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Applied Physics Research

A Surprising Physical Quantity Involved in the Phase Velocity and Energy Levels of the Electron in a Hydrogen Atom

Koshun Suto

Published Aug 15, 2022• 5 citations
Publisher

Abstract

The phase velocity of a material wave is given by the following equation: If this formula is rewritten using the Planck-Einstein relation and the de Broglie formula , it becomes: Next, the values for E and p, obtained from a relation applicable to the electron in a hydrogen atom derived by the author, are substituted into this equation. When that is done, multiple formulas relating to the phase velocity of the electron wave are derived. These formulas contain still-unknown ultra-low energy levels of the hydrogen atom and electron orbital radii corresponding to those energy levels. also appear in the formula for energy levels of the hydrogen atom derived in this paper. This serves as grounds for the existence of ultra-low energy levels in addition to the already-known energy levels of the hydrogen atom.

DOI
10.5539/apr.v14n2p1

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Applied Physics Research•Oct 23, 2026•0 citations
The phase velocity of a material wave is given by the following equation: If this formula is rewritten using the Planck-Einstein relation and the de Broglie formula , it becomes: Next, the values for E and p, obtained from a relation applicable to the electron in a hydrogen atom derived by the author, are substituted into this equation. When that is done, multiple formulas relating to the phase velocity of the electron wave are derived. These formulas contain still-unknown ultra-low energy levels of the hydrogen atom and electron orbital radii corresponding to those energy levels. also appear in the formula for energy levels of the hydrogen atom derived in this paper. This serves as grounds for the existence of ultra-low energy levels in addition to the already-known energy levels of the hydrogen atom.
Einstein’s energy-momentum relationship, holds when the energy absorbed by a body is all converted to kinetic energy of that body. However, an electron in an atom acquires kinetic energy through emission of photon energy. Therefore, Einstein’s relationship cannot be applied to an bound electron in an atom. The author has already derived an energy-momentum relationship for an electron in a hydrogen atom. This paper incorporates de Broglie’s formula into the above relationship previously derived by the author. Also, if the momentum of an electron in a hydrogen atom is considered using an ellipse, it can be predicted that the electron has two types of momentum and two types of waves. This paper concludes that a moving electron has two types of waves.
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.
Bohr assumed a quantum condition when deriving the energy levels of a hydrogen atom. This famous quantum condition was not derived logically, but it beautifully explained the energy levels of the hydrogen atom. Therefore, Bohr’s quantum condition was accepted by physicists. However, the energy levels predicted by the eventually completed quantum mechanics do not match perfectly with the predictions of Bohr. For this reason, it cannot be said that Bohr’s quantum condition is a perfectly correct assumption. Since the mass of an electron which moves inside a hydrogen atom varies, Bohr’s quantum condition must be revised. However, the newly derived relativistic quantum condition is too complex to be assumed at the beginning. The velocity of an electron in a hydrogen atom is known as the Bohr velocity. This velocity can be derived from the formula for energy levels derived by Bohr. The velocity v of an electron including the principal quantum number n is given by αc/n. This paper elucidates the fact that this formula is built into Bohr’s quantum condition. It is also concluded in this paper that it is precisely this velocity formula that is the quantum condition that should have been assumed in the first place by Bohr. From Bohr’s quantum condition, it is impossible to derive the relativistic energy levels of a hydrogen atom, but they can be derived from the new quantum condition. This paper proposes raising the status of the previously-known Bohr velocity formula.
This paper presents a geometric and wave interpretation of energy quantization in the hydrogen atom, based on the de Broglie closure condition of the electron wave in a circular orbit. In this concept, energy quantization is a secondary phenomenon resulting from the fact that the electron wave in each orbit consists of exactly n full periods, and the transition to level n+1 corresponds to the addition of one full period. Combining the wave condition with the classical equilibrium of the Coulomb and centripetal forces leads to values of the orbital radii and a discrete energy spectrum consistent with solutions of the Schrödinger equation for the hydrogen atom. It is shown that energy quantization can be interpreted as a consequence of the resonant nature of the electron's wave nature and the condition of phase uniqueness after a complete orbit around the nucleus.1 The model under consideration is semiclassical in nature and serves as an intuitive representation of known results from quantum mechanics. It is assumed that the allowable states correspond to configurations in which the de Broglie wave forms a standing wave containing an integer number of full periods around the orbital circumference. This condition leads directly to the quantization of angular momentum according to the Bohr model. The waveband model provides a one-dimensional analogy of the full quantum description and can serve as a teaching tool to facilitate understanding the geometric aspects of energy quantization in the hydrogen atom. It demonstrates that energy quantization is a natural consequence of the standing wave geometry and the addition of successive full periods along the orbit as the system transitions to the next energy eigenstate after activation. The electron is a stable eigenstate of a quantum field whose behavior in bound systems can be geometrically interpreted as the self-resonance of a de Broglie wave satisfying the condition of single-valent phase, rather than as a local particle with a classical trajectory. Self-resonance of a wave means that the condition of single-valent phase of the wave function after a complete orbit around the nucleus is satisfied, i.e., the requirement that the phase change be an integer multiple of 2π; this is equivalent to the Bohr–de Broglie condition for the closure of the de Broglie wave.
de Broglie’s concept of particles as sources of phase is extended to the analysis of systems in which the potential is not constant. The treatment assumes that the phase information propagates as waves according to classical electrodynamics including reflection with phase changes from potential energy barriers. The energies and weighting functions in such cases are obtained by comparison of the phase of the accelerated particle relative to the phases of reflections from the change in potential. Application to the chemically interesting systems of the harmonic oscillator and hydrogen atom are considered using the time evolution of the relative phases of the reflected de Broglie phase waves, whereby the weighting functions are obtained from the distance dependence of the phase differences between the reflections and the source. This is accomplished conveniently by taking time derivatives of the relative de Broglie phases and considering the periodicity of the motion. Optimum trajectories are determined by minimization of the action 2i(mv2-r∂V/∂r)/ħγ which depends on the potential V and the velocity v. The treatment leads to the Newtonian force law for circular orbits and affords the same energies as derived using the Schrödinger equation.
Bohr assumed a quantum condition when deriving the energy levels of a hydrogen atom. However, this quantum condition was not a logically derived formula. The support of the de Broglie’s condition was necessary to give physical meaning to Bohr’s quantum condition. However, contemporary physics denies the existence of the waves which de Broglie thought actually existed. Since the theory of relativity are not taken into account in Bohr’s theory, the derived energy levels do not have the correct values. Thus, the author has previously proposed , a quantum condition superior to Bohr’s quantum condition. However, although this quantum condition was mathematically valid, it could not provide physical meaning. This paper has improved that point. When the ratio of the momentum vector of an electron moving inside a hydrogen atom and the momentum vector of the photon possessed by the electron satisfy a certain condition, the two are orthogonal. The condition for these two vectors to be orthogonal is the quantum condition derived in this paper. It is thought that the state where these two vectors are orthogonal is stable, and this state becomes the steady state of the hydrogen atom.