Exact Energy Eigenvalues of the Generalized Dirac–Coulomb Equation via a Modified Similarity Transformation

Abstract

With the aid of a modified similarity transformation we have obtained exact energy eigenvalues of the generalized Dirac -Coulomb equation. This equation consists of the time component of the Lorentz 4-vector potential V v (r) = −A 1 /r, and a Lorentz scalar potential V s (r) = −A 2 /r. The transformed radial equations are so simple so that their solutions are inferred from the conventional solutions of the Schrödinger -Coulomb equation.

The Quantum Spectrum of the 2D Dirac Equation with a Coulomb Potential: Power Series Approach

“…The Dirac equation in a central potential [14][15][16] and the qualitative properties of this case as well as the discrete eigenvalues of the radial Dirac operator [17,18] have been discussed. The Dirac equation with the Coulomb potential and the case with the complex Coulomb field have been carried out by Mustafa et al with the aid of a similarity transformation [19]. Recently, with the interest of the higher-dimensional field theory, we have studied the Dirac equation with the Coulomb potential in D þ 1 dimensions [20].…”

Dirac and Klein Gordon particles in complex Coulombic fields: a similarity transformation

“…The key idea is that instead of solving Dirac-Coulomb equation directly, one can solve the second-order Dirac equation [16][17][18][19][20][21][22] In what follows we recycle the modified similarity transformation ( used by Mustafa et al [17] ) and obtain exact solutions for the non-Hermitian generalized Dirac and Klein-Gordon Coulomb Hamiltonians. Although this problem might be seen as oversimplified, it offers a benchmark for the yet to be adequately explored non-Hermitian relativistic Hamiltonians.…”

“…The ordinary ( Hermitian) Dirac Hamiltonian is exactly solvable in this case ( cf. e.g., [16,17]). In fact the exact solution to Dirac equation for an electron in a Coulomb field was first obtained by Darwin [18] and Gordon [19].…”

“…Obviously, unlike the ordinary ( Hermitian) Sommerfeld fine structure formula, equation (17) suggests that a continuous increase of the coupling strength Zα from zero pushes up the electron states into the positive energy continuum, avoiding herby the energy gap. Nevertheless, for states with n = j + 1/2 one obtains…”

“…The key idea is that instead of solving Dirac-Coulomb equation directly, one can solve the second-order Dirac equation [16][17][18][19][20][21][22] which is obtained by multiplying the original equation, from the left, by a differential operator. The second-order equation is similar to Klein-Gordon equation in a Coulomb field.…”