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European Journal of Physics

The magnetic field produced by a hydrogen atom

W Gough

Published Jul 1, 1996• 7 citations
Publisher

Abstract

The magnetic field in a hydrogen atom is calculated for a number of energy eigenstates. The contributions from both the orbital and spin motion of the electron are considered. The field follows the expected magnetic dipole behaviour only at large distances from the proton, and there may also be contributions due to magnetic octupole, 32-poles... moments. The field pattern is illustrated for a number of cases, and these are briefly discussed.

DOI
10.1088/0143-0807/17/4/011

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The magnetic field in a hydrogen atom is calculated for a number of energy eigenstates. The contributions from both the orbital and spin motion of the electron are considered. The field follows the expected magnetic dipole behaviour only at large distances from the proton, and there may also be contributions due to magnetic octupole, 32-poles... moments. The field pattern is illustrated for a number of cases, and these are briefly discussed.
Basic electromagnetism and relativistic quantum theory are used to evaluate the magnetic field due to the electron in a one-electron atom. The field patterns are displayed and discussed for several eigenstates with low values of the quantum numbers. The effect of increasing the number of protons in the nucleus is shown explicitly. Comparison with the non-relativistic result illustrates the effects of relativity in atoms. There is a brief discussion of the meaning of the terms spin and orbital magnetic moment of an atom in the light of these results.
The hydrogen atom is the simplest of all atoms consisting of proton of charge e and the electron of charge −e. It is a two particle system. A two-body problem can always be reduced to a one-body problem by choosing the center of mass coordinate and the relative coordinates. In the hydrogen atom the attractive force exerted by the proton on electron prevents it from escape. The potential energy V of the system is −e2/r where r is the distance between the electron and the proton. Suppose the nucleus is held fixed at the origin of coordinates. V (r) is independent of θ and φ, that is, insensitive to the direction of r and depends only on the scalar magnitude r. The potential energy remains invariant under rotation and is said to be spherically symmetric. Figure 12.1 shows V (r) as a function of r. In this chapter the hydrogen atom problem in threedimension is solved and momentum eigenfunction is obtained. The problem in D-dimension where D may be noninteger is analysed. The magnetic field produced by a hydrogen atom is worked out. Finally, the system in parabolic coordinates is discussed.
The mechanical angular momentum and magnetic moment of the electron and proton spin have been calculated semiclassically with the aid of the uncertainty principle for energy and time.The spin effects of both kinds of the elementary particles can be expressed in terms of similar formulae.The quantization of the spin motion has been done on the basis of the old quantum theory.It gives a quantum number n = 1/2 as the index of the spin state acceptable for both the electron and proton particle.In effect of the spin existence the electron motion in the hydrogen atom can be represented as a drift motion accomplished in a combined electric and magnetic field.More than 18,000 spin oscillations accompany one drift circulation performed along the lowest orbit of the Bohr atom.The semiclassical theory developed in the paper has been applied to calculate the doublet separation of the experimentally well-examined D line entering the spectrum of the sodium atom.This separation is found to be much similar to that obtained according to the relativistic old quantum theory.
Abstract In this chapter we discuss the intrinsic angular momentum called spin. Starting from the magnetic dipole moment related to orbital angular momentum, we focus on the hydrogen atom and consider the effect of an external magnetic field, resulting in Larmor precession and Zeeman splitting. We then turn to the discovery of spin in the Stern-Gerlach experiment. After providing a mathematical description of spin-1/2 systems in terms of two-component spinors and Pauli matrices, we illustrate projective measurements for the example of spin. The relationship between the Lie groups SU(2) and SO(3) is discussed in the context of the transformation of spinors under rotations. We analyse how the presence of spin alters the description of the hydrogen atom. Discussing the addition of angular momenta and the Pauli exclusion principle leads us to the structure of electronic orbitals. Finally, we discuss the occupation of electronic energy levels in heavier nuclei