Some Exact Particle-Wave Solutions for Power-Law Energies

Within the framework of special relativity, the author has developed an extended in vacuo Lorentz invariant particle-wave theory of Newton’s mechanics that takes account of the particle energy as a major driver of the motion that may be of comparable magnitude to any applied external potential. Lorentz invariance ensures that the proposed theory provides a more fundamental representation of applied force than the conventional notion of applied force as the rate of change of momentum. The purpose of this paper is to generalise an existing exact solution of this theory, originally derived for Einstein’s particle energy expression, to a family of Lorentz invariant power-law particle energies characterised by an arbitrary constant κ. By Lorentz invariance, we refer to invariance under the combined special relativistic space-time and energy–momentum transformations. The power-law energy expressions are important because they underpin and bear the same relationship with an extension of the Planck–de Broglie energy–momentum relations in exactly the same manner as Einstein’s energy expression does for the conventional Planck–de Broglie relations. The de Broglie wave energy as a function of particle velocity is determined as an integral expression. This integral is evaluated explicitly for a number of illustrative values of κ, and the known solution corresponding to the Einstein energy is included through the special case κ=0.