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Journal of Anhui Normal University

Calculation of the Energy Levels for Hydrogen Atom in a Magnetic Field of Intermediate-Strength

Huang Shi-zhong

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

With the help of angular momentum coupling theory,the general expression of diagonal and non-diagonal matrix elements of hydrogen atom Hamilton in a magnetic field of intermediate-strength is further worked out in coupling representation,a general theory that describes the interaction between hydrogen atom and uniformity magnetic field of intermediate-strength is established,and the energy level structure of n=5 for hydrogen atom is calculated.

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With the help of angular momentum coupling theory,the general expression of diagonal and non-diagonal matrix elements of hydrogen atom Hamilton in a magnetic field of intermediate-strength is further worked out in coupling representation,a general theory that describes the interaction between hydrogen atom and uniformity magnetic field of intermediate-strength is established,and the energy level structure of n=5 for hydrogen atom is calculated.
Abstract The problem of the hydrogen atom in a magnetic field can be treated satisfactorily by approximate methods which are quite accurate for very low or very high magnetic field strengths. Before this paper the only useful treatments for intermediate field strengths were variational methods giving upper bounds to the energies. As described in the paper the diffusion QMC method was applied in the same way as in earlier solutions for the hydrogen atom, but with additional magnetic terms added to the Hamiltonian expression. These include complex expressions and the required wavefunction includes a corresponding factor. With the magnetic field aligned along one axis, integration over the angle ϕ in (r, θ, ϕ) coordinates reduces the additional terms to real functions of r and z as well as the magnetic quantum number m. In the diffusion/multiplication processes of a random walk procedure these may be simply included with the potential energy term. In this study the energy at each step was determined from the average multiplication factor for the walkers. Importance sampling was not used. Results were obtained for quantum numbers m = 0, 1 at each of three magnetic field strengths. A comparison of results showed good agreement with the lowest values obtained in several of the earlier analytic variational calculations.
The energy levels of hydrogen and helium atoms in strong magnetic fields are calculated in this study. The current work contains estimates of the binding energies of the first few low-lying states of these systems that are improvements upon previous estimates. The methodology involves computing the eigenvalues and eigenvectors of the generalized two-dimensional Hartree-Fock partial differential equations for these one- and two-electron systems in a self-consistent manner. The method described herein is applicable to calculations of atomic structure in magnetic fields of arbitrary strength as it exploits the natural symmetries of the problem without assumptions of any basis functions for expressing the wave functions of the electrons or the commonly employed adiabatic approximation. The method is found to be readily extendable to systems with more than two electrons.