532 Quantum time-dependent statistical mechanics
into the free-field Maxwell equations, then the vector potential isseen to satisfy the
classicalwave equation
∇^2 A(r,t)−1
c^2∂^2
∂t^2A(r,t) = 0, (14.1.4)provided the gauge functionχ(r,t) is also chosen to be a solution of Laplace’s equation
∇^2 χ(r,t) = 0. This choice, known as theCoulomb gauge, is tantamount to requiring
thatA(r,t) satisfy
∇·A(r,t) = 0. (14.1.5)In the Coulomb gauge, the scalar potential satisfies Laplace’s equation∇^2 φ= 0, which,
in free space with no charges present, means thatφis, at most, a linear function ofr
and independent of time. If we require thatφ→0 as|r|→∞, then the only possible
choice isφ= 0, which we can take with no loss of generality. in the Coulomb gauge,
we can simply chooseφ= 0.^2
The appropriate solution to the wave equation forA(r,t) for an electromagnetic
field in a vacuum describe freely propagating waves of frequencyω, wavelengthλ, and
wave vectork(|k|= 2π/λ) prescribing the direction of propagation is
A(r,t) =A 0 cos (k·r−ωt+φ 0 ). (14.1.6)Here,ω=c|k|,φ 0 is an arbitrary phase, andA 0 is the amplitude of the wave. Since
∇·A= 0, it follows thatk·A=k·A 0 = 0, andAis perpendicular to the direction
of propagation. From eqns. (14.1.2), the electric and magnetic fields that result are
E(r,t) =ω
c
A 0 sin (k·r−ωt+φ 0 )≡E 0 sin (k·r−ωt+φ 0 )B(r,t) =−k×A 0 sin (k·r−ωt+φ 0 )≡B 0 sin (k·r−ωt+φ 0 ). (14.1.7)Note that, sincek×Ais perpendicular tokandA 0 , it follows thatE 0 ⊥B 0 ⊥k. A
snapshot in time of the free electromagnetic wave described by eqns. (14.1.7) is shown
in Fig. 14.1.
In general, when an electromagnetic field interacts with matter, the field must be
included in the Hamiltonian of the physical system, which complicates the mathemat-
ical treatment considerably. Therefore, in the present discussion, we will work in an
approximation in which the field can be treated as an external perturbation whose
degrees of freedom do not need to be explicitly included in the quantum description of
the system, i.e., they do not need to be included in the state vector of the system. In
this approximation, the Hamiltonian for a system ofNparticles with chargesq 1 ,..,qN
(^2) These choices are particular to the free-field theory. For time-dependent fields in the presence of
sources, a more appropriate gauge choice is theLorentz gauge:
∇·A(r,t) +
1
c
∂
∂t
φ(r,t) = 0.