 Welcome to Quantum Mechanics 8, SPIN. In many ways, this video is about the number 2, how it crops up an atomic behavior, and the seemingly impossible explanation for this. We've already seen in video 6 that the number of hydrogen orbitals per energy level, 1, 4, 9, and 16, when multiplied by 2, gives a sequence of numbers corresponding to the periods of the periodic table, 2, 8, 18, and 32. In 1887, the same year their famous Etherdrift experiment helped lay the foundations for relativity theory. Michelson and Morley discovered the so-called fine structure of the hydrogen spectrum. In video 4 we discussed the central role that the hydrogen spectrum played in the development of the quantum mechanical theory of the hydrogen atom. One of the most notable hydrogen lines is the so-called H-alpha line in the red region of the visible spectrum. In normal resolution, this looks like a single line of wavelength 656 nanometers, but by constructing an extremely precise optical system, Michelson and Morley were able to show that it actually consists of two lines, about .02 nanometers apart. That's a split of only about 30 parts in a million. Further precise observations of other atomic spectra showed more examples of this fine structure of spectral lines. With the development of the quantum theory of the hydrogen atom, it became clear that this implied a tiny splitting of hydrogen energy levels. In video 6 we saw that the Schrodinger solution predicts that the different orbital types for a given energy level have exactly the same energy. But in fact, precise measurement shows that the S orbitals have single energy values, while the P and D orbitals are split into two closely spaced energy levels. Why single energies for the S orbitals and double energies for the others? Recall from video 7, S orbitals have angular momentum number L equals 0. These are states of zero orbital angular momentum. In these states the electron does not on average go around the nucleus. The other orbitals have quantum number L equal to 1, 2, 3, and so on. In these orbitals the electron does have angular momentum. So apparently in orbitals with angular momentum, the energy level splits in two, but not in the S orbitals with no angular momentum. Why? Also in video 7 we saw how an atom in a magnetic field can produce the Zeeman effect, where the electron orbitals energy changes for different Z components of angular momentum. Since there are always an odd number of possible components, the Zeeman effect shows up as a splitting into an odd number of lines. But in some cases, the so-called anomalous Zeeman effect, we see a splitting into an even number of lines, that is, a multiple of the number 2. One of the most striking experiments characterized by the number 2 was the Stern-Gerlach experiment of 1922 using silver atoms. A similar experiment five years later by Phipps and Taylor used hydrogen atoms. These experiments employed a non-uniform magnetic field created by a magnet with different sized north and south poles. This causes the magnetic field to spread out, hence get weaker as we move from top to bottom. Suppose we now put a little magnetic dipole inside with its south pole upward. Magnetic poles attract, so the dipole feels an upward force at its south pole, and a downward force at its north pole. If the magnetic field was uniform, these would cancel, and there would be no net force on the dipole. But the magnetic field is stronger at the top, so the upward force is a little stronger than the downward force, and there is a net upward force on the entire dipole. If we flip the dipole over, like poles repel, so the north pole feels a downward force, and the south pole an upward force. Again, the top force is stronger due to the non-uniform magnetic field, and the dipole feels a net downward force. Now consider a side view of our magnetic field. We put a screen at right and shoot uncharged atoms through the non-uniform magnetic field. If an atom has a magnetic dipole with south pole up, the atom will curve upward. If the dipole is horizontal, there will be no net force, and the atom will travel straight. If the dipole has its north pole upward, the atom will curve downward. Each dipole orientation will produce a spot on the screen. If these dipoles are due to orbital motion of an electron, then as we've seen, there should be an odd number of spots. For silver and hydrogen we should expect to see a single spot, however both experiments found two spots. After stirring Gerlach's results for silver atoms, their beam was more of a ribbon. Along its center, where the magnetic field had the greatest non-uniformity, we clearly see the splitting into two. Phipps and Taylor found similar results for hydrogen, although the experiment was more difficult and the effect was less pronounced. In 1924, Wolfgang Pauly proposed that these experimental results are evidence that the electron has a two-valuedness, that is, quote, not describable classically. He was being very careful to avoid any analogy with classical phenomena. A year later, graduate students George Ulinbeck and Samuel Gusschmitt proposed that the two-valuedness was due to the electron spinning. Classically, a spinning charged particle should have angular momentum and produce a magnetic dipole. As we've already discussed, the motion of a particle around an orbit results in orbital angular momentum. If the particle is charged, as the electron is, this will also produce a small magnetic dipole. We might also envision the electron spinning about an axis, analogous to the way a planet rotates. This would create another component of angular momentum that will label S for spin angular momentum. Since the electron's charge would also be spinning, the resulting electric current should also produce another little magnetic dipole. The quantum mechanical theory of angular momentum, which we briefly described at the end of video 7, tells us how a two-valuedness of angular momentum can arise. If lowercase S is the spin quantum number, then the magnitude of spin angular momentum is uppercase S equals square root of S times S plus 1 times h bar, and recall h bar is Planck's constant over 2 pi. The Z, or up and down component, is quantum number M sub S h bar. M sub S takes on values minus S minus S plus 1 up to S. S can be zero, a positive integer, or a half integer. If it's one half, then M sub S takes on two values, minus one half and plus one half. A classical picture of this situation would be the electron as a spinning sphere with mass and charge and angular momentum of square root of 3 over 2 times h bar. The angular momentum has a Z component of minus h bar over 2, or plus h bar over 2, which we can refer to as spin down and spin up. So it seems there is a classical phenomenon, the spinning of a body, that when placed in a quantum context can explain the two-valuedness that appears in several experiments. However, Pauli was justified in avoiding reference to classical phenomena. If we imagine an electron with mass M traveling with velocity v around a circle of radius r equal to the size of an atom, and we set the orbital angular momentum, M v r, equal to h bar, we find a velocity that is about 1% the speed of light, that's fast but physically plausible. However, if a spherical electron of radius r and mass M spins such that the velocity of the surface is v, and we require the angular momentum to be h bar, then for even the largest plausible radius we find a surface velocity that is hundreds of times the speed of light. There is no classical model of a spinning particle that can come close to explaining the angular momentum of the electron. One quote, spin, is truly not describable classically. Still, the term two-valuedness not describable classically is an abstract mouthful. It's so much easier to just say spin. And that's the term people have settled on, while offering the caveat that quantum mechanical spin does not correspond to any classical phenomenon. Previously we discussed Bohr's correspondence principle, or for ever larger quantum numbers quantum mechanics corresponds ever more closely to classical mechanics. Although small quantum number orbitals don't look anything like a classical orbit, for large enough quantum numbers we can start to see a correspondence. However the spin quantum number s is fixed at one half, and that in itself is weird. In both the classical and quantum cases we can change orbital angular momentum, but spin angular momentum cannot be changed. It's an intrinsic property of the particle, like its mass and electric charge. And it never gets big so we can't observe quantum mechanical spin transitioning into a corresponding classical phenomenon. We'll continue to use the word spin that at least gives us something visual to relate to, but it's actually a strictly quantum mechanical phenomenon associated with the intrinsic angular momentum of a particle.