154
Chapter 3. Wave optics
is H = hν, from which it follows that ν is a constant. In this case
Fermat’s principle is equivalent to
ds
δ
= 0.
(3.117)
λ
Physically, all hypothetical rays in an infinitesimal neighborhood
surrounding the physical ray interfere constructively. In this context, Fermat’s principle is fundamentally wave-mechanical.
Separately, the physical trajectory of a classical point particle with
mass m obeys the principle of least action,
δ p ds = 0,
(3.118)
where we have assumed that the magnetic vector potential A is
zero, and the electrostatic potential φ(x) has no explicit time dependence. Substituting for the kinetic momentum p, this is equivalent in one dimension to
δ
2m[H − U (x)] ds = 0,
(3.119)
where U (x) = q φ(x) is the potential energy, and H is the conserved total energy. The integrand can be regarded as an index of
refraction in the mechanical analog of Fermat’s principle in classical mechanics. Thus we have two alternative expressions for the
index of refraction. They are not equivalent, since one is wavemechanical, and the other is derived from classical mechanics.
Following Fermi, equations (3.116) and (3.119) guide us to form a
working assumption, namely, the phase velocity v p can be written
in the analogous functional form
1 = f (ω) H(ω) − U (x),
(3.120)
v p
where f (ω) and H(ω) are arbitrary functions of the angular frequency ω, yet to be determined. We will investigate the validity
of this assumption in the following.
Chapter 3. Wave optics
is H = hν, from which it follows that ν is a constant. In this case
Fermat’s principle is equivalent to
ds
δ
= 0.
(3.117)
λ
Physically, all hypothetical rays in an infinitesimal neighborhood
surrounding the physical ray interfere constructively. In this context, Fermat’s principle is fundamentally wave-mechanical.
Separately, the physical trajectory of a classical point particle with
mass m obeys the principle of least action,
δ p ds = 0,
(3.118)
where we have assumed that the magnetic vector potential A is
zero, and the electrostatic potential φ(x) has no explicit time dependence. Substituting for the kinetic momentum p, this is equivalent in one dimension to
δ
2m[H − U (x)] ds = 0,
(3.119)
where U (x) = q φ(x) is the potential energy, and H is the conserved total energy. The integrand can be regarded as an index of
refraction in the mechanical analog of Fermat’s principle in classical mechanics. Thus we have two alternative expressions for the
index of refraction. They are not equivalent, since one is wavemechanical, and the other is derived from classical mechanics.
Following Fermi, equations (3.116) and (3.119) guide us to form a
working assumption, namely, the phase velocity v p can be written
in the analogous functional form
1 = f (ω) H(ω) − U (x),
(3.120)
v p
where f (ω) and H(ω) are arbitrary functions of the angular frequency ω, yet to be determined. We will investigate the validity
of this assumption in the following.