nature at the origin was investigated particularly for singular potentials by Khelashvili et al [50,51]. While the Laplace operator is portrayed in spherical coordinates, the radial wave equation's exact derivation demonstrates the perspective of a delta function term.…”

Analytical bound-state solutions of the Klein-Fock-Gordon equation for the sum of Hulthén and Yukawa potential within SUSY quantum mechanics

“…The numerous research works reveal the SUSY QM method's power and simplicity in solving wave equations of the central and non-central potentials for arbitrary l states. [42][43][44][45][46][47][48][49] In principle, the radial function nature at the origin was investigated particularly for singular potentials by Khelashvili et al [50,51]. While the Laplace operator is portrayed in spherical coordinates, the radial wave equation's exact derivation demonstrates the perspective of a delta function term.…”