Previously Unknown Formulas for the Relativistic Kinetic Energy of an Electron in a Hydrogen Atom

Einstein’s energy-momentum relationship, which holds in an isolated system in free space, contains two formulas for relativistic kinetic energy. Einstein’s relationship is not applicable in a hydrogen atom, where potential energy is present. However, a relationship similar to that can be derived. That derived relationship also contains two formulas, for the relativistic kinetic energy of an electron in a hydrogen atom. Furthermore, it is possible to derive a third formula for the relativistic kinetic energy of an electron from that relationship. Next, the paper looks at the fact that the electron has a wave nature. Five more formulas can be derived based on considerations relating to the phase velocity and group velocity of the electron. This paper presents eight formulas for the relativistic kinetic energy of an electron in a hydrogen atom.

Electron Mass in an Atom Is Less than Rest Mass

Einstein’s energy-momentum relationship is a formula that typifies the special theory of relativity (STR). According to the STR, when the velocity of a moving body increases, so does the mass of the body. The STR asserts that the mass of a body depends of the velocity at which the body moves. However, when energy is imparted to a body, this relation holds because kinetic energy increases. When the motion of an electron in an atom is discussed at the level of classical quantum theory, the kinetic energy of the electron is increased due to the emission of energy. At this time, the relativistic energy of the electron decreases, and the mass of the electron also decreases. The STR is not applicable to an electron in an atom. This paper derives an energy-momentum relationship applicable to an electron in an atom. The formula which determines the mass of an electron in an atom is also derived by using that relationship.

The Physical Constant Called the Rydberg Constant Does Not Exist

In classical quantum theory, the Rydberg constant is a fundamental physical constant that plays an important role. It comes into play as an indispensable physical constant in basic formulas for describing natural phenomena. However, relativity is not taken into account in this Rydberg formula for wavelength. If the special theory of relativity is taken into account, R∞ can no longer be regarded as a physical constant. That is, we have continued to conduct experiments to this day in an attempt to determine the value of a physical constant, the Rydberg constant, which does not exist in the natural world.

An Energy Formula That Is Physically Easier to Understand than E = hν

This paper rewrites the famous energy formula of quantum theory, E = hν, as a formula that is physically easier to understand. If we let me be the rest mass of the electron, c the speed of light in a vacuum, and λc the Compton wavelength of the electron, then the product of the three physical constants, mecλc, matches the value of the Planck constant. In the usual interpretation, h is regarded as a universal constant on a par with c. However, this paper holds that, contrary to the historical viewpoint, the Planck constant is logically nothing more than replacement of mecλc with the alphabetic letter h. Thus, this paper looks for an energy formula that does not contain h. E = hν is a formula that was assumed at the beginning, and then subsequently verified through experiment. The formula was not derived logically. In contrast, the energy formula derived in this paper can be derived logically. The formula derived in this paper also has a clear physical meaning, and it can be concluded that it is a superior formula to E = hν.

The Incompleteness of Quantum Mechanics Demonstrated by Considerations of Relativistic Kinetic Energy

The energy levels of a hydrogen atom, derived by Bohr, are known to be approximations. This is because the classical quantum theory of Bohr does not take the theory of relativity into account. In this paper, the kinetic energy and momentum of an electron in a hydrogen atom are treated relativistically. A clearer argument is developed while also referring to papers published in the past. The energy levels of a hydrogen atom predicted by this paper almost match the theoretical values of Bohr. It is difficult to experimentally distinguish the two. However, this paper predicts the existence of an n = 0 energy level that cannot be predicted even with Dirac’s relativistic quantum mechanics. The only quantum number treated in this paper is n. This point falls far short of a finished quantum mechanics. However, even in discussion at the level of this paper, it can be concluded that quantum mechanics is an incomplete theory.

A Violation of the Special Theory of Relativity Demonstrated Using the Correlation between Two Pair-Generated Photons

When Einstein developed the special theory of relativity (STR), he assumed the principle of relativity, i.e. that all inertial frames are equivalent. Einstein thought it was impossible to differentiate inertial frames into classically stationary frames where light propagates isotropically, and classically moving frames where light propagates anisotropically. However, the author has previously pointed out that classically moving frames have a velocity vector attached, and presented a thought experiment for determining the size of that velocity vector. The author has already shown a violation of the STR, but this paper presents a violation of the STR using different reasoning. More specifically, this paper searches for a coordinate system where light propagates anisotropically. This is done by using the correlation of two photons pair-generated from a photon pair generator. If the existence of such a coordinate system can be ascertained, it will constitute a violation of the STR.

The Planck Constant Was Not a Universal Constant

According to traditional classical quantum theory, due to the prior existence of the Planck constant, considered a universal constant, it is thought that the energy of a photon can be determined if its frequency is known, and the wavelength of a quantum can be determined if its momentum is known (E = hv and λ = h/p). In this paper, however, the Planck constant only comes into existence when mecλC is replaced with h. There is no problem with introducing h to simplify equations, but quantum mechanics is not affected even if there is no symbol h. The physicists at the beginning of the 20th century overestimated the Planck constant, and this gave rise to universal constants that do not exist in the natural world in itself.

Dark Matter and the Energy-Momentum Relationship in a Hydrogen Atom

Einstein derived the energy-momentum relationship which holds in an isolated system in free space. However, this relationship is not applicable in the space inside a hydrogen atom where there is potential energy. Therefore, in 2011, the author derived an energy-momentum relationship applicable to the electron constituting a hydrogen atom. This paper derives that relationship in a simpler way using another method. From this relationship, it is possible to derive the formula for the energy levels of a hydrogen atom. The energy values obtained from this formula almost match the theoretical values of Bohr. However, the relationship derived by the author includes a state that cannot be predicted with Bohr’s theory. In the hydrogen atom, there is an energy level with n = 0. Also, there are energy levels where the relativistic energy of the electron becomes negative. An electron with this negative energy (mass) exists near the atomic nucleus (proton). The name “dark hydrogen atom” is given to matter formed from one electron with this negative mass and one proton with positive mass. Dark hydrogen atoms, dark hydrogen molecules, other types of dark atoms, and aggregates made up of dark molecules are plausible candidates for dark matter, the mysterious type of matter whose true nature is currently unknown.

Theoretical Prediction of Negative Energy Specific to the Electron

If an electron emits all of its rest mass energy mec2, the relativistic energy of the electron will become zero. According to the special theory of relativity, an electron whose relativistic energy is zero does not have photon energy. In this paper, however, an electron is regarded as having photon energy mec2 and negative energy −mec2, even when its relativistic energy is zero. The state where relativistic energy is zero is achieved due to the positive energy and negative energy canceling each other out. Relativistic energy becomes zero for an electron in a hydrogen atom when the principle quantum number n is zero. The author has already pointed out the existence of an energy level with n=0. If this model is used, it is possible for an electron in the state with n=0 to emit additional photons, and transition to negative energy levels. The existence of negative energy specific to the electron has previously been nothing more than a conjecture. However, this paper aims to theoretically show the existence of negative energy based on a discussion using an ellipse. The results show that the electron has latent negative energy.

Relativistic Correction of the Rydberg Formula

The relationship E = −K holds between the energy E and kinetic energy K of the electron constituting a hydrogen atom. If the kinetic energy of the electron is determined based on that relationship, then the energy levels of the hydrogen atom are also determined. In classical quantum theory, there is a formula called the Rydberg formula for calculating the wavelength of a photon emitted by an electron. In this paper, in contrast, the formula for the wavelength of a photon is derived from the relativistic energy levels of a hydrogen atom derived by the author. The results show that, although the Rydberg constant is classically a physical constant, it cannot be regarded as a fundamental physical constant if the theory of relativity is taken into account.

Dark Matter Interacts with Electromagnetic Waves

The large-scale structure of DM cannot be directly seen, but it is believed it can be inferred from the distribution of visible galaxies formed from ordinary matter. Bright visible galaxies that emit the Lyman α (Lyα) emission line of the hydrogen atom (Lyα galaxies) are used to observe the large-scale structure of distant space. However, recently Momose et al. have reported cases where the large-scale structures of DM indicated by Lyα galaxies and other galaxies fail to match. This raises the possibility that Lyα galaxies may not correctly indicate the large-scale structure of DM. In the currently accepted cold DM model, DM and neutral hydrogen gas are thought to interact only through the mutual effects of gravity. However, according to Suto, DM and ordinary matter are like two sides of the same coin. By giving and receiving approximately 2mec2 (1.022 MeV), it is possible to mutually convert between the two. If, in future observations of the density distribution of interstellar gas using Lyα emission lines, unexpected data is obtained that cannot be explained based only on absorption by neutral hydrogen gas, then the author believes the problem can be solved with Suto’s DM model.