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Physical Review

The Motion of a Dirac Electron in a Magnetic Field

L. D. Huff

Published Aug 1, 1931• 37 citations
Publisher

Abstract

Solutions of the Dirac wave equation representing a free electron moving in a uniform magnetic field are obtained. The functions are similar to those obtained by Landau and by Uhlenbeck and Young as solutions of the Schr\"odinger equation. A wave packet is constructed representing a beam of electrons passing through a slit. The results agree with the classical predictions to terms of the order of the de Broglie wave-length of the electron divided by the radius of curvature of its classical path. For experimental cases this ratio is of the order ${10}^{\ensuremath{-}8}$ to ${10}^{\ensuremath{-}10}$. Hence it is concluded that the difference between magnetic deflection measurements of $\frac{e}{m}$ and other determinations cannot be explained as a quantum effect.

DOI
10.1103/physrev.38.501

Related papers

The Motion of a Dirac Electron in a Magnetic Field.

L. D. Huff

Solutions of the Dirac wave equation are obtained representing the motion of an electron in a uniform magnetic field. These solutions are used to calculate the maximum penetration of the electrons into the field and also the average x coordinate of the current. Both results agree with the classical result to distances of the order of a de Broglie wave length. The solutions were then combined to represent a beam of electrons passing thru a slit. It was shown that the deviation of this beam from the classical path was also of the order of a de Broglie wave length. It was necessary to impose the condition that the slit be wide compared to an electron wave length - a condition amply fulfilled in all applications. This means that the slit must be so wide that diffraction of the electrons can be neglected. The conclusion is drawn that in all experiments which have been performed the differences between the classical and the quantum predictions will be too small to be observed. The differences in paths predicted will in all cases be of the order of an electron wave length.

•Aug 1, 1931•1 citation

Deflection of Electrons by a Magnetic Field on the Wave Mechanics

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The wave equation for a stream of electrons passing through a uniform magnetic field is solved and the characteristic functions discussed. The motion at right angles to the field is found to be completely quantized.The current density is investigated, and the mean square and mean fourth power radii of curvature found for each quantum state. The mean square radius of curvature is found to be the same for all quantum states of the same energy and identical with that predicted by the classical theory.The mean radius of curvature is determined by interpolation and is found to be less than that predicted by the classical theory. Therefore values of $\frac{e}{\ensuremath{\mu}}$ calculated by means of the classical formula from measured values of the mean radius of curvature should be too large. The error is estimated and found to be of the same general magnitude as the discrepancy between the results of deflection experiments and those of spectroscopic measurements.

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To clarify the discussion as to whether or not a quantum mechanical treatment of the motion of electrons in magnetic fields will lead to a formula for $\frac{e}{m}$, differing from that obtained by classical electro-dynamics the following problem was solved. A uniform magnetic field in the $z$-direction exists in the half-space $x>0$. A plane monochromatic de Broglie wave, travelling in the positive $x$-direction representing electrons of arbitrary energy, impinges normally on the plane $x=0$. Solutions of the wave equation were found fulfilling appropriate boundary conditions at the plane $x=0$. Currents are calculated quantum mechanically and compared with the corresponding classical expressions. It was found that for electrons possessing energies of the order of magnitude used in deflection experiments, no observable deviations from classical results are predicted. Another quantum mechanical effect is diffraction at slits. Simple approximate calculations show that this effect can produce a fractional error in $\frac{e}{m}$ of the order of the de Broglie wave-length divided by the slit width. These results are opposite to the conclusions reached by Page. We may remark that he solved a problem of "stationary states" which does not represent the actual experiments.

Physical Review•Dec 15, 1930•9 citations

Relativistic Wave Mechanics of Electrons Deflected by a Magnetic Field

MILTON S. PLESSET

It is shown that the relativistic wave equation for electrons in a uniform magnetic field leads to the same wave function as that already deduced by Page from the non-relativistic equation. As in the latter case the motion at right angles to the field is quantized.An expression is found for the current density from the relativistic wave equation. The relativistic expression differs from the non-relativistic only by a constant factor which does not affect the calculation of the mean radii of curvature of the electron current.Hence, for the relativistic case, as for the non-relativistic, the mean radius of curvature is less than that expected on the classical theory. It follows that the classical relativistic relation between $\frac{\ensuremath{\epsilon}}{\ensuremath{\mu}}$ and the mean radius of curvature upon deflection gives a value of $\frac{\ensuremath{\epsilon}}{\ensuremath{\mu}}$ which is too large.

Physical Review•Dec 15, 1930•20 citations

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Hideo Tsuru

A motion of a Gaussian wave packet of a single electron in a uniform magnetic field is obtained by solving the time dependent Schrödinger equation analytically. The wave packet moves changing its width and central position periodically. The period of the change of the width is the same as that of the central position. The interpretation of these periodic changes in terms of the Green's function which can be derived by the path integration formulation is also discussed.

Journal of the Physical Society of Japan•Jul 15, 1992•4 citations

Motion of a Wave Packet Constructed from Landau Gauge Solutions

D. M. Thomson

The eigensolutions of the Schrödinger equation for the motion of a charged particle in a uniform magnetic field depend on the choice of electromagnetic gauge. In the Landau gauge it is hard to discern a relationship between the eigensolutions and the circular or helical orbits of classical theory. In this paper a wave packet is constructed from the Landau gauge eigensolutions which simulates the classical motion in the plane perpendicular to the magnetic field. Some properties of the wave packet are discussed.

American Journal of Physics•Nov 1, 1972•2 citations
Solutions of the Dirac wave equation representing a free electron moving in a uniform magnetic field are obtained. The functions are similar to those obtained by Landau and by Uhlenbeck and Young as solutions of the Schr"odinger equation. A wave packet is constructed representing a beam of electrons passing through a slit. The results agree with the classical predictions to terms of the order of the de Broglie wave-length of the electron divided by the radius of curvature of its classical path. For experimental cases this ratio is of the order 10\ensuremath−8{10}^{\ensuremath{-}8}10\ensuremath−8 to 10\ensuremath−10{10}^{\ensuremath{-}10}10\ensuremath−10. Hence it is concluded that the difference between magnetic deflection measurements of em\frac{e}{m}me​ and other determinations cannot be explained as a quantum effect.
Solutions of the Dirac wave equation are obtained representing the motion of an electron in a uniform magnetic field. These solutions are used to calculate the maximum penetration of the electrons into the field and also the average x coordinate of the current. Both results agree with the classical result to distances of the order of a de Broglie wave length. The solutions were then combined to represent a beam of electrons passing thru a slit. It was shown that the deviation of this beam from the classical path was also of the order of a de Broglie wave length. It was necessary to impose the condition that the slit be wide compared to an electron wave length - a condition amply fulfilled in all applications. This means that the slit must be so wide that diffraction of the electrons can be neglected. The conclusion is drawn that in all experiments which have been performed the differences between the classical and the quantum predictions will be too small to be observed. The differences in paths predicted will in all cases be of the order of an electron wave length.
The wave equation for a stream of electrons passing through a uniform magnetic field is solved and the characteristic functions discussed. The motion at right angles to the field is found to be completely quantized.The current density is investigated, and the mean square and mean fourth power radii of curvature found for each quantum state. The mean square radius of curvature is found to be the same for all quantum states of the same energy and identical with that predicted by the classical theory.The mean radius of curvature is determined by interpolation and is found to be less than that predicted by the classical theory. Therefore values of $\frac{e}{\ensuremath{\mu}}$ calculated by means of the classical formula from measured values of the mean radius of curvature should be too large. The error is estimated and found to be of the same general magnitude as the discrepancy between the results of deflection experiments and those of spectroscopic measurements.
To clarify the discussion as to whether or not a quantum mechanical treatment of the motion of electrons in magnetic fields will lead to a formula for em\frac{e}{m}me​, differing from that obtained by classical electro-dynamics the following problem was solved. A uniform magnetic field in the zzz-direction exists in the half-space x>0x>0x>0. A plane monochromatic de Broglie wave, travelling in the positive xxx-direction representing electrons of arbitrary energy, impinges normally on the plane x=0x=0x=0. Solutions of the wave equation were found fulfilling appropriate boundary conditions at the plane x=0x=0x=0. Currents are calculated quantum mechanically and compared with the corresponding classical expressions. It was found that for electrons possessing energies of the order of magnitude used in deflection experiments, no observable deviations from classical results are predicted. Another quantum mechanical effect is diffraction at slits. Simple approximate calculations show that this effect can produce a fractional error in em\frac{e}{m}me​ of the order of the de Broglie wave-length divided by the slit width. These results are opposite to the conclusions reached by Page. We may remark that he solved a problem of "stationary states" which does not represent the actual experiments.
It is shown that the relativistic wave equation for electrons in a uniform magnetic field leads to the same wave function as that already deduced by Page from the non-relativistic equation. As in the latter case the motion at right angles to the field is quantized.An expression is found for the current density from the relativistic wave equation. The relativistic expression differs from the non-relativistic only by a constant factor which does not affect the calculation of the mean radii of curvature of the electron current.Hence, for the relativistic case, as for the non-relativistic, the mean radius of curvature is less than that expected on the classical theory. It follows that the classical relativistic relation between $\frac{\ensuremath{\epsilon}}{\ensuremath{\mu}}$ and the mean radius of curvature upon deflection gives a value of $\frac{\ensuremath{\epsilon}}{\ensuremath{\mu}}$ which is too large.
The eigensolutions of the Schrödinger equation for the motion of a charged particle in a uniform magnetic field depend on the choice of electromagnetic gauge. In the Landau gauge it is hard to discern a relationship between the eigensolutions and the circular or helical orbits of classical theory. In this paper a wave packet is constructed from the Landau gauge eigensolutions which simulates the classical motion in the plane perpendicular to the magnetic field. Some properties of the wave packet are discussed.