APPENDIX TO UNIT 18: Dispersion in the Hydrogen Spectrum
The dependence of the index of refraction N upon wavelength λ for transparent materials is a property that is called dispersion. Dispersion in a material may be understood
with a purely classical model although it is, in fact, a purely quantum mechanical effect.
For the prominent visible lines in the Hydrogen spectrum, you have measured the dispersion of a glass prism and found that N decreases with increasing λ (or decreasing frequency
ω). Longer wavelength light travels faster than shorter wavelength light. I will show you
how to understand this trend and estimate from it the frequency at which glass will begin
to absorb EM radiation—that is, lose its transparency.
Glass is transparent because the molecules making up the glass cannot absorb photons
with such a small energy (∼ 1 eV). The first excited state of the glass molecule, SiO 2 , is
up much higher—maybe 9 eV. So why does glass do anything at all to visible photons?
Clearly, it is able to bend or refract the light. If we want to get the right answer treating
light as a particle with a discrete quantized energy as above, we would have to do some
quantum mechanics. But if we treat light as an electric field wave, we will get an equivalent
answer classically.
Recall that when an electric field, E, is applied to a nonconducting medium, a small
dipole moment is induced and the medium becomes slightly electrically polarized. For
small applied fields, the dipole moment per unit volume, P , is proportional to the applied
field itself:
P = χE
where χ is called the polarizability or susceptibility. A large χ means that medium undergoes large polarization and depolarization every time an electric field wave crest passes by.
Just as the transverse wave speed in a wire decreases if the density of the wire is increased,
a large χ effectively increases the “inertia” of the medium and the wave speed is slower.
Dispersion, then, is a symptom of a polarizability that depends upon the frequency of the
wave. This effect is familiar from the theory of forced harmonic oscillators.
Let us model the molecules making up the glass as an electric charge (the valence
electrons) attached by a spring to the center of mass of the nuclei. (I will regard the nuclei
as being fixed in position—the glass is frozen after all.) Consider a harmonic electric field
wave driving one of these charge/springs systems. The equation of motion is:
d2 x
λ dx
e
+
+ ω02 x = E cos ωt
2
dt
m dt
m
λ is a frictional damping coefficient, and e and m are the charge and mass of the electron.
The natural frequency of the oscillator ω0 plays the role of the absorption frequency of the
molecule.
[Note: It turns out that in the limit of λ → 0+ , the results of the classical model agree
exactly with the quantum mechanical results as long as the fraction of atoms in the excited
1
state is very small. λ = 0+ is necessary so that the oscillator actually absorbs and frictionally dissipates energy when the applied frequency is tuned to resonance. Surprisingly,
friction is necessary in the classical theory to mimic the irreversible quantum behavior of
a photon being absorbed.]
The equation of motion has a solution:
x(t) = Re
ω2
−e/m
Eeiωt
− ω02 − iλω/m
This solution behaves in the expected way: the oscillation develops a huge amplitude
as the resonance condition ω = ω0 is reached. For frequencies far away from resonance
(technically, ω0 − ω >> λ/m), taking the real part of this expression amounts to ignoring
the imaginary term in the denominator. Each oscillator contributes a dipole moment ex(t)
and if we assume that the density of identical oscillators in the glass is n, the dipole moment
per unit volume becomes:
P (t) = χE(t) =
ne2 /m
E cos ωt
ω02 − ω 2
It turns out that the index of refraction N that you measure is related to χ in the
following way:
N 2 = 1 + 4πχ
The “1” in the expression reflects the fact that the index of refraction is the ratio of the
speed of light in a vacuum to that in a nonconducting medium: N ≡ c/v. Now the index
of refraction is related to ω or λ by
1
1
1
4π 2 c2 1
2
2
= 2 (ω0 − ω ) =
( 2 − 2)
2
2
N (ω) − 1
Ω
Ω
λ0
λ
where Ω2 ≡ 4πne2 /m. Getting back to the experiment that you performed, if you plot the
left hand side versus −1/λ2 you should get a reasonably straight line. From the slope and
the intercept, you can determine λ0 which, when converted to an energy, is an estimation
of the absorption energy of glass.
2