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American Journal of Physics

An exactly soluble quantum model of an atom in an arbitrarily strong uniform magnetic field

Mark P. Silverman

Published Jun 1, 1981• 4 citations
Publisher

Abstract

The isotropic oscillator in a uniform magnetic field is an exactly soluble quantum problem which can serve as a useful model of an atom in a magnetic field of arbitrary strength. The exact eigenvalue spectrum is derived, compared with perturbation theory for low field strengths, and shown to yield a quasi-Landau spectrum for high field strengths. The frequency, polarization, and relative intensity of the spectral lines of the Zeeman effect are determined as a function of the field strength. The effect of the field on atomic size and on the atomic wave functions is discussed; the three-dimensional system is shown to collapse into a needlelike one-dimensional system for a sufficiently intense field.

DOI
10.1119/1.12670

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Physical Review•Dec 10, 1967•87 citations
The isotropic oscillator in a uniform magnetic field is an exactly soluble quantum problem which can serve as a useful model of an atom in a magnetic field of arbitrary strength. The exact eigenvalue spectrum is derived, compared with perturbation theory for low field strengths, and shown to yield a quasi-Landau spectrum for high field strengths. The frequency, polarization, and relative intensity of the spectral lines of the Zeeman effect are determined as a function of the field strength. The effect of the field on atomic size and on the atomic wave functions is discussed; the three-dimensional system is shown to collapse into a needlelike one-dimensional system for a sufficiently intense field.
The energy and eigenstate spectrum of a charged particle in the electric field of a 2D anisotropic oscillator and in a uniform magnetic field is considered. The exact analytic solution to the problem is obtained for an arbitrary magnetic field strength. The characteristic features of variation of the energy spectrum depending on the magnetic field strength are analyzed. The results of this study are of interest for the quantum-mechanical theory of magnetism and can be used to simulate the magnetic properties of atoms and molecules.
The spectrum of a Hamiltonian for NNN harmonically interacting two-dimensional particles in a constant perpendicular magnetic field is determined exactly by the transformation to normal coordinates. The energy of the ground state consistent with Fermi-Dirac statistics is calculated in the limit of a strong magnetic field. The Fermi energy is found to exhibit de Haas---van Alphen---like oscillations.
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.
Contrary to popular belief, the Zeeman effect can be treated exactly in single-electron systems, for arbitrary magnetic field strengths, as long as the term quadratic in the magnetic field can be ignored. These formulas were actually derived already around 1927 by Darwin, using the classical picture of angular momentum, and presented in their proper quantum-mechanical form in 1933 by Bethe, although without any proof. The expressions have since been more or less lost from the literature; instead, the conventional treatment nowadays is to present only the approximations for weak and strong fields, respectively. However, in fusion research and other plasma physics applications, the magnetic fields applied to control the shape and position of the plasma span the entire region from weak to strong fields, and there is a need for a unified treatment. In this paper we present the detailed quantum-mechanical derivation of the exact eigenenergies and eigenstates of hydrogen-like atoms and ions in a static magnetic field. Notably, these formulas are not much more complicated than the better-known approximations. Moreover, the derivation allows the value of the electron spin gyromagnetic ratio g s to be different from 2. For completeness, we then review the details of dipole transitions between two hydrogenic levels, and calculate the corresponding Zeeman spectrum. The various approximations made in the derivation are also discussed in details.
In this paper we solve exactly the problem of the spectrum and Feynman propagator of a charged particle submitted to both an anharmonic oscillator in the plane and a constant and homogeneous magnetic field of arbitrary strength aligned with the perpendicular direction to the plane. As we shall see in the beginning of the letter, the Hamiltonian, being a quadratic form, is easily diagonalizable and the Classical Action can be used to construct the exact Feynman Propagator using the Stationary Phase Approximation. The result is useful for the treatment of quasi two dimensional samples in the field of magnetic effects in nano-structures and quantum optics. The presented solution, after minor extensions, can also be used for motion in three dimensions, and in fact it has been used for years in such cases. Also it can be used as a good exercise of a Feynman Path Integral that can be calculated easily with just the help of the Classical Action.
The general theory of a Zeeman laser given in part I is used to analyze laser operation in axial and transverse magnetic fields. The analysis is illustrated by graphs of intensities and beat frequencies versus cavity detuning and magnetic field strength. Among the topics discussed are frequency-locking effects due to cavity anisotropy, the dependence of competition between polarization modes on magnetic field strength for various atomic angular momenta, and the measurement of atomic decay rates and ggg values using a laser in a magnetic field.