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Journal of Guangdong University of Petrochemical Technology

The Fine Structure of a Hydrogen Atom Energy Levels in a Uniform High Magnetic Field

Xi Wei

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

There are many interactions in a uniform high magnetic field for a hydrogen atom. When the B~2 proportional term in the Hamiltonian of the hydrogen atom is considered and according to the perturbation theory in degenerate state and properties of spherical harmonics function and angular momentum, a calculated method of energy level fine structure for hydrogen atom in a high magnetic field is derived. In accordance with this method, energy levels of the hydrogen atom in n=2 state is calculated.

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There are many interactions in a uniform high magnetic field for a hydrogen atom. When the B~2 proportional term in the Hamiltonian of the hydrogen atom is considered and according to the perturbation theory in degenerate state and properties of spherical harmonics function and angular momentum, a calculated method of energy level fine structure for hydrogen atom in a high magnetic field is derived. In accordance with this method, energy levels of the hydrogen atom in n=2 state is calculated.
The article studies a simple method for any energy level of three dimensional harmonic oscillator in uniform magnetic field.According to the perturbation theory in degenerate state,wave function of three dimensional harmonic oscillator in the uniform magnetic field,and the popular formula of perturbation matrix elements of three dimensional harmonic oscillator in the uniform magnetic field,the energy levels of first-order for n = 5 are calculated.A formula of energy level for three dimensional harmonic oscillator in the uniform magnetic field is given.
A hydrogen atom moving across a magnetic field is considered in a wide region of magnitudes of magnetic field and atom momentum. We solve the Schr\"odinger equation of the system numerically using an imaginary time method and find wave functions of the lowest states of atom. We calculate the energy and the mean electron-nucleus separation as a function of atom momentum and magnetic field. All the results obtained could be summarized as a phase diagram on the ``atom-momentum -- magnetic-field'' plane. There are transformations of wave-function structure at critical values of atom momentum and magnetic field that result in a specific behavior of dependencies of energy and mean interparticle separation on the atom momentum P.P.P. We discuss a transition from the Zeeman regime to the high magnetic field regime. A qualitative analysis of the complicated behavior of wave functions vs PPP based on the effective potential examination is given. We analyze a sharp transition at the critical momentum from a Coulomb-type state polarized due to atom motion to a strongly decentered (Landau-type) state at low magnetic fields. A crossover occurring at intermediate magnetic fields is also studied.
Electron spin and orbital interaction outside the nucleus is a classical and important problem in quantum mechanics. In this paper, a simplified model is constructed for the phenomenon of atomic energy level splitting in weak magnetic field in quantum mechanics, and the wave function of the fine structure of hydrogen-like atoms under perturbation is derived. According to the probability interpretation of quantum mechanics, we obtain the probability density function of the fine structure electron cloud of hydrogen-like atoms. The Monte Carlo selection method is used to sample, and the electron cloud images under different quantum numbers are drawn by Mathematica software. The experimental results are explained regularly, so that people can intuitively understand the distribution of electron cloud in three-dimensional space.
For pt.I see ibid., vol.17, p.29-52 (1984). Wavelengths, dipole strengths, oscillator strengths and transition probabilities for electromagnetic transitions between low-lying states of the hydrogen atom embedded in a magnetic field of arbitrary strength (0<B<or=4.70*10 8 T) are calculated with high numerical accuracy for a sufficiently dense mesh of B values using the wavefunctions and energy values determined in paper I by expanding the wavefunctions in terms of spherical harmonics (weak and moderate fields), or in terms of Landau states (strong and very strong fields). Effects caused by the finiteness of the proton mass are taken into account, and the scaling laws are quoted by the aid of which the results presented can be utilised to cover quantitatively the whole hydrogenic sequence in intense magnetic fields; finally, the quality of the results of previous calculations is checked.