 Welcome to the second video in this series on quantum mechanics. In the first video we saw that classical physics failed to explain the relationship between temperature and electromagnetic radiation, such as arises in an incandescent light bulb. The classical prediction is that radiation intensity will always increase with frequency. Planck was able to derive a radiation law that matches what is observed. Intensity reaches a peak at a certain frequency and then rapidly drops off at higher frequencies. But to achieve this, Planck had to assume that radiated energy comes in discrete chunks, or quanta. At frequency new, the quantum of energy is Planck's constant h times nu. He called this assumption an act of desperation, and although it produced the correct formula he was highly skeptical of it. The assumption worked because at a temperature T there is an average thermal energy, kT, available to excite a radiation mode. Temperature K is called Boltzmann's constant. For low frequencies the energy quantum is much less than kT. The mode can have an energy of kT which is the classical prediction by containing some number of quanta. But at high enough frequencies the energy quantum will be more than kT. These modes either have to have no energy, or more energy than on average is thermally available. So at higher frequencies the probability becomes greater that a mode will have no energy at all. This is how the quantum hypothesis explains the drop-off of radiation intensity with increasing frequency. Planck's skepticism about the quantum hypothesis was understandable. Radiation takes the form of electromagnetic waves, and the energy of a wave is distributed throughout space. How could this energy exist in discrete chunks? Discreteness is something we associate with particles, not waves, and nobody had ever seen direct evidence of any type of radiation particle. But then it was pointed out that such evidence had indeed been observed in a phenomenon called the photoelectric effect. A solid material contains electrons, and these could be pulled loose by supplying electrical energy, as is thought in a sparklet. Energy supplied in the form of light should also be able to remove electrons. But if a material is illuminated with low-frequency light represented in red here, no electrons are ejected, even if the light is very intense. On the other hand, if the illumination is high-frequency light represented here in blue, electrons are emitted, even if the light is very weak. It was Einstein who in 1905 explained this by considering Planck's quanta as particles, moving at the speed of light, what we now call photons. If W is the energy needed to remove an electron, only a photon with an energy greater than W can eject an electron. The total light energy, which represents the number of photons striking the material, will determine the number of electrons ejected, but not whether electrons are ejected, and the kinetic energy of the ejected electrons will simply be the energy left over from this photon electron process. On the other hand, a photon of low-frequency light does not have enough energy to free an electron. So even if there are many such photons, that is the light is very bright, no electrons will be ejected from the material. Thus, Einstein explained how Planck's quanta manifest themselves as, quote, real particles of light. A more prosaic example is how photons explain sunburn in sunscreen. If you look at the spectrum of solar radiation, you find that the great majority of the energy is in the visible and infrared regions. Only a tiny fraction is in the high-frequency, short-wavelength ultraviolet region. That sunscreen protects us from sunburn by blocking only this small amount of UV radiation. Let's assume the chemical reaction responsible for sunburn requires a certain energy W, and light is composed of discrete photons. If a molecule absorbs a photon of energy less than W, there will be no chemical reaction and no sunburn. If a molecule absorbs a photon of energy greater than W, then there will be a chemical reaction leading to sunburn. For sunburn, the required energy corresponds to photons with frequencies in the ultraviolet region of the spectrum. So by blocking only a tiny fraction of the sunlight's energy, our skin is protected against sunburn. The great majority of the sunlight's energy still reaches our skin, but no single photon corresponding to these frequencies and wavelengths has enough energy to cause sunburn-inducing chemical changes. We've seen that there is strong evidence that light consists of discrete particles we call photons, having energies given by Planck's formula. But it had long been firmly established that light is a wave phenomenon. If you put a small pinhole in an opaque screen, place another screen to its right, and shine a beam of light from the left, then on the right screen the light intensity will display a so-called diffraction pattern that is the unmistakable signature of a wave. If you open a second pinhole so that another diffraction pattern appears on the screen, a very rapid variation of intensity, called interference fridges, will appear. These observations are a slam dunk for the wave theory of light. Let's look at this experimentally. First we take an image immediately after the pinhole screen. This shows light emerging from the two pinholes, one large and one small. Next, we take an image some distance away from the pinholes. Here we see that the light from the pinholes has spread out into ring diffraction patterns. Notice how the rings for the large hole are more closely spaced than are the small hole rings. These features are precisely explained by the wave theory of light. Moving even farther away from the pinholes, the ring patterns grow in size and overlap more. And looking at this in more detail, we see closely spaced vertical lines superimposed on the diffraction patterns. These are interference fridges, and there's no reasonable way to explain these observations without invoking the concept of waves. Now, we think of a particle as a little thing at a single point in space, possibly moving in some direction at some velocity. And we think of a wave as something spread out in space, oscillating with some frequency. If light is so precisely described as a wave, but also displays unmistakable particle properties, then what is it exactly? What is a photon? How do we picture it? The answer is, not very well. You can draw a little wiggle indicating a wave-like oscillation with an arrow indicating a particle-like thing moving in some direction. But the fact is, we simply don't have any day-to-day experience with something that has the properties of both a wave and a particle. So we don't have a frame of reference in which to understand such a thing in any intuitive sense. We've seen images that clearly show the wave nature of light. We've also presented evidence that argues for the particle nature of light, but we haven't actually seen light particles with our own eyes. So reasonably we might still be skeptical about that, especially since the implications are a bit mind-bending. Fair enough? What if we repeat the pinhole experiment, but with light so weak that at any given time there should be no more than a single photon in the system? What will we see? Doing this successfully requires extremely sensitive equipment, but it is possible. Here we're going to look at frames from a truly beautiful video titled From Photons to Waves that shows the result of this experiment. The link is given below. After a short time we see a seemingly random pattern of dots. These are the images of individual photons. A while later more photons have arrived and there is a hint of some sort of pattern emerging. As time goes on the pattern becomes clearer while the contributions of individual photons are no longer so apparent. We finally have the fringe pattern that characterizes wave interference. Yet we've seen that it's actually built up from a great many point events. The light energy does indeed arrive at the image in discrete photon packets at specific places at specific times. Yet somehow the photons quote know how to arrange their arrivals so that after some time the average effect of a great many photons is to create a wave interference pattern. So we have a wave picture in which interference patterns are the result of a wave passing through two pinholes or slits creating two diffraction patterns that spread out and overlap. But we also have a very convincing particle picture where light energy is concentrated in photons which occupy a particular place space at a particular time. We typically think of a particle as being at one point at time t1 and at another point at time t2 and therefore having followed some specific path between the points. But we've seen that photons arrive at the image screen in accordance with an interference pattern. Do single photons interfere with themselves in the presence of two slits? If so, then how does a single particle pass through two slits at the same time? Or if the photon only goes through one slit then how does the existence of the other slit influence how the photon arrives at the image screen? This is one of those nobody understands it moments. But the problem is in our own heads. There's a precise mathematical description of this whole process. Light energy manifests itself as discrete photons, each with a certain energy and arriving at a certain place at a certain time. This occurs at random and the classical wave intensity is just the probability that you will find a photon at some place and time. What paths do the photons follow? That's not part of the theory, and in fact it doesn't even appear to be a meaningful question. Asking and simply trying to force our limited concepts of particles and waves onto a realm of nature where they don't seem to be as valid as they are in our macroscopic world. Let's continue to try and push back against the idea of particles of light. We might argue that real particles carry momentum and you can exert force by scattering them off of other particles. If light is composed of real particles then you should be able to push an object around by shining light on it. Well, it turns out you can't. In classical theory light has two components, an electric field and a magnetic field shown here is red and blue arrows. The length of an arrow signifies the strength of the field at that point. The fields are perpendicular to each other and to the direction in which light is moving. Now suppose the light passes by a charged particle. Classical theory predicts the phenomenon of radiation pressure. This slide is a little technical, but in the end the important thing is the final result. The steps are just for completeness and there's no need to follow them. Radiation pressure works like this. Suppose at a particular time the charge q sees an electric field e and a magnetic field b which are moving in the direction s. The magnetic field strength is the electric field strength divided by the speed of light. Suppose the charge is moving with a velocity v. The electric field pulls on the charge with force q times e. The energy the field transfers to the charge, which we call the work it does, is this force times the distance the charge moves. The distance is velocity times elapsed time so the field does work eq vt. The magnetic field will also exert a force on the charge. A force exerted for a given time transfers a momentum of force times time. The force exerted by a magnetic field on a moving charge is the magnetic field b times the charge q times the velocity v. For an elapsed time t this transfers momentum bq vt to the charge. The magnetic force is peculiar in that it pushes in a direction perpendicular to both the magnetic field and the charge's velocity. This is in the direction s. We see that the momentum expression is the same as the work expression except the e field is replaced by the b field. But the b field equals the e field over the speed of light. So we have the following result. If light transfers some energy to a charged particle, it also transfers a momentum equal to the energy over the speed of light. And the transferred momentum is in the direction that the light is traveling. The reality of radiation pressure was spectacularly demonstrated by a recent Japanese space mission to Venus, which deployed a so-called solar sail to use radiation pressure from the sun to accelerate and navigate the spacecraft. One more detail before we summarize all this. If a light wave propagates at the speed of light, c, for one second, it travels a distance c. We call the distance between wave peaks the wavelength lambda. If the light frequency is new, then in one second there are new wave cycles. So new wavelengths must add up to the distance c. Therefore the wavelength is the speed of light divided by the frequency, that is c over nu. So the takeaway from this discussion is that if light consists of photons of energy e equals h nu, then the existence of radiation pressure requires that those photons must have momentum p equals e over c, which is h nu over c, which is h over the wavelength lambda. So photons do indeed carry momentum, just like we'd expect for, quote, real particles. Later we'll see that this relationship between momentum and wavelength will provide a critical insight when we try to understand the behavior of electrons. The proverbial final nail in the coffin of skepticism towards the photon concept was provided by observations of so-called Compton scattering, discovered by Arthur Compton. He found that if an electron at rest is illuminated by high frequency light, the electron scatters off in some direction, carrying kinetic energy and momentum, while the light is scattered off in another direction and with a longer wavelength. Using the photon concept, the longer wavelength implies the scattered photon has lost energy and momentum, and these are found to be precisely the energy and momentum transferred to the electron. Thus a photon can be considered to literally bounce off an electron analogous to a collision between two billiard balls. There is rock solid evidence that light is neither a classical wave nor a classical particle, but has both wave-like and particle-like properties simultaneously. This wave particle duality represents, in Richard Feynman's words, the wonderfully different behavior of things on very small scales, a behavior that is unlike anything we have direct experience with, yet which countless experiments tell us is simply how nature is. As compelling as this new picture may be, it is also quite unsettling. In the next video we'll see that wave particle duality undermines classical views of determinism and causality, and of our ability to, in principle, determine physical quantities with arbitrary precision.