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Wuli yu gongcheng.

A COMMENT ON THE WORK AND POTENTIAL ENERGY OF A MAGNETIC MOMENT IN AN EXTERNAL MAGNETIC FIELD

Chun LIU

Published Oct 1, 2024• 0 citations
Publisher

Abstract

In classical electrodynamics, the work of a magnetic dipole (current loop) done by external magnetic field is related to electromagnetic induction, and there is generally no potential energy conceptionally. In quantum mechanics, a magnetic dipole (electron spin) does have potential energy in an external magnetic field. We discuss the subtle differences between classical and quantum situations, and try to understand their relation in a natural way.

DOI
10.26599/phys.2024.9320501

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In classical electrodynamics, the work of a magnetic dipole (current loop) done by external magnetic field is related to electromagnetic induction, and there is generally no potential energy conceptionally. In quantum mechanics, a magnetic dipole (electron spin) does have potential energy in an external magnetic field. We discuss the subtle differences between classical and quantum situations, and try to understand their relation in a natural way.
The behavior of an atom in a nonuniform magnetic field is analyzed, as well as the motion of a classical magnetic dipole (a spinning charged ball) and a rotating charged ring. For the atom it is shown that, while the magnetic field does no work on the electron-orbital contribution to the magnetic moment (the source of translational kinetic energy being the internal energy of the atom), whether or not it does work on the electron-spin contribution to the magnetic moment depends on whether the electron has an intrinsic rotational kinetic energy associated with its spin. A rotational kinetic energy for the electron is shown to be consistent with the Dirac equation. If the electron does have a rotational kinetic energy, the acceleration of a silver atom in a Stern-Gerlach experiment or the emission of a photon from an electron spin flip can be explained without requiring the magnetic field to do work. For a constant magnetic field gradient along the z axis, it is found that the classical objects oscillate in simple harmonic motion along the z axis, the total kinetic energy -- translational plus rotational -- being a constant of the motion. For the charged ball, the change in rotational kinetic energy is associated only with a change in the precession frequency, the rotation rate about the figure axis remaining constant.
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.
The force on and the energy of a ''di-monopole'', which is the limiting case of a dipole made from two monopoles at zero separation and finite magnetic moment, interacting with an externally fixed magnetic field resulting from an electric current, is considered. A model involving only a monopole is used to illustrate the physical principles involved when magnetic sources move in a solenoidal field whose source is an electric current. The problems encountered in Hamiltonian theory are discussed. 5 refs., 3 figs. (LEW)
The concept of electron spin has a broad application prospect. However, after a century of efforts, the essence and specific form of electron spin have not been revealed. The reason is probably that it is constrained by the concept of point particle material structure. In order to solve the problem, one way to initiate the quantum mechanics revolution on the concept of material structure is to establish the concept of wave element material structure. Assuming that the composition of electrons is waves, the inherent motion of electrons (the spin of free electrons) is "the rotation of mass equivalent to the energy of waves". This rotation is not the rotation of a sphere, but similar to the rotation of a ring-shaped substance along a ring. This assumption breaks the concept of point particle structure. According to it, the angular kinetic energy and spin magnetic moment of electrons can be accurately calculated by classical force electrodynamics method. Other applications are also very successful, which has led to the revolution of quantum mechanics and material structure theory. The electron spin magnetic moment and electron spin angular momentum both originate from the motion of the waves that make up the electrons along the phase trajectory, providing a highly unified explanation for the fact that "discrete waves make up localized electrons and electrons have spin magnetic moments".
The classical equations of motion of a charged point-particle with intrinsic spin under the influence of an external electromagnetic field are restated and compared with the Heisenberg equations of motion derived from the Dirac theory. The partition of angular momentum between particle and field in the classical theory is contrasted to the Dirac theory of electron spin. The analogy between the Dirac equation and the theory of parametric amplification is pointed out.A free spinning point particle moving according to the laws of classical relativistic point-particle mechanics may move along a helix. The sum of the intrinsic spin $\ensuremath{\sigma}$ and the angular momentum of the helical motion in this classical analog of zitterbewegung is an effective spin vector S which is a constant of the motion. Because of this internal motion, the effective mass MMM of the particle differs from the mass mmm which is ascribed to it in the equations of motion. Solutions are found in which S is parallel or antiparallel to the momentum, and the sign of MMM is determined by the helicity. When placed in a uniform electromagnetic field, the particle behaves as if it had a rest mass MMM and a magnetic moment $\frac{e\ensuremath{\sigma}}{\mathrm{Mc}}$, in addition to any explicit magnetic moment that may be ascribed to it.