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Physical Review A

Diamagnetism of a free particle in classical electron theory with classical electromagnetic zero-point radiation

Timothy H. Boyer

Published Jan 1, 1980• 33 citations
Publisher

Abstract

Landau's diamagnetism for a free point charge is shown to exist within classical electron theory with classical electromagnetic zero-point radiation. The system considered is a nonrelativistic classical point charge bound harmonically in three dimensions and situated in a magnetic field when random classical electromagnetic zero-point radiation or thermal radiation is present. The average energy, angular momentum, and magnetic moment are calculated at finite temperature, and then carried to the limit at which the harmonic binding vanishes so as to obtain the behavior for a free particle. In the presence of the Rayleigh-Jeans radiation spectrum one finds that all diamagnetic effects vanish, in agreement with the results of traditional classical statistical mechanics. In the Planck radiation spectrum one finds exactly the results of quantum theory involving a Langevin function for the temperature dependence. In particular, the average angular momentum and magnetic moment for a classical point charge in classical zero-point radiation are $〈{L}_{z}〉=\ensuremath{-}\ensuremath{\hbar}$ and $〈{M}_{z}〉=\ensuremath{-}|e|\frac{\ensuremath{\hbar}}{2\mathrm{mc}}$, where the orientation of the $z$ axis is given by the magnetic field direction. Thus the classical results in the presence of classical zero-point radiation are entirely different from those of traditional classical electron theory, and they suggest possible classical explanations of the space quantization appearing in quantum theory. The results also suggest a derivation of Planck's spectrum from traditional classical statistical mechanics by applying Boltzmann statistics to the orientation of the average magnetic moment caused by the zero-point radiation.

DOI
10.1103/physreva.21.66

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Physical Review A•Oct 16, 2020•11 citations
Landau's diamagnetism for a free point charge is shown to exist within classical electron theory with classical electromagnetic zero-point radiation. The system considered is a nonrelativistic classical point charge bound harmonically in three dimensions and situated in a magnetic field when random classical electromagnetic zero-point radiation or thermal radiation is present. The average energy, angular momentum, and magnetic moment are calculated at finite temperature, and then carried to the limit at which the harmonic binding vanishes so as to obtain the behavior for a free particle. In the presence of the Rayleigh-Jeans radiation spectrum one finds that all diamagnetic effects vanish, in agreement with the results of traditional classical statistical mechanics. In the Planck radiation spectrum one finds exactly the results of quantum theory involving a Langevin function for the temperature dependence. In particular, the average angular momentum and magnetic moment for a classical point charge in classical zero-point radiation are 〈Lz〉=\ensuremath−\ensuremathℏ〈{L}_{z}〉=\ensuremath{-}\ensuremath{\hbar}〈Lz​〉=\ensuremath−\ensuremathℏ and 〈Mz〉=\ensuremath−∣e∣\ensuremathℏ2mc〈{M}_{z}〉=\ensuremath{-}|e|\frac{\ensuremath{\hbar}}{2\mathrm{mc}}〈Mz​〉=\ensuremath−∣e∣2mc\ensuremathℏ​, where the orientation of the zzz axis is given by the magnetic field direction. Thus the classical results in the presence of classical zero-point radiation are entirely different from those of traditional classical electron theory, and they suggest possible classical explanations of the space quantization appearing in quantum theory. The results also suggest a derivation of Planck's spectrum from traditional classical statistical mechanics by applying Boltzmann statistics to the orientation of the average magnetic moment caused by the zero-point radiation.
A classical spinning magnetic dipole is considered within classical electrodynamics with classical electromagnetic zero-point radiation. The stationary probability distribution for the angle of alignment between the spinning magnetic dipole and an external magnetic field is calculated when the system is located in an arbitrary spectrum of random classical radiation. It is found that for the Rayleigh-Jeans spectrum the probability distribution for alignment is just the Boltzmann distribution. However, in classical zero-point radiation the alignment probability distribution is independent of the magnetic field causing alignment. Moreover, for large classical spin angular momentum $S>>\ensuremath{\hbar}$, the average component of the spin in the direction of alignment is given by $〈{S}_{z}〉=S\ensuremath{-}\frac{1}{2}\ensuremath{\hbar}$, where $\ensuremath{\hbar}$ is Planck's constant (divided by $2\ensuremath{\pi}$) used to set the scale of the classical zero-point radiation spectrum. The results seem suggestive of the idea of space quantization in quantum theory. The model discussed here was first considered by S. Sachidanandam (unpublished).
An external magnetic field is applied to the classical hydrogen atom treated in classical electromagnetic theory including classical electromagnetic zero-point radiation. In an earlier article, it was shown that the average value of each of the electron's action variables appears as a discrete value, corresponding to a representation of the rotation group, because of resonance between the periodic orbit of the electron and the random classical zero-point radiation. Here it is shown that, in the presence of a magnetic field and because of resonance, the classical orbital motion of the electron is in resonance with random classical zero-point radiation only for orientations of the orbit which take integer values for the angle made with the direction of the magnetic field, but excluding the m=0 orientation where the magnetic field is parallel to the orbital plane of the electron. Classical electromagnetic explanations are given for the Stern-Gerlach result, and for the Zeeman effect.
1. In a recent paper Landau has shown that, when electrons are moving freely in a magnetic field, they exhibit, in addition to the paramagnetic effect of their spin, a diamagnetic effect due to their motion. This result is rather unexpected, since it is quite contrary to the classical case. There it might appear as though the circles described by the electrons must produce a magnetic moment, but the error was long ago pointed out by Bohr. The motion of the electrons must be confined to some region by means of a boundary wall, and the electrons near the wall describe a succession of circular arcs, repeatedly bouncing on the wall, and slowly creeping round it in the direction opposite to that of the uninterrupted circles; when the moment of these electrons is taken into account, it exactly cancels out that due to the free circles. In Landau's work it is of course necessary to consider the boundary, but he shows how allowance is to be made for it by an appropriate process. The complete justification is rather subtle, and so it may be worth considering a special case, admitting of exact solution, which takes the boundary into account, and so makes it possible to follow more closely the analogy between the classical and quantum problems. With regard to the general case with a boundary wall of any type, we shall only observe that the different results arise, because in the wave problem ψ must vanish at the bounding “potential wall,” and so will be small near it; this upsets the balance of the electric current near the wall, and yields the magnetic moment.
We study some aspects of the quantum theory of a charged particle moving in a time-independent, unidirectional magnetic field. When the field is uniform, we make a few clarifying remarks on the use of angular momentum eigenstates and momentum eigenstates with the diamagnetism of a free electron gas as an example. When the field is nonuniform but weakly varying, we discuss both perturbative and nonperturbative methods for studying a quantum mechanical system. As an application, we derive the quantized energy levels of a charged particle in a Helmholtz coil, which go over to the usual Landau levels in the limit of a uniform field.
We make three remarks about the quantum theory of a charged particle moving in a time-independent, uni-directional magnetic field (which we take to be parallel to the z-axis). First, when the field is uniform, we comment on the relation between the wave-function on the one hand if it is chosen to be an eigenstate of momentum (leaving aside the trivial z-dependence) and hence non-normalizable, and on the other hand if it is chosen to be an eigenstate of angular momentum. Second, we discuss the form of the wave-function when the magnetic field had a weak z-dependence. Finally, we make a short comment on the derivation of the diamagnetism of a free electron gas.
We consider a quantum charged particle moving in the xyxyxy plane under the action of a uniform perpendicular constant magnetic field, in the presence of a parabolic binding potential. The magnetic-moment operator has a continuous spectrum, and its eigenfunctions in the momentum representation are expressed in terms of modified Bessel functions. The probability distribution of the magnetic moment in the thermodynamic equilibrium state is calculated. At zero temperature, it has a simple exponential form in the diamagnetic region, with a sharp jump to zero at the origin. With an increase of temperature, a paramagnetic wing of the distribution becomes more and more pronounced. In the high-temperature regime, the diamagnetic and paramagnetic wings of the distribution have almost identical forms, described with a high precision by simple exponential functions with very large extensions. Therefore, diamagnetic and paramagnetic contributions almost cancel each other. The remaining non-zero diamagnetic value is due to a small asymmetry of the distribution nearby the origin, where some nonexponential fine structure is observed. The total width of the distribution function strongly depends on the strength of the binding potential. Strong fluctuations of the magnetic moment (described in terms of the variance) are discovered in all temperature regimes.