The Dirac Hamiltonian with a superstrong Coulomb field

Abstract

We consider the quantum-mechanical problem of a relativistic Dirac particle moving in the Coulomb field of a point charge Ze. In the literature, it is often declared that a quantum-mechanical description of such a system does not exist for charge values exceeding the so-called critical charge with Z = α −1 = 137 based on the fact that the standard expression for the lower bound state energy yields complex values at overcritical charges. We show that from the mathematical standpoint, there is no problem in defining a self-adjoint Hamiltonian for any value of charge. What is more, the transition through the critical charge does not lead to any qualitative changes in the mathematical description of the system. A specific feature of overcritical charges is a non uniqueness of the self-adjoint Hamiltonian, but this non uniqueness is also characteristic for charge values less than the critical one (and larger than the subcritical charge with Z = ( √ 3/2)α −1 = 118). We present the spectra and (generalized) eigenfunctions for all self-adjoint Hamiltonians. The methods used are the methods of the theory of self-adjoint extensions of symmetric operators and the Krein method of guiding functionals. The relation of the constructed one-particle quantum mechanics to the real physics of electrons in superstrong Coulomb fields where multiparticle effects may be of crucial importance is an open question.

Bound states of massive fermions in Aharonov–Bohm-like fields

“…Physically, they show that the probability current density is equal to zero at the origin. The spectrum of the radial Hamiltonian is determined by [25,31] dσ…”

“…We find all self-adjoint Dirac Hamiltonians as well as their spectra in the AB potential in 2+1 dimensions using the socalled form asymmetry method developed in Refs. [25,26]. In particular, expressions for the wave functions and bound-state energies are obtained as functions of the magnetic flux, spin, and extension parameters.…”

Planar Massless Fermions in Coulomb and Aharonov–bohm Potentials

“…This means that the operator h 0 is essentially self-adjoint, i.e., its unique self-adjoint extension is its closure h =h 0 , which coincides with the adjoint operator h = h * = h † . If (8) is not satisfied then the self-adjoint operator h = h † can be found as the narrowing of h * on the so-called maximum domain [20]. The needed solution of (5) is…”

“…By means of solutions U 1 (r) and U 2 (r) any doublet of D * can be represented in the form (see, [20])…”

“…This Dirac Hamiltonian is symmetric operator so the problem arises to construct all the self-adjoint extensions of a given symmetric operator and then to choose correct self-adjoint extensions by means of physical conditions. We construct the self-adjoint radial Dirac Hamiltonians on the above background by the asymmetry form method [20] originated from von Neumann theory of self-adjoint extensions.…”