Impulsive waves superposed incessantly part I

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.

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.