 In this lecture, we will learn the following things. We will learn how the black body radiation spectrum was finally understood. We'll learn about the possibility that's implied by that solution, that energy may come in discrete units. We'll learn about a phenomenon known as the photoelectric effect, and we'll learn how Albert Einstein resolved the puzzle of the photoelectric effect. In the last lecture, we saw how the Rayleigh-Jean's power spectrum prediction utterly failed to model nature correctly. Given a black body heated to a certain temperature T, the Rayleigh-Jean's model predicted that more and more energy should be emitted in shorter and shorter wavelengths, leading to some kind of natural catastrophe merely heating up a body to a few thousand degrees Kelvin. However, matter heated to a temperature T simply does not radiate according to that prediction, because if it did, the effects would be catastrophic. The shorter the wavelength and the more damaging the electromagnetic radiation, the more of it would have been emitted from such a body as predicted by the Rayleigh-Jean's model. It simply did not comport with reality. This mismatch between reality and the prediction of classical physics has been called the ultraviolet catastrophe. Correspondingly, the mismatch between reality and the prediction of Wien's model is known as the infrared catastrophe. Now, historically, this problem was not considered threatening or really so important than anyone truly panicked, although at least one individual did take this problem extremely seriously, and that's Max Planck. You've got to admit though, ultraviolet catastrophe is a lovely and exciting name. I should note that the term ultraviolet catastrophe actually doesn't date to the exact period when the Rayleigh-Jean's prediction or Planck and his work were established, but it actually appears to date to much later, about 1911, and seems to have been coined by the physicist Paul Ehrenfest. Now in the last lecture we saw that there was a model by a man named Planck, Max Planck, that did seem to have gotten the right answer. So what was it that Max Planck did? Well, he started from a mathematical model of a perfect black body, a simple model, a cavity fully enclosed on all sides, except for a tiny hole in the cavity. An ideal black body, to remind you, is one that absorbs all incident radiation on it, and then it re-emits its own radiation with some spectrum. It's got perfect emissivity, so it maximally radiates given its other physical properties. Now once one hypothesizes that such a system exists, one then has to apply the laws of physics to predict or describe that emitted radiation spectrum, the amount of energy emitted, for instance per unit solid angle, per unit time, per unit wavelength, and per unit area, per unit many things, but your bottom line is you're attempting to predict how much of each wavelength interval of radiation is present in the emitted bulk of radiation. Now a cavity with a single small hole in it is actually a really good model for a perfect black body. If you shoot radiation at the hole, 100% of it, incident on the hole, will enter the cavity and be lost to the outside world. That radiation is absorbed by the cavity. Now it then enters the cavity and it begins bouncing around inside the cavity, striking the walls and therefore hitting the bits of matter that make up the walls of the cavity, and fundamentally as we've seen in physics too, matter is made from electric charges. Now as we also know, as these electric charges get struck by radiation, they're going to begin to gain kinetic energy, which will cause them to heat up the material surface of the cavity inside the cavity. A hotter object emits radiation in a different way than a cooler object. So again, the question we want to boil this down to is, what will that spectrum of emitted radiation due to the heating of the walls of the cavity from the incident radiation actually look like? Now we can boil the black body problem down to just a very simple collection of phenomena that we can conceptualize of using information from physics too. This is a very simple model of an electromagnetic wave, which would be what the radiation impinging on the surface of the cavity walls would look like. It's got an oscillating electric field and perpendicular to it, it's got an oscillating magnetic field, and it's traveling perpendicular to both of those fields. This wave then strikes a charge in the wall of the cavity, so for instance an electron. The electron feels the electric and magnetic fields of the wave, and it will respond to those by accelerating. This is what we learned in physics too. The wave, with its increasing and then decreasing electric field strength, for example, will cause an electron to accelerate more than less. It will oscillate. It will wave like a bit of matter in a rope that's wiggled, or in a chain that shook, or in a string that's plucked. The electron will oscillate. So radiation enters the cavity with any number of possible frequencies or wavelengths that can compose that incident radiation, and all of it is taken in by the cavity through the hole. The electric charges that make up the matter in the walls of the cavity will either scatter, they'll be knocked off of their parent atoms for instance, or maybe they'll wiggle in response to the electromagnetic wave that strikes them, and thus absorb some of the electromagnetic radiation as motion. Now absorbing an electromagnetic wave causes the charges to oscillate, and an oscillating electric charge is a source of an electromagnetic wave, so these newly oscillating electric charges can emit their own electromagnetic radiation. This is the source of the emission spectrum from the black body. So what will that re-radiated energy look like when it escapes the cavity? That radiation too will bounce around inside the cavity, but some of it will make it out of the hole. What will it look like, and how much of each frequency is found in its power spectrum? Well, recall that the Rayleigh-Jeans model, using a purely classical model of all of this system, mechanics and electromagnetism via Maxwell's equations, determined that the spectrum should look something like this, that the energy emitted per unit time, taking into account the surface area and the whole viewing solid angle of the black body, will basically go as the temperature of that body over the wavelength to the fourth of a particular wavelength of light that we're considering as part of the outgoing spectrum. But as we can see, as you decrease the wavelength, that is increase the frequency of the radiation, more and more and more power is emitted by the black body. Now a key assumption that lay underneath the building of the Rayleigh-Jeans model was that all frequencies are possible for oscillating charges. A charge stuck in an atom in the wall of this cavity model can oscillate at any frequency it likes. All frequencies are possible. And that led to the Rayleigh-Jeans model. Let's make a very simple model of a system where we can cause oscillations to occur in electric charges and then those oscillating electric charges in turn emit electromagnetic radiation, radio waves or light. That light then travels across a gap, striking another electric charge and setting it into oscillatory motion. To illustrate what I mean by this, imagine we have the ability to wiggle an electric charge over here at a transmitter site and watch a sympathetic wiggle over here at a receiver site when an electromagnetic wave from the transmitter reaches the receiver. To illustrate this, let me start oscillating the electric charge on the left. What you're seeing here is the full electric field around that charge as it changes in time as the charge moves in space. The changing electric field propagates out at the speed of light and causes an oscillatory pattern in space. Some places have strong electric fields pointing in one direction, some places have weak electric fields, some places have electric fields that point in the opposite direction. We can better see this by looking at the amplitude of the electromagnetic wave as a function of position away from the oscillator and we see the rising and falling in time of the wave as it travels to where the receiver is. The oscillator in this model is a charge that has been set in motion by radiation that was absorbed by the cavity walls. The absorption of the radiation causes the charge to oscillate and the oscillating charge in turn emits its own electromagnetic radiation. So we're watching a charge that's been set into oscillatory motion by external radiation emitting its own radiation here on the left and then causing another charge to oscillate over on the right. That would in turn of course cause that secondary oscillation to generate its own radiation. And you can see how the black body problem is a very complex interplay not only of mechanics but electromagnetism and getting the details of this right are essential to correctly predicting the radiation from a black body. Now the fatal flaw that people like Rayleigh and Jean's made when constructing their prediction for the energy emitted per unit solid angle per unit time and per unit wavelength from say a black body was that they assumed that any oscillatory frequency was possible for the charges. It seems a natural assumption electromagnetic waves originate on oscillating electric charges. If I change the frequency and I can change it to anything I like in classical physics I expect a different kind of electromagnetic wave with its own frequency to be emitted and in classical physics I can pick any frequency I want, any one at all, because in classical physics they're all possible they're all allowed and this was the fatal flaw it turns out in the Rayleigh Jean's calculation of the black body spectrum. They assumed that those oscillating charges in the walls of the cavity could emit any frequency of radiation they wanted as they sympathetically begin to oscillate having been struck by external radiation. It turns out that this leads to the Rayleigh Jean's prediction of the power spectrum which is utterly wrong. The Planck model on the other hand which arrived at the correct answer results in a power spectrum that looks like this. It goes as one over lambda to a power in this case lambda to the fifth but there's an overall multiplicative factor and that's where the temperature dependence shows up. It's also where a wavelength dependence shows up as well and this extra piece has the effect of cutting off the power spectrum at high frequencies. In other words as you go to higher frequencies you actually see there's a turnover in the prediction of the model and it drops off to zero as you go to shorter and shorter wavelengths higher and higher frequencies you don't emit more energy you wind up emitting less. Now what was the difference between the Rayleigh Jean's model and Planck's effort to model the black body spectrum? Well one key assumption was that Planck did not allow all frequencies to be possible for oscillating charges and I'll return to that assumption in a bit looking at some of the historical context of Planck's own work. To give you a better sense of what atoms and molecules actually do when they are struck by electromagnetic radiation let's look at this simulation incorporating the modern understanding of the interaction of radiation and matter. We have here a water molecule two hydrogens bonded to one oxygen and we can shoot radiation at it. Let's begin by shooting microwaves long wavelength electromagnetic waves somewhere between visible light and radio. If we start shooting microwaves at the water molecule we see that many of the microwaves will pass through the water molecule but some of them will be absorbed and cause rotational motion of the molecule which then scatters the microwave. This is in fact how a microwave oven works. Microwaves at the right frequency will cause water molecules to rotate and collide with each other and kinetic energy is added to the system and as we know kinetic energy is related to the temperature of material. If you add kinetic energy to the water molecules in the system you will heat it up. Let's change the wavelength of the radiation to infrared. We are now shooting much shorter wavelength light at the water molecule. No longer are we able to make it rotate rather we are able to make it oscillate. The hydrogen atoms that are bonded to the oxygen will occasionally be struck by an infrared photon that then causes them to jiggle around a little bit before scattering off the photon. If we shoot visible radiation which has even shorter wavelengths at our water molecule we see that it is effectively transparent to the visible light. All the visible light, all the visible light radiation is passing through the water molecule as if it's not even there and that shouldn't come as a surprise to us. Water is transparent to light. So it makes sense that visible light should be able to make it through a body of water and we see that modeled here. If we shorten the wavelength of the radiation even more to ultraviolet we see that this also tells us something about water. That water doesn't respond to this wavelength of radiation. Ultraviolet radiation passes through the water molecule essentially unscathed. This kind of little simulation incorporates our modern understanding of electric charge chemicals, bonding, and the ways that energy can and cannot be absorbed and re-radiated by atoms and molecules. We see that not all radiation causes a water molecule in this case to do anything. Only certain light frequencies or wavelengths have an effect on the charges of the water molecule and thus can cause them to vibrate, oscillate, or rotate in such a way that might result in subsequent, later, re-radiation of energy. Now as I showed you in the last lecture video, this model accurately describes the shape of a black body spectrum, but it comes at one small cost. Planck's effort resulted in the need for a new physical constant which he labeled H and eventually came to be known as Planck's constant. It is related to the degree of the discretization of the oscillation of the charges in the cavity. In other words, not all frequencies of oscillation are allowed and H tells us something about the gap between allowed frequencies. Things in between in the gaps are not allowed. This is known as the quantization of the oscillatory motion of charges in the cavity walls. Radiation coming from quantum, a Latin word for how much, implying not an unlimited set of values that are possible for a system, but rather a discrete, well-defined, and finite set of values that are allowed for a system with no values in between the allowed ones. Now the reason that the spectrum winds up cutting off at short wavelengths or high frequency is that electromagnetic radiation as a consequence of Planck's model requires a specific amount of energy to make a specific wavelength. In other words, if you want to make ultraviolet light, you've got to put in a minimum amount of energy to do that. If you want to make something with a shorter wavelength than ultraviolet light like x-rays or gamma rays, you have to put in even more energy and not all of those energies are possible inside the oscillating charges of the cavity walls. So if you don't have that energy, you can't make that wavelength, and the spectrum naturally cuts itself off. This implied also that the energy of the radiation is quantized and itself can come in units or packets. Now this new constant, Planck's constant H, ultimately had to be determined from experiment. It wasn't predicted by Planck's model. It was a parameter in the model that had to be determined. And it has units of joules times seconds, which if you flip back to physics one and play around with those units a little bit, you'll realize that they correspond to units of angular momentum. This actually has deep implications for the universe, but we're not going to get to them right now. Now its value was originally determined by Max Planck. By simply changing the value of H around in his calculations, until at a specific temperature for a black body, he had a value that yielded a shape for the black body spectrum that best described that particular heated black body. Now that's how he did it, and in fact by doing this, by fitting the parameter to the data and determining the value of the parameter itself, he came to within a few percent of the currently accepted value of Planck's constant, which is already a remarkable achievement. But in science, if you build a model by tuning it to existing experiment, the true test of a model is whether or not it correctly predicts new phenomena that have not yet been either explained or observed. So Planck's constant by itself being determined from the black body may just be tuning a mathematical model to the data to get the answer you wanted in the first place. That's the first step in describing nature. But if you want to see whether or not you've learned something deep about nature, you need to find the next thing that you can test by applying the same idea with the same constant and see if you get answers that are consistent with nature. Now the currently accepted value of Planck's constant is 6.626 and a bunch of other decimal places times 10 to the negative 34 joule seconds. That is a number worth memorizing. On par with the speed of light, 2.998 times 10 to the 8th meters per second were the mass of the electron, 9.11 times 10 to the minus 31 kilograms. Planck's constant is one of those fundamental numbers that when committed to memory can be busted out when you need it to do a quick calculation and can be very handy when doing things like engineering new systems like in electronics for instance. Now this constant is crucially important in the modern world. I can't understate its value any more than I can understate the value of the speed of light. Its value is now the basis of the system international definition of the kilogram. The definition of the kilogram used to be based on the size and mass of a platinum iridium bar that was kept under glass in France. There are many flaws with that. For instance if atoms of that bar flake off over time and you don't notice it then over time your definition of the kilogram using that as a reference changes. Weights and measures are crucial to things like economies and standards and so forth and so you don't want your definition of the kilogram drifting over time. So far as we know Planck's constant is stable over vast periods of time certainly over many billions of years and so it was wise to redefine the kilogram using something that itself can be determined independently and is stable and it turned out that a particular way of measuring Planck's constant lends itself to defining the kilogram and that change went into effect only in 2018. Planck's constant also plays a fundamental in key role in all electronic devices certainly all modern microelectronics. Those devices rely on the exact properties of semiconducting materials and semiconductors can be precisely engineered thanks to the quantization the discretization of radiation and matter and ultimately all of this stems in its scale size and control from the value of Planck's constant. Now as I've hinted before Planck's work had a consequence built into it that if true would radically change our view of radiation electromagnetic waves. He realized in his paper on the subject that as part of the only way he could find to describe the black body spectrum he was forced to assume that radiation had to come in quantized units whose sizes were controlled by the constant H and proportional by that constant to the frequency of the electromagnetic waves. This equation relates the energy and the frequency of electromagnetic radiation E equals H Planck's constant times F the frequency of the radiation and since frequency and wavelength are related by the speed of light this also implies a relationship to the wavelength of that light. Let me give you some of the context of Max Planck and his work. He concluded this effort in 1900 after many desperate years of working on the problem but he himself did not fully accept the implication of what his newly developed constant H implied and the consequences of his solution to the black body spectrum problem. Basically his solution implied, if correct, that matter and energy can be quantized into discrete units and that units in between those are simply not realized in nature, they're forbidden by the system somehow by the parameters of the system. Now he assumed that this was all some kind of convenient math trick that he had played, that it wasn't really describing nature at a fundamental level and that someone else would come along, really solve this problem using the correct description of nature and one day explain why the trick worked. If you look at some things that Planck himself has said over the history of his own life from the year in which he published his black body spectrum paper to decades later as he reflected on that period of his life, you can gain some insights into his psychology as a scientist at the time. And in the paper that he published in 1900 he states, moreover, it is necessary to interpret the total energy of a black body radiator not as a continuous infinitely divisible quantity but as a discrete quantity composed of an integral number of finite equal parts. You can see here in sort of the tone and writing of his sentence that he finds something necessary to do but he doesn't necessarily take away from that that it implies reality follows from this assumption. The assumption that the total energy of a black body radiator is discretized and not continuous may merely be a mathematical assumption but nonetheless he found it necessary to make this assumption in order to interpret the data. Now many decades later in a letter that he wrote to RW Wood he reflected back on this period and one famous quote from this letter is often repeated wherein he said the whole procedure was an act of despair because a theoretical interpretation had to be found at any price no matter how high that may be and you get a real taste of his professional desperation where others had failed to describe the black body spectrum. Planck was desperate to figure out what avenue would lead to the correct description. He didn't necessarily accept that the mathematical steps required to follow that avenue implied anything about nature but it worked and he published it even if he didn't fully embrace the implications of his own work. Now another famous quotation from Max Planck whose source I simply couldn't track down but it is attributed to him by many other sources was that he was ready to sacrifice any of his previous convictions about physics in order to solve this problem. Now this last quote especially was motivated by another thing that Planck had to do to solve this problem and that was to employ a statistical description of matter and radiation. Many physicists found statistics distasteful because under the hood statistics tells you that you can't know for sure the outcome of a particular system but you can know the probabilities of all possible outcomes even if you don't know which one will be realized in the next experiment. Many physicists who believed that the universe was deterministic that is that if you know exactly the initial conditions you can find the exact outcome of the system every time found the use of statistics to describe nature distasteful. Distasteful doesn't necessarily mean wrong and that's why the hard work of the scientist is to use observations of nature to assess the assumptions that we have made in trying to describe and predict nature. Now as I said before the burden in science of a new idea falls not on your ability to describe the things that came before but to explain the things that come after without changing any of the assumptions of the idea. A truly successful theory a theory that is not only built on facts but predicts the existence of new ones is ultimately forged in the fire of experimental science married with mathematical effort. This lands us on the subject of the photoelectric effect. Now the photoelectric effect was known in the late 1800s but could not be described using what was known in the late 1800s. It was observed by physicist Heinrich Hertz. Now he was the first person to definitively demonstrate the existence of electromagnetic waves. These had been a phenomenon predicted by Maxwell's equations and in that same prediction captured the essence of light that light itself is an electromagnetic wave. Hertz realized that if you were going to test the prediction that electromagnetic waves are real independent of light you would need to demonstrate their existence by transmitting them from one place in a laboratory receiving them at another and showing that the wave induces an oscillating electric charge at the target location. So what he ultimately showed was that an oscillating charge at one place in a room, a laboratory, could induce an oscillating charge elsewhere in the room with no physical contact and this established the reality of electromagnetic waves beyond light. In fact you could think of this as the first radio transmission. Now he was also the first person to demonstrate an intriguing physical phenomenon, the photoelectric effect. Light which is an electromagnetic wave at heart at least in the Maxwell view of nature shown on a metal can liberate electrons from the metal. So take a beam of light, shine it on the surface of a metal, look for an electric current and under the right conditions you will see an electric current develop in the metal. Now Maxwell's equations predict that the intensity of a light beam an electromagnetic wave is proportional to the squared strength of its electric field that is E not squared if E not is the base maximal electric field value of a particular wave. Now because of that prediction attempts were made to describe and predict and explain the observed features of the photoelectric effect. So let me use an analogy combining mechanics and the laws of electromagnetism Maxwell's equations to attempt to predict the set of phenomena that you would expect to arise in the photoelectric effect. Think of the charges in a metal as a ball that's stuck in a pond in a patch of lily pads or weeds. What you want to do is liberate the ball, you would like to knock the ball out of the lily pads free it so that it floats over to the shore and you can get it because you don't want to step in all of these weeds, who knows what's swimming around in this thing. Fine. So you and your friends devise a sort of classical photoelectric effect experiment. You get a bunch of empty buckets that you might have around to keep ice, keep your beverages cool while you're playing that day. You empty out the buckets and you carry them over to the shore of the pond. And one of you kneels down at the edge of the pond and starts using the bucket to push on the surface of the pond. This generates water waves. So you're pushing on the surface of the pond and the water waves are making the ball and the lily pads wiggle up and down but it's not knocking the ball loose. No problem. You're at the limit of your strength but you've got lots of friends so your friends all also kneel down at the edge of the pond near you and they start pushing on the surface of the pond and you're not very coordinated so these waves have different amplitudes at different times but eventually if you're patient enough some waves will pass through the ball they'll add up an amplitude constructively interfering and they'll deliver enough energy to the ball to knock it out of the lily pads. So the photoelectric effect in analogy to this ball stuck on a pond and a bunch of lily pads should be behaving as follows. If you send in light waves even feeble light waves that don't themselves have enough energy to liberate a charge from a metal. If you send in enough of those light waves at the metal you will begin liberating charges. The light wave amplitude should add up they go as the electric field squared of each wave and if you wait long enough you'll start knocking electrons out of the metal. That's what people expected from the classical theory of mechanics and electromagnetism but what was actually observed in the close study of the photoelectric effect? So what was observed was that the intensity of the light you shine on the metal has no effect on initiating the effect itself. The photoelectric effect can't be induced by simply cranking up and up and up and up the intensity of light if that light doesn't already seem to have the ability to make a current flow in the metal. We can simulate the observed photoelectric effect using this FET simulation that's available on the web. For example I can start by trying to shine long wavelength light onto a metal. I've selected a copper plate which is located on the left side of the apparatus. I have a representative light source at the top of the apparatus and as you can see I can control the intensity of the electromagnetic radiation or light that I can shine on the copper. I'm going to go ahead and crank this red light source up to 100% of its intensity. And as you can see there is no observed current in the graph on the right. The graph shows on the y-axis the electric current that's observed in the system and on the horizontal axis the intensity of the light which is currently pegged at 100%. Even if you wait 1 minute, 10 minutes, 100 minutes, you think you're allowing the amplitude to build up and occasionally knock an electron out of the metal but you see nothing. Now instead if you change the wavelength or frequency of the light you maybe can see what happens to the effect. Do you make an electric current flow? Well if you start from a particular wavelength of light that doesn't cause the photoelectric effect and then you change it gradually to a longer wavelength say start with red light and then change it to microwaves or radio. You'll also notice moving the intensity of the light up and down doesn't cause the photoelectric effect to start. But if you shorten the wavelength from the ineffective wavelength to something shorter, higher frequency, shorter wavelength, at some point you'll suddenly notice that electrons will begin to flow through and off the metal. You can induce the effect as you shorten the wavelength. I'm going to begin to lower the wavelength of the light from red at about 750 nanometers down to orange down to yellow and we still see that nothing is happening. I've definitely switched to a shorter wavelength of electromagnetic radiation but we still see no current versus intensity on the graph. I'm going to continue to shorten the wavelength of the light. Now we're into the green. We're approaching light blue or blue. Now we're going to the more richer blues and we're heading toward violet. Now I'm definitely down at the shortest visible wavelength range of light and yet the copper is doing nothing and I'm blasting it with 100% intensity. But watch what happens when I push this simulation into the ultraviolet, very short wavelength radiation. Once I cross below a threshold wavelength or frequency for the radiation, suddenly electrons begin to get shot off the copper by the light. Now over here you'll notice that the current has gone up a little bit on the vertical axis. I'm at 100% intensity and I've moved up about one tick mark on this axis. Now once you've set the photoelectric effect in motion, you might hypothesize that if you crank down the intensity of the light to some sufficiently low level, then the waves won't be able to add up enough anymore and no more charge will flow even before the intensity gets to zero. But what you find is that the electric current that you induce in the metal declines to zero as the intensity goes to zero and the electric current only goes to exactly zero when the intensity is also zero, that is you switch the light off. What I'm going to do now is I'm going to go ahead and lower the intensity of the light. Now you'll notice that of course the current is decreasing. As the intensity of the light decreases, I'm still knocking electrons off, but not as many. And of course if I bring the intensity all the way down to zero, then the photoelectric effect switches off. There was no point in the intensity and current plot where the effect suddenly switched off before I got to zero intensity. In fact, if there's even a little bit of intensity, you'll notice that electrons start boiling off of the copper, not many, but they come off some very fast and some very slow. At a particular threshold wavelength and frequency, the photoelectric effect simply begins. Raising and lowering the intensity of the light seems to have no effect on the maximum kinetic energy of an ejected electron. Even very weak intense light, but with the correct wavelength or frequency, will rapidly eject an electron occasionally with a high kinetic energy, despite the fact that the intensity scales as the square of the electric field strength. And shouldn't more electric field produce more acceleration? That's what all of that stuff from Coulomb's law and Physics 2 and Maxwell's equations says should be happening. I can lower the intensity down even more, down to just one percent of the source, and yet nonetheless electrons will come shooting off of this thing with lots of kinetic energy. It's as if the kinetic energy of the ejected electrons from the copper have nothing to do with the intensity of the light, but only to do with the wavelength or frequency of that light. Now I can bring the radiation up in intensity a little bit so that we can see a few more electrons boiling off the metal, and what I'm going to do is I'm going to begin to increase the wavelength of the light just very gradually, just a nudge at a time. At some point we're going to cross a threshold where the light simply does not have sufficient wavelength to induce the photoelectric effect, and it seems that I've gotten to it at about 270 nanometers. Now I can go ahead and crank up the intensity to a hundred percent now that I've moved just past the wavelength threshold to induce the photoelectric effect, and yet again we see that intensity does not suddenly cause the photoelectric effect to switch on. Now you can see how frustrating this must have been for the physicist of the late 1800s. This set of observational facts defied explanation using all the battle-tested notions of classical waves and the laws of electromagnetism. Does this sound familiar? Does this sound like the moment that led to special relativity? Because if it does, you're on the right trail. You've found a place where the theory of motion and the theory of electricity and magnetism, which were largely developed on macroscopic things, then encounters a new microscopic phenomenon where it utterly fails to make accurate predictions, and that smacks of opportunity. So how was the photoelectric effect explained? Well it was our old friend Albert Einstein who cracked the photoelectric effect in one of his 1905 so-called miracle year publications. This was the year that catapulted him into at least physics academic fame and allowed him to finally secure a faculty position after years of toil at the patent office in Bern, Switzerland. Now to explain the phenomenon, Einstein reached back to Planck's 1900 paper on the black body spectrum. Recall that a consequence of Planck's solution to the problem, desperate though the remedy may have been, was that light has an energy that's given not by the intensity of the electric field of the wave, but rather by the frequency or wavelength of the light. That is, E is equal to H Planck's constant times F the frequency of the light. Now since the speed of light is equal to the frequency times the wavelength, one can substitute into this to get the corresponding relationship with wavelength. Shorten the wavelength, increase the energy, increase the frequency, increase the energy. Those are the relationships between frequency and wavelength and the energy of a light packet, a light quantum. So Einstein embraces the implication of Max Planck's work that radiation can be quantized into discrete units and therefore a single unit of light is hypothesized to carry or cost to produce H times F for light of a certain frequency F. So even if one unit of light of a certain frequency strikes an electron and therefore strikes it with a certain amount of energy given by H times F, the liberation of the electron is immediately possible independent of the intensity of the light. More light quanta striking more electrons per second means more electric current. Fewer light quanta striking fewer electrons per second means less current but if you have even one you will liberate a charge and that's consistent with the observations of the photoelectric effect that once you make it happen it happens all the way down to even very low intensity until you switch off the light source. So what are the equations describing the photoelectric effect that were worked up by Einstein in 1905? So he reasoned that it takes a certain minimum amount of energy to remove an electron from a metal. A metal isn't just going to give up its electrons without a fight. I mean otherwise it would be really easy to just reach out and strip electrons off a metal but it takes energy. So there's some minimum amount of energy that's required to liberate one charge from a metal and this is called the work function and it's denoted by the lowercase Greek letter phi. Now if a quantum of light with a given energy strikes the electron and has energy that exceeds the work function then it's possible to transfer energy to the electron and remove it from the metal. It can scatter the electron or even be fully absorbed by the electron. The maximum amount of energy that can be transferred to the electron by such an interaction of matter and light is given by the following equation that the maximum energy that an electron can get when struck by a quantum of light is given by the energy of the quantum of light minus the work function. It takes some energy to remove the electron. If there's extra energy left over after that it goes into the energy of the electron in motion. And finally we arrive from Planck's hypothesis about the energy of these light quanta at the equation that the maximum energy you can transfer to an electron removing it from a metal is hf minus phi. Now if hf is less than phi the electron can't receive sufficient energy to be removed from the metal. hf must equal or exceed the work function in order to liberate an electron from a metal. And metals of different kinds take different amounts of energy to remove charges from them. Now where does that energy go? Well it goes into the kinetic energy of the electron. The electron will gain kinetic energy as a result of this interaction with a quantum of light. And so finally we arrive at the following equation that the energy we're talking about here is really the maximum amount of kinetic energy that any given electron can receive in this collision process. And that's going to be equal to Planck's constant times the frequency of the light that struck the electron minus the work function of the metal. The minimum energy required to remove the electron from the metal. This ultimately leads to the birth of the concept of the photon and implies that light has both particle-like and wave-like aspects that need to be taken into account. Now the classical description of light from Maxwell's equations imagines that light is an electromagnetic wave with an electric field that oscillates in time and space, a magnetic field that oscillates perpendicular to the electric field in time and space, and that the wave travels perpendicular to both the electric and magnetic fields. Each wave will have an energy per unit area given by this equation. This is what I said before that the intensity of the radiation is proportional to the electric field squared. This is all a wave description. But Einstein's special relativistic description of massless phenomena, which light seems to be, says that the energy of a massless phenomenon is equal to its momentum times the speed of light. Now recall that special relativity did not tell us where the momentum itself for light comes from, but that thanks to Max Planck and Albert Einstein, Planck quantizing radiation and the oscillations of matter in order to explain the black body spectrum, and Einstein adopting the quantization of radiation in order to explain the photoelectric effect and doing so perfectly, then leads to the following description of light interacting with matter, that the energy of a light quantum is equal to Planck's constant times the frequency, or Planck's constant times the speed of light divided by the wavelength of the radiation. And we see that the energy of the light is related to the frequency, and that the momentum is also related to the frequency, or the wavelength. The origin of the numerical value of a light quantum's energy is wavelength and frequency. Those tune and control the energy and momentum of a light quantum. Now the wave-like aspects of light, like diffraction and interference, oscillating charges making electromagnetic waves, and electromagnetic waves then also sympathetically causing charges to oscillate, these are all very wave-like things that had all been very well confirmed prior to the early 1900s. But the black body problem and the photoelectric effect couldn't be solved with those wave-like aspects. You needed particle-like aspects of light, and these phenomena began to hint that those were needed. Light energy comes in units, that energy is defined by frequency and wavelength, and light is a massless phenomenon. An electron is a massive particle-like thing that can travel through space. Light is a massless thing that travels through space, and we see from the resolution of the black body problem and the photoelectric effect that light has these quantum discrete behaviors in the same way that particulate matter has a quantum or discrete nature. Now Einstein referred to these packets of light energy as light-quanta, and again that comes from the Latin quantum, meaning how much. Now in a letter in 1926, physical chemist Gilbert Lewis coined the more common term, the one we use today, photon, implying a quantum of light from the Greek for light. Now in science it's it's not enough to describe a phenomenon, it's important that that description have testable consequences, and that there is a test that could falsify the explanation and show that it's wrong. Now if your explanation survives a test it lives another day and gets to make more predictions, and over time if it keeps surviving it gets adopted as an accurate description of nature, perhaps even as a law of nature. You can imagine that Einstein's explanation was not readily accepted of course, and much as Planck had met his own work with serious scientific skepticism Einstein's adoption of the quantum nature of radiation to explain the photoelectric effect with all of these interesting consequences was also met with serious scientific skepticism. The American physicist Robert Millican, who is one of the sort of few well-known American physicists in this early part of the 1900s, and his famous especially to high school chemistry students for the oil drop experiment that established the fundamental unit of electric charge although that experiment is a whole fascinating story in and of itself. Millican did not take the claims of Einstein's explanation about the so-called you know the maximum kinetic energy of an ejected electron and so forth very seriously. He wanted to test this claim to see if it was possible to refute Einstein's explanation of the photoelectric effect. Now we're going to do a reproduction of this famous experiment by Millican in our class, but I'll tease the conclusion of this and it's the following. That Millican in 1914 after careful experimentation confirmed Einstein's description of the photoelectric effect all stemming from the quantum hypothesis of radiation. In the end, the photoelectric effect paper that appeared in 1905 during this amazingly productive year of work from Albert Einstein won the day and it's no accident therefore that Einstein went on to receive a Nobel Prize in Physics in 1921. Interestingly, it's for this work that Einstein received the Nobel Prize in Physics, not for special relativity, not for general relativity, but for this niche effect in experimental physics. Now, Einstein had extended Planck's work to an entirely separate space of experimental effort, not the black-body spectrum where Planck determined the value of his constant and while he made a satisfactory explanation of that spectrum didn't accept the implications of his own explanation. Einstein embraced those implications and then predicted all the aspects of the photoelectric effect, not only correctly describing what was known of the phenomena, but then leading to the experiments of Millican who confirmed that description as accurate fully in its mathematical formulation. This set the stage for an entirely new other perspective on nature, not the theory of space and time and the speed of light and gravity, the theory of the very fast, but the quantum view of matter and radiation, the correct theory of the very small. So to review, in this lecture we have learned the following things. We've learned how the black-body radiation spectrum was finally understood and about the possibility implied by this resolution that energy may come in discrete units. We've also learned about the photoelectric effect and how Einstein resolved the puzzle of the photoelectric effect by embracing the conclusion of Max Planck's work on the black-body radiation spectrum, applying them to the photoelectric effect to make predictions about that phenomenon, which ultimately proved to be the correct description of nature, and that all of this has set the stage for a new view of radiation and matter.