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Basic principles of the photon model (2013)

Page history last edited by Joe Redish 6 years, 11 months ago

Working Content > Three models of light > The photon model

 

Prerequisites:

 

In the 1700's two models of light competed fiercely: Newton's particle model, which seemed to explain ray optics -- light, shadow, the properties of lenses and mirrors, and Huygens' wave model, which was harder to apply but also worked. In the early 1800's the work of Young, Fresnel, and others made a much more convincing case for the wave model -- it described complex interference phenomena in exquisite detail. And in the later 1800's Maxwell and Hertz made powerful theoretical and experimental demonstrations that light had to correspond to electromagnetic oscillations.

 

This sounds as if light had been figured out once and for all. But as often happens, nature had some more surprises for us. The problems arose when people began to consider how light interacted with matter. Since light was an electromagnetic oscillation that was produced by moving electric charges, and since matter was made up of complex combinations of electric charges, the internal electric motions of matter ought to be able to exchange energy with light.

 

The thermodynamics of light: Equipartition and the black-body radiation puzzle

Once researchers were convinced that they understood what light was, and how it might be absorbed and emitted by matter, it was natural to consider how light played into thermodynamics and statistical physics. Suppose you took a box made of matter and filled it with a gas and let it come to thermodynamic equilibrium. The box and the gas would exchange energy at random and eventually come to the same temperature. It was understood that this meant that the gas molecules would have a certain average speed and that the distribution of speeds would be governed by the Maxwell-Boltzmann distribution (A Boltzmann factor, weighted by the energy of the molecule: Ee-E/kT.) This has proved to work extremely well.

 

Now suppose instead you took a box made of matter and emptied everything out -- "filled the box with vacuum". The matter in the walls contain vibrating charges so they should emit electromagnetic radiation into the box. Eventually, the electromagnetic radiation should exchange energy with the matter of the walls and come to a thermodynamic equilibrium -- a common temperature. The question asked was: What does it mean for a box of light to have a particular temperature? Electromagnetic radiation can be made up of lots of different frequencies. What would be the distribution of electromagnetic energy in the box as a function of frequency?

 

This may sound like a silly question, but if you heat the box up until it starts to glow (to a few thousand degrees Kelvin), it's clear that it emits light. If you now cut a small hole in the box and let light come out from inside, you could take the light coming out and put it through a prism (or diffraction grating) to get the spectrum. If the light comes into thermal equilibrium with the matter, the spectrum should represent the distribution of the wavelengths of light "when it has a particular temperature" and should be independent of the material the box was made of. This proved to be true. (A hole in a box is referred to as a "black-body" since it is a perfect absorber: any light that falls on it will be absorbed.) The natural assumption was made that there would be an equipartition of energy -- and equal amount of energy (on the average) in each degree of freedom. The result is a simple function of the wavelength and the scale of the distribution is determined by the temperature.

 

The result of the classical theory (Rayleigh-Jeans law) is shown for 5000 K in the figure at the right along with the observational results at three different temperatures.

 

It's clear that the classical result is terrible. It agrees for very long wavelengths, but the experimental curves cut of for short wavelengths. The experimental curves have a peak that moves to shorter wavelengths as the temperature goes up. What's going on?


Source: Wikimedia Commons

Planck's proposal: Quantization

Researchers struggled with this problem. Some hypothesized that ordinary matter could not confine short wavelength radiation in a box and so there were necessarily leaks. Models of the leakage didn't do too badly. But the turn of the century brought a hypothesis that wound up transforming our view of matter entirely (and having powerful implications for the technology of the 20th century).

 

Planck did some playing with the theoretical thermodynamics of electromagnetism. He began with the relation between the entropy, the internal energy, and the temperature. In one form, the result for classical electromagnetism was linear in the energy: E. Planck tried adding a quadratic term: E +aE2. Using some other results of classical physics he showed that his "a" had to be proportional to frequency. After some transformations he was able to construct a function for the energy distribution that looked right and, adjusting his single constant, a, was able to fit the curves for all observations. His result was that the density of energy in light at thermodynamic equilibrium depended on frequency and temperature in a more complex way then just a Boltzmann factor.  It looked like this:

 

where kB is Boltzmann's constant, f is the frequency, and h is a new constant -- Planck's constant. Note that it has units of energy times time, since kBT has units of energy and f has units of 1/time. It sort of looks like a Boltzmann factor if you dropped the "-1" in the denominator. Then it would reduce to the classical law.

 

Interestingly enough, Planck did not stop there but tried to figure out what this new structure might mean physically. It's a complex story, but in the end, Planck reasoned that the walls of the box could only absorb or emit light in packets of particular sizes. (For details for the technically inclined, I recommend W. H. Cropper's book, The Quantum Physicists and an Introduction to Their Physics.) This was reasonable, given our understanding of resonance. Various bits of connected matter can oscillate at various natural frequencies and they preferentially absorb and emit energies at those frequencies.

 

The next step: Einstein's photons

Einstein made the next big leap in 1905: he suggested that the limitation to absorption and emission didn't reside in the matter but in the light itself. Interestingly, Einstein also began with thermodynamics. Planck's curve yielded a function for the entropy of the electromagnetic energy. Einstein played with that and showed that it could be rearranged to look like the entropy for an ideal gas of particles if you took the number of particles corresponding to a frequency of light, f, as being

 

N = (amount of energy in frequency f ) / (hf )

 

where h was Planck's new constant. This could be rewritten

 

energy in frequency f = N x (hf )

 

This is easy to interpret: the light having frequency f comes in packets that have energy proportional to f. Using Maxwell's theory of electromagnetic waves he was able to come up with what is the fundamental principle of the photon model of light:

 

  • Energy in electromagnetic waves comes in packets (photons) that have energy E = hf and momentum p = h/λ.
  • Only whole photons can be emitted or absorbed by matter.

 

Einstein would go on to make predictions about how high energy photons (ultraviolet) might knock electrons out of a metal, the photoelectric effect. These predictions seemed bizarre to many physicists of the day, and one, Albert Michelson, decided to test it out carefully. His 1916 paper confirmed all of Einstein's surprising predictions and Einstein was awarded the 1921 Nobel prize for his photon model (not for his relativity, also published in 1905).

 

This was rather an extraordinary leap as Einstein was using contradictory ideas. A wave having a well defined wavelength goes on forever. It is not restricted in space. The idea of a "packet" seems localized. But the packet's energy and momentum -- something seeming very "particle-like" -- is determined by the frequency and the wavelength -- something seeming very "wave-like".

 

The photon model requires a kind of "suspension of disbelief" of the "blind men and the elephant" sort. We treat light as packets and use wave properties when we need and particle properties when we need. The critical place we use particle properties is in emission and absorption where the model tells us how energy conservation works.

 

Where the photon model stands today

Einstein's photon model gave critical hints that led the way to the development of the quantum theory, and his relativity work provided the basis for the creation of a theory of matter that encompasses both electrically charged matter and electromagnetic radiation: quantum field theory. This theory is our current overarching theoretical model for the behavior of all of everyday matter. (Sub-nuclear processes require an extension of quantum field theory: what is currently called the Standard Model of particle physics.) This theory is a grand unified structure that in principle described everything about all matter we typically encounter. In practice, you can only calculate the simplest things, but it provides a framework for understanding and for modeling more complex systems. For the most basic systems, it allows us to calculate properties of systems, say the Hydrogen atom or the Helium molecule, to extraordinary accuracy (12 significant figures). And when we get to photons, it provides the structure that allows us to figure out what some of the interesting and surprising properties lie;  Such as:

 

  • Photons don't interfere with each other, they interfere with themselves, appearing to know about all possible paths they might take.
  • The number of photons is not necessarily a fixed number. Photons can exist in states that are mixed -- part of the time there is one number of photons, part of the time there is a different number.
  • Photons don’t interact with each other: they just pass right through, like small pulses propagating on an elastic string.* 

 

Right now, the complex properties of photons predicted by quantum field theory are being utilized in biology to probe small systems in extraordinary ways, such as two-photon confocal microscopy.  There are even possibilities that such basic properties as photosynthesis rely on some of these subtleties. It is likely that the next decade or so that biology will find much value in our sophisticated understanding of the complex nature of light.

 

 

* This is the photon version of the statement that the electric field due to multiple sources is just the sum of the fields from the individual sources. The presence of additional fields doesn’t change what electric field a source produces. While these are very good approximations under most circumstances, there are specialized circumstances where there are interesting corrections. An electric field in matter polarizes the matter and this polarization can effect how a second electric field is modified by the matter (dielectric constant). The study of this is called non-linear optics. Very high frequency photons (gamma rays) polarize the vacuum slightly, pulling apart electron and positron pairs that travel with them. Two gamma rays can interact (very weakly) through the electric forces between these bits of polarized vacuum.

 

Joe Redish 5/2/12

 

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