Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Scientific Reports

Exact solutions of Dirac equation for hydrogen atom using the linear combination of orthogonal Laguerre basis functions

Shudong Wu

Published May 8, 2025• 1 citation
Publisher

Abstract

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.

DOI
10.1038/s41598-025-01139-3

Related papers

Relativistic Schrödinger Wave Equation for Hydrogen Atom Using Factorization Method

Mohammad Reza Pahlavani, Hossein Rahbar, Mohsen Ghezelbash

In this investigation a simple method developed by introducing spin to Schrodinger equation to study the relativistic hydrogen atom. By separating Schrodinger equation to radial and angular parts, we modify these parts to the associated Laguerre and Jacobi differential equations, respectively. Bound state Energy levels and wave functions of relativistic Schrodinger equation for Hydrogen atom have been obtained. Calculated results well matched to the results of Dirac’s relativistic theory. Finally the factorization method and supersymmetry approaches in quantum mechanics, give us some first order raising and lowering operators, which help us to obtain all quantum states and energy levels for different values of the quantum numbers n and m.

Open Journal of Microphysics•Jan 1, 2013•4 citations

Algebra of the spinor invariants and the relativistic hydrogen atom

A. A. Eremko, Larissa S. Brizhik, В. М. Локтев

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.

arXiv:2211.01857•Annals of Physics•Mar 2, 2023•6 citations

Nondegenerate, Lamb-shift Solution of the Dirac Hydrogen Atom

Roberts, Nathan S.

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.

arXiv:2504.18477•Apr 25, 2025•0 citations

Hydrogen Atom

Ram Yatan Prasad, Pranita

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).

CRC Press eBooks•Jan 15, 2021•0 citations

General solution vs spin invariant eigenstates of the Dirac equation with the Coulomb potential

Larissa S. Brizhik, A. A. Eremko, В. М. Локтев

Solutions of the Dirac equation for an electron in the Coulomb potential are obtained using operator invariants of the equation, namely the Dirac, Johnson-Lippmann and recently found new invariant. It is demonstrated that these operators are the spin invariants. The generalized invariant is constructed and the exact general solution of the Dirac equation are found. In particular, the explicit expressions of the bispinors corresponding to the three complete sets of the invariants, their eigenvalues and quantum numbers are calculated. It is shown that the general solution of one center Coulomb Dirac equation contains free parameters. Changing one or more of these parameters, one can transform one solution of the Dirac equation into any other. It is shown for the first time that these invariants determine electron spatial probability amplitude and spin polarization in each quantum state. Electron probability densities and spin polarizations are explicitly calculated in the general form for several electron states in the hydrogen-like energy spectrum. Spatial distributions of these characteristics are shown to depend essentially on the invariant set, demonstrating, in spite of the accidental degeneracy of energy levels, physical difference of the states corresponding to different spin invariants.

arXiv:2111.08552•Nov 16, 2021•1 citation

Hydrogen states described by solutions of the Dirac equation: Role of spinor invariants

A. A. Eremko, Larissa S. Brizhik

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.

Low Temperature Physics•Dec 1, 2022•0 citations
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.
In this investigation a simple method developed by introducing spin to Schrodinger equation to study the relativistic hydrogen atom. By separating Schrodinger equation to radial and angular parts, we modify these parts to the associated Laguerre and Jacobi differential equations, respectively. Bound state Energy levels and wave functions of relativistic Schrodinger equation for Hydrogen atom have been obtained. Calculated results well matched to the results of Dirac’s relativistic theory. Finally the factorization method and supersymmetry approaches in quantum mechanics, give us some first order raising and lowering operators, which help us to obtain all quantum states and energy levels for different values of the quantum numbers n and m.
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.
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.
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).
Solutions of the Dirac equation for an electron in the Coulomb potential are obtained using operator invariants of the equation, namely the Dirac, Johnson-Lippmann and recently found new invariant. It is demonstrated that these operators are the spin invariants. The generalized invariant is constructed and the exact general solution of the Dirac equation are found. In particular, the explicit expressions of the bispinors corresponding to the three complete sets of the invariants, their eigenvalues and quantum numbers are calculated. It is shown that the general solution of one center Coulomb Dirac equation contains free parameters. Changing one or more of these parameters, one can transform one solution of the Dirac equation into any other. It is shown for the first time that these invariants determine electron spatial probability amplitude and spin polarization in each quantum state. Electron probability densities and spin polarizations are explicitly calculated in the general form for several electron states in the hydrogen-like energy spectrum. Spatial distributions of these characteristics are shown to depend essentially on the invariant set, demonstrating, in spite of the accidental degeneracy of energy levels, physical difference of the states corresponding to different spin invariants.
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.