Page 18 - Photonics Essentials an introduction with experiments

Electrons and Photons

12 Introductory Concepts

this sensitivity is yellow. It appears just before the dark red glow of
the heating element appears in the visible range as it warms up. In
my classes, this effect is seen by about one out of thirty students. Sen-
sitivity does not appear to depend on age or sex.

Planck’s proposition was that temperature is proportional to fre-
quency. But Boltzmann already knew that temperature is proportion-
al to energy. Therefore, we conclude that color is proportional to ener-
gy. As the energy goes up, how does the frequency change?
Remembering that f = c, as the energy gets larger, does the wave-
length increase or decrease? As the energy gets larger, does the fre-
quency increase or decrease?
So, of the two things that characterize light, and f, which one is
proportional to the energy? As the energy goes up, the wavelength
gets shorter or smaller. However, the frequency has to increase be-
cause f = c. Thus, energy is proportional to frequency:

E = hf (2.6)

h, of course, is Planck’s constant.
Energy in a monochromatic beam of red light equal to n·h·f(red
light), where n is the amplitude, or the number of vibrations, each one
of which carries hf of energy:

energy = hf·n f over all frequencies
f
where n f is the number of photons distributed according to Bose–Ein-
stein statistics:
1
n f = const · (2.7)
e hf/k B T – 1
When hf > k B T, such as in the case of an incandescent body like a
stove element, n f is distributed to a good approximation by Boltz-
mann’s law.
Some important results obtained so far are:

1. Boltzmann’s law. For a group of electrons at equilbrium,

n(E 2 )
= e –(E 2 –E 1 )/k B T
n(E 1 )
2. Energy is proportional to frequency: E = hf, where h is Planck’s
constant, equal to 6.63 × 10 –34 joule-sec.

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