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Derivation of the Schr\"odinger equation from QED

Spyros Efthimiades
Dec 2022
摘要
The Schr\"odinger equation relates the electron wavefunction and the electricpotential, which are emergent physical quantities. At that emergent level, theSchr\"odinger equation is either postulated as a principle of quantum physicsor obtained heuristically. However, the Schr\"odinger equation is a low energycondition we can derive from the foundations of QED. Due to the small value ofthe electromagnetic coupling constant, we show that, in low energyinteractions, the electric potential accurately represents the contributions ofthe intermediate photon exchanges. Then, we see that the dominant term of theelectron 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 kineticenergy term of the Schr\"odinger equation does not represent the kinetic energyof the interacting electron. We analyze and clarify the dynamics of theSchr\"odinger equation.
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