TITLE:
Formulas for the Kinetic Energy and Momentum of an Electron in a Hydrogen Atom, and the Relationship between the Two
AUTHORS:
Koshun Suto
KEYWORDS:
Einstein’s Energy-Momentum Relationship, Energy-Momentum Relationship in a Hydrogen Atom, Relativistic Kinetic Energy, Potential Energy, Phase Velocity, Group Velocity
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.8,
August
13,
2026
ABSTRACT: The formula for kinetic energy in classical mechanics holds when the velocity of a body is low. However, the formula no longer holds when the velocity of the body increases, and the theory of relativity can no longer be ignored. In classical mechanics, the mass of a body is constant and does not depend on the velocity of the body. Einstein and Sommerfeld defined the kinetic energy of a body as the difference between the relativistic energy of the body
m
c
2
and its rest mass energy
m
0
c
2
. However, Einstein’s energy-momentum relationship, which holds in an isolated system in free space, is not applicable in a hydrogen atom, where potential energy is present. The author has already derived an energy-momentum relationship for an electron in a hydrogen atom, but this paper derives previously unknown formulas for the kinetic energy and momentum of an electron by taking that previous relationship as a departure point.