Hi everyone. Today we'll resolve the paradox posed by the so called anomalous Zeeman effect in atomic spectroscopy, by looking at the consequences of having both orbital and spin angular momentum at play in the atomic electron Hamiltonian. But first I want to tell you a little bit more about spin and its interaction with magnetic fields. First, let's review the elements of our theory of electron spin. The spin vector components are represented as operators by the two by two Pauli matrices. In direct analogy to orbital angular momentum, the square of the total spin is h bar squared times the half integer spin quantum number, little s, times s plus 1. Eigenvalues of the individual components are simply the spin magnetic quantum number, m sub s. Times the quantum of action h bar. In a magnetic field along z, we have a trivial 2 by 2 Hamiltonian whose eigenvalues are given by the diagonal elements. With corresponding eigenvectors which are purely one spin component or the other. We see that the first moves up in energy which we expect from a magnetic moment pointed anti-parallel to the magnetic field. Because of spin angular momentum and its magnetic moment or antiparallel, however, this corresponds to spin parallel to the field. And so we call it spin up. The other moves down in energy, consistent with a magnetic moment parallel to the field, which has angular momentum opposite to the field. It can be called spin down. Note that we can compactly represent the eigen energies by this simple function of the spin magnetic quantum number, m sub s. We can use the stationary states of the simple spin Hamiltonian to calculate the state evolution of a spin in a magnetic field in which it is not in eigenstate. For example, let's say we prepare a spin as an eigenstate of Sx. For example, by placing it in a magnetic field long x and waiting for slow relaxation processes that we ignored in our Hamiltonian, to take the system to its lowest energy state. For a magnetic field along the negative x direction, that's the state with a positive eigen value of S X. The ket labeled plus. Now, let's suddenly rotate the magnetic field along Z. The state is not a eigen state of the new Hamiltonian, but we can decompose it into a super position of eigen states that diagonalize the Hamiltonian. And which each evolve in time with a different complex exponential given by its energy. We can now use this to determine how the expectation value of spin along x changes with time. We simply sandwich the operator by the state. Remembering to not only transpose but also complex conjugate the bra on the left converting it from a column vector to a row vector. The result is a quantity which oscillates in time. But how is spin conserved then? If s x is changing, where does the spin go? By calculating the expectation value of spin along the y direction, s y we see that it too is oscillating in time, 90° out of phase with spin along x. This is simply the quantum mechanical version of a process we are all familiar with, the Larmor precession of an object with angular momentum subject to a torque, such as a gyroscope. In this classical example, it's a mechanical torque due to gravity. Whereas the spin undergoes procession due to a mechanic, magnetic torque, caused by a magnetic moment in a magnetic field. Note that the procession frequency is controlled by the Zeeman splitting from magnetic field magnitude. This field control over spin procession Larmor frequency is precisely how we can spacially image using magnetic resonance imaging. Powerful electromagnets are used to create strong magnetic fields and field gradients. So that only a small volume will the Zeeman splitting that corresponds to a resonant exotation from narrow band radio frequency electromagnetic waves. This selective auscultation is then sensed by inductive detectors. Pickup coils. By scanning the resonance slice around in space by changing the electromagnetic coil currents. Images of deeply buried structures are created. MRI works by acting on the spin of the nuclei, not electrons. Which is enabled by the much smaller nuclear magneton due to much larger mass of the nuclei. And correspondingly small zamon energies and radio frequencies for excitation. Although impractical for imaging human biology, electrons spin resonance at around 20 gigahertz per tesla for the free electron. Is, however, very useful for study of inanimate samples, such as organic molecules and solid state materials. Now we can get back to the issue of anomalous Zeeman effect of atomic spectra in a magnetic field. Both the orbital and spin angular momentum of the electron have magnetic moments which couple to the magnetic field. But as we saw, the relativistic nature of the spin makes inclusion of a non-unity Thomas G factor necessary. We can of course, diagonalize the Hamiltonian, and determine the Zeeman splitting for each term separately. But how do we go about diagonalizing this expression with vectors l and s, that are, in general, not parallel? The key is to notice that the total angular momentum, l plus s, is conserved. We expect that the energetic splitting will be on the same form as before, but with an effective g factor. Let's see how this works out. First we project both vectors onto the conserved angular momentum axes by taking the inner product with the unit vector along j l plus s. For a magnetic field along z, this then picks out the z component of j, which has eigenvalues h bar times total magnetic quantum number, m sub j. Carrying out the dot product yields an expression that is mostly known quantities in terms of the individual quantum numbers. But what's the dot product between the spin and the orbital angular momentum? This can be seen by once again looking at the conserve quantity, j squared. Substituting this quantity into our Hamiltonian entirely converts it into a scalar energy quantity. Which is proportional to the magnaton times the magnetic field times the magnetic quantum number, m sub j. With the coefficient out front, it depends on little l, s and j. This is the so called Lande G-factor. Here's a few useful examples. When the orbital quantum number is zero. We just have the magnetic field interacting with the spin. Which we know has a g factor of two. Lande correctly predicts this. For l equals one, the total angular momentum j can be one plus one half, three halves. Or minus one half, yielding one half. In the first case, the Lande G-factor is four thirds. And in the second, two thirds. So from the six l equals one states, that's three orbital projections times two spin projections. Four of the j equal to three halves states, where m sub j is minus three halves minus one half plus one half and plus three halves, have one G factor and the two remaining, j is equal one half states with m sub j equal to plus and minus one half have another G factor of half its value. In a magnetic field, the six split like this. Now, let's look at transitions to the l = 0 derived states. The selection rules with Landau Spin Orbit Coupling are analogous to the orbital selection rules. Changes in quantum number j, are plus or minus 1, and changes in the total magnetic quantum number, m sub j are 0 and plus 1 or minus 1. This means that only the four j equal three halves states can radiatively transition to the j equals one half states. And only the m sub j equal to plus or minus one half states can transition to both l equals zero states. This leaves six distinct transitions. The energy of the spectroscopic shifts that we observe by, by dispersing the emitted photos through the spectrometer are proportional to the difference in the products of lambda g factor times the total magnetic quantum number m sub j for initial and final states. We can now see how multiplates more than the Lorenz triplet are created. In this example, spectral, spectroscopic line splits into six in a magnetic field. Each separated by two thirds of the [UNKNOWN] splitting, an all line shift with magnetic field. It is this spin orbit coupling, that explains the Anomalous Zeeman effect.