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Journal of Modern Physics

Phase Velocity and Rest Energy of Schrödinger Equation Seen from Neutrino Oscillations

Shan Dan Zhou

Published Jan 1, 2016• 1 citation
Publisher

Abstract

The phase velocity is discussed using de Broglie relations and Schrödinger equation. We argue that in non-relativistic quantum mechanics the Hamiltonian should be added by a rest-energy term when Schrödinger equation is used to study the interference between particles of different masses. From neutrino oscillation experiments, we find that the phase velocity can be related to a measurable quantity (the flavor transition probability), therefore, the phase factor originated from the rest energy can’t be omitted. Correspondingly, the energy in de Broglie relations should always be total energy rather than kinetic energy, contrary to some textbooks of quantum mechanics.

DOI
10.4236/jmp.2016.76049

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The phase velocity is discussed using de Broglie relations and Schrödinger equation. We argue that in non-relativistic quantum mechanics the Hamiltonian should be added by a rest-energy term when Schrödinger equation is used to study the interference between particles of different masses. From neutrino oscillation experiments, we find that the phase velocity can be related to a measurable quantity (the flavor transition probability), therefore, the phase factor originated from the rest energy can’t be omitted. Correspondingly, the energy in de Broglie relations should always be total energy rather than kinetic energy, contrary to some textbooks of quantum mechanics.
Abstract This work revisits the apparent paradox that emerges from some results found in many basic textbooks which address the problem of calculating the phase velocity of free particles in the relativistic regime and its relation with the same quantity in the non-relativistic regime. Because some books use the Newtonian kinetic energy and others use the total relativistic energy (including the rest energy), the non-relativistic limit of the relativistic phase velocity differs from the Newtonian phase velocity, which can be confusing for many students. This paper proposes expressing the phase velocity by first subtracting the rest energy from the relativistic energy before imposing the usual de Broglie relations. In addition to providing the corresponding Newtonian phase velocity, it also provides a relativistic Schrödinger equation which intuitively reduces to the non-relativistic Schrödinger equation in the appropriate regime. The problem of the equivalence between the relativistic Schrödinger equation and the Klein–Gordon equation is also addressed by using arguments which can be understood by undergraduate students.
It is well known that the kinetic energy formula of classical mechanics has a limited range of applicability. That formula breaks down when the mass of a moving body increases and the effects of relativity can no longer be ignored. More than a hundred years ago, Sommerfeld and Einstein defined relativistic kinetic energy as the difference between a body’s relativistic energy and its rest mass energy. This paper, however, derives a formula for relativistic kinetic energy that is not a definitional formula. The formula derived in this paper incorporates the phase velocity of a wave. It further finds that the phase velocity of the wave is an important physical quantity linking a body’s kinetic energy and momentum. In current standard physics, the de Broglie wave is not regarded as having physical reality. Nevertheless, this paper concludes that the phase velocity of a matter wave is an essential physical quantity.
In this article, it is shown that if a particle is at rest and has some harmonic oscillation, then a traveling particle with a constant speed corresponds to a De Broglie Wave.In 1924, De Broglie proposed that particles of matter exhibit wave like behavior, the expression for which became fundamental in the creation of wave mechanics (quantum mechanics).The main equation for this is the Schrödinger equation, which in the case of free motion of a particle becomes the expression for the De Broglie wave.This hypothesis is supported by many early experiments on the diffraction of electrons, and later the development of modern physics.However until now, the origin of this hypothesis remains unclear.We show that the expression for the De Broglie wavelength can be obtained from a more general well-known concept, that any particle at rest has a corresponding harmonic oscillation that transforms into a wave of a moving particle with the help of relativistic change of coordinates (the Lorentz transformations).In this way, a De Broglie wave can be explained as a spacetime wave.
We set up the classical wave equation for a particle formed of an oscillatory zero-rest-mass charge together with its resulting electromagnetic waves, traveling in a potential field VVV in a susceptible vacuum. The waves are Doppler-displaced upon the source motion, and superpose into a total, traveling- and in turn a standing- beat wave, or de Broglie phase wave, described by a corresponding total classical wave equation. By back-substitution of the explicitly known total, standing beat wave function and upon appropriate reductions at classic-velocity limit, we separate out from the total a component wave equation describing the kinetic motion of particle, which is equivalent to the Schrödinger equation. The Schrödinger wave function follows to be the envelope function of the standing beat wave at classic-velocity limit.
Simple rigorous quantum mechanics with no hand waving nor loopholes clarifies the confusion between three contradictory descriptions of the coherence between different neutrino mass eigenstates that can give rise to oscillations: (1) The standard textbook description of oscillations in time produced by coherence between states with different masses and different energies. (2) Stodolsky's proof that interference between states having different energies cannot be observed in realistic experiments. (3) The description of a pion decay at rest into an observed muon and unobserved neutrino as a “missing mass” experiment where coherence between different neutrino mass eigenstates is not observable. The known position in space of all realistic detectors is rigorously shown to provide the quantum-mechanical ignorance of the neutrino momentum needed to produce coherence between amplitudes from neutrino states with the same energy and different masses. Conditions are precisely formulated for the loss of coherence when mass eigenstate wave packets moving with different velocities separate. The example of Bragg scattering shows how quantum-mechanically imposed ignorance produces coherence.