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Wave Mechanics

Keith Hannabuss

Published Mar 20, 1997• 0 citations
Publisher

Abstract

Abstract According to Planck and Einstein the energy and frequency of light are related by E = ђw. De Broglie observed that to be consistent with the theory of relativity the momentum p of a wave should be ђk where k is the wave vector of magnitude 2π /wavelength, perpendicular to the wave front. (This is because the special theory of relativity combines E and p and also ω and k into four-dimensional vectors, and linear relations between particular components force the same relations for all other components as well. This is explained in more detail in Section 17.1, which is independent of the rest of the book.)

DOI
10.1093/oso/9780198537946.003.0002

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Abstract According to Planck and Einstein the energy and frequency of light are related by E = ђw. De Broglie observed that to be consistent with the theory of relativity the momentum p of a wave should be ђk where k is the wave vector of magnitude 2π /wavelength, perpendicular to the wave front. (This is because the special theory of relativity combines E and p and also ω and k into four-dimensional vectors, and linear relations between particular components force the same relations for all other components as well. This is explained in more detail in Section 17.1, which is independent of the rest of the book.)
Special Relativity sets tight constraints on the form of the possible relations between the four-momentum of a particle and the wave four-vector. In fact, we demonstrate that there is just one way, according to Special Relativity, to relate the energy and the momentum of a corpuscle with the characteristics of a plane wave, frequency and wave vector, if the momentum has to flow in the same direction of the wave propagation: the laws must be of direct proportionality like de Broglie and Planck-Einstein equations.
Whatever was going to replace classical physics in the description of radiation and atomic phenomena had to confront the difficult task of somehow reconciling the wave-like and particle-like aspects of light in a single structure. An important clue would come from Einstein’s E=mc2. We tend to want to associate mass (and linear momentum) with material particles. But the Planck–Einstein relation E=hν connects energy with frequency, a determinedly wave-like property. So, here are two very simple yet fundamental equations connecting energy to mass and energy to frequency. Can they be combined? Louis de Broglie thought so, and in 1923 he generalized the discovery made by Einstein in 1905 by extending it to all material particles, and most notably to electrons. The de Broglie relation λ=h/p can be derived quite straightforwardly by comparing the Lorentz transformations for energy and momentum with those for a system of plane waves.
Dispersion dynamics applies wave-particle duality, together with Maxwell’s electromagnetism, and with quantization E = hν = ħω (symbol definitions in footnote) and p = h/λ = ħk, to special relativity E2 = p2c2 + m2c4. Calculations on a wave-packet, that is symmetric about the normal distribution, are partly conservative and partly responsive. The complex electron wave function is chiefly modelled on the real wave function of an electromagnetic photon; while the former concept of a “point particle” is downgraded to mathematical abstraction. The computations yield conclusions for phase and group velocities, vp⋅vg = c2 with vp ≥ c because vg ≤ c, as in relativity. The condition on the phase velocity is most noticeable when p≪mc. Further consequences in dispersion dynamics are: derivations for ν and λ that are consistently established by one hundred years of experience in electron microscopy and particle accelerators. Values for vp = νλ = ω/k are therefore systematically verified by the products of known multiplicands or divisions by known divisors, even if vp is not independently measured. These consequences are significant in reduction of the wave-packet by resonant response during interactions between photons and electrons, for example, or between particles and particles. Thus the logic of mathematical quantum mechanics is distinguished from experiential physics that is continuous in time, and consistent with uncertainty principles. [Footnote: symbol E = energy; h = Planck’s constant; ν = frequency; ω = angular momentum; p = momentum; λ = wavelength; k = wave vector; c = speed of light; m = particle rest mass; vp = phase velocity; vg = group velocity].