Quantum Mechanics Mini-Lesson

Shawn Carlson
Executive Director, Society for Amateur Scientists

Recently the following letter arrived at SAS:

Dr. Shawn,

I’m very confused about the properties of photons. When I used de Broglie’s equation, I found that a photon would have a mass inversely proportional to its wavelength, which I can’t understand because I thought photons were not supposed to have mass. What’s more, when I plug the energy of a particular photon into E=mc², I get the same mass as in the first equation. Why is it that multiple sources have said that photons have no mass, yet both these equations disagree? I know photons are supposed to have wave-particle duality; is it related to this? Thanks for answering this question for me!

Eric Johnston

Here is my reply:

Well, Eric, it goes like this.

Energy is the ability to do work, which means the ability to apply a force to a mass over a distance. (I tell my younger students that something has work if it has the ability to "knock something else around." This is not as rigorous a definition as the "work" definition, but it's a bit easier to understand physically.) Einstein discovered that everything that has energy can also be thought of as having mass. Thus, the greater something's kinetic energy, the harder it is to accelerate.

Newton would have been astonished at that result. Why should it matter how fast something is moving past you? A given force should produce the same acceleration. But Einstein realized that the laws of nature conspired to make it impossible to accelerate things to the speed of light, and that means that as you get closer to the speed of light it gets harder and harder to go faster. Since we define the mass of an object in terms of the acceleration it experiences when it gets pushed on by a given force, this cosmic speed limit requires an object's mass to increase as it gets accelerated closer and closer to the speed of light.

Physicists might not have been too bothered by this if the explanation were merely mathematical. There's a cosmic speed limit and, because of the way we define mass, mass increases as you get closer to this speed limit. The greater the kinetic energy, the greater the mass. Sure. So what?

But Einstein discovered that if the laws of nature are the same in all reference frames (meaning there is no such thing as absolute motion, and you can't do an experiment inside a space ship that will tell you how fast you are traveling), then the relationship between kinetic energy (in fact, all energy) and mass has real consequences. But before we can consider those consequences, you need understand a few very important facts about photons.

First, a photon doesn't have "mass" in our common sense definition of that term. A photon can't be stopped and weighed. It exists entirely as a self-propagating bundle of electromagnetic fields. As the electric field collapses, it generates a magnetic field that's in a slightly different position. The electric field disappears when it transfers all its energy to the growing magnetic field. When that transfer has taken place, there is no longer anything to build the magnetic field. It then begins to collapse and to create an electric field in a slightly different position. When the cycle repeats, this bundle of electromagnetic energy has moved a bit and it turns out that the distance moved divided by the time it took for the cycle to take place is always exactly the same. This is the speed of light. The laws of physics determine where the magnetic field appears when the electric field collapses. You can't keep the electric and magnetic fields oscillating in place, because that violates the laws of physics. You can transfer the energy out of the field and into something else--that is, you and absorb and destroy a photon--but you can't stop it.

That means photons can never be at rest. They have no existence at rest, and so they have no "mass" or any other property at rest. We physicists say that a photon has no "rest mass."

And yet a photon does have mass, because the mass-energy relationship still holds. For instance, if a photon has greater than about 2 MeV of energy, I can turn it into two electrons (well, an electron and a positron). And I can reverse the process as well. I can bang an electron and positron together and have them disappear and be replaced by a photon that contains all the energy--mass energy, potential energy, and kinetic energy-- of the original particles. Furthermore, photons are affected by gravitational fields, and you can calculate the acceleration they experience (not change in speed, but change in direction), as well as the change in energy they undergo as they move through a gravitational field, by using E/c² as their mass.

So photons do have mass through the mass-energy relations. However, they have no rest mass, because they can not be stopped without being destroyed. I hope that answers your question.

Now, and this is the really cool thing about the speed of light, it doesn't matter what the wavelength is! If a photon has a long wavelength, it moves farther each cycle, but the cycles take longer. It all works out to ensure that long wavelength photons move at exactly the same speed as short wavelength photons. If you understand that, then the step to relativity is a short one. Why? Well, suppose you and I are sitting on a beach at the Kennedy Space Center looking at Mars, the red planet. We say goodbye, and you then climb into a rocket and get launched on your mission to become the first person to go to Mars. As you rush toward the planet, you would pass through the electromagnetic fields faster than I would while watching your progress from earth. This means that you would see the photons coming from Mars as having shorter wavelengths. But, since you must experience the same laws of physics that I do, they would still have to look like perfectly good photons. They have to look like any other photon you've ever seen that had that particular wavelength.

In particular, those photons have to move at the speed of light for you just as they do for me. Everything in your space ship is moving at a different speed for you than for me. If you throw a ball, it's moving at one speed relative to me and at a different speed relative to you. That's true for everything except the photons. We see them as having different wavelengths and different frequencies, but you and I will both see them as moving at exactly the same speed. Photons are very special things. And, it turns out that only by using photons as measuring tools can we define space and time in a completely self-consistent way. And since photons are special things, their special natures are reflected in the behavior of space and time itself.

Length contraction, time dilation, stellar aberration, mass and energy relationship--it all stems from the special properties of photons and the fact that we need to use them to define space and time. If you can wrap you mind around that, you will find special relativity will become much easier to understand.

I hope you find this useful.

Yours for great science,

Dr. Shawn