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The Coulomb gauge (also known as the transverse gauge) is used in quantum chemistry and condensed matter physics and is defined by the gauge condition (more precisely, gauge fixing condition) (,) =. It is particularly useful for "semi-classical" calculations in quantum mechanics, in which the vector potential is quantized but the Coulomb ...
The Lorenz gauge hence contradicted Maxwell's original derivation of the EM wave equation by introducing a retardation effect to the Coulomb force and bringing it inside the EM wave equation alongside the time varying electric field, which was introduced in Lorenz's paper "On the identity of the vibrations of light with electrical currents".
Position vectors r and r′ used in the calculation. The starting point is Maxwell's equations in the potential formulation using the Lorenz gauge: =, = where φ(r, t) is the electric potential and A(r, t) is the magnetic vector potential, for an arbitrary source of charge density ρ(r, t) and current density J(r, t), and is the D'Alembert operator. [2]
There is gauge freedom in A in that of the three forms in this decomposition, only the coexact form has any effect on the electromagnetic tensor F = d A {\displaystyle F=dA} . Exact forms are closed, as are harmonic forms over an appropriate domain, so d d α = 0 {\displaystyle dd\alpha =0} and d γ = 0 {\displaystyle d\gamma =0} , always.
The solutions of Maxwell's equations in the Lorenz gauge (see Feynman [5] and Jackson [7]) with the boundary condition that both potentials go to zero sufficiently fast as they approach infinity are called the retarded potentials, which are the magnetic vector potential (,) and the electric scalar potential (,) due to a current distribution of ...
The gauge-fixed potentials still have a gauge freedom under all gauge transformations that leave the gauge fixing equations invariant. Inspection of the potential equations suggests two natural choices. In the Coulomb gauge, we impose ∇ ⋅ A = 0, which is mostly used in the case of magneto statics when we can neglect the c −2 ∂ 2 A/∂t ...
As in the Gaussian system (G), the Heaviside–Lorentz system (HL) uses the length–mass–time dimensions. This means that all of the units of electric and magnetic quantities are expressible in terms of the units of the base quantities length, time and mass. Coulomb's equation, used to define charge in these systems, is F = q G 1 q G
Choosing the Coulomb gauge, for which ∇⋅A = 0, makes A into a transverse field. The Fourier expansion of the vector potential enclosed in a finite cubic box of volume V = L 3 is then
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