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Wave Equations

Electromagnetic Wave Equation

The field equations for electromagnetic radiation is a product of the Victorian Era in the 19th century. It was originated from a set of equations in electromagnetism. Maxwell's contribution is to add a "displacement current" term, i.e., the (1/c)E term in Eq.(40), and thus put the equations into a consistent set which implied new physical phenomena, at that time unknown but subsequently verified in all details by experiments. In terms of the electric field E and magnetic field B the Maxwell's equations are four first order differential equations :

EM Eqs.           ---------- (39)



          ---------- (40)



          ---------- (41)



          ---------- (42)
,
the i, j, k, are unit vectors along the x, y, z axis respectively, J is the current density with current I, is the charge density with charge Q, and c is the velocity of light. The conversion of the laws from differential to integral form is accomplished by applying the Gauss's and the Stoke's theorems as shown below via a much simplified derivation with either a cube or square (the circle on the integral sign denotes either a closed surface or loop).



For regions of space where charge and current are absent, the right-hand sides vanish in all the equations. These first order differential equations can be translated into second order equations more suitable for describing the electromagnetic wave in relativistic invariant form. By virtue of the mathematical identity ( A)=0, Eq.(41) can be defined by the vector potential A:
B = A ---------- (43)
By substituting Eq.(43) into Eq.(42) and using another mathematical identity ()=0, we can define E in terms of A and the scalar potential as :
E = - - (1/c)A ---------- (44)
which combines with Eq.(39) yields:
2 + (1/c)(A) = 0 ---------- (45)
Substituting Eqs.(43), (44) into (40) and using one more identity ( ) A = -2A + (A), we get another second order differential equation :
2A - (1/c2)A - (A + (1/c)) = 0 ---------- (46)

The U(1) symmetry on the spin of A adds an extra degree of freedom, which can be fixed by taking the Lorenz condition :
A + (1/c) = 0,
then Eqs.(45), and (46) can be written in simpler form :
2A - (1/c2)A = 0 ---------- (46a),
and    2 - (1/c2) = 0 ---------- (45a).

The vector potential A and the scalar potential form a four potential, whcih are prescribed by the Lorentz invariant wave equations of Eqs. (46a), (45a). The formulation is now suitable for use in high energy particle physics. The Lorenz condition is also manifest Lorentz invariance (note the two names differ by a "t"), but is incomplete in the sense that the definition of B in Eq.(43) is unchanged by adding a term in the form of the "gradient of scalar function " because of the identity () = 0. It follows that E, B and Eqs.(45a), (46a) are unchanged by the transformation :

A A' = A + ---------- (47)
if     ' = - (1/c) ---------- (48)
and    2 - (1/c2) = 0.
This is known as the gauge transformation of the second kind (or Lorenz gauge).

    Other choice for involves setting 2 = -A, and eliminating the longitudinal component with = (1/c) in the transformation Eqs.(47), and (48). In this way, the transversality condition A' = 0 is ensured. This condition is also known as "Coulomb gauge". It follows that :

  1. Eq.(45) is reduced to the Laplace's equation for time-independent electrostatic potential in free space, that's why it is also called "Coulomb gauge".
  2. The longitudinal component is eliminated, i.e., there is no component along the direction of motion. This is ensured by the condition A' = 0.
  3. The two transversal components correspond to the two polarizations as observed in electromagnetic radiation.
  4. Since ' = 0 in free space without the presence of any source, the wave equation Eq.(46) become :
    2A' - (1/c2)A' = 0 ---------- (49)
  5. The magnetic field is related to A by B = A' ---------- (50)
  6. The electric field is related to A by E = - (1/c)A' ---------- (51)
  7. From Eqs.(49), (51), the wave equation for E can be derived :
    2E - (1/c2)E = 0 ---------- (50a)
  8. From Eqs.(49), (50), the transverse gauge, and the identity leading to Eq.(46), we obtain the wave equation for B :
    2B - (1/c2)B = 0 ---------- (51a)
  9. It is now clear that the introduction of the vector potential A is mainly to simplify computations. For the case of transverse electromagnetic wave two components of A is required to describe the wave, while the E and B fields together introduce four components.
  10. The Coulomb gauge is not covariant under a Lorentz transformation.
  11. See more in "Quantum Electrodynamics".
Eqs.(49), (50a), (51a) are in a form very similar to the Klein-Gordon equation in Eq.(18) except that the mass term vanishes (because the photon has no rest mass) and the field is a vector (instead of scalar) with two transversal components (polarization) perpendicular to each other. The Maxwell's equations were derived more than one hundred years ago without any knowledge about quantum theory, the similarity in form is perhaps because all kinds of wave (including the gravitational wave for spin 2 particle) are governed by the same kind of equation. Only the details (such as number of components) and interpretations are slightly different (note the omission of the Planck constant h in the Maxwell's equations has no consequence in the comparison as m = 0).

The solution of Eq.(49) is :

kx, ky, kz = 2n/L (n = 1, 2, ...) with kx = kx - t = kx - |k|ct. The ck coefficients is linked to the spectrum associated with the electromagnetic radiation. It will assume the role of q-number in second quantization, which endows particle attributes to the wave.

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