Abstract
Bohr’s quantum condition is an indispensable assumption for classical quantum theory. However, strictly speaking, Bohr's quantum condition does not hold when deriving the energy of an electron forming a hydrogen atom from the perspective of the theory of relativity. In this paper, it is thought that the relationship enfolded in Bohr's quantum condition, i.e.,  is suitable as a new quantum condition to replace Bohr’s quantum condition. Also, in quantum mechanics, the energy of an electron is derived based on the theory of relativity, as exemplified in the theory of Sommerfeld. However, this paper points out that the previous energy formula based on the theory of relativity is mistaken. It also proposes a previously unknown formula for the kinetic energy of an electron.
Highlights
When the obtained physical quantities are substituted into the left side of Equation (35), the following values are obtained
It is undesirable to assume these formulas at the beginning as a quantum condition to replace Bohr's condition Equation (23)
If non-relativistic physical expressions are written on the left side of Equation (113), a non-relativistic approximation appears on the right side (Equations (32) and (34))
Summary
If r is taken to be the radius of a circular orbit, and v is taken to be the speed of the electron, the following Newtonian equation of motion holds. If Equation (16) is assumed, Equation (13) becomes as follows This made it possible to find the energy levels of the hydrogen atom. According to de Broglie, Bohr's quantum condition was able to acquire a substantive meaning, and it came to be that the energy levels of the hydrogen atom in Equation (17), found by assuming Equation (16), were believed to be correct. If En in Equation (17) is substituted into Equation (10), the following formula can be derived as the orbital radius of the electron.
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