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Physical Review

The Value of em by Deflection Experiments

G. E. Uhlenbeck, Lloyd A. Young

Published Dec 15, 1930• 9 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.36.1721

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Notas de Física•Apr 10, 2018•0 citations
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.
The validity of de Broglie's equation is checked experimentally for electrons of 24 to 64 electron-kilovolts energy. The experiment consists essentially of deflecting a beam of cathode rays by an electrostatic field and, keeping the deflection constant, determining the relation between the intensity of the deflecting field, VVV, and the de Broglie wave-length, $\ensuremath{\lambda}$, of the deflected electrons. On the assumption that the motion of the electrons is in accordance with the laws of relativistic mechanics, and that the de Broglie wave-length is inversely proportional to the momentum, it is shown theoretically that the expression ${\ensuremath{\lambda}}^{2}V{(1+\frac{{h}^{2}}{{{m}_{0}}^{2}{c}^{2}{\ensuremath{\lambda}}^{2}})}^{\frac{1}{2}}$ should be independent of $\ensuremath{\lambda}$. Within the limits of experimental error this expression is found to be independent of $\ensuremath{\lambda}$, thereby demonstrating that the de Broglie equation, $\ensuremath{\lambda}=\frac{h}{\mathrm{mv}}$, combined with the relativity expression for mmm, is adequate to describe the facts.
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.
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 Bohm-de Broglie quantum mechanics has made possible to calculate the trajectories of electrons in a typical double-slit experiment[C. Philippidis et al., Il Nuovo Cimento, 52 B, 15-28 (1979)]. The trajectories do not correspond to an uniform movement but to an accelerated one. The acceleration is caused by the quantum potential. From the quantum theoretical point of view, the accelerated electrons should, with a certain probabil-ity, emit photons during its movement from the slits to the screen. According to the Copenhagen interpretation we found that this quantity is strictly zero because the electron moves as a free particle along its path after it leaves the slit and before reach the screen. Then, there is no emission of photons. On the other hand, by using the Bohm-de Broglie (BdB) approach we calculate the emission power, giving a general formula and which results, for a concrete real experiment, in a very tiny but not a vanishing value. We give an idea of the type of spectrum that could be measured. Therefore we have shown that the theoretical predictions of the usual quan-tum mechanics and that of Bohm-de Broglie’s interpretation are different. An experiment could determine the correct prediction.