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Physical Review

Deflection of Electrons by a Magnetic Field on the Wave Mechanics

Leigh Page

Published Aug 1, 1930• 49 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.36.444

Related papers

Relativistic Wave Mechanics of Electrons Deflected by a Magnetic Field

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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.

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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.

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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.

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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.

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Double wave quantum theory is applied to describe the motion of charged particle in a uniform magnetic field. This description of the motion of the particle is complete. One can tell the value of an arbitrary physical quantity at any time. The probability and the formula to calculate mean values in ordinary quantum mechanics comes from the average of double wave description over a certain ensemble. The rule of gauge transformation in quantum mechanics can be seen from the average over an ensemble.

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Quantum Wave Mechanics as the Magnetic Interaction of Dirac Particles

David C. Lush

It is shown that a wave mechanical quantum theory can be derived from relativistic classical electrodynamics, as a feature of the magnetic interaction of Dirac particles modeled as relativistically circulating point charges. The magnetic force between two classical point charges, each undergoing relativistic circulatory motion of small radius compared to the separation between their centers of circulation, and assuming a time-symmetric electromagnetic interaction, is modulated by a factor that behaves similarly to the Schrödinger wavefunction. The magnetic force between relativistically-circulating charges has been shown previously to have a radially-directed inverse-square part of similar strength to the Coulomb force, and sinusoidally modulated by the phase difference of the charges' circulatory motions. The magnetic force modulation in the case of relatively moving centers of charge circulation solves an equation formally identical to the time-dependent free-particle Schrödinger equation, apart from a factor of two on the partial time derivative term. Considering motion in a time-independent potential obtains that the modulation also satisfies an equation formally similar to the time-independent Schrodinger equation. Using a formula for relativistic rest energy advanced by Osiak, the time-independent Schrödinger equation is solved exactly by the resulting modulation function. The significance of the quantum mechanical wavefunction follows straightforwardly from these observations. After considering the modification of Wheeler-Feynman absorber theory required by the adoption of Minkowski-Osiak relativity, the model is extended to obtain the full complex Schrödinger wavefunction.

arXiv:1609.04446•Sep 15, 2016•1 citation
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 e\ensuremathμ\frac{e}{\ensuremath{\mu}}\ensuremathμe​ 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.
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.
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.
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 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.
Double wave quantum theory is applied to describe the motion of charged particle in a uniform magnetic field. This description of the motion of the particle is complete. One can tell the value of an arbitrary physical quantity at any time. The probability and the formula to calculate mean values in ordinary quantum mechanics comes from the average of double wave description over a certain ensemble. The rule of gauge transformation in quantum mechanics can be seen from the average over an ensemble.
It is shown that a wave mechanical quantum theory can be derived from relativistic classical electrodynamics, as a feature of the magnetic interaction of Dirac particles modeled as relativistically circulating point charges. The magnetic force between two classical point charges, each undergoing relativistic circulatory motion of small radius compared to the separation between their centers of circulation, and assuming a time-symmetric electromagnetic interaction, is modulated by a factor that behaves similarly to the Schrödinger wavefunction. The magnetic force between relativistically-circulating charges has been shown previously to have a radially-directed inverse-square part of similar strength to the Coulomb force, and sinusoidally modulated by the phase difference of the charges' circulatory motions. The magnetic force modulation in the case of relatively moving centers of charge circulation solves an equation formally identical to the time-dependent free-particle Schrödinger equation, apart from a factor of two on the partial time derivative term. Considering motion in a time-independent potential obtains that the modulation also satisfies an equation formally similar to the time-independent Schrodinger equation. Using a formula for relativistic rest energy advanced by Osiak, the time-independent Schrödinger equation is solved exactly by the resulting modulation function. The significance of the quantum mechanical wavefunction follows straightforwardly from these observations. After considering the modification of Wheeler-Feynman absorber theory required by the adoption of Minkowski-Osiak relativity, the model is extended to obtain the full complex Schrödinger wavefunction.