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Total energy is the sum of rest energy = and relativistic kinetic energy: = = + Invariant mass is mass measured in a center-of-momentum frame. For bodies or systems with zero momentum, it simplifies to the mass–energy equation E 0 = m 0 c 2 {\displaystyle E_{0}=m_{0}c^{2}} , where total energy in this case is equal to rest energy.
The following notations are used very often in special relativity: Lorentz factor = where = and v is the relative velocity between two inertial frames.. For two frames at rest, γ = 1, and increases with relative velocity between the two inertial frames.
For example, for a speed of 10 km/s (22,000 mph) the correction to the non-relativistic kinetic energy is 0.0417 J/kg (on a non-relativistic kinetic energy of 50 MJ/kg) and for a speed of 100 km/s it is 417 J/kg (on a non-relativistic kinetic energy of 5 GJ/kg). The relativistic relation between kinetic energy and momentum is given by
Relativistic kinetic energy: The relativistic kinetic energy relation takes the slightly modified form: = ...
Quantity (common name/s) (Common) symbol/s Defining equation SI unit Dimension Wavefunction: ψ, Ψ : To solve from the Schrödinger equation: varies with situation and number of particles
The fine structure energy corrections can be obtained by using perturbation theory.To perform this calculation one must add three corrective terms to the Hamiltonian: the leading order relativistic correction to the kinetic energy, the correction due to the spin–orbit coupling, and the Darwin term coming from the quantum fluctuating motion or zitterbewegung of the electron.
Due to kinetic energy and binding energy, this quantity is different from the sum of the rest masses of the particles of which the system is composed. Rest mass is not a conserved quantity in special relativity, unlike the situation in Newtonian physics.
A fundamental prediction of special relativity is the relativistic energy–momentum relation; for a particle of rest mass m, and in a particular frame of reference with energy E and 3-momentum p with magnitude in terms of the dot product =, it is: [10]