Planck’s relation (E=hf) is a result of a transverse wave, from the vibration of a particle due to a difference in wave amplitude. Particles move to minimize their displacement (minimize amplitude) on a wave. This happens during motion, including annihilation of a particle or when a particle transitions between orbitals in an atom. Its vibration causes a secondary, transverse wave and the energy is based on the frequency of this vibration.







The derivation starts with a difference in longitudinal wave energy from the Energy Wave Equation, as the particle’s vibration creates a secondary, transverse wave. However, it also requires explanation about the derivation of a transverse wave that can be found in the Photons section. Further details can be found in the Key Physics Equations and Experiments paper.

E=hf derived




Planck Constant: Solving for the wave constants in Eq. 2.3.9 for Planck constant yields the accurate numerical value and units.

Planck Constant Derived


Hydrogen Frequency (Ground State): Solving for Eq. 2.3.4 at the Bohr radius (a0) for a hydrogen atom (amplitude factor is one – d=1) yields the correct frequency.  Hydrogen Ground State Frequency Derived


Rydberg Unit of Energy: Solving for the energy of a hydrogen atom at the Bohr radius (a0) in Eq. 2.3.6 yields the Rydberg unit of energy.

Rydberg Energy Derived