In classical physics, matter is composed of tiny particles called atoms. It you assume this, it explain a lot of chemical and physical properties, the laws of chemical combination, the behavior of gases, and so on, On the other hand, light in classical physics, is a phenomenon of continuous electromagnetic waves. How do we know that light is a wave phenomenon? We know because we can observe constructive and destructive interference of light waves in Thomas Young's two-slit experiment, which we actually performed, to see for ourselves that light indeed does travel as waves.
So now we're going to continue the story into the 20th century, where two revolutionary thinkers, Max Planck and Albert Einstein, began to question the 19th century synthesis and introduce quantum ideas into physics. Why would they do that? What was wrong with classical physics?
The problem was that there were just a few leftover experimental puzzles about light and matter. To solve them, they needed to change the entire structure of physics. The first puzzle was the problem of thermal radiation. When we heat up a solid object, it gives radiation in the form of long wavelength light. An everyday example of that is a light bulb. When we run an electric current through the tungsten filament of a light bulb, that filament gets very hot, so that it emits light.
How much radiation is emitted? What frequencies are emitted? They might be in the visible range, that narrow range of frequencies we can perceive without eyes. There could be lower frequency light than that, the infrared light. There could be higher frequency light, the ultraviolet light.
When we analyze the thermal radiation from an object, the simplest case actually turns out to be when the object itself is black in color, whats called a blackbody. The reason is kind of paradoxical and surprising, since it turns out the more efficient an object is at absorbing light, the more efficient it is at emitting thermal radiation when you heat it up. So a blackbody would be a perfect radiator of thermal radiation.
It turns out that this thermal radiation, called blackbody radiation, is quite interesting. At a given temperature, all blackbodies, no matter what they're made of, radiate in exactly the same way. So at the end of the 19th century, many scientists tried to apply the classical theory of heat, to explain blackbody radiation. It turns out that this works pretty well for explaining the low frequency light, the infrared radiation that comes off of a blackbody.
Yet the classical theory predicts way too much high frequency radiation, way too much ultraviolet radiation. So back to the light bulb. We run the electric current through the tungsten filament, which gets very hot, more than 2000 degrees Celsius in fact. What kind of light comes out? Well mostly its infrared light that is at too low a frequency for us to see.
Yet there's also a good deal of visible light, and there's almost no ultraviolet light from an incandescent light bulb. This is why you don't need to wear sunscreen around them, since they don't emit enough UV light to give you a sunburn. So lets try to apply the classical theory of heat to this. We then find out that UV light coming from the bulb, should in fact be much more intense than the visible light, which should in turn be much more intense than the infrared light. So we get it exactly wrong. There should be immense amounts of UV light coming out of the bulb.
This prediction is so radically wrong, that it came to be called the ultraviolet catastrophe. It your theory cannot explain how a light bulb works, you need a better theory! That's where Max Planck 1858-1947 came in. At the time of our story, Planck is a German physicist at age 42, quite old for a revolutionary in fact, and he's been working on the blackbody problem for some time. Planck realizes that this is a really important problem, which will teach him something very fundamental
Why does he think this? It's because all blackbodies, whatever they are made of, radiate in exactly the same way. So if he can figure out why that is, he'll learn something fundamental about the relationship of heat and light. So he's been working on it for a long time, and finally in desperation, by 1900, he proposes a strange hypothesis. Planck says that suppose light is not emitted and absorbed continuously, but only in discrete amounts? These discrete amounts, he called light quanta, which is the quantum hypothesis, the quantum in quantum mechanics. The energy in one light quantum, is related to the frequency of the light. The equation is:
E = hf
where h is a new fundamental constant of nature, later called Planck's constant. This has a tiny value, 6.6 times ten to the -34th joule seconds. That's some 33 zeros after the decimal point, before we got to the first six. So we multiply Planck's constant with the frequency of the light, and that tells us the size of the quantum of light energy.
Now since Planck's constant is so very small, light quanta are very small as well. They're very tiny, which is why energy appears to be so continuous. It comes in discrete units, yet they are so very small. The light bulb, under Planck's hypothesis, would emit something like ten to the 20th power, a hundred million trillion light quanta every second.
How does the light quanta hypothesis, explain blackbody radiation? Well, higher frequency light has quanta that have higher energy. If an object is going to emit a quantum of light, it has to get together in one place, enough energy to do so. So it's just harder for an object to emit an ultraviolet photon. The UV photons have high frequency, higher energy, so it's just harder to emit them. So Planck's hypothesis predicts no ultraviolet catastrophe. The quantum hypothesis in fact, exactly accounts for all the details of blackbody radiation.
Now Planck's idea is pretty radical indeed. A continuous wave should carry any amount of energy, since the amount of that energy is just related to how strong the wave is, and you can make a continuous wave of any strength. Planck himself recognizes how radical his quantum hypothesis was. On the very day that we worked out that his quantum hypothesis solved the blackbody radiation problem, he said to his son:
"Today, I have made a discovery as important as Newton's discovery."
That's quite a statement to make. As we'll see, it's entirely justified.
That brings us to the second puzzle, the problem of the photoelectric effect. Consider the following experiment. Suppose we take a metal surface and we polish it to remove any layer of oxides off it, and put it in a vacuum. Now we shine light onto the polished metal surface. What happens? It turns out that electrons, tiny electrically charged particles, constituents of atoms, are emitted from the surface. So this is why it's called the photoelectric effect, "photo" for the light, and "electric" because you emit electrons.
This was discovered by the late 1800s, and very soon, scientists had done experiments on it, making some curious observations about the nature of the photoelectric effect. They could measure both the number of electron emitted by the surface, and the energy of the electron, how fast they were going. They discovered that the energy of the electrons, how fast they were going, was not dependent at all on the intensity of the light that you shined. If you shine a brighter light, you get more electrons coming out of the surface, but every electron is moving at the same speed as previously. This is quite amazing.
If you imagine that the electrons are sort of being shaken out of the metal by the oscillations of the light, then if the oscillations are stronger, surely the electrons should be more shaken by the light? Yet even if you shined a light of a thousand times greater intensity on the metal surface, you won't have more energy in the electrons, you'll just have more electrons. That's very strange.
What does determine the energy of the electron then? It turns out that's dependent on the frequency of the light. If the frequency is too low in fact, you get no photoelectrons at all. Infrared light may not produce any photoelectrons from your metal surface. At higher frequency, you get electrons out, and at even high frequency, you get faster, higher energy electrons out. So the observation is that the energy of the electrons does not depend on light intensity, but only on the frequency of the light. That seems very puzzling indeed.
So along comes Albert Einstein, a very romantic figure in the history of science, being justly famous for all kinds of discoveries. This is the year of 1905, Einstein's year of miracles when he discovers Relativity theory, demonstrates E=mc², among all kinds of other things. Yet he also provides a new theory to explain the photoelectric effect.
By day in 1905, he's a technical expert third class in the Swiss patent office. By night, he's a physics revolutionary! Einstein is aware of Planck's work, so he realizes that Planck's quantum hypothesis amounts to thinking of light as a stream of discrete particles, later called photons. So he realized, somehow that all meant light was made of photons. So the energy of one photon, is determined by Planck's formula:
E=hf
That's sort of a strange picture, so how does it explain the photoelectric effect? Einstein says that every photoelectron, each and every one that comes out of the metal, gets its energy from a single photon striking the metal. Now some of the energy of the photon is used up in prying the electron loose from the metal, since the electrons are bound to the metal and it just takes a little energy to pry them loose. Yet whatever energy is left over afterward, the electron gets.
So photons in bright or dim light of a given color, have the same energy. Yet if you change the color, the frequency of the light, then the photons have different energies. The same intensity of light with higher frequencies, should have photons with higher energy, so the photoelectrons that are produced will also have a higher energy. So Einstein is then able to give a detailed account of the photoelectric effect in term of electrons.
Now this is a very radical idea, and in fact, several experimental physicists were very skeptical. So they went off and did the experiments very carefully. They measured all kinds of things about the photoelectric effect, and they found the experimental results agreed exactly with Einstein's theory in every detail. So in fact, he won the Nobel Prize in 1921 for the photoelectric effect, not for Relativity Theory or any of the many other things he did, the photoelectric effect was that important.
So with Planck's explanation of blackbody radiation, and Einstein's explanation of the photoelectric effect, we sort of have ourselves in a weird situation for the theory of light. The two-slit experiment we did last lecture, seems to prove that light is a continuous wave. It seems to be a conclusive experiment. On the other hand, when we do the photoelectric effect, that seems to prove that light is a stream of discrete photons. So there's something really funny going on.
Yet there's not just something funny about light, which brings us to the third puzzle, the problem of heat capacities. This is the heat energy needed to raise an object's temperature by one degree Celsius. How much energy do we have to add to the object, so that its temperature will go up by one degree Celsius? If we have an object with a very high heat capacity, then it's hard to heat or cool, since it takes a lot of energy to put in to make it hotter, and a lot taken out to make it cooler.
On the other hand, if the object has a very low heat capacity, it's relatively easy to heat it or cool it. You don't have to add much energy to heat it up, and you don't have to extract much energy to cool it down. So heat capacity is a very significant quantity in the theory of heat, which was one of the aspects of classical 19th century physics.
There was a long-standing puzzle in the 19th century about pure solids, a solid made of one type of atom, of one particular element. So to explain this puzzle, we'll introduce a couple of examples. Platinum metal and diamond, which is made in carbon. We figure as long as we're doing thought experiments, we might just as well use the most expensive materials.
So classical heat theory of the 19th century, predicts that all pure solids should have about the same heat capacity, for the same number of atoms. This works where at a given temperature, all atoms should vibrate with the same energy on average. Heat energy is the energy of microscopic, random vibrations of the atoms. Platinum atoms are very heavy, so vibrate very slowly at a given temperature. Carbon atoms are much lighter atoms, where in diamond are held together by very stiff covalent bonds. So they vibrate much faster at a given temperature.
Yet the amount of energy in a platinum atoms or carbon atom, should be about the same at a given temperature. That means that the prediction is all substances made of one kind of atom, should have about the same heat capacity, for a given number of atoms.
So what do the experiments show? They show it's not quite true. So lets consider doing this experiment, at three different temperatures. We can imagine doing it at a very high temperature of 1000 degrees Celsius, hotter than any oven, then at an intermediate temperature, such as room temperature of 20 degrees Celsius, or at a very cold temperature like -200 degrees Celsius, about that of liquid Nitrogen.
So we can measure the heat capacities of materials at different temperatures, and when we do it at the high value, we find that both platinum and diamond behave about as expected. The classical theory actually predicts that all solid objects should have a heat capacity of about 25 in joule degrees Celsius per mole. So both metals have heat capacities of about that amount. Maybe they're not exactly right, but they are pretty close to that number.
Now we imagine the experiment at room temperature and find platinum to behave roughly as expected, yet diamond's heat capacity is much too low. It takes much too little energy to change the temperature of diamond.
Again, we find, if we do the experiment at liquid Nitrogen temperatures, that both metals have unexpectedly low heat capacities. That of diamond is very tiny. So this doesn't agree at all with classical physics.
Code:
Temp 1000 20 -200
Pt 25 24 16
C 22 4 0.1
Along comes Einstein again, and does something really radical, strange, and amazing. He applies quantum ideas, not to light, but to the vibrations of atoms. He says that suppose atomic vibration energy comes in discrete quanta of size E=hf. In other words, the Planck formula applies not just to energy and light waves, but also to the energy in atomic vibrations. This is the first application of quantum physics to matter. There's no light anywhere in the problem, so it's irrelevant to measuring the heat capacity of these materials.
This is a direct challenge to Newton's mechanics, the most successful theory of physics that had stood for more than 200 years for the gold standard of accurate and far-reaching theories. It's a direct challenge to Newton's mechanics, because a vibration atom in this mechanics, can have any energy whatsoever.
So what does Einstein say? Well consider these pure solids. At very high temperature there's a lot of heat energy for all the atoms to vibrate just as expected. Yet suppose we consider the same solid at low temperature. Then there's not enough heat energy for all the atoms to have at least one quantum of energy, because the atoms can have zero, one or two quanta, and so on.
So if there's not enough for everyone to have one quantum, that means a lot of the atoms will not be vibrating at all. So Einstein will say if a lot aren't even vibrating, the heat capacity of the material will be lower than expected. So at high temperatures, they should behave about as expected, as there is plenty of heat energy. At low temperatures, a lot of the atoms are fixed in place without any quanta of vibration energy and only a few are vibrating, so the heat capacity is lower.
So what's the difference between high and low temperatures? What counts as a low temperature for a given substance? It depends on the substance, on the speed of vibration of the substance, since it depends on how big the quantum of energy is, therefor the frequency of vibration. So for diamond, you have very high frequency vibrations. That means that the quantum of vibration energy is relatively high. That means that both 200 degrees below zero, and 20 degrees above zero, count as low temperatures for diamond. Even at room temperature, most atoms id diamond are not vibrating, donot have one quantum of vibration.
For platinum, on the other hand, those atoms are much more massive. The vibrations are much slower, and so room temperature of 20 degrees Celsius counts as high for the platinum vibrations. There is plenty of energy to have energy quanta for all the atoms. Yet 200 degrees below zero, that's low temperature, even for platinum.
Einstein's idea, with a few refinements, actually precisely explains the observed heat capacities of pure solids of all kinds at all temperatures. It's a remarkable agreement with experiment.
OK, what's happened in the first decade of the 20th century? Max Planck and Albert Einstein have introduced quantum ideas to explain a series of otherwise baffling observations, blackbody radiation in the case of Planck, the photoelectric effect and later the heat capacity of pure solids explained by Einstein.
All three of these were puzzles of classical physics, yet in each case, Planck and Einstein find that energy comes in discrete units, discrete quanta, the quantum in quantum physics. The size of the quantum is E=hf, where h is the same Plank's constant for everything, including radiation, the photoelectric effect, and the heat capacities of pure solids.
This is one of those really interesting moments in the history of science, where we're on the verge of a new point of view. You can tell there's a great new principle of physics in the offing. A principle that is universal, because the quantum hypothesis appears to apply both to light and to matter, yet we don't quite see the details yet, or the whole idea. We just see a little bit.
Now the universal quantum principle that applies to both light and matter, is a direct challenge to the classical physics that prevailed at the end of the 19th century, because in classical physics, energy can come in any amount, it's a continuous quantity. So this means that Plank's and Einstein's discoveries tell us that the universe is discrete in ways we never expected. That the old resolution of the universe into matter made of discrete things and light made of continuous, wavy stuff, that old distinction must not be quite right.
Well, next time of course, we're going to probe the other half of the mystery. Not only is nature discrete in ways we never expected, but it will turn out that nature is also continuous in ways that we never expected. So far, we've seen that light, electromagnetic waves, can also act as a stream of particles, of photons. Yet next time, we'll find out that particles of matter that we find out were completely discrete, can also act as waves. So every sort of stuff has a little bit of things, and every sort of thing has a little bit of stuff in it.
