 Welcome to the ninth video in this series on quantum mechanics, where we'll discuss photon spin and the quantum paradox known as Schrödinger's cat. In the previous video we saw that when a hydrogen atom emits a photon, it transitions between states of different energy and angular momentum. In order to conserve both energy and angular momentum, we concluded that not only does the photon carry energy, but it must also carry angular momentum. The photon must be a particle with spin s equals one. Back in video two we saw that classical electromagnetic theory explains how absorbed radiation exerts pressure, and this is consistent with photons having linear momentum. Here we want to see if there is an analogous way to understand photon angular momentum. Consider a radio transmitter. By pushing electrons back and forth along its antenna, this generates an electric field which oscillates parallel to the antenna and travels away at the speed of light. Imagine being able to see this electric field oscillate as it comes towards you. You would see the field oscillate between pointing up and pointing down. The path it traces out over each oscillation period is a vertical line, and the wave is said to have vertical polarization. The energy of this field is carried in discrete photons, and we'll say that these photons are in the, quote, quantum state v. We'll denote this state by the letter v inside some strange looking brackets. This strange bracket is called a ket, and the notation is due to a fellow named Iraq. A ket with the Greek letter psi could denote the quantum state corresponding to a wave function of psi. Or we could just list quantum numbers in the ket, or we could even be more abstract and use happy face and sad face kets to talk about live and dead ket states. It's an abstract and general way to talk about quantum concepts without having to get into much mathematical detail. So the photons that are vertically polarized wave are in the v state. Now imagine we rotate our antenna 90 degrees so it's parallel to the ground. We'll now see the electric field oscillate left and right. The wave is horizontally polarized. We'll say the photons in this wave are in the h state. Let's now combine these two quantum states. Here we have the v state on the right and the h state on the left. In the middle we show the sum of the two electric fields. As time goes on this composite electric field traces out a diagonal line, which we could call the 45 degree state corresponding to a wave polarized at 45 degrees. If we form the difference of the v and h fields, which is the sum of the v state and a flipped version of the h state, we get a wave polarized at minus 45 degrees. The photons in this wave are in the minus 45 degree quantum state. By combining the v and h polarizations in different ways, we can produce a wave polarized at any angle theta. We simply need to add appropriately scaled v and h states. The scaling coefficients are the cosine and sine of the angle theta. We say that any polarization can be represented as a superposition of v and h polarizations. In this sense we can think of there being two fundamental polarizations, the v and h states, and that these form a basis for all possible polarizations. We can also imagine delaying the oscillations of a field. On top we have our horizontal polarization and below it a version delayed by one quarter oscillation period. In the mathematical bookkeeping, this delay is represented as multiplication by the imaginary unit i, the square root of negative one that we introduced in video five. In both cases the electric field traces out the same horizontal line, but in the lower case the field is zero when the upper field is largest and vice versa. Let's see what happens when we combine the v state with the delayed h state. As time goes on, instead of changing its length, the electric field of the superposition traces out a counterclockwise circle. We call this a circular polarization, as opposed to the linear polarization of the v and h states. Specifically this is right handed circular polarization. If you point your right thumb in the direction the photon is traveling, out of the screen, your right fingers curl in the direction of the field rotation. We're referred to this as quantum state r. A superposition of the v state and the negative of the delayed h state is also circularly polarized, but the electric field rotates in the clockwise direction. We call this left handed circular polarization and denote it as state l. As an aside, at radio frequencies circularly polarized waves can be generated using helical antennas and are very important in satellite communications. While circularly polarized states r and l can be considered superpositions of the linearly polarized states v and h, the superposition of the r and l states produced the v state. Moreover, a superposition of the v and minus l states produced the time delayed h state. So we can just as well think of the linear states as superpositions of the circular states. So a quantum superposition is simply a sum or difference of two quantum states. The right hand state is the sum of the vertical state and the delayed horizontal state. To have all the states represent the same photon energy, we need to scale the sum by one over the square root of two. By shifting the plus sign to a minus sign, we get the left hand state. Other superpositions can be worked out with simple arithmetic. Consider one over square root of two times the sum of the r and l states. Simply substitute the first two expressions, and one over square root of two times itself is one-half. The two h terms have opposite signs and cancel, leaving one-half v plus one-half v, which equals the v state. Similarly, one over square root of two times the difference of the r and l states is the delayed h state. The circular polarizations can be represented as superpositions of the linear polarizations, and the linear polarizations as superpositions of the circular polarizations. And either the linear polarizations or the circular polarizations can be used as a basis to represent any polarization state whatsoever. OK. Good enough so far. But what about photon spin and angular momentum? Let's see how different polarizations can interact with the charged particles in a material. In this simulation, the blue and red circles are opposite charges, which attract each other. The blue charges are fixed, but the red charges are free to move. The electric field is represented by a magenta line, and a green line traces the free charge paths. On the left, the field polarization is vertical, while on the right, it's right-handed circular. Not surprisingly, the v-polarization causes the red charge to move up and down along a vertical line. The r-polarization causes the red charge to spiral away from the blue charge, and clearly imparts angular momentum. We can work out the precise angular momentum carried by a photon of right-handed circularly polarized light. Consider our blue and red charges, and assume the red charge, q, is moving in a circle around the blue charge, at least momentarily, at a distance r. The electric field E has component ET tangential to the circle, the direction of charged motion. Assume the field pulls on the charge through an angle d theta. The work done, or the energy transferred from, the field is the force q ET times the distance traveled, r d theta. The angular momentum transferred is the torque q ET r times the time dt. If you think of the green line as a wrench, the torque it exerts is the force q ET times the length of the wrench, r. Looking at the ratio of these expressions, the q ET r expression cancels, leaving d theta over dt. Once around a circle is 2 pi radians, and the field goes around at a frequency new times per second. So the ratio of energy to angular momentum of a photon is 2 pi nu. We know that a photon's energy is h nu, so the angular momentum is h nu over 2 pi nu equals h over 2 pi, which we denote by h bar. For left-handed light, we simply obtain the negative of this. So the quantum mechanical concept of photon spin corresponds to the classical concept of circular polarization. A photon of frequency nu carries energy h nu, linear momentum h over lambda, as we showed in video 2, and angular momentum plus or minus h bar, plus for the r state and minus for the l state. As imprecise as it necessarily is, we might visualize photons of a given frequency as coming in two little spiraling spring-like forms, mirror images of each other, corresponding to left and right circular polarizations. There's been so much of quantum mechanics, as soon as we start looking into the implications of the inherent wave-particle duality of this picture, we'll find that we've opened a conceptual can of worms.