1 00:00:00,770 --> 00:00:03,560 Hi everyone. Today we'll resolve the paradox posed by 2 00:00:03,560 --> 00:00:06,910 the so called anomalous Zeeman effect in atomic spectroscopy, by looking at the 3 00:00:06,910 --> 00:00:10,110 consequences of having both orbital and spin angular momentum at play in the 4 00:00:10,110 --> 00:00:15,222 atomic electron Hamiltonian. But first I want to tell you a little bit 5 00:00:15,222 --> 00:00:19,342 more about spin and its interaction with magnetic fields. 6 00:00:19,342 --> 00:00:24,205 First, let's review the elements of our theory of electron spin. 7 00:00:24,205 --> 00:00:27,743 The spin vector components are represented as operators by the two by 8 00:00:27,743 --> 00:00:32,511 two Pauli matrices. In direct analogy to orbital angular 9 00:00:32,511 --> 00:00:37,339 momentum, the square of the total spin is h bar squared times the half integer spin 10 00:00:37,339 --> 00:00:43,780 quantum number, little s, times s plus 1. Eigenvalues of the individual components 11 00:00:43,780 --> 00:00:47,390 are simply the spin magnetic quantum number, m sub s. 12 00:00:47,390 --> 00:00:52,020 Times the quantum of action h bar. In a magnetic field along z, we have a 13 00:00:52,020 --> 00:00:55,501 trivial 2 by 2 Hamiltonian whose eigenvalues are given by the diagonal 14 00:00:55,501 --> 00:00:59,284 elements. With corresponding eigenvectors which are 15 00:00:59,284 --> 00:01:03,019 purely one spin component or the other. We see that the first moves up in energy 16 00:01:03,019 --> 00:01:06,724 which we expect from a magnetic moment pointed anti-parallel to the magnetic 17 00:01:06,724 --> 00:01:10,445 field. Because of spin angular momentum and its 18 00:01:10,445 --> 00:01:14,705 magnetic moment or antiparallel, however, this corresponds to spin parallel to the 19 00:01:14,705 --> 00:01:17,500 field. And so we call it spin up. 20 00:01:20,350 --> 00:01:23,430 The other moves down in energy, consistent with a magnetic moment 21 00:01:23,430 --> 00:01:28,920 parallel to the field, which has angular momentum opposite to the field. 22 00:01:28,920 --> 00:01:34,110 It can be called spin down. Note that we can compactly represent the 23 00:01:34,110 --> 00:01:37,510 eigen energies by this simple function of the spin magnetic quantum number, m sub 24 00:01:37,510 --> 00:01:43,440 s. We can use the stationary states of the 25 00:01:43,440 --> 00:01:46,455 simple spin Hamiltonian to calculate the state evolution of a spin in a magnetic 26 00:01:46,455 --> 00:01:52,417 field in which it is not in eigenstate. For example, let's say we prepare a spin 27 00:01:52,417 --> 00:01:56,273 as an eigenstate of Sx. For example, by placing it in a magnetic 28 00:01:56,273 --> 00:01:59,384 field long x and waiting for slow relaxation processes that we ignored in 29 00:01:59,384 --> 00:02:03,740 our Hamiltonian, to take the system to its lowest energy state. 30 00:02:05,440 --> 00:02:09,094 For a magnetic field along the negative x direction, that's the state with a 31 00:02:09,094 --> 00:02:14,800 positive eigen value of S X. The ket labeled plus. 32 00:02:14,800 --> 00:02:18,300 Now, let's suddenly rotate the magnetic field along Z. 33 00:02:18,300 --> 00:02:21,784 The state is not a eigen state of the new Hamiltonian, but we can decompose it into 34 00:02:21,784 --> 00:02:26,490 a super position of eigen states that diagonalize the Hamiltonian. 35 00:02:26,490 --> 00:02:29,703 And which each evolve in time with a different complex exponential given by 36 00:02:29,703 --> 00:02:33,672 its energy. We can now use this to determine how the 37 00:02:33,672 --> 00:02:37,530 expectation value of spin along x changes with time. 38 00:02:37,530 --> 00:02:39,900 We simply sandwich the operator by the state. 39 00:02:39,900 --> 00:02:43,995 Remembering to not only transpose but also complex conjugate the bra on the 40 00:02:43,995 --> 00:02:48,600 left converting it from a column vector to a row vector. 41 00:02:49,630 --> 00:02:52,050 The result is a quantity which oscillates in time. 42 00:02:53,850 --> 00:02:58,931 But how is spin conserved then? If s x is changing, where does the spin 43 00:02:58,931 --> 00:03:03,694 go? By calculating the expectation value of 44 00:03:03,694 --> 00:03:08,406 spin along the y direction, s y we see that it too is oscillating in time, 90° 45 00:03:08,406 --> 00:03:15,540 out of phase with spin along x. This is simply the quantum mechanical 46 00:03:15,540 --> 00:03:19,560 version of a process we are all familiar with, the Larmor precession of an object 47 00:03:19,560 --> 00:03:24,650 with angular momentum subject to a torque, such as a gyroscope. 48 00:03:25,860 --> 00:03:30,180 In this classical example, it's a mechanical torque due to gravity. 49 00:03:30,180 --> 00:03:34,156 Whereas the spin undergoes procession due to a mechanic, magnetic torque, caused by 50 00:03:34,156 --> 00:03:40,080 a magnetic moment in a magnetic field. Note that the procession frequency is 51 00:03:40,080 --> 00:03:45,110 controlled by the Zeeman splitting from magnetic field magnitude. 52 00:03:45,110 --> 00:03:48,782 This field control over spin procession Larmor frequency is precisely how we can 53 00:03:48,782 --> 00:03:52,140 spacially image using magnetic resonance imaging. 54 00:03:53,200 --> 00:03:56,450 Powerful electromagnets are used to create strong magnetic fields and field 55 00:03:56,450 --> 00:03:59,537 gradients. So that only a small volume will the 56 00:03:59,537 --> 00:04:03,194 Zeeman splitting that corresponds to a resonant exotation from narrow band radio 57 00:04:03,194 --> 00:04:08,009 frequency electromagnetic waves. This selective auscultation is then 58 00:04:08,009 --> 00:04:11,370 sensed by inductive detectors. Pickup coils. 59 00:04:12,560 --> 00:04:15,416 By scanning the resonance slice around in space by changing the electromagnetic 60 00:04:15,416 --> 00:04:18,250 coil currents. Images of deeply buried structures are 61 00:04:18,250 --> 00:04:21,365 created. MRI works by acting on the spin of the 62 00:04:21,365 --> 00:04:25,790 nuclei, not electrons. Which is enabled by the much smaller 63 00:04:25,790 --> 00:04:30,130 nuclear magneton due to much larger mass of the nuclei. 64 00:04:30,130 --> 00:04:36,850 And correspondingly small zamon energies and radio frequencies for excitation. 65 00:04:36,850 --> 00:04:40,546 Although impractical for imaging human biology, electrons spin resonance at 66 00:04:40,546 --> 00:04:44,520 around 20 gigahertz per tesla for the free electron. 67 00:04:44,520 --> 00:04:47,875 Is, however, very useful for study of inanimate samples, such as organic 68 00:04:47,875 --> 00:04:52,514 molecules and solid state materials. Now we can get back to the issue of 69 00:04:52,514 --> 00:04:58,430 anomalous Zeeman effect of atomic spectra in a magnetic field. 70 00:04:58,430 --> 00:05:01,236 Both the orbital and spin angular momentum of the electron have magnetic 71 00:05:01,236 --> 00:05:04,580 moments which couple to the magnetic field. 72 00:05:04,580 --> 00:05:08,399 But as we saw, the relativistic nature of the spin makes inclusion of a non-unity 73 00:05:08,399 --> 00:05:12,200 Thomas G factor necessary. We can of course, diagonalize the 74 00:05:12,200 --> 00:05:16,880 Hamiltonian, and determine the Zeeman splitting for each term separately. 75 00:05:16,880 --> 00:05:19,935 But how do we go about diagonalizing this expression with vectors l and s, that 76 00:05:19,935 --> 00:05:24,768 are, in general, not parallel? The key is to notice that the total 77 00:05:24,768 --> 00:05:30,754 angular momentum, l plus s, is conserved. We expect that the energetic splitting 78 00:05:30,754 --> 00:05:35,040 will be on the same form as before, but with an effective g factor. 79 00:05:35,040 --> 00:05:39,161 Let's see how this works out. First we project both vectors onto the 80 00:05:39,161 --> 00:05:43,717 conserved angular momentum axes by taking the inner product with the unit vector 81 00:05:43,717 --> 00:05:49,430 along j l plus s. For a magnetic field along z, this then 82 00:05:49,430 --> 00:05:53,720 picks out the z component of j, which has eigenvalues h bar times total magnetic 83 00:05:53,720 --> 00:05:59,666 quantum number, m sub j. Carrying out the dot product yields an 84 00:05:59,666 --> 00:06:02,498 expression that is mostly known quantities in terms of the individual 85 00:06:02,498 --> 00:06:06,472 quantum numbers. But what's the dot product between the 86 00:06:06,472 --> 00:06:11,237 spin and the orbital angular momentum? This can be seen by once again looking at 87 00:06:11,237 --> 00:06:20,799 the conserve quantity, j squared. Substituting this quantity into our 88 00:06:20,799 --> 00:06:24,920 Hamiltonian entirely converts it into a scalar energy quantity. 89 00:06:24,920 --> 00:06:28,214 Which is proportional to the magnaton times the magnetic field times the 90 00:06:28,214 --> 00:06:32,570 magnetic quantum number, m sub j. With the coefficient out front, it 91 00:06:32,570 --> 00:06:37,160 depends on little l, s and j. This is the so called Lande G-factor. 92 00:06:37,160 --> 00:06:45,150 Here's a few useful examples. When the orbital quantum number is zero. 93 00:06:45,150 --> 00:06:47,810 We just have the magnetic field interacting with the spin. 94 00:06:47,810 --> 00:06:53,410 Which we know has a g factor of two. Lande correctly predicts this. 95 00:06:53,410 --> 00:06:57,354 For l equals one, the total angular momentum j can be one plus one half, 96 00:06:57,354 --> 00:07:01,781 three halves. Or minus one half, yielding one half. 97 00:07:03,230 --> 00:07:05,880 In the first case, the Lande G-factor is four thirds. 98 00:07:05,880 --> 00:07:09,830 And in the second, two thirds. So from the six l equals one states, 99 00:07:09,830 --> 00:07:14,630 that's three orbital projections times two spin projections. 100 00:07:14,630 --> 00:07:18,164 Four of the j equal to three halves states, where m sub j is minus three 101 00:07:18,164 --> 00:07:22,194 halves minus one half plus one half and plus three halves, have one G factor and 102 00:07:22,194 --> 00:07:25,914 the two remaining, j is equal one half states with m sub j equal to plus and 103 00:07:25,914 --> 00:07:32,680 minus one half have another G factor of half its value. 104 00:07:32,680 --> 00:07:38,430 In a magnetic field, the six split like this. 105 00:07:38,430 --> 00:07:42,160 Now, let's look at transitions to the l = 0 derived states. 106 00:07:42,160 --> 00:07:44,844 The selection rules with Landau Spin Orbit Coupling are analogous to the 107 00:07:44,844 --> 00:07:49,922 orbital selection rules. Changes in quantum number j, are plus or 108 00:07:49,922 --> 00:07:54,794 minus 1, and changes in the total magnetic quantum number, m sub j are 0 109 00:07:54,794 --> 00:08:00,540 and plus 1 or minus 1. This means that only the four j equal 110 00:08:00,540 --> 00:08:04,260 three halves states can radiatively transition to the j equals one half 111 00:08:04,260 --> 00:08:08,608 states. And only the m sub j equal to plus or 112 00:08:08,608 --> 00:08:16,420 minus one half states can transition to both l equals zero states. 113 00:08:16,420 --> 00:08:22,578 This leaves six distinct transitions. The energy of the spectroscopic shifts 114 00:08:22,578 --> 00:08:26,274 that we observe by, by dispersing the emitted photos through the spectrometer 115 00:08:26,274 --> 00:08:29,970 are proportional to the difference in the products of lambda g factor times the 116 00:08:29,970 --> 00:08:36,380 total magnetic quantum number m sub j for initial and final states. 117 00:08:36,380 --> 00:08:40,400 We can now see how multiplates more than the Lorenz triplet are created. 118 00:08:40,400 --> 00:08:46,250 In this example, spectral, spectroscopic line splits into six in a magnetic field. 119 00:08:46,250 --> 00:08:49,598 Each separated by two thirds of the [UNKNOWN] splitting, an all line shift 120 00:08:49,598 --> 00:08:53,400 with magnetic field. It is this spin orbit coupling, that 121 00:08:53,400 --> 00:08:55,470 explains the Anomalous Zeeman effect.