1 00:00:00,008 --> 00:00:02,762 Hello everybody, my name is Ian Applebaum, and I'm an associated 2 00:00:02,762 --> 00:00:06,960 professor of physics at the University of Maryland. 3 00:00:06,960 --> 00:00:10,015 Today I'll be giving a guest lecture in this exciting online course exploring 4 00:00:10,015 --> 00:00:13,774 quantum physics. I'm thrilled to be delivering this topic 5 00:00:13,774 --> 00:00:16,834 in part because my own research deals with experimental aspects of spin 6 00:00:16,834 --> 00:00:21,440 polarized electron transport in semi conductor materials. 7 00:00:21,440 --> 00:00:24,512 And I feel strongly that understanding the historical background I'm going to 8 00:00:24,512 --> 00:00:27,488 tell you about today is absolutely essential in making future progress in 9 00:00:27,488 --> 00:00:33,124 this modern field. I want to start our discussion of the 10 00:00:33,124 --> 00:00:36,820 discovery of electron spin by reminding you of what was known around the turn of 11 00:00:36,820 --> 00:00:41,916 the 19th century about atomic spectra. High voltage across a discharge tube 12 00:00:41,916 --> 00:00:45,836 filled with low pressure hydrogen causes emission of electromagnetic radiation, 13 00:00:45,836 --> 00:00:49,140 optical photons. And a spectrometer can be used to 14 00:00:49,140 --> 00:00:52,500 disburse the different wavelengths across a detector. 15 00:00:54,130 --> 00:00:57,681 The discreet wavelength scene were known long before quantum theory to satisfy 16 00:00:57,681 --> 00:01:01,040 Rydberg's formula. Which told us that the photon energy is 17 00:01:01,040 --> 00:01:05,765 proportional to the difference between the reciprocals of two squared integers. 18 00:01:05,765 --> 00:01:08,900 Bohr's theory, which included quantization of angular momentum in 19 00:01:08,900 --> 00:01:13,130 classical electron orbits, captured this famous result. 20 00:01:13,130 --> 00:01:16,850 But it was wrong for many reasons we now know, such as failure to predict the 21 00:01:16,850 --> 00:01:22,359 ground state absence of angular momentum. And importantly the degeneracy of each 22 00:01:22,359 --> 00:01:28,000 principal electronic level. From the solution of Schodringer's 23 00:01:28,000 --> 00:01:31,068 equation in spherical symmetric potentials, we know not only that the 24 00:01:31,068 --> 00:01:35,680 ground state has zero angular momentum. But also that the excited levels where 25 00:01:35,680 --> 00:01:39,060 the principal quantum number is not one are actually degenerate meaning that 26 00:01:39,060 --> 00:01:42,329 several states have the same eigen energy. 27 00:01:43,840 --> 00:01:48,056 This follows from the multiple spherical harmonics labeled by the polar quantum 28 00:01:48,056 --> 00:01:51,820 number, L, and the azimuth of quantum number, M. 29 00:01:51,820 --> 00:01:57,798 A spectroscopic notation is to label l equals 0 as S, l equals 1 P, l equals 2 30 00:01:57,798 --> 00:02:05,290 D, and so forth. Each level of principle quantum number n 31 00:02:05,290 --> 00:02:09,930 has a maximum l equal to n minus 1 and states with a given value for l have 32 00:02:09,930 --> 00:02:14,890 values of m given by minus l, minus l plus 1, in integer steps through 0 and up 33 00:02:14,890 --> 00:02:24,279 to l minus 1 and l. This gives a total degeneracy of 2l plus 34 00:02:24,279 --> 00:02:27,656 1. Therefore, each set labeled by a given 35 00:02:27,656 --> 00:02:31,630 principle quantum number n has degeneracy of n squared. 36 00:02:32,940 --> 00:02:36,132 A natural question to ask then is how can we observe this degeneracy 37 00:02:36,132 --> 00:02:40,238 experimentally? We need to break the degeneracy, split 38 00:02:40,238 --> 00:02:43,639 the levels and alter the photon emission spectrum. 39 00:02:44,660 --> 00:02:49,450 We can do this with a magnetic field. Here's how. 40 00:02:49,450 --> 00:02:53,014 Let's imagine that we have a classical electron orbit, with angular momentum, 41 00:02:53,014 --> 00:02:59,214 given by this expression. Now, this circulating charged particle 42 00:02:59,214 --> 00:03:06,220 comprises a current, and so the orbit has, necessarily, a magnetic moment. 43 00:03:06,220 --> 00:03:10,064 Classically we know that the magnetic moment, that absolute value of that 44 00:03:10,064 --> 00:03:14,094 magnetic moment is given by just the current times the area that it circulates 45 00:03:14,094 --> 00:03:19,820 around. Now the current in this case is one 46 00:03:19,820 --> 00:03:25,270 electron circulating around in a given period. 47 00:03:25,270 --> 00:03:30,790 So this is the charge of the electron, minus e times the frequency of its orbit. 48 00:03:30,790 --> 00:03:35,290 The area of course for circular orbit is just pie times the radius squared. 49 00:03:37,040 --> 00:03:41,380 Now, the frequency of the orbit is just given by the velocity divided by the 50 00:03:41,380 --> 00:03:45,832 circumference, 2 pie r. And that gives rise to a simple 51 00:03:45,832 --> 00:03:52,383 expression for the magnetic moment. But here we're going to play a little bit 52 00:03:52,383 --> 00:03:58,990 of a trick and multiply and divide by the electron mass and also planck's constant. 53 00:03:58,990 --> 00:04:04,532 And the reason why we do this, is that we can see mvr. 54 00:04:04,532 --> 00:04:12,155 Here is the absolute value of the total angular momenteum which has the same 55 00:04:12,155 --> 00:04:20,436 units as the action h bar. That means that this fraction out front 56 00:04:20,436 --> 00:04:26,130 carries all of the units of the magnetic moment. 57 00:04:26,130 --> 00:04:30,946 In fact, it has a special name. It's called the Bohr magneton. 58 00:04:30,946 --> 00:04:37,282 So, with this magnetic moment, we know that in a magnetic field, each state is 59 00:04:37,282 --> 00:04:43,880 going to acquire an energy, due to the interaction. 60 00:04:43,880 --> 00:04:47,207 Now, if the magnetic field is along the z axis. 61 00:04:47,207 --> 00:04:51,827 Then we can simply write this interaction energy as minus the z component of the 62 00:04:51,827 --> 00:04:56,410 magnetic moment times the magnetic field along z. 63 00:04:57,570 --> 00:05:02,262 So, if different states, with the same principle quantum number n have different 64 00:05:02,262 --> 00:05:06,678 values for the z component of the magnetic moment, there energy eigenvalues 65 00:05:06,678 --> 00:05:13,980 will shift differently in a magnetic field, and the degeneracy will be broken. 66 00:05:15,900 --> 00:05:19,150 So we need to calculate the magnetic moment associated with each orbital from 67 00:05:19,150 --> 00:05:23,385 the preceding discussion. We know the relationship between angular 68 00:05:23,385 --> 00:05:26,946 momentum and magnetic moment. And if the field is aligned to the z axis 69 00:05:26,946 --> 00:05:30,430 we only need to calculate the vector component along z. 70 00:05:30,430 --> 00:05:33,406 But this means we need to know the angular momentum vector component along 71 00:05:33,406 --> 00:05:35,738 z. So we need to calculate its expectation 72 00:05:35,738 --> 00:05:38,700 value and for that we need an operator representation. 73 00:05:39,900 --> 00:05:42,812 We know that the linear momentum operators are h bar over I times the 74 00:05:42,812 --> 00:05:47,490 derivative with respect to the conjugate real space variable. 75 00:05:47,490 --> 00:05:51,396 So it's easy to see that the angular momentum along z takes this form because 76 00:05:51,396 --> 00:05:55,760 phi, the actual total angle, is its concrete variable. 77 00:05:57,130 --> 00:06:00,310 The expectation value then is this matrix element. 78 00:06:00,310 --> 00:06:04,256 The operator sandwiched by the state. The only part of the wave function that 79 00:06:04,256 --> 00:06:08,960 matters here is the phi dependence, which we know from the spherical harmonics. 80 00:06:08,960 --> 00:06:12,340 The derivative brings down a factor of I times m and what's left is the original 81 00:06:12,340 --> 00:06:16,836 normalized state. We see, then, that our answer is integer 82 00:06:16,836 --> 00:06:20,430 units of h bar determined by the azimuthal quantum number m. 83 00:06:21,440 --> 00:06:24,620 This is why m is commonly called the magnetic quantum number. 84 00:06:25,740 --> 00:06:30,386 Now we can complete our calculation. The energy added due to the interaction 85 00:06:30,386 --> 00:06:34,354 of the orbit's magnetic moment with magnetic field along z is simply the Bohr 86 00:06:34,354 --> 00:06:38,074 magneton times the magnetic field strength times the magnetic quantum 87 00:06:38,074 --> 00:06:42,934 number. This result is pleasing enough, but some 88 00:06:42,934 --> 00:06:45,443 understanding of the scale of the effect is helpful. 89 00:06:45,443 --> 00:06:51,570 The Bohr magneton is small, about 60 microelectron volts per Tesla. 90 00:06:51,570 --> 00:06:55,166 And a Tesla is a huge magnetic field, about 20,000 times the strength of the 91 00:06:55,166 --> 00:06:59,528 Earth's geomagnetic field. The largest fields in the lab created 92 00:06:59,528 --> 00:07:04,230 with super conducting coils are several tens of Tesla. 93 00:07:04,230 --> 00:07:08,451 The energies are then small in comparison to the electronic transitions, so any 94 00:07:08,451 --> 00:07:13,982 observed shifts of the spectral lines are going to be proportionately small. 95 00:07:13,982 --> 00:07:17,876 So here's what happens, a non-zero magnetic field induces a splitting 96 00:07:17,876 --> 00:07:22,562 between degenerate states, adding energy to states with positive magnetic quantum 97 00:07:22,562 --> 00:07:26,540 number m. And subtracting from states with negative 98 00:07:26,540 --> 00:07:29,062 m. All states with m equal 0, including the 99 00:07:29,062 --> 00:07:34,870 sole ground-state level, are un-affected. The splitting energy is named after 100 00:07:34,870 --> 00:07:38,390 Pieter Zeeman who won the Nobel Prize in 1902 for observing the spectral line 101 00:07:38,390 --> 00:07:42,765 splitting when gas tubes were placed in a magnetic field. 102 00:07:42,765 --> 00:07:47,187 Now we have to remember that this energy spectrum we calculate is not the same as 103 00:07:47,187 --> 00:07:53,237 the optical spectrum of emitted photos. Which are only due to transitions between 104 00:07:53,237 --> 00:07:56,870 the levels and not all transitions are allowed. 105 00:07:56,870 --> 00:08:01,292 The expressions which determine when the transitions are allowed are known as 106 00:08:01,292 --> 00:08:05,425 selection rules. We're going to calculate them next. 107 00:08:05,425 --> 00:08:08,963 During an electronic transition, the electron wave function forms a 108 00:08:08,963 --> 00:08:14,039 superposition of the initial state and the final state with lower energy. 109 00:08:15,050 --> 00:08:17,360 Each component evolves differently in time. 110 00:08:17,360 --> 00:08:21,530 So the electron probability distribution centre of mass can move. 111 00:08:21,530 --> 00:08:25,080 This oscillating dipole is what radiates electromagnetic energy. 112 00:08:26,420 --> 00:08:31,014 A simple calculation of the expectation value of position yields this expression. 113 00:08:31,014 --> 00:08:34,542 Where we can see the first two direct terms from the interproduct of the wave 114 00:08:34,542 --> 00:08:40,155 function are symmetric. Whereas r, the radio, radial variable, is 115 00:08:40,155 --> 00:08:45,270 antisymmetric, so integration over them benefits. 116 00:08:45,270 --> 00:08:48,432 The cross terms which oscillate at the frequency determined by the energy 117 00:08:48,432 --> 00:08:52,630 difference of initial and final states may survive. 118 00:08:52,630 --> 00:08:55,668 Now, though our wave functions have previously been written in spherical 119 00:08:55,668 --> 00:08:58,960 coordinates. We want to convert this integral into 120 00:08:58,960 --> 00:09:02,376 Cartesian coordinates, so we may see along which axis we might get charge 121 00:09:02,376 --> 00:09:05,932 oscillation and electromagnetic radiation. 122 00:09:05,932 --> 00:09:10,807 Here's the transformation we need in terms of the polar and azimuthal angles, 123 00:09:10,807 --> 00:09:14,910 theta and phi. It allows us to split the vector integral 124 00:09:14,910 --> 00:09:19,150 from the previous slide into individual Cartesian components. 125 00:09:20,210 --> 00:09:23,260 First lets consider the x and y components of the integral over azimuthal 126 00:09:23,260 --> 00:09:26,838 angle. We'll have to evaluate something like 127 00:09:26,838 --> 00:09:31,020 this. Two complex exponents containing phi and 128 00:09:31,020 --> 00:09:35,806 a sign of cosine over phi. Using the euler formulas you see that 129 00:09:35,806 --> 00:09:39,782 this will involve integrals over integer periods of the osci-, oscillating complex 130 00:09:39,782 --> 00:09:44,805 exponential. Which is identically equal to 0, except 131 00:09:44,805 --> 00:09:49,638 when the exponent is 0. Since the z component does not depend on 132 00:09:49,638 --> 00:09:51,956 phi. It's even easier to see that in this 133 00:09:51,956 --> 00:09:55,540 case, the integral will be equal to 0 only if the magnetic quantum numbers for 134 00:09:55,540 --> 00:10:01,474 initial and final states are the same. Taking together we see that there is no 135 00:10:01,474 --> 00:10:06,585 dipole transition and no corresponding emission of photons. 136 00:10:06,585 --> 00:10:12,240 Unless the change in magnetic quantum number is plus or minus 1 or 0. 137 00:10:12,240 --> 00:10:16,658 This is our selection rule. By considering these kinds of arguments 138 00:10:16,658 --> 00:10:20,142 about periodic symmetry of the intergrand and the integral over the polar angel 139 00:10:20,142 --> 00:10:23,970 theta. We can likewise derive another selection 140 00:10:23,970 --> 00:10:27,834 rule for transitions such that the change in orbital quantum number must be plus or 141 00:10:27,834 --> 00:10:33,153 minus 1. So which transitions are allowed. 142 00:10:33,153 --> 00:10:36,995 l must change by 1 and m can change at most by 1. 143 00:10:36,995 --> 00:10:41,590 The 2P states can decay to the 1S ground state. 144 00:10:41,590 --> 00:10:45,118 And in a magnetic field, all 3 of these transitions from m equals 1, 0 minus 1, 145 00:10:45,118 --> 00:10:48,982 have different energies giving different emitted photon wavelengths that we can 146 00:10:48,982 --> 00:10:55,294 analyze with a spectrometer. The 3S state decays only to the 32P 147 00:10:55,294 --> 00:11:04,175 states, and 3P decays to 2S, also with 3 distinct transition energies. 148 00:11:04,175 --> 00:11:08,790 For higher values of principal quantum number n, we can have transitions from 149 00:11:08,790 --> 00:11:13,495 the l equals 2d states with a broken degeneracy of 5. 150 00:11:13,495 --> 00:11:17,968 Which can make transitions the lower p states, this involves many possible 151 00:11:17,968 --> 00:11:22,654 transitions that satisfy the selection rules, however there are always only 3 152 00:11:22,654 --> 00:11:31,842 transition energies. Delta m equals minus 1, 0, and plus 1. 153 00:11:31,842 --> 00:11:38,505 So each spectral line splits into a triplet in a magnetic field. 154 00:11:38,505 --> 00:11:43,010 Now, I want to point out something extremely improtant in atomic 155 00:11:43,010 --> 00:11:47,390 spectroscopy. Recall the energy time uncertainty 156 00:11:47,390 --> 00:11:49,922 principle. It says that there's a reciprocal 157 00:11:49,922 --> 00:11:53,107 relationship between the lifetime of a state and the resulting spectral line 158 00:11:53,107 --> 00:11:57,820 width, which limits resolution of high precision measurements. 159 00:11:57,820 --> 00:12:01,460 If a transition is forbidden by dipole selection rule, other processes may be 160 00:12:01,460 --> 00:12:04,840 allowed. But are typically far less efficient, and 161 00:12:04,840 --> 00:12:09,372 result in exceptionally long lifetimes. Here, we see that the transition to the 162 00:12:09,372 --> 00:12:14,150 ground state, from the 2S state is forbidden by the selection rules. 163 00:12:14,150 --> 00:12:18,505 And other processes yield a lifetime, of over 100 milliseconds in comparison to 164 00:12:18,505 --> 00:12:25,910 the nearly equal transition. From 2p to 1s, in about 1 ns. 165 00:12:25,910 --> 00:12:28,925 The line width of this transition is therefore extremely narrow, allowing very 166 00:12:28,925 --> 00:12:33,910 high precision measurements of exquisite quantum effects, such as the lamb shift. 167 00:12:33,910 --> 00:12:37,531 Due to small corrections of energy levels from quantum electrodynamical effects of 168 00:12:37,531 --> 00:12:41,654 the electron interacting with short-lived excitations in the vacuum. 169 00:12:41,654 --> 00:12:45,218 In addition to the selection rules, we can use the different forms of the dipole 170 00:12:45,218 --> 00:12:50,234 vector components to explain another feature of Zeeman split spectral lines. 171 00:12:50,234 --> 00:12:54,350 They're optical polarizations and directional dependents. 172 00:12:54,350 --> 00:12:58,900 For instance, we know that the transition corresponding to delta N is 0. 173 00:12:58,900 --> 00:13:02,676 No change in the magnetic quantum number during the transition is called by a 174 00:13:02,676 --> 00:13:08,670 dipole along the magnetic field axis Z. However we also know from classical 175 00:13:08,670 --> 00:13:13,978 electrodynamics that an oscillating dipole does not radiate along its axis. 176 00:13:13,978 --> 00:13:18,610 Therefore this spectral line is absent when observed along this orientation. 177 00:13:19,750 --> 00:13:23,715 The other two are present and do have the dipoles oscillating 90 degrees out of 178 00:13:23,715 --> 00:13:29,550 phase along X and Y, yielding left and right handed circularly polarized light. 179 00:13:29,550 --> 00:13:35,055 If we observe from a direction perpendicular to the magnetic field axis. 180 00:13:35,055 --> 00:13:39,402 Then, the delta m equal to zero line in the middle can be seen, and has a linear 181 00:13:39,402 --> 00:13:45,220 polarization along z. The other two are polarized perpendicular 182 00:13:45,220 --> 00:13:49,540 to the field axis because they are, again, due to dipoles along x and y. 183 00:13:51,540 --> 00:13:55,892 Hendrik Lorentz won the Nobel Prize in 1902 along with Zeeman for explaining 184 00:13:55,892 --> 00:14:01,065 this polarization dependence. He used only classical physics, a theory 185 00:14:01,065 --> 00:14:05,522 which we now know is wrong. Despite this perceived success in 186 00:14:05,522 --> 00:14:10,340 explaining the Zeeman effect, a serious problem remained, some special lines 187 00:14:10,340 --> 00:14:18,645 split into triple lines as predicted. But others look into a multiplex, four, 188 00:14:18,645 --> 00:14:23,485 six, etc. Here's a few examples. 189 00:14:23,485 --> 00:14:28,785 Now you haven't made a silly mistake. We just need to re-examine the 190 00:14:28,785 --> 00:14:32,886 ingredients of our theory, namely the Schrodinger equation. 191 00:14:34,610 --> 00:14:37,823 Converting the classical kinetic energy into an operator is correct in the 192 00:14:37,823 --> 00:14:41,240 absolutely non-relativistic case, as we've done here with construction of the 193 00:14:41,240 --> 00:14:45,593 Schrodinger equation. But we're clearly leaving out an 194 00:14:45,593 --> 00:14:48,932 essential piece of physics. We're not even using the relativistically 195 00:14:48,932 --> 00:14:54,342 covariant expression. Now, can we fix the problem by starting 196 00:14:54,342 --> 00:14:57,422 from scratch, constructing a wave equation, by starting with the 197 00:14:57,422 --> 00:15:01,980 relativistic expression for kinetic energy that's given here. 198 00:15:03,680 --> 00:15:05,230 That's what we're going to see in the next slides. 199 00:15:07,180 --> 00:15:10,597 By the way, those of you who haven't seen this expression before, might want to see 200 00:15:10,597 --> 00:15:13,963 that It's asymptotically equivalent to the classical expression in the limit 201 00:15:13,963 --> 00:15:19,723 that the momentum p is small. Then we can expand the square root in a 202 00:15:19,723 --> 00:15:23,221 Taylor series and see that the dominant terms are the familiar mc squared, rest 203 00:15:23,221 --> 00:15:27,000 mass energy, and the classical kinetic energy. 204 00:15:28,110 --> 00:15:31,230 Everything else is small, although not negligible, as we will see. 205 00:15:31,230 --> 00:15:35,632 If we take the relativistic expression, and try to use it as an operator on a 206 00:15:35,632 --> 00:15:40,600 wave function. We immediately encounter a problem. 207 00:15:40,600 --> 00:15:42,870 Our momentum operators are within the square root. 208 00:15:42,870 --> 00:15:45,979 And it's not clear at all whether this makes any mathematical sense. 209 00:15:47,110 --> 00:15:50,230 The problem disappears if the expression inside the square root is, itself, a 210 00:15:50,230 --> 00:15:54,509 perfect square. If we write the rest mass energy and 211 00:15:54,509 --> 00:15:58,108 kinetic energy components in x, y, and z here, 1, 2, and 3 with arbitrary 212 00:15:58,108 --> 00:16:02,368 coefficients. Then we can make this a perfect square, 213 00:16:02,368 --> 00:16:06,085 if these coefficients satisfy what appears, at first, to be an unusual 214 00:16:06,085 --> 00:16:10,529 constraint. They give unity when squared, but they 215 00:16:10,529 --> 00:16:16,430 anti commute with each other. These coefficients clearly are not scalar 216 00:16:16,430 --> 00:16:21,955 values, however this fact didn't frighten Paul Dirac from wriiting down this 217 00:16:21,955 --> 00:16:27,410 relativisticly and variant equation in 1928. 218 00:16:27,410 --> 00:16:30,580 The alpha coefficients are clearly not scalar values. 219 00:16:30,580 --> 00:16:34,420 But matrices which satisfy the anti commutation relations. 220 00:16:34,420 --> 00:16:38,600 And form a so called Clifford Algebra. Now, we can write down many matrices 221 00:16:38,600 --> 00:16:41,942 which satisfy the algebra. But it makes sense to first look at the 222 00:16:41,942 --> 00:16:45,160 simplest case with the smallest dimension matrices. 223 00:16:45,160 --> 00:16:49,260 It turns out this can be done with matrices as small as four by four. 224 00:16:49,260 --> 00:16:53,541 Here's one choice of basis. I2 is a 2 by 2 identity, and o2 is a 2 by 225 00:16:53,541 --> 00:17:00,132 2 matrix of all zeroes. The 2 by 2 Pauli matrices, the sigmas, 226 00:17:00,132 --> 00:17:07,835 form the off diagonal blocks of the alpha 1, 2, and 3, 4 by 4 matrices. 227 00:17:07,835 --> 00:17:12,671 Now importantly this converts the wave equation into a four by four matrix 228 00:17:12,671 --> 00:17:17,610 equation. And the wave function into a four 229 00:17:17,610 --> 00:17:21,970 component vector. Two of these correspond to a rest mass 230 00:17:21,970 --> 00:17:26,170 energy of mc squared, when the momentum p is equal to 0, as we expect for an 231 00:17:26,170 --> 00:17:31,422 electron. But an actual question to ask is why two 232 00:17:31,422 --> 00:17:35,395 values? Degeneracies like this are a signature of 233 00:17:35,395 --> 00:17:39,830 symmetry, but which one? Which degree of freedom do these two 234 00:17:39,830 --> 00:17:43,330 values correspond to? We're going to look at an experiment for 235 00:17:43,330 --> 00:17:44,070 a clue