In this chapter, we consider a quantum particle wave function with a bound
spectrum of velocity c, and obtain the relativistic momentum based on the group
velocity of this wave function. With a space-time isometry condition, the Lorentz
transformation and the relativistic dynamics were obtained. Considering a field
interacting with a quantum particle as a four-vector conjugated to the space-time vector
in the time-dependent phase, we obtain the Lorenz force and the Maxwell equations. It
is interesting that only the Ampère-Maxwell equation of a magnetic circuit is specific
to the electromagnetic field, while the other equations are general for a field interacting
with a charged quantum particle. Considering the time-dependent phase of a quantum
particle interacting with an electromagnetic field with a space-time homogeneity
condition, we obtain Lorentz transformations for this field. For a quantum particle at a
non-relativistic velocity, we obtain a wave function with a very rapidly-varying factor,
of a frequency proportional to the rest energy of this particle. From the Schrödinger
equation of a particle with a relativistic Hamiltonian, we obtain a split of the wave
function into four components, describing a proper rotation of this particle with an
angular momentum called spin (Dirac’s relativistic electron theory). Moreover, we also
calculate electron potential in the magnetic field, and two-electron interaction potential.
Keywords: Action, Ampère-Maxwell equation, Angular momentum, Bohr
magneton, Dirac matrices, Electric field, Electric potential, Faradey-Maxwell
equation, Four-vector, Gauss equation, Giro-magnetic ratio, Group velocity,
Hamiltonian, Kinetic energy, Lagrangian, Liénard-Wiechert potentials, Lorentz
transformation, Magnetic field, Magnetic moment, Momentum, Pauli matrices,
Potential energy, Rest energy, Rest mass, Spin, Vector potential, Wave function,
Wave-packet.
