In my recent post about the quantum
vacuum, I said that
nobody can picture a photon. But I do think we should try to
imagine what a photon might look like.
Lately I've been studying three books in particular: Intro to the
Theory of Coherence and Polarization, Probability and
Stochastic Processes: With A View Toward Applications, and Intro to Laser Physics. Some of the common subjects in
these books are stationary
random processes,
the auto-correlation function and power spectra, and in the case of the two
"intro" books, the characteristics of different types of light, such
as the Planck spectrum of light from a thermal source (incandescent bulb, the
Sun) versus the quasi-monochromatic light output of a laser.
Here is a general description from Intro to Laser Physics
of light waves: "... the light waves emerging from the light source
are not a long continuation of harmonic waves, but rather a series of waves of
shorter duration. The reason why light from a light source appears
steadily bright, on the other hand, is because these short impulsive waves appear
one after the other and are superposed incessantly. The energy of a light wave
emitted from an excited atom is a constant (ħω), and the corresponding
light waves may be considered a damped oscillation, with the amplitude
decreasing in time..."
I like that description, but it does have a problem: if the
amplitude is decreasing in time, the light wave ought to be completely
extinguished after a certain time period. The constant energy would not
really be constant! So, instead of saying the light waves are damped
oscillations, I would say the motion of an electron in an atom as it emits light is a damped oscillation.
You can imagine a weight hanging on a spring as an
analogy. Start it oscillating up and down, and it will sooner or later
stop oscillating. That's what a damped oscillation is. It is
interesting (and useful) that for not-too-large initial amplitudes, the
frequency of the spring-and-weight system will not decrease. It remains a
single frequency wave (sine wave), decreasing in amplitude. The same is true of
a pendulum undergoing small oscillations, and this is why a pendulum can be
used as a timing device: Its frequency (and therefore its period, or time of a
complete oscillation) is constant. Approximately. Also, it does lose energy, so
it needs a slight kick occasionally to keep it going.
In the quantum mechanical world, however, energy (E) is essentially the same thing as frequency (ω). They are related by E = ħω, where ħ is Planck's constant. So here is the conundrum: To decrease the energy, doesn't the frequency of the light wave need to decrease, not just the amplitude, since a constant frequency means a constant energy?
The answer is that the energy in a beam of light is dependent on both the frequency and the amplitude of the beam. A greater amplitude can be interpreted as a greater number of photons. This means a more intense beam, which means more energy per surface area. One way to achieve this is merely by focusing the beam, as when you use a magnifying glass to focus sunlight. The original beam has a diameter or width equal to that of the magnifying glass. The focused beam brings all those photons together in a very small area, almost just a point of light. Which is enough concentrated energy to ignite a dry leaf or a piece of paper, or burn your skin.
In the quantum mechanical world, however, energy (E) is essentially the same thing as frequency (ω). They are related by E = ħω, where ħ is Planck's constant. So here is the conundrum: To decrease the energy, doesn't the frequency of the light wave need to decrease, not just the amplitude, since a constant frequency means a constant energy?
The answer is that the energy in a beam of light is dependent on both the frequency and the amplitude of the beam. A greater amplitude can be interpreted as a greater number of photons. This means a more intense beam, which means more energy per surface area. One way to achieve this is merely by focusing the beam, as when you use a magnifying glass to focus sunlight. The original beam has a diameter or width equal to that of the magnifying glass. The focused beam brings all those photons together in a very small area, almost just a point of light. Which is enough concentrated energy to ignite a dry leaf or a piece of paper, or burn your skin.
This kind of increase of amplitude by focusing takes the
given beam of light and increases its energy-per-area. The total energy in the beam is not
increased. Now consider a beam whose
energy you can increase at the source. You
can think of doing this with a dimmer switch on an incandescent light bulb, or
you can imagine doing it with a beam of red laser light from a helium-neon
laser with some kind of brightness control on it. This would be controlling the number of
photons leaving the source.
What about controlling the frequency of photons leaving the
source? This is the other way of increasing
or decreasing the energy in the beam at its source. And a dimmer switch on an incandescent bulb
is one way of doing this. The initial dim
red glow from the bulb has fewer photons and each photon has a lower energy
than when the bulb is glowing brightly.
If you could tune a red laser’s frequency up into the blue part of the
visible spectrum, you could increase the energy in the beam without increasing
the number of photons. There are such
things as tunable lasers.
Next time: incandescent bulbs and Planck’s discovery in 1900 of
the quantum harmonic oscillator relation E = ħω, and Einstein’s discovery of photons in 1905 by applying Planck’s
relation to light as a way of explaining the photoelectric effect.
