Sunday, February 28, 2010

Light Gains AND Loses Momentum At The Same Time

Well, it is only appropriate since we're dealing with quantum optics some time! :)

Just when you thought that the issue has been resolved and one side won, the same problem comes back into center stage. And in the latest twist of the story, it turns out to be a dead tie - BOTH sides are correct. That is, assuming that the latest theoretical explanation present is itself correct.

The question is about the momentum of light going through a dense medium with an index of refraction n.

It is well known than when light enters a material medium it slows down in proportion to the refractive index, n, of that medium. Minkowski and Abraham wanted to know how light's momentum changes as a result. Abraham calculated that the momentum of a single photon within the light is also reduced by a factor n, a result which agrees with our experience of everyday objects – as their speed drops, so too does their momentum. Indeed, a number of powerful arguments have been put forward over the years in support of this position. Prominent among these has been a simple proof based on Newton's first law of motion and Einstein's equivalence of mass and energy, which considers what happens when a single photon travels through a transparent block and transfers some of its momentum to the block, given that the motion of the system's centre of mass-energy must remain constant.

Minkowski's formulation, on the other hand, seems more natural from the point of view of quantum mechanics. As light slows down inside a medium its wavelength also decreases, but quantum mechanics tell us that shorter wavelengths are associated with higher energies, and therefore higher momenta. In fact, Minkowski's approach suggests that the momentum of a single photon of light increases by a factor n as it passes through a medium. This result can also be supported by strong theoretical arguments, among them one that considers what happens when an atom moving at some speed through a medium absorbs a photon and experiences an electronic transition.

Lately, it is thought that the Minkowski's version was pulling ahead, especially with the latest set of results. But hold on to your horses.

According to Barnett, however, both formulations are correct. He says that the one put forward by Abraham corresponds to a body's "kinetic momentum" – its mass multiplied by its velocity. Minkowski's momentum, on the other hand, is a body's "canonical momentum" – Planck's constant divided by its de Broglie wavelength. "These two formulations reflect the fact that in different situations momentum does different things," he adds. "In free space they coincide, but not when inside a medium."

Hum... I think I need to think a little bit more about this one.


1 comment:

Arjen Dijksman said...

The active feature of a photon's motion is its periodical "waving", not its velocity. If one defines the momentum as the power residing in its motion, one therefore has to consider the wavenumber of the photon, and not its velocity, as a measure of its momentum. My preference therefore goes to Minkowski's version.