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#YourDailyEquation with Brian Greene offers brief and breezy discussions of the most pivotal equations of the ages. Even if your math is a bit rusty, these accessible and exciting stories of nature and numbers will allow you to see the universe in a new way.
The series includes live Q+As that explore many of the big questions that have occupied some of the greatest thinkers of our age and yielded some of the deepest insights into the nature of reality.
Episode 08 #YourDailyEquation: Einstein won the Nobel Prize for explaining a puzzle to do with light and electrons known as the photoelectric effect. In today’s Your Daily Equation, Brian discussed the key formula in this explanation, an insight that helped launch the quantum revolution.
Brian Greene is a professor of physics and mathematics at Columbia University, and is recognized for a number of groundbreaking discoveries in his field of superstring theory. His books, The Elegant Universe, The Fabric of the Cosmos, and The Hidden Reality, have collectively spent 65 weeks on The New York Times bestseller list.Read More
Hi everyone. Welcome to Your Daily Equation. Today I’m going to focus on one of the key equations that leads us to quantum physics, quantum mechanics. This is an equation that Albert Einstein came up with, and he came up with it in trying to unravel a puzzle that had been around for, I don’t know, probably a couple of decades. We need to cast our minds back again to the year 1905, the same year that Einstein came up with the special theory of relativity. But now he is thinking about a different puzzle, and the puzzle has to do with the photoelectric effect. What is that? Well, I think it was in the late 1800s, somebody will correct my history of science if I have this wrong, and I think it was Heinrich Hertz who realized that if you shine a light on a metallic surface in the right way, then the light can actually cause electrons to be emitted from that surface.
I guess I can probably do even a little show and tell. I have a lot of junk around here. You wouldn’t think so based on what you see behind me. It looks nice and neat, but I throw everything on this side of the camera so you can’t see it. But I think I do. Yes I do. I have a flashlight here. I just need something metallic that I can use. The radon detector? No. I guess I can use this, I don’t know, the back of a measuring device, here a tape measure. Imagine if this is my metallic surface, and I’m shining this flashlight on the surface. The idea is that if I do this in the right way, in the right experimental setup, then the light from the source can cause electrons from the surface to be ejected outward. This in itself is not a particular puzzle because, after all, light is an electromagnetic wave.
An idea that we will also discuss subsequent to the discussion today, one of our other discussions of Maxwell’s equations. But light carries energy, and so energy slams into the metallic surface. Electrons are loosely bound to that surface, and the energy from the wave can knock the electrons free. Not particularly puzzling at all. But what is puzzling is, when you look at the details of the data, because you would think, or at least most people would think, that the kinetic energy, the energy that the electrons have, their speed when they leave the surface, should be determined by the intensity of the light. Right? After all, light is this wave and the intensity of a wave, the intensity of an ocean wave, is given by its amplitude, the ups and downs of the wave.
Similarly, the ups and downs of the electric and magnetic fields that comprise the electromagnetic wave that is light, the ups and downs, the amplitude, that should determine the energy of the light, and that should determine the energy of the electrons that are ejected. But when you look at the data, that’s not at all the case. You know what determines the kinetic energy of the electrons that are not freed from the surface? The color of the light. It’s frequency. That is, how quickly it oscillates up and down determines at least the maximum kinetic energy of the ejected electrons. The intensity of the light does determine something else. It determines the number of electrons that are ejected from the surface, but their energy comes from the color of the light.
This was a puzzle that Albert Einstein starts to think about, and he ultimately comes up with a solution, and that solution … I can actually show you the paper right here. This is his 1905 paper on the photoelectric effect. 1905 is often described as Einstein’s miracle year. He writes a handful of papers, any two or three of which could have themselves received the Nobel Prize, but it’s actually this paper, not his paper on special relativity, not his paper on E=mc2, it’s this paper for which you receives the 1921 Nobel Prize in Physics. It is in this paper that he unravels this paradox of the photoelectric effect, and let me just describe to you what he finds.
Let me just bring up my iPad here. Good. The picture that we have, at least that we’re trying to figure out here, imagine that this is my metallic surface, and let me just describe light as a wave coming in. This is the usual picture. You’ve got this electromagnetic wave slamming into the surface, and you’ve got say, let’s say little electrons in here, and these electrons are flying out, and surprisingly their energy is determined by the color of the light. How does Einstein explain this? Well, Einstein makes use of a different image of light, a different picture, a different description, of what a beam of light actually is. He actually goes back to an idea that we can trace to Isaac Newton himself, where Newton thought that light was actually made of a torrent of particles. We call those particles of light now photons, so let me use that language.
A torrent of photons as opposed to some kind of wave-like phenomenon. But that idea was dropped when people like Thomas and Maxwell showed, apparently, that light is an electromagnetic wave. But Einstein goes back to the old idea of light as a stream of particles. In fact, I can show you in this fancier version of the demonstration, now I’ve done an animation. You see that from the flashlight, that beam of light, Einstein says is actually a stream of particles. Now, how does this resolve the problem? Well, let me go back to this picture over here. Let me erase this idea of light as a wave, and in its place let me describe it as a collection of particles, each of which is flying down to the surface, and let me focus on one of them. This guy over here.
Imagine what’s going on when a photon hits the surface and ejects an electron is a collision between the photon and the electron, and that’s one-to-one collision is what ejects the electron. Clearly then, the energy of the ejected electron, the energy of the electron, will be determined by the energy of the photon that hits it. Now, Einstein says in order to match the data, that the energy of that photon must be proportional to the color of the light, which is the frequency of its oscillations. Indeed, he can go further and make that proportionality into an equality, which is today’s daily equation, by using a number called h, which is known as Planck’s constant, after Max Plank, and the equation therefore that he comes to is E = hv.
This idea of light as a collection of particles explains why it would be that the kinetic energy of the ejected electron would depend on the color of the light because the energy of each individual photon via this equation, is dependent upon the frequency of the light, dependent therefore on the color of the light. It can go even further. Why would it be that the number of these electrons that are ejected is dependent upon the intensity of the light? Well, now that’s quite obvious. The intensity of the light is nothing but the number of photons. Higher intensity, greater number of photons. Greater number of photons, greater number of collisions with electrons. Greater number of collisions, greater number of electrons that will be emitted.
That’s why the number of ejected electrons is determined by the intensity of the light, because the intensity is just the number of photons and the kinetic energy of each of those electrons, at least the maximum kinetic energy that any of them can have, is determined by the color of the light because the energy of each photon is proportional to the frequency of the light. So it’s a beautiful blending of wave-like ideas. I mean, frequency after all is a notion that has to do with a wave, and Einstein says, take that wave-like idea and blend it into a particle description of light. So it’s not quite going back to the Newtonian picture of particles of light, it’s not quite using the pure wave-like description of light that came to us from James Clerk Maxwell and previous analyses and experiment.
Einstein blends them together using a wave-like concept, the frequency of the light, but using it to define a quality of the particulate ingredients making up light, namely the energy of each individual photon. This really is a profound move toward the quantum mechanical description of energy and matter. These are ideas that we will take up further as we continue on in our description of the fundamental equations of quantum mechanics. But for today, that’s all I wanted to cover, this fantastically profound equation E = hv, introduced to explain the photoelectric effect, which launches the quantum revolution. That’s today’s equation in Your Daily Equation. Looking forward to carrying on this discussion next time. But for today, that’s all. Take care.