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The magnetic field generated by an electron bound in angular-momentum eigenstates

K. Ayuel, P.F. de Châtel

Published Jan 1, 1999• 3 citations
OpenAlex

Abstract

The magnetic field generated by an electron bound in a spherically symmetric potential is calculated for eigenstates of the orbital and total angular momentum. General expressions are presented for the current density in such states and the magnetic field is calculated through the vector potential, which is obtained from the current density by direct integration. The method is applied to the hydrogen atom, for which we reproduce and extend known results.

Related papers

The magnetic field generated by an electron bound in angular-momentum eigenstates

K Ayuel, P.F. de Châtel

The magnetic field generated by an electron bound in a spherically symmetric potential is calculated for eigenstates of the orbital and total angular momentum. General expressions are presented for the current density in such states and the magnetic field is calculated through the vector potential, which is obtained from the current density by direct integration. The method is applied to the hydrogen atom, for which we reproduce and extend known results.

arXiv:quant-ph/9901003•European Journal of Physics•Jan 1, 1999•3 citations

Vector operators and spherical harmonics in quantum mechanics

M. Andrews

The usual explicit expressions for spherical harmonics are derived by a method which does not involve the solution of differential equations or the use of angular momentum operators in spherical polar form. Instead a raising operator constructed from an arbitrary vector is used together with the usual angular momentum raising and lowering operators. The method is first presented as simply as possible in Schrödinger representation using position as the vector. Then it is generalized to an arbitrary vector and arbitrary integral angular momentum without using a particular representation. Finally it is shown how a conserved vector (other than angular momentum) of a central system can be used to reveal properties of the energy eigenstates. As an example, eigenstates of the hydrogen atom are found using the Runge-Lenz vector.

American Journal of Physics•Mar 1, 1979•5 citations

Magnetically bound states of electronic and positronic hydrogen ions: The case of zero orbital angular momentum

Daniel H. E. Dubin

This paper considers the binding of an electron or a positron to a ground-state hydrogen atom in a uniform magnetic field $B$, concentrating on states with zero orbital angular momentum $J$. It is shown that the affinity of a positron to a hydrogen atom is $2.23{b}^{2}{e}^{2}∕a$, where $a$ is the Bohr radius, $b\ensuremath{\equiv}B∕{B}_{0}$, ${B}_{0}=2.3505\ifmmode\times\else\texttimes\fi{}{10}^{5}\phantom{\rule{0.3em}{0ex}}\mathrm{T}$, $b⪡1$ is assumed, and the proton is assumed to have infinite mass. With these assumptions it is also shown that no $J=0$ singlet or triplet excited state of ${\mathrm{H}}^{\ensuremath{-}}$ exists. These results are compared to a previous estimate of the electron affinity of $J=0$ magnetically bound ions [V. G. Bezchastnov, P. Schmelcher, and L. S. Cederbaum, Phys. Rev. A 61, 052152 (2000)].

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Induced angular momentum

G. W. Parker

The angular momentum induced in the bound motion of an electron by an external magnetic field B is evaluated classically and quantum mechanically. The quantum-mechanical model used is one in which the angular momentum is quenched when B=0 so that it is representative of a molecule rather than an atom. The current density and its associated magnetic moment are calculated and discussed. Two different methods are used to solve the first-order perturbation theory equation for the required eigenfunction.

American Journal of Physics•Sep 1, 1978•0 citations

Bound States in Three Dimensions: The Atom

Duncan G. Steel

Abstract In this chapter, we go to three dimensions in space and look at the solution of the time independent Schrödinger equation for the hydrogen atom. The Hamiltonian is then the kinetic energy plus the potential energy due to the Coulomb coupling between the positively charged nucleus and the electron. We construct the angular momentum operator and find that the partial differential equation for the angular momentum eigenfunctions of the spherical coordinates θ,ϕ is the same as the angular part of the ∇2 operator in spherical coordinates. The angular momentum eigenfunctions are the spherical harmonics, with two quantum numbers, l and m, and the solution to the radial part of the Hamiltonian including the Coulomb potential are Laguerre polynomials with one quantum number, called the principle quantum number, n. The hydrogen wave function is the product of a Laguerre polynomial and a spherical harmonic with three quantum numbers. Since these are two- and three-dimensional functions for angular momentum and hydrogen respectively, they are best understood in a series of plots. The chapter concludes by giving the historical letter names to specific orbitals, since they continue to be used today.

•Apr 30, 2021•0 citations

Exact Solution of the Zeeman Effect in Single-Electron Systems

A Blom

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.

arXiv:physics/0406141•Physica Scripta•Jan 1, 2005•4 citations
The magnetic field generated by an electron bound in a spherically symmetric potential is calculated for eigenstates of the orbital and total angular momentum. General expressions are presented for the current density in such states and the magnetic field is calculated through the vector potential, which is obtained from the current density by direct integration. The method is applied to the hydrogen atom, for which we reproduce and extend known results.
The usual explicit expressions for spherical harmonics are derived by a method which does not involve the solution of differential equations or the use of angular momentum operators in spherical polar form. Instead a raising operator constructed from an arbitrary vector is used together with the usual angular momentum raising and lowering operators. The method is first presented as simply as possible in Schrödinger representation using position as the vector. Then it is generalized to an arbitrary vector and arbitrary integral angular momentum without using a particular representation. Finally it is shown how a conserved vector (other than angular momentum) of a central system can be used to reveal properties of the energy eigenstates. As an example, eigenstates of the hydrogen atom are found using the Runge-Lenz vector.
This paper considers the binding of an electron or a positron to a ground-state hydrogen atom in a uniform magnetic field BBB, concentrating on states with zero orbital angular momentum JJJ. It is shown that the affinity of a positron to a hydrogen atom is 2.23b2e2∕a2.23{b}^{2}{e}^{2}∕a2.23b2e2∕a, where aaa is the Bohr radius, $b\ensuremath{\equiv}B∕{B}_{0}$, ${B}_{0}=2.3505\ifmmode\times\else\texttimes\fi{}{10}^{5}\phantom{\rule{0.3em}{0ex}}\mathrm{T}$, b⪡1b⪡1b⪡1 is assumed, and the proton is assumed to have infinite mass. With these assumptions it is also shown that no J=0J=0J=0 singlet or triplet excited state of ${\mathrm{H}}^{\ensuremath{-}}$ exists. These results are compared to a previous estimate of the electron affinity of J=0J=0J=0 magnetically bound ions [V. G. Bezchastnov, P. Schmelcher, and L. S. Cederbaum, Phys. Rev. A 61, 052152 (2000)].
The angular momentum induced in the bound motion of an electron by an external magnetic field B is evaluated classically and quantum mechanically. The quantum-mechanical model used is one in which the angular momentum is quenched when B=0 so that it is representative of a molecule rather than an atom. The current density and its associated magnetic moment are calculated and discussed. Two different methods are used to solve the first-order perturbation theory equation for the required eigenfunction.
Abstract In this chapter, we go to three dimensions in space and look at the solution of the time independent Schrödinger equation for the hydrogen atom. The Hamiltonian is then the kinetic energy plus the potential energy due to the Coulomb coupling between the positively charged nucleus and the electron. We construct the angular momentum operator and find that the partial differential equation for the angular momentum eigenfunctions of the spherical coordinates θ,ϕ is the same as the angular part of the ∇2 operator in spherical coordinates. The angular momentum eigenfunctions are the spherical harmonics, with two quantum numbers, l and m, and the solution to the radial part of the Hamiltonian including the Coulomb potential are Laguerre polynomials with one quantum number, called the principle quantum number, n. The hydrogen wave function is the product of a Laguerre polynomial and a spherical harmonic with three quantum numbers. Since these are two- and three-dimensional functions for angular momentum and hydrogen respectively, they are best understood in a series of plots. The chapter concludes by giving the historical letter names to specific orbitals, since they continue to be used today.
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.