## ABSTRACT

It follows from Eqs. (10.44) and (10.45), and the orthonormality condition for the polarization unit vectors [Eq. (2.52)] that the amplitude of the transverse vector potential belonging to index s, viz.,

a (+) T,s (q; t) = ε

∗ s(κ) · a(+)T,s (q; t), (11.1)

satisfies the wave equation (s = 1, 2)

i~ ∂

∂t a (+) T,s (q; t) = c~qa

(+) T,s (q; t). (11.2)

In free space, and in the Lorenz gauge, the analytical amplitudes of the longitudinal vector potential and the scalar potential satisfy form-identical Schro¨dinger-like dynamical equations, as we now shall see. In free space the longitudinal part of the electric field vanishes everywhere, so that [Eq. (10.27)]

∂t AL(r, t) + c∇A

0(r, t) = 0. (11.3)

In the Lorenz gauge, the potentials entering Eq. (11.3) are related by

∇ ·AL(r, t) + 1 c

∂t A0(r, t) = 0, (11.4)

see Eq. (10.15). By combining Eqs. (11.3) and (11.4) one can obtain the two wave equations

AL(r, t) = 0, (11.5)

A0(r, t) = 0. (11.6)

The reader may reach Eq. (11.5) by taking the gradient of Eq. (11.4), and then eliminating ∇A0 using Eq. (11.3). Hence, she gets

∇∇ ·AL(r, t)− 1 c2

∂2

∂t2 AL(r, t) = 0. (11.7)

Since 0 =∇× (∇×AL) =∇∇ ·AL −∇2AL, the first term in Eq. (11.7) can be replaced by ∇2AL, thus giving Eq. (11.5). By taking the divergence of Eq. (11.3), and eliminating thereafter ∇ ·AL by means of Eq. (11.4), Eq. (11.6) is obtained.