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arXiv:1901.00074

A Fully Relativistic Treatment of Confined Hydrogen-Like Atoms

Jacob Noon

Published Jan 1, 2019• 0 citations
arXivHTMLPDF

Abstract

The Dirac equation is used to provide a relativistic calculation of the binding energy of a hydrogen-like atom confined within a penetrable spherical barrier. We take the potential to be Coulombic within the barrier and constant outside the barrier. Binding energies are derived for the ground state of hydrogen for various barrier heights and confining radii. In addition, it is shown that without the introduction of the principle quantum number $n$, all energy states of the confined relativistic hydrogen atom, determined by a single quantum number $k$, transfer into the known energy states of the free relativistic hydrogen atom as the radius of confinement becomes large.

DOI
10.48550/arxiv.1901.00074

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The Dirac equation is used to provide a relativistic calculation of the binding energy of a hydrogen-like atom confined within a penetrable spherical barrier. We take the potential to be Coulombic within the barrier and constant outside the barrier. Binding energies are derived for the ground state of hydrogen for various barrier heights and confining radii. In addition, it is shown that without the introduction of the principle quantum number nnn, all energy states of the confined relativistic hydrogen atom, determined by a single quantum number kkk, transfer into the known energy states of the free relativistic hydrogen atom as the radius of confinement becomes large.
We make use of the Dirac equation to provide a relativistic calculation of the binding energy of a hydrogen-like atom confined within a penetrable spherical barrier. We take the potential to be Coulombic within the barrier and constant outside the barrier. The derived energy values are compared with the energy values obtained from a non-relativistic calculation based on the Schrodinger equation, where significant differences are found for small values of the barrier radius. In addition, we show that without the introduction of the principle quantum number nnn, all energy states of the confined atom, determined by a single quantum number kkk, transfer into the known energy states of the free atom as the radius of the barrier becomes large.
Since the pioneering works of Michels et. al., followed by Sommerfeld and Welker, over eighty years ago, there is much interest in modelling confinement of the ground state hydrogen atom by an impenetrable sphere of confining radius, Rc. The literature “truncated probability density” model assumes that, upon confinement, the nodeless free hydrogen atom ground state probability density is replaced by the innermost node of an excited state probability density, fitted to Rc by a non-quantum parameter, λ. Since the literature model is an excited state, the confined electron energy, E1c, is inversely proportional to the excited state Bohr radius, Rλ (or aoλ2), which becomes larger than Rc as λ increases. However, due to the infinite potential barrier at radii greater than Rc, such large values of Rλ are not allowed. In contrast, for the Particle in an Infinite Box (PIB), Harmonic Oscillator (HO) and the proposed compressed probability density ground state models, the particle energy is inversely proportional to the confined length (Lc or Rc), the maximum length allowed. Additionally, the Dirac/Sommerfeld fine structure formula calculation of E1c is in agreement with the proposed model, but does not support the literature truncated probability density model, indicating the literature model is incorrect.
In 1928, the brilliance of Paul Dirac granted him the insight to develop a particle equation that was fully consistent with both the principles of quantum mechanics and the theory of special relativity. In this bachelor thesis we have studied the fundamentals of relativistic quantum mechanics including the derivation of the Dirac equation in a similar fashion to the way originally done by Paul Dirac. Additionally we have studied the analytical solution for hydrogen-like atoms. Furthermore we have numerically analysed the energy spectrum, radial characteristics and correctional terms using programs written in Python and MATLAB. Finally we have investigated the implications and applications ofthe Dirac Hamiltonian and its use to effectively describe the attributes of gold atoms.We find that relativistic contribution scales approximately by Z2 for large Z. In addition, we conclude that the Dirac Hamiltonian is of vital importance for an accurate description of chemical and physical properties of large atoms. Furthermore, we have completed calculations of the radial solution for the Coulomb potential and also for a free particle.
It is shown that the Dirac equation with the Coulomb potential can be solved using the algebra of the three spinor invariants of the Dirac equation without the involvement of the methods of supersymmetric quantum mechanics. The Dirac Hamiltonian is invariant with respect to the rotation transformation, which indicates the dynamical (hidden) symmetry SU(2) of the Dirac equation. The total symmetry of the Dirac equation is the symmetry SO(3)⊗SU(2). The generator of the SO(3) symmetry group is given by the total momentum operator, and the generator of SU(2) group is given by the rotation of the vector-states in the spinor space, determined by the Dirac, Johnson–Lippmann, and the new spinor invariants. It is shown that using algebraic approach to the Dirac problem allows one to calculate the eigenstates and eigenenergies of the relativistic hydrogen atom and reveals the fundamental role of the principal quantum number as an independent number, even though it is represented as the combination of other quantum numbers.
The solution of the Dirac equation with the Coulomb potential is used to analyze bound electron states in a hydrogen atom. The analysis is based on the fact that such states are characterized by a set of quantum numbers which describe definite values of the complete set of physical quantities that can be determined simultaneously. This set includes the energy, square of the total angular momentum, one of its component and a spinor invariant. The latter, spinor invariant gives two-valued quantum number which determines the sign of its eigenvalue. In addition to the known Dirac and Johnson–Lippman invariant, there exists a new one. Operators of these three spinor invariants do not commute between themselves which results in the degeneracy of the energy levels with respect to the two-valued quantum number. Three different systems of the eigenbispinor corresponding to the three spinor invariants are obtained and the generalized solution with free parameters is calculated. Variation of the free parameters transforms one particular solution into any other. It is shown that the electron probability densities and spin polarizations in an electron cloud depend essentially on the invariant set, demonstrating physical difference of the states corresponding to different spinor invariants.