Derivation of the Schr\"odinger equation from QED

Abstract

The Schr\"odinger equation relates the electron wavefunction and the electric potential, which are emergent physical quantities. At that emergent level, the Schr\"odinger equation is either postulated as a principle of quantum physics or obtained heuristically. However, the Schr\"odinger equation is a low energy condition we can derive from the foundations of QED. Due to the small value of the electromagnetic coupling constant, we show that, in low energy interactions, the electric potential accurately represents the contributions of the intermediate photon exchanges. Then, we see that the dominant term of the electron wavefunction is a superposition of plane (but not free) waves which, by fulfilling the total energy relations, satisfies the Schr\"odinger, Pauli, and Dirac equations. Furthermore, we show that what is considered the kinetic energy term of the Schr\"odinger equation does not represent the kinetic energy of the interacting electron. We analyze and clarify the dynamics of the Schr\"odinger equation.