Abstract
The author has previously derived an energy-momentum relationship applicable in a hydrogen atom. Since this relationship is taken as a departure point, there is a similarity with the Dirac’s relativistic wave equation, but an equation more profound than the Dirac equation is derived. When determining the coefficients  and β of the Dirac equation, Dirac assumed that the equation satisfies the Klein-Gordon equation. The Klein-Gordon equation is an equation which quantizes Einstein's energy-momentum relationship. This paper derives an equation similar to the Klein-Gordon equation by quantizing the relationship between energy and momentum of the electron in a hydrogen atom. By looking to the Dirac equation, it is predicted that there is a relativistic wave equation which satisfies that equation, and its coefficients are determined. With the Dirac equation it is necessary to insert a term for potential energy into the equation when describing the state of the electron in a hydrogen atom. However, in this paper, a potential energy term is not introduced into the relativistic wave equation. Instead, potential energy is incorporated into the equation by changing the coefficient  of the Dirac equation.
Highlights
One of the most important relationships in the Special Theory of Relativity (STR) is as follows:( ) ( ) m0c2 2 + p 2c 2 = mc2 2 . (1)Here, mc2 is the relativistic energy of an object or a particle, and m0c2 is the rest mass energy.Currently, Einstein’s relationship (1) is used to describe the energy and momentum of particles in free space, but for explaining the behavior of bound electrons inside atoms, opinion has shifted to quantum mechanics as represented by equations such as the Dirac’s relativistic wave equation.For reasons such as these, there was no search for a relationship between energy and momentum applicable to an electron in a hydrogen atom.Here, let's consider Einstein's energy-momentum relationship, that holds for isolated systems in free space
If the photonic energy released when an electron is drawn into a hydrogen atom is taken to be hν, and the kinetic energy acquired by the electron is taken to be K, the following relationship holds
The author presented the following equation as an equation indicating the relationship between the rest mass energy and potential energy of the electron in a hydrogen atom (Suto, 2009): V (r) = −Δmec2
Summary
One of the most important relationships in the Special Theory of Relativity (STR) is as follows:. Einstein’s relationship (1) is used to describe the energy and momentum of particles in free space, but for explaining the behavior of bound electrons inside atoms, opinion has shifted to quantum mechanics as represented by equations such as the Dirac’s relativistic wave equation. If the photonic energy released when an electron is drawn into a hydrogen atom is taken to be hν , and the kinetic energy acquired by the electron is taken to be K, the following relationship holds. The author presented the following equation as an equation indicating the relationship between the rest mass energy and potential energy of the electron in a hydrogen atom (Suto, 2009):
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