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arXiv:2504.18477

Nondegenerate, Lamb-shift Solution of the Dirac Hydrogen Atom

Roberts, Nathan S.

Published Apr 25, 2025• 0 citations
arXivHTMLPDF

Abstract

When the Dirac equation was first published in 1928, three solutions appeared immediately within the same year, each describing the most important problem in physics at that time: the hydrogen atom. These solutions lifted some of the degeneracy from earlier atomic models, but not all of it--they still predicted the same degenerate energy levels for the 2s1/2 and 2p1/2 states, for example. In this paper, we introduce a new solution of the Dirac equation, which finally removes all degeneracy from the hydrogen atom. We work in terms of dimensionless quantities and use the Lamb shift to give each atomic state in our model a unique, nondegenerate energy level. We obtain radial eigenfunctions in terms of the Laguerre polynomials, demonstrate how they can be reduced to the Schrodinger wavefunctions by applying limits, and plot our results.

DOI
10.48550/arxiv.2504.18477

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When the Dirac equation was first published in 1928, three solutions appeared immediately within the same year, each describing the most important problem in physics at that time: the hydrogen atom. These solutions lifted some of the degeneracy from earlier atomic models, but not all of it--they still predicted the same degenerate energy levels for the 2s1/2 and 2p1/2 states, for example. In this paper, we introduce a new solution of the Dirac equation, which finally removes all degeneracy from the hydrogen atom. We work in terms of dimensionless quantities and use the Lamb shift to give each atomic state in our model a unique, nondegenerate energy level. We obtain radial eigenfunctions in terms of the Laguerre polynomials, demonstrate how they can be reduced to the Schrodinger wavefunctions by applying limits, and plot our results.
The associated Laguerre polynomials are very popular in the Schrödinger equation of nonrelativistic quantum mechanics. However, the situation is quite different in the case of the Dirac equation of relativistic quantum mechanics. The exact solutions of the Dirac equation for three-dimensional (3D) hydrogen atom have been obtained using the wavefunction expansion method. The exact solutions of the radial wavefunctions are found in the form of a linear combination of the orthogonal Laguerre basis function spinors. According to the equality of coefficients in Laguerre series on both sides of the equations, the exact eigenenergies and exact expressions of the corresponding eigenstates of the Dirac equation for 3D hydrogen atom are obtained, which are the same as the power series solutions. Additionally, a comparison with the results from Dirac and Schrödinger equations for 3D hydrogen atom is discussed. Due to the relativistic effects, the Dirac bispinor occurs, and there are the tiny differences of the results between Dirac and Schrödinger equations. For the Dirac states with n r = 0, there is one set of the Dirac bispinor. However for the Dirac states with n r ≠ 0, there is a combination of two sets of the Dirac bispinor. For degenerate Dirac states, their exact expansion coefficients are commutative symmetric. These results show that the linear combination of the orthogonal Laguerre basis functions is the exact solutions of the Dirac equation for 3D hydrogen atom.
The Schrödinger, Klein–Gordon, and Dirac equations for the hydrogen atom are solved in D dimensions, where D may be noninteger. For D=1 the nondegenerate ground state with infinite binding energy is obtained as a limit as D tends to 1 for the Schrödinger equation. The corresponding solution of the Klein–Gordon equation is shown to be unacceptable, contradicting previous work. Unexpectedly, no bound state solutions of the Dirac equation with D=1 are found at all. A few comments are made on the D=3 and 2 cases.
This chapter begins with the simple solution of the Schrödinger equation for hydrogen atom for which the energy value has been derived. The Bohr radius of H atom has also been computed. After this, this chapter provides the generalised solution of the Schrödinger equation for H atom/hydrogen-like species in which F equation, R equation, and T equations have been derived in a simple and lucid way. This chapter also deals with the Laguerre differential equation, Laguerre polynomials, Rodrigues formula for the Laguerre polynomials, Laguerre associated equation and its solution, associated Laguerre polynomials, Rodrigues formula for the associated Laguerre polynomials, and solution of the radial equation. It also provides normalisation of the radial wave function and complete wave function; for the hydrogen atom, hydrogenic atomic orbital; electron density of 1 s electron; mathematical background of shape of orbitals; the most probable distance of electron from nucleus of hydrogen atom, and average distance of electron from the nucleus of hydrogen atom. The end of this chapter describes the references and adequate computational problems (solved problems and question on concepts).
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.
Abstract In this chapter, we go to three dimensions in space and look at the solution of the time independent Schrödinger equation for the hydrogen atom. The Hamiltonian is then the kinetic energy plus the potential energy due to the Coulomb coupling between the positively charged nucleus and the electron. We construct the angular momentum operator and find that the partial differential equation for the angular momentum eigenfunctions of the spherical coordinates θ,ϕ is the same as the angular part of the ∇2 operator in spherical coordinates. The angular momentum eigenfunctions are the spherical harmonics, with two quantum numbers, l and m, and the solution to the radial part of the Hamiltonian including the Coulomb potential are Laguerre polynomials with one quantum number, called the principle quantum number, n. The hydrogen wave function is the product of a Laguerre polynomial and a spherical harmonic with three quantum numbers. Since these are two- and three-dimensional functions for angular momentum and hydrogen respectively, they are best understood in a series of plots. The chapter concludes by giving the historical letter names to specific orbitals, since they continue to be used today.
In the scientific community dominates the conviction that the hydrogen atom is the simplest, and existing theories based on equations of Schrodinger and Dirac fully describe all the subtleties of behavior of an electron in a hydrogen atom. In this paper and in the following articles will be shown that in addition to the known state-based solutions of the Schrodinger and Dirac equations, there are states which were not studied and they were not known, or often simply were discarded with the phrase have no physical meaning. Further it will be shown that the existing equations of quantum mechanics, namely the Dirac equation, Klein-Gordon equation and especially the Schrodinger equation, can not accurately describe all the subtleties of behavior of an electron in a hydrogen atom. As a result there will be given a reasonable basis for the feasibility of an experimental search or targeted synthesis of new unknown states of the hydrogen atom.