Hello everybody, my name is Ian Applebaum and I'm an associate professor of physics at the University of Maryland. Today I'll be giving a guest lecture in this exciting online course exploring quantum physics. I'm thrilled to be delivering this topic, in part because my own research deals with experimental aspects of spin polarized electron transport in semiconductor materials. And I feel strongly that understanding the historical background I'm going to tell you about today is absolutely essential in making future progress in this modern field. I want to start our discussion of the discovery of electrons spin, by reminding you of what was known around the turn of the 19th century about atomic spectra. High voltage across a discharge tube filled with low pressure hydrogen causes the emission of electromagnetic radiation. Optical photons and a spectrometer can be used to disperse the different wavelengths across a detector. The discrete wavelength seen were known long before quantum theory to satisfy Rydberg's formula which told us that the photon energy Is proportional to the difference between the reciprocals of two squares of integers. Bohr's theory which included quantization of angular momentum in classical electron orbits captured this famous result but it was wrong for many reasons we now know such as failure to predict the ground state absence of angular momentum and importantly the degeneracy of each principal electronic level. From the solution of Shrodinger's equation in spherical metrical potentials, we know not only that has the ground stayed zero angular momentum but also that the excited levels were the principal quantum levels not one, we're actually degenerate. Meaning that several states have same iogin energy g. This follows from the multiple spherical harmonics, labeled by the polar quantum number l, and as quantum number m. A spectroscopic notation is to label l equals zero as s, l equals one, p, l equals two, d, and so forth each level of principle quantum number n has a maximum l equal to n minus 1 and states with a given value for l, have values of m given by minus l, minus l plus 1, in integer steps through zero and up to l minus one and l. This gives a total degeneracy of 2l plus one. Therefore each set labeled by a given principle quantum number n has degeneracy of n squared. A natural question to ask then is how can we observe this degeneracy experimentally? We need to break the degeneracy, split the levels and alter the photon emission spectrum. We can do this with a magnetic field. Here's how. Let's imagine that we have a classical electron orbit with total angular momentum given by this expression. Now, this circulating charge particle comprises a current, and so, the orbit has, necessarily, a magnetic moment. Classically we know that the magnetic moment that absolute value of that magnetic moment is given by just the current times the area that it circulates around. Now the current in this case is one electron circulating around in a given period. So this is the charge of the electron minus e times the frequency of it's orbit. And the area off course for circular orbit is just PI times the radius squared. Now the frequency of the orbit is just given by the velocity divided by the circumference two PI R. And that gives rise to a simple expression for the magnetic moment. But here, we're going to play a little bit of a trick and multiply and divide by the electron mass, and also Planck's constant. And the reason why we do this Is that we can see m, v, r. [NOISE] here, is the absolute value of the total angular momentum. Which has the same units as the action h bar. That means that this fraction of out front carries all of the units of the magnetic moment. In fact, it has a special name. It's called the Bohr magneton. So with this magnetic moment we know that in a magnetic field, each state is going to acquire an energy due to the interaction. Now if the magnetic field is along the z axis, then we can simply write this interaction energy as minus the z component of the magnetic moment times the magnetic field along z. So, if different states with the same principle quantum number n have different values for the z component of the magnetic moment, their energy eigenvalues will shift differently in a magnetic field, and the degeneracy will be broken. So we need to calculate the magnetic moment associated with each orbital. From the preceding discussion, we know the relationship between angular momentum and magnetic moment. And if the field is aligned to the z-axis, we only need to calculate the vector component along z. But this means we need to know the angular momentum vector along z. So we need to calculate is expectation value, and for that we need an operator representation. We know that the linear momentum operators are h bar over i, times the derivative with respect to the conjugate real space variable. So it's easy to see, that the angular momentum along z takes this form. Because phi, the azimuthal angle, is its conjugate variable. The expectation value then is this matrix element, the operator sandwiched by the state. The only part of the way function that matters here is the phi dependents which we know from the spherical harmonics. The derivative brings down a factor if i x m and what's left is the original normalized state. We see then that our answer is integer units of h bar determined by the azimuthal quantum number m. This is why m is commonly called the magnetic quantum number. Now we can complete our calculation. The energy added due to the interaction of the orbit's magnetic moment with magnetic field along z is simply the Bohr magneton times the magnetic field strength times the magnetic quantum number. This result is pleasing enough but some understanding of the scale of the effect is helpful. The Bohr magneton is small, about 60 micro electron volts per Tesla. And a Tesla is a huge magnetic field, about 20,000 times the strength of the Earth's geomagnetic field. The largest fields in the lab, created with superconducting coils, are several tens of Tesla. The energies are then small in comparison to the electronic transitions. So any observed shifts of the spectral lines, are going to be proportionately small. So here's what happens. A non zero magnetic field induces a splitting between degenerate states, adding energy to states with positive magnetic quantum number m, and subtracting from states with negative m. All states with m equal zero Including this all ground state level are unaffected. The splitting energy is name after Pieter Zeeman who want to now bill price in 1902 for observing this Petron line splitting when gas discharge tubes were placing in the negative field. Now we have to remember that this energy spectrum we calculate is not the same as the optical spectrum of emitted photons, which are only due to transitions between the levels. And not all transitions are allowed. The expressions which determine when the transitions are allowed are known as selection rules. We're going to calculate them next. During an electronic transition electron wave function forms a super position of the initial state and the final state with lower energy. Each component evolves differently in time so the electron probability distribution center of mass can move. This oscillating dipole is what radiates electromagnetic energy. A simple calculation of the expectation value of position yields this expression. Where we can see the first two direct terms from the inner product of the wave function are symmetric, whereas r, the radial variable, is antisymmetric. So integration over them vanishes. The cross terms, which oscillate at the frequency determined by the energy difference of initial and final states, may survive. Now, those away functions have previously been writing in spherical coordinates. We want to convert interval into cartesian coordinates so we can see along which axis we might get a charge oscillation and electromagnetic radiation. Here's the transformation we need in terms of the polar and azimuthal angles veta and phi. It allows us to split the vector integral from the previous slide into individual cartesian components. First, let's consider the x and y components of the integral over angle. We'll have to evaluate something like this, two complex exponents containing phi and sin of cosine over phi. Using the Euler formulas, we see that this will involve integrals over integer periods of the oscillating complex exponential. Which is identically equal to 0 except when the exponent is 0. Since the z component does not depend on phi, it's even easier to see that in this case the integral will be equal to 0. Only if the magnetic quantum numbers for initial and final states are the same. Taking together, we see that there is no dipole transition and no corresponding emission of photons unless the change in magnetic quantum number is plus or minus one or zero. This is our selection rule. By considering these kinds of arguments about periodic symmetry of the integrand and the integral over the polar angle theta, we can likewise derive another selection rule for transitions, such that the change in orbital quantum number must be + or -1. So which transitions are allowed? l must change by 1 and m can change at most by 1. The 2 piece stage can decay to the 1s ground stage and in a magnetic field all 3 of these transitions from m equals 1 zero minus 1 have different energies giving different emitted photon wave lengths that we can analyze with the spectrometer. The 3s state. Decays only to the three 2p states. And 3p decays to 2s, also with three distinct transition energies. For higher values of principle quantum number n, we can have transitions from the l = 2 d states with a broken degeneracy of 5, which can transitions to the lower p states. This involves many possible transitions that satisfy the selection rules. However, there are always only three different transition energies, delta m equals minus one, zero and plus one. So each spectral line splits into a triplet in a magnetic field. Now I want to point out something extremely important in atomic spectroscopy. Recall the energy time uncertainty principle. It says that there's a reciprocal relationship between the lifetime of a state and the resulting spectral line width, which limits resolution of high precision measurements. If a transition is forbidden by selection rules, other processes maybe allowed, but typically far less efficient and result in exceptionally long life times. Here, we see that the transition to the bound state from the 2s state, is forbidden by the selection rule. And other processes yield a lifetime of over a hundred milliseconds in comparison to the nearly equal transition from 2p to 1s in about one nano second. The line went through this transition, is therefore extremely narrow. Allowing very high-precision measurements of exquisite quantum effects such as the Lamb shift. Due to small corrections of energy levels from quantum electrodynamical effects of the electron interacting with short lived excitations in the vacuum. In addition to the selection rules, we can use the different forms of the dipole vector components to explain another feature of zamon splits electrical lines. They're optical polarization and directional dependants. For instance, we know that the transition corresponding to delta m equal to 0, no change in the magnetic quantum number during the transition is caused by a dipole along the magnetic field axis, z. However, we also know from classical electrodynamics that an oscillating dipole does not radiate along its axis. Therefore, this spectral line is absent when observed along this orientation. The other two are present, and due to dipoles oscillating 90 degrees out of phase along x and y, yielding right and left handed circularly polarized light. If we observe from a direction perpendicular to the magnetic field access, then the delta m equal to zero line in the middle can be seen and has a linear polarization along c. The other two are polarized perpendicular to the field axis. Because they are again due to dipole x and y. Hendrik Lorentz win the Nobel Prize in 1902 along with [INAUDIBLE]. For explaining this polarization dependence. He used only classical physics, a theory which we now know is wrong. Despite this perceived success in explaining the Zeeman effect, a serious problem remained. Some spectral lines split into triplets as predicted but others split into multiplets of four, six, etc. Here's a few examples. Now, we haven't made a silly mistake, we just need to re-examine the ingredients of our theory, namely the equation. Converting the classical kinetic energy into an operator is correct in the absolutely non-relativistic case as we've done here with the construction of the Schroedinger equation, but we're clearly leaving out an essential piece of physics by not even using a relativistically invariant expression. Now, can we fix the problem by starting from scratch, constructing a wave equation by starting with the relativistic expression for kinetic energy that's given here? That's what we're going to see in the next slides. By the way, those of you who haven't seen this expression before might want to see that It's asymptotically equivalent to the classical expression, in the limit that the momentum p is small. Then we can expand the square root and it's Taylor series, and see that the dominant terms are the familiar mc2, rest-mass energy, and the classical kinetic energy. Everything else is small, all though not negligible, as we will see. If we take a relativistic expression and try to use it as an operator on a wave function, we immediately encounter a problem. Our momentum operators are within the square root. And it's not clear at all whether this makes any mathematical sense. The problem disappears if the expression inside the square root is itself a perfect square. If we write the rest mass energy and kinetic energy components in x, y, and z here, 1, 2, and 3 with arbitrary coefficients, then we can make this a perfect square if these coefficients satisfy what appears, at first, to be an unusual constraint. It give unity when squared, but they anti commute with each other. These coefficients clearly are not scaler values. However, this fact didn't frighten Paul Dirac from writing down this relativistically invariant equation in 1928. The alpha coefficients are clearly not scaler values. But matrices, which satisfy the anti commutation relations, and form a so called Clifford algebra. Now we can write down many matrices which satisfy the algebra, but it makes sense to first look at the simplest case, with the smallest dimension matrices. It turns out, this can be done with matrices as small as four by four. Here's one choice of basis. I two is the two by two identity and o two is a two by two matrix of all zeros. The two by two poly matrices, the sigmas form the off diagonal blocks of the alpha one two and three four by four matrices. Now importantly, this converts the wave equation into a four-by-four matrix equation and the wave function into a four-component vector. Two of these correspond to arrest mass energy of mc squared, when the momentum p is equal to zero, as we expect for an electron. But a natural question to ask is why two values? Degeneracies like this are a signature of symmetry, but which one? Which degree of freedom do these two values correspond to? We're going to look at an experiment for a clue.
