Photon Equations


Photon Energy

Photons are transverse waves of energy as a result of particle vibration. The equation to calculate photon energy uses the energy wave equation and the longitudinal energy difference between two points measured as a distance (r) from the atom’s nucleus. The difference in longitudinal wave energy creates a new transverse wave (photon).  An explanation is provided in the section below.


Classical Constant Form

Planck relation in electron constant terms

Wave Constant Form

Transverse Energy Equation

Transverse Energy Equation 

Using classical constants Using energy wave constants



  • f: The variable in classical constants is frequency – typically expressed as E=hf but h replaced with classical constants for comparisons with other equations.
  • δ: amplitude factor variable for constructive wave interference. A single proton and electron is one. All other configurations are found in the Amplitude Factors table.
  • r: distance variable for starting position (r0) and ending position (r) of electron in an atom (in meters).


Photon Frequency & Wavelength 

Photons can also be expressed in terms of frequency or wavelength. Beyond hydrogen, many photon experiments present results as energy instead of wavelengths so the Transverse Energy Equation is used.  The equation is not represented in classical terms as there is no equivalent of a variable for constructive wave interference.

Photon Frequency Equation

Photon Frequency 

Photon Wavelength Equation

Photon Wavelength 



Explanation of Equation

The Transverse Energy Equation begins with the fundamental equation for calculating energy in a volume:

Fundamental Energy equation

Energy Wave Equation


A photon is generated by the vibration of a particle. It is a traveling wave originating from the particle, which is constantly reflecting longitudinal waves. Now, it also has a transverse component during its vibration before the particle comes to rest. Thus, it has a longitudinal component (l) and transverse component (t) that will become its electric and magnetic components.  The vibration creates two photons traveling in opposite directions. In the creation of a photon in an atom, one of the photons will be absorbed by the nucleus and cause recoil as explained here.


Photon Transverse Wave


The transverse component has a frequency that depends on the speed of the vibration of the particle, which may happen during orbital transitions in an atom until a particle like the electron settles into an orbital. The faster the vibration, the higher the frequency of the transverse component.

Particle Vibration



Detailed Assumptions

The photon frequency, wavelength and energy equations are built and derived from these assumptions. The orbital distance calculations are found in the tables for orbital distances, with further detail in the Atomic Orbitals paper. An explanation of how photons are created and absorbed, including incident and scattered angles are presented in the Photons paper.

The following assumptions were made when understanding particle interaction, including atomic orbitals:

  • Particle vibration creates a transverse wave. A particle may vibrate upon annihilation, when transitioning between orbitals in an atom, or when an entire atom vibrates due to kinetic energy.
  • Longitudinal amplitude difference creates particle motion as particles seek to minimize amplitude.
  • The difference in longitudinal energy is transferred to transverse energy in two photons that travel in opposite directions, perpendicular to particle vibration.
  • Particles and their antimatter counterparts attract because of destructive waves between the particles; like particles (e.g. electron-electron) repel due to constructive waves, seeking to minimize amplitude.
  • Electrons in an atomic orbital are both attracted and repelled by the nucleus. A positron is assumed to be at its core to attract the orbital electron; opposing forces in the nucleus repel the orbital electron as explained in the Atoms section.


Conservation of Energy – Longitudinal Waves to Transverse Waves

The transverse wave energy for the photon is a result of a difference in longitudinal wave energy as two opposite phase particles destructively interfere, which is the case with a proton and electron. The energy difference in longitudinal wave form in classical terms is:

Orbital Transition Eq 2


The values Ee and re are the energy of the electron and classical electron radius, respectively. Q is the particle group count that is also used in force equations. These values can be replaced by their wave constants form.

Orbital Transition Eq 3

There are two photons created, thus each photon (Et) is half of the longitudinal energy above. For two particle groups (Q) at distance (r), the equation works. But in an atom, this is limited to hydrogen or ionized atoms that have only one electron. For all other atoms, there are multiple electrons at varying distances.  As a result, a variable called the amplitude factor (δ), replaces Q1 and Q2 as the method to calculate constructive wave interference on an electron from multiple particles at varying distances.  The calculations of amplitude factor is explained here.  Substituting for this variable, and taking half of the longitudinal energy for one photon, leads to the Transverse Energy Equation.

Transverse Energy Equation


Calculations and Examples

A summary of calculations and some examples using the equations are provided here. The remainder of the calculations and examples are detailed in the Particle Energy and Interaction paper. A summary of photon interactions in atoms is detailed in the Photons paper.