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Physical Review

An Experimental Investigation of de Broglie's Equation

J. G. Tappert

Published Dec 15, 1938• 0 citations
Publisher

Abstract

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, $V$, 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 $m$, is adequate to describe the facts.

DOI
10.1103/physrev.54.1085

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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, \ensuremathλ\ensuremath{\lambda}\ensuremathλ, 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 \ensuremathλ2V(1+h2m02c2\ensuremathλ2)12{\ensuremath{\lambda}}^{2}V{(1+\frac{{h}^{2}}{{{m}_{0}}^{2}{c}^{2}{\ensuremath{\lambda}}^{2}})}^{\frac{1}{2}}\ensuremathλ2V(1+m0​2c2\ensuremathλ2h2​)21​ should be independent of \ensuremathλ\ensuremath{\lambda}\ensuremathλ. Within the limits of experimental error this expression is found to be independent of \ensuremathλ\ensuremath{\lambda}\ensuremathλ, thereby demonstrating that the de Broglie equation, \ensuremathλ=hmv\ensuremath{\lambda}=\frac{h}{\mathrm{mv}}\ensuremathλ=mvh​, combined with the relativity expression for mmm, is adequate to describe the facts.
As for DE Broglie's hypothesis of matter waves, many experimental physicists have experimentally verified it by using cathode ray tube and proved that matter waves do exist, so there is a mysterious hypothesis of wave-particle duality.In this paper, according to the electric field force of electrons in the cathode ray tube and the continuity equation of electrons, Under assumptions such as inelastic approximation, it is proved that the wave-particle duality is actually the macroscopic motion of real particles in the form of waves.In this paper, it is proved theoretically that the component of electron velocity in the tube satisfies the wave equation, and the wave velocity varies with the change of electric potential in the tube.From the discussion in this paper, it can be seen that there are still some problems to be studied about the motion of electrons in the cathode ray tube, so that the special motion law of electrons in the cathode ray tube can be further explored from both theoretical and experimental aspects.At the same time, it is expected that the theory in this paper can provide theoretical guidance for the manufacture and debugging of electron microscopes and the design of large-scale integrated circuits, and thus influence other related fields...This will involve a vast area of science and technology!This paper is devoted to the in-depth analysis of the objective truth of the wave-particle duality presented by microscopic particles.
To members of the British Institute of Radiology the cathode ray is probably most familiar as the necessary antecedent of X rays. You are interested in them chiefly as a means to an end, but the cathode rays had a very real intrinsic interest in the nineties of last century. It was the study of the beam of light in a low-pressure discharge tube which led to the recognition of the existence of a universal constituent of matter—the electron—and I hope to show that the importance of cathode rays for the study of fundamental problems in physics is not yet exhausted. The cathode rays still form in many respects the most convenient way of studying the properties if electrons. Of recent years our views on the nature of these electrons have undergone a striking change, as a result of the brilliant inspiration of Prince Louis de Broglie, who has been rewarded with a Nobel Prize. He founded the theory which is now dominant in physics under the name of wave mechanics, and which is a general theory of the behaviour of any form of particle. According to this view all particles are associated with some form of wave, the properties of which determine much of the behaviour of the particle. This theory was originally devised by de Broglie from considerations based upon relativity.
Abstract 1. M. L. de Broglie has introduced a theory of mechanics according to which a moving particle behaves as a group of waves whose velocity and wave-length are governed by the speed and mass of the particle. In fact if m0 is the mass for slow speed and v the speed of a freely moving particle, the wave-length is given by λ = h√1-v2/c2 / m0v, and the wave velocity V by V = c2/v, the group velocity being v, the velocity of the particle. Here c is the velocity of light and it will be seen that the wave velocity is greater than c. There is nothing impossible in this because the waves are regarded as purely geometrical —“phase waves”—not as carrying energy. Compare, in ordinary optical theory, the case of substances, such as sodium, for which the refractive index is less than unity. The above is for free space; in the presence of a field of force V varies, and the consequent bending of the waves by refraction corresponds, on the new theory, to the deviation of the path of the particle by the field of force, on the old. The consequences of this theory have been worked out by de Broglie, Schrödinger and others and applied to problems in spectroscopy where they have provided the solution of several outstanding difficulties left by the older theory of orbits. In view, however, of the extremely fundamental nature of the theory it is highly desirable that it should rest on more direct evidence, and, in particular, that it should be shown capable of predicting as well as of merely explaining. Dymond has obtained some remarkable results on the scattering of slow electrons in helium which are of the general nature to be expected in this theory, but our knowledge of the structure of helium, together with the mathematical difficulties of the problem have so far prevented any exact comparision of the theory with experiment. Davisson and Kunsman and Davisson and Germer have obtained results on the reflection of slow electrons from the surfaces of crystals, especially nickel, which show good qualitative agreement with the theory but a discrepancy of 30 per cent. in certain magnitudes. It is hoped that the experiments described in this paper will advance the matter a stage further. They are a development of some experiments of which a preliminary account appeared recently in ‘Nature.’