1 00:00:00,050 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,820 under a Creative Commons license. 3 00:00:03,820 --> 00:00:06,540 Your support will help MIT OpenCourseWare continue 4 00:00:06,540 --> 00:00:10,120 to offer high quality educational resources for free. 5 00:00:10,120 --> 00:00:12,700 To make a donation, or to view additional materials 6 00:00:12,700 --> 00:00:16,590 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,590 --> 00:00:17,282 at ocw.mit.edu. 8 00:00:21,600 --> 00:00:25,940 PROFESSOR: OK, today's lecture will 9 00:00:25,940 --> 00:00:30,770 begin with photon states, which is 10 00:00:30,770 --> 00:00:33,710 a very interesting application of what 11 00:00:33,710 --> 00:00:36,860 we've learned about coherent states. 12 00:00:36,860 --> 00:00:41,530 And in a sense, it's an advanced topic. 13 00:00:41,530 --> 00:00:46,675 Photon states are states of the electromagnetic field. 14 00:00:46,675 --> 00:00:50,530 They are quantum states of the electromagnetic field. 15 00:00:50,530 --> 00:00:54,250 You A photon, this particle, is a quantum 16 00:00:54,250 --> 00:00:55,570 of the electromagnetic field. 17 00:00:55,570 --> 00:01:00,350 There's a discrete piece of energy and momentum carried 18 00:01:00,350 --> 00:01:02,160 by this particle. 19 00:01:02,160 --> 00:01:06,000 So when we talk about photon states, 20 00:01:06,000 --> 00:01:09,430 we're really doing quantum field theory. 21 00:01:09,430 --> 00:01:12,690 So in some sense, this lecture, you 22 00:01:12,690 --> 00:01:17,580 will see how quantum field theory works. 23 00:01:17,580 --> 00:01:22,430 A first introduction to quantum field theory. 24 00:01:22,430 --> 00:01:27,100 And it's interesting that the harmonic oscillator 25 00:01:27,100 --> 00:01:30,080 plays such an important role in that. 26 00:01:30,080 --> 00:01:34,460 So a key identity that we are going to use, of course, 27 00:01:34,460 --> 00:01:38,050 is this coherent states that were 28 00:01:38,050 --> 00:01:41,990 defined as displacements of the vacuum. 29 00:01:41,990 --> 00:01:46,140 For D, if I remember right, was e 30 00:01:46,140 --> 00:01:52,385 to the alpha a dagger minus alpha star a. 31 00:01:56,080 --> 00:02:01,730 And it had the property that a acting on alpha 32 00:02:01,730 --> 00:02:08,889 was equal to alpha-- alpha, the operator a. 33 00:02:08,889 --> 00:02:10,780 So these were the coherent states 34 00:02:10,780 --> 00:02:13,260 we've been talking about. 35 00:02:13,260 --> 00:02:16,425 And today we're going to talk about photon states. 36 00:02:25,840 --> 00:02:30,090 So that will be probably about half of the lecture. 37 00:02:30,090 --> 00:02:32,050 And in the second half of the lecture, 38 00:02:32,050 --> 00:02:39,640 we will begin a more systematic study of two-state systems. 39 00:02:39,640 --> 00:02:42,710 Two-state systems, of course, are our spin states, 40 00:02:42,710 --> 00:02:45,660 are the classical two-state system. 41 00:02:45,660 --> 00:02:50,310 And we're going to sort of put it all together. 42 00:02:50,310 --> 00:02:54,460 We'll understand the general dynamics of a two-spin system, 43 00:02:54,460 --> 00:02:57,415 what is the most general Hamiltonian you can have, 44 00:02:57,415 --> 00:03:00,600 and therefore the most General Dynamics. 45 00:03:00,600 --> 00:03:03,160 And then we'll solve that. 46 00:03:03,160 --> 00:03:05,650 And we'll have two physical examples, 47 00:03:05,650 --> 00:03:09,680 one having to deal with the ammonia molecule. 48 00:03:09,680 --> 00:03:12,930 And another having to do with nuclear magnetic resonance. 49 00:03:12,930 --> 00:03:15,660 Both are applications of two-state systems. 50 00:03:15,660 --> 00:03:21,170 So till it's the end of the lecture, we'll be doing that. 51 00:03:21,170 --> 00:03:25,010 So about photon states. 52 00:03:25,010 --> 00:03:29,980 Well, photon states have to do with electromagnetic fields. 53 00:03:29,980 --> 00:03:31,980 That's electric and magnetic fields. 54 00:03:31,980 --> 00:03:34,200 And one important quantity that you 55 00:03:34,200 --> 00:03:40,020 know about the electromagnetic fields is the energy. 56 00:03:40,020 --> 00:03:42,800 If you have an electromagnetic field, you have an energy. 57 00:03:42,800 --> 00:03:45,660 And remember, energies have to do with Hamiltonians. 58 00:03:45,660 --> 00:03:47,930 So we're going to try to do a quantum 59 00:03:47,930 --> 00:03:50,500 description of the electromagnetic field. 60 00:03:50,500 --> 00:03:55,210 Therefore, knowing the energy would be a good place to start. 61 00:03:55,210 --> 00:03:58,570 So the energy, as you know, in an electromagnetic field 62 00:03:58,570 --> 00:04:03,620 goes like e squared times some epsilon and b squared. 63 00:04:03,620 --> 00:04:05,610 And you add the two of them. 64 00:04:05,610 --> 00:04:07,140 So here we go. 65 00:04:07,140 --> 00:04:09,340 Let me write a precise formula. 66 00:04:09,340 --> 00:04:13,830 The energy is equal to 1/2 the integral 67 00:04:13,830 --> 00:04:24,040 over volume, epsilon 0, the electric field squared, 68 00:04:24,040 --> 00:04:27,040 plus c squared times the magnetic field squared. 69 00:04:34,120 --> 00:04:35,695 So this is our energy. 70 00:04:38,420 --> 00:04:41,300 And we're going to try to describe 71 00:04:41,300 --> 00:04:45,640 a configuration of electromagnetic fields. 72 00:04:45,640 --> 00:04:50,190 We will to focus on one mode of the electromagnetic field. 73 00:04:50,190 --> 00:04:55,540 So I will imagine I have some sort of cavity, finite volume. 74 00:04:55,540 --> 00:04:58,850 And in there I have one electromagnetic field, 75 00:04:58,850 --> 00:05:06,110 what you usually called in 802 or in 8022 or 807 a wave. 76 00:05:06,110 --> 00:05:09,690 A single wave with some wavelength, some frequency, 77 00:05:09,690 --> 00:05:11,420 and that's all we have. 78 00:05:11,420 --> 00:05:13,970 So we're going to simplify to the case 79 00:05:13,970 --> 00:05:18,170 where we have a single one consistent with Maxwell's 80 00:05:18,170 --> 00:05:23,950 equations and some boundary conditions that we need not 81 00:05:23,950 --> 00:05:26,500 worry about. 82 00:05:26,500 --> 00:05:33,120 And I will normalize them as follows with a V in here. 83 00:05:33,120 --> 00:05:37,590 That this is the volume of the system. 84 00:05:37,590 --> 00:05:38,240 So, volume. 85 00:05:42,550 --> 00:05:45,250 And that could be the volume of the cavity that 86 00:05:45,250 --> 00:05:47,220 has this electromagnetic field. 87 00:05:47,220 --> 00:05:49,090 Or some large box. 88 00:05:49,090 --> 00:05:53,550 Or you can let it almost be infinite and work with that 89 00:05:53,550 --> 00:05:55,130 as well. 90 00:05:55,130 --> 00:05:59,210 So we'll have a wave. 91 00:05:59,210 --> 00:06:01,360 Omega would be the frequency. 92 00:06:01,360 --> 00:06:07,140 K is omega over c for a electromagnetic wave. 93 00:06:07,140 --> 00:06:13,480 So we'll have this times omega sine 94 00:06:13,480 --> 00:06:19,250 of kz, a spatial distribution. 95 00:06:19,250 --> 00:06:22,580 And there will be a function of time, as you know. 96 00:06:22,580 --> 00:06:24,570 But this function of time, I want 97 00:06:24,570 --> 00:06:30,150 to leave it a little ambiguous at this moment-- or, general. 98 00:06:30,150 --> 00:06:31,470 Not ambiguous, general. 99 00:06:31,470 --> 00:06:37,420 So I'll call it q of t, some function of time 100 00:06:37,420 --> 00:06:38,165 to be determined. 101 00:06:41,020 --> 00:06:44,900 There's going to be an electromagnetic and magnetic 102 00:06:44,900 --> 00:06:47,442 component to this field, By. 103 00:06:47,442 --> 00:06:51,320 c times By will also depend on z and t 104 00:06:51,320 --> 00:06:54,050 and will have the same pre-factor. 105 00:06:54,050 --> 00:06:55,820 I put the c here. 106 00:06:55,820 --> 00:06:59,760 So your c squared b squared also works well. 107 00:06:59,760 --> 00:07:02,260 Epsilon 0 v. 108 00:07:02,260 --> 00:07:10,105 This time I'll put another function, p of t cosine of kz. 109 00:07:13,630 --> 00:07:18,790 It's another function of time and I just call them that way. 110 00:07:18,790 --> 00:07:20,996 There is a question there. 111 00:07:20,996 --> 00:07:25,830 STUDENT: Why is your frequency outside your function of time? 112 00:07:25,830 --> 00:07:30,220 PROFESSOR: It's just another constant here. 113 00:07:30,220 --> 00:07:32,000 STUDENT: What would that mean then? 114 00:07:32,000 --> 00:07:34,770 PROFESSOR: No particular meaning to it. 115 00:07:37,600 --> 00:07:40,840 At this moment, whatever this constant 116 00:07:40,840 --> 00:07:43,950 is you would say probably it's useful 117 00:07:43,950 --> 00:07:46,410 because you somehow wanted the q here. 118 00:07:46,410 --> 00:07:48,050 That has some meaning. 119 00:07:48,050 --> 00:07:51,860 So you probably would put the same constants here 120 00:07:51,860 --> 00:07:53,230 in first trial. 121 00:07:53,230 --> 00:07:55,010 You wouldn't have this omega here. 122 00:07:55,010 --> 00:07:58,480 But if you put it, this is just another way 123 00:07:58,480 --> 00:08:00,720 of changing their own manifestation of q. 124 00:08:00,720 --> 00:08:04,980 So it doesn't have a profound meaning so far. 125 00:08:07,790 --> 00:08:10,460 Any other questions about this? 126 00:08:10,460 --> 00:08:13,810 This is and electromagnetic field configuration. 127 00:08:13,810 --> 00:08:19,240 And this q of t and p of t are functions of time. 128 00:08:19,240 --> 00:08:21,490 You know your Maxwell's equations. 129 00:08:21,490 --> 00:08:26,490 And you will check things related to Maxwell's equations 130 00:08:26,490 --> 00:08:28,670 for this configuration in the homework. 131 00:08:28,670 --> 00:08:35,669 But at this moment, it's not too crucial. 132 00:08:35,669 --> 00:08:38,549 The thing that this important is that we 133 00:08:38,549 --> 00:08:42,110 can try to calculate the energy now. 134 00:08:42,110 --> 00:08:47,560 And if you do it, well, the squares, the epsilon 0's 135 00:08:47,560 --> 00:08:49,369 are going to disappear. 136 00:08:49,369 --> 00:08:50,910 And you're going to have to integrate 137 00:08:50,910 --> 00:08:57,980 over the box, this integral of sine squared of kz or cosine 138 00:08:57,980 --> 00:08:59,210 squared of kz. 139 00:08:59,210 --> 00:09:01,400 The functions of time don't matter-- 140 00:09:01,400 --> 00:09:04,080 this energy could depend on time. 141 00:09:04,080 --> 00:09:08,570 And the way we've prepared is when you integrate over 142 00:09:08,570 --> 00:09:11,970 sine squared of kz, if the box is big, 143 00:09:11,970 --> 00:09:16,490 it's a good situation where you can replace that for 1/2, which 144 00:09:16,490 --> 00:09:20,070 is the average, and 1/2 for the average of this. 145 00:09:20,070 --> 00:09:27,180 Or you could define where the box extends, 146 00:09:27,180 --> 00:09:30,990 from what values of z to what other values of z's. 147 00:09:30,990 --> 00:09:37,390 And so the integral, in fact, is not any complicated integral. 148 00:09:37,390 --> 00:09:40,240 And we have immediately the answer 149 00:09:40,240 --> 00:09:50,640 that energy is 1/2 p squared of t plus omega squared q 150 00:09:50,640 --> 00:09:52,700 squared of t. 151 00:09:56,890 --> 00:09:59,775 And that was why this omega was here. 152 00:10:03,790 --> 00:10:06,090 There's not really much to this. 153 00:10:06,090 --> 00:10:09,240 Except that when you square it and you take the integral 154 00:10:09,240 --> 00:10:12,340 over the volume, you replace the sine squared by 1/2 155 00:10:12,340 --> 00:10:13,900 and the cosine squared by 1/2. 156 00:10:13,900 --> 00:10:17,260 And that's it. 157 00:10:17,260 --> 00:10:21,340 So actually, the labels that we've chosen here 158 00:10:21,340 --> 00:10:23,460 are pretty good. 159 00:10:23,460 --> 00:10:28,290 This starts to look like a harmonic oscillator. 160 00:10:28,290 --> 00:10:32,400 Except that the mass is gone. 161 00:10:32,400 --> 00:10:39,920 1 over 2m p squared should be plus 1/2 m omega squared 162 00:10:39,920 --> 00:10:42,170 q squared. 163 00:10:42,170 --> 00:10:47,300 So the units are wrong here. 164 00:10:47,300 --> 00:10:52,240 p squared over 2m has units of energy. 165 00:10:52,240 --> 00:10:55,520 But p squared doesn't have units of energy. 166 00:10:55,520 --> 00:11:00,760 And 1/2 m omega squared q squared has units of energy 167 00:11:00,760 --> 00:11:01,980 but this one doesn't. 168 00:11:01,980 --> 00:11:08,100 So the units are a little off for a harmonic oscillator. 169 00:11:08,100 --> 00:11:14,430 So it's interesting to notice now. 170 00:11:14,430 --> 00:11:17,190 But you couldn't have done better. 171 00:11:17,190 --> 00:11:21,490 Because photons have no mass. 172 00:11:21,490 --> 00:11:24,250 And we're trying to describe the electromagnetic field. 173 00:11:24,250 --> 00:11:25,350 It has photons. 174 00:11:25,350 --> 00:11:31,820 So there's no way this could have showed up a mass there. 175 00:11:31,820 --> 00:11:34,440 There's no such thing. 176 00:11:34,440 --> 00:11:37,720 And that's why it doesn't show up. 177 00:11:37,720 --> 00:11:40,720 On the other hand, you can say, well, OK, 178 00:11:40,720 --> 00:11:46,380 let's see if this makes a minimum of sense. 179 00:11:46,380 --> 00:11:48,600 How do we deal with this unit? 180 00:11:48,600 --> 00:11:58,835 So p has units of square root of energy. 181 00:12:03,830 --> 00:12:17,072 And q has units of time times square root of energy. 182 00:12:17,072 --> 00:12:18,380 Why is that? 183 00:12:18,380 --> 00:12:21,800 Because omega has units of 1 over time. 184 00:12:21,800 --> 00:12:26,510 So q over time squared is energy. 185 00:12:26,510 --> 00:12:32,630 So q is t times square root of energy. 186 00:12:32,630 --> 00:12:38,420 And therefore p doesn't have the right units to deserve the name 187 00:12:38,420 --> 00:12:40,550 p. 188 00:12:40,550 --> 00:12:44,520 And q doesn't have the right units to deserve the name q. 189 00:12:44,520 --> 00:12:53,520 But p times q has the units of time times energy, 190 00:12:53,520 --> 00:12:56,145 which are the units of h bar. 191 00:12:59,330 --> 00:13:01,990 So that's good. 192 00:13:01,990 --> 00:13:08,110 This p and q have the right units in some sense. 193 00:13:08,110 --> 00:13:16,990 So this thing could be viewed as an inspiration for you. 194 00:13:16,990 --> 00:13:18,440 And you say at this moment, well, 195 00:13:18,440 --> 00:13:23,080 I don't know what is a quantum of an electromagnetic field. 196 00:13:23,080 --> 00:13:27,220 But here I have a natural correspondence 197 00:13:27,220 --> 00:13:30,235 between one mode of vibration, classical, 198 00:13:30,235 --> 00:13:34,160 of the electromagnetic field, and an energy functional 199 00:13:34,160 --> 00:13:37,780 that looks exactly like a harmonic oscillator. 200 00:13:37,780 --> 00:13:46,036 So I will declare these things to be a Hamiltonian and this p 201 00:13:46,036 --> 00:13:51,430 of t and q of t to be the Heisenberg operators 202 00:13:51,430 --> 00:13:54,580 of the electromagnetic field. 203 00:13:54,580 --> 00:13:58,720 So what we're saying now is that I'm 204 00:13:58,720 --> 00:14:04,840 going to just call the Hamiltonian 1/2 p 205 00:14:04,840 --> 00:14:10,050 hat squared plus omega squared q hat squared. 206 00:14:13,240 --> 00:14:15,830 This is a time independent Hamiltonian. 207 00:14:15,830 --> 00:14:20,020 If you're doing Heisenberg, it's the same thing 208 00:14:20,020 --> 00:14:24,790 as the Hamiltonian that would have p hat square of t 209 00:14:24,790 --> 00:14:30,270 plus omega q hat squared of t. 210 00:14:30,270 --> 00:14:36,250 Now, at this moment, this might sound to you 211 00:14:36,250 --> 00:14:37,750 just too speculative. 212 00:14:37,750 --> 00:14:43,890 But you can do a couple of checks that this is reasonable. 213 00:14:43,890 --> 00:14:51,050 So one check, remember that the Hamiltonian-- quantum equations 214 00:14:51,050 --> 00:14:53,800 of motion, of Heisenberg operators 215 00:14:53,800 --> 00:14:57,710 should look like classical equations of motion. 216 00:14:57,710 --> 00:15:03,460 So I can now compute what are the Heisenberg 217 00:15:03,460 --> 00:15:05,700 equation of motions for the operators. 218 00:15:05,700 --> 00:15:15,010 Remember something like v dt of p Heisenberg of t dt 219 00:15:15,010 --> 00:15:20,670 is related to h with p Heisenberg. 220 00:15:20,670 --> 00:15:26,680 And you can calculate the Heisenberg equations of motion. 221 00:15:26,680 --> 00:15:29,520 I may have signs wrong here. 222 00:15:29,520 --> 00:15:33,070 Nevertheless, you know those for the harmonic oscillator 223 00:15:33,070 --> 00:15:35,280 and you can write them. 224 00:15:35,280 --> 00:15:39,600 But you also know Maxwell's equations. 225 00:15:39,600 --> 00:15:42,330 And you can plug into Maxwell's equations. 226 00:15:42,330 --> 00:15:47,080 And that's one check you will do in homework, in which you will 227 00:15:47,080 --> 00:15:49,760 take Maxwell's equations and see what 228 00:15:49,760 --> 00:15:52,390 equations you have for q of t p of t. 229 00:15:52,390 --> 00:15:55,920 And then they will be exactly the same 230 00:15:55,920 --> 00:15:59,700 as the Heisenberg equations of motion of this Hamiltonian, 231 00:15:59,700 --> 00:16:03,660 giving you evidence that this is a reasonable thing to do. 232 00:16:03,660 --> 00:16:09,710 That we can think of this dynamical system with q and p 233 00:16:09,710 --> 00:16:13,260 being quantum operators. 234 00:16:13,260 --> 00:16:18,480 So let's accept that this is a Hamiltonian for this quantum 235 00:16:18,480 --> 00:16:21,280 system that we want to work with. 236 00:16:21,280 --> 00:16:28,010 And therefore, write the operators that we have. 237 00:16:28,010 --> 00:16:29,300 And what are they? 238 00:16:29,300 --> 00:16:34,820 Well, we had formulas with masses. 239 00:16:34,820 --> 00:16:38,290 But now mass goes to 1. 240 00:16:38,290 --> 00:16:41,850 So know the units. 241 00:16:41,850 --> 00:16:46,220 You cannot let in general in a formula mass going to 1 unless 242 00:16:46,220 --> 00:16:48,840 you're going to do something with the units. 243 00:16:48,840 --> 00:16:54,690 But we agreed already that these p's and q's have funny units. 244 00:16:54,690 --> 00:16:58,130 So those units are in fact consistent with a mass 245 00:16:58,130 --> 00:17:00,530 that has no units. 246 00:17:00,530 --> 00:17:03,390 And you can set it equal to 1. 247 00:17:03,390 --> 00:17:08,869 So I claim that you can take all the formulas we had with m 248 00:17:08,869 --> 00:17:14,190 and just put m equals to 1 and nothing would go wrong. 249 00:17:14,190 --> 00:17:15,770 Nothing goes funny. 250 00:17:15,770 --> 00:17:18,910 So in particular, you had an expression 251 00:17:18,910 --> 00:17:26,050 for x that now is called q terms of creation and annihilation 252 00:17:26,050 --> 00:17:34,500 operators and now that reads-- And you 253 00:17:34,500 --> 00:17:36,190 have an expression for p. 254 00:17:41,820 --> 00:17:47,256 And that one reads now a minus a dagger. 255 00:17:51,470 --> 00:17:54,680 These formulas used to have m's in there. 256 00:17:54,680 --> 00:17:57,180 And I've just set m equals to 1. 257 00:17:57,180 --> 00:18:00,570 And that should be the right thing. 258 00:18:00,570 --> 00:18:04,640 Unit-wise, indeed h bar omega has units of energy. 259 00:18:04,640 --> 00:18:09,430 And we claim that p has units of energy, square root of energy. 260 00:18:09,430 --> 00:18:11,085 So this is fine. 261 00:18:14,160 --> 00:18:19,640 So what else do we get from this Hamiltonian? 262 00:18:19,640 --> 00:18:23,960 Well, we can write it in terms of the number operators. 263 00:18:23,960 --> 00:18:30,890 So this Hamiltonian now, it's equal to h bar omega 264 00:18:30,890 --> 00:18:33,410 a dagger a plus 1/2. 265 00:18:36,470 --> 00:18:42,470 And this is just because this p and q written in this way 266 00:18:42,470 --> 00:18:44,530 corresponds to m equals to 1. 267 00:18:44,530 --> 00:18:48,770 And m doesn't show up anyway in this formula. 268 00:18:48,770 --> 00:18:54,930 So no reason to be worried that anything has gone wrong. 269 00:18:54,930 --> 00:19:05,200 And this is H equals to h bar omega, N hat plus 1/2. 270 00:19:05,200 --> 00:19:09,220 And this is a number operator. 271 00:19:09,220 --> 00:19:11,880 And then you get the interpretation, 272 00:19:11,880 --> 00:19:17,990 the physical interpretation that if you have states 273 00:19:17,990 --> 00:19:23,200 with some number operator, the energy is the number times h 274 00:19:23,200 --> 00:19:27,840 omega, which is exactly what we think about photons. 275 00:19:27,840 --> 00:19:32,510 If you have N photons in a given state, 276 00:19:32,510 --> 00:19:37,680 you would have an energy N times h bar omega. 277 00:19:37,680 --> 00:19:42,790 So it may look a little innocent what we've done here. 278 00:19:42,790 --> 00:19:48,010 But this is a dramatic assumption. 279 00:19:48,010 --> 00:19:54,570 You've really done something that took physicists 30 years 280 00:19:54,570 --> 00:19:58,030 to figure out, how to do quantum field theory. 281 00:19:58,030 --> 00:20:01,510 And of course, this is just the very beginning. 282 00:20:01,510 --> 00:20:03,730 And there's lots of things to learn about it. 283 00:20:03,730 --> 00:20:05,560 But the first thing that is happening 284 00:20:05,560 --> 00:20:10,430 is that somehow-- look what's happening. 285 00:20:10,430 --> 00:20:19,600 In normal quantum mechanics, x and p became quantum operators. 286 00:20:19,600 --> 00:20:23,940 In a sense here, this q and p are like x and p. 287 00:20:23,940 --> 00:20:29,190 But they have nothing to do with usual position and momentum. 288 00:20:29,190 --> 00:20:31,950 Nothing absolutely. 289 00:20:31,950 --> 00:20:34,820 q is like E really. 290 00:20:34,820 --> 00:20:36,750 And p is like B. 291 00:20:36,750 --> 00:20:40,490 So who has become a quantum operator? 292 00:20:40,490 --> 00:20:43,590 Not x and p, in a sense. 293 00:20:43,590 --> 00:20:47,980 E and B have become quantum operators. 294 00:20:47,980 --> 00:20:52,170 Quantum field theory is the idea that the fields 295 00:20:52,170 --> 00:20:54,890 become operators. 296 00:20:54,890 --> 00:20:56,720 That's what's really happening. 297 00:20:56,720 --> 00:21:00,390 And it seems to be right in the sense 298 00:21:00,390 --> 00:21:04,380 that our intuition that the state with N photos 299 00:21:04,380 --> 00:21:07,080 would be viewed as a state of a harmonic 300 00:21:07,080 --> 00:21:11,410 oscillator, an usual one with mass equals 1. 301 00:21:11,410 --> 00:21:13,960 So that this really is not a momentum 302 00:21:13,960 --> 00:21:16,010 and this is not a position. 303 00:21:16,010 --> 00:21:17,940 But they behave as that. 304 00:21:21,020 --> 00:21:33,670 So we can turn now this formula to its Heisenberg form 305 00:21:33,670 --> 00:21:45,340 so that q of t is square root of h bar over 2 omega. 306 00:21:45,340 --> 00:21:50,190 Remember a as a function of time becomes e to the minus 307 00:21:50,190 --> 00:21:54,720 i omega t a hat-- that's the Heisenberg 308 00:21:54,720 --> 00:22:03,100 version of a-- plus e to the plus i omega t a hat dagger. 309 00:22:10,540 --> 00:22:14,970 So given that, we can substitute back 310 00:22:14,970 --> 00:22:22,100 to our electric field that has this omega here, 311 00:22:22,100 --> 00:22:24,690 that has this factor in there. 312 00:22:24,690 --> 00:22:28,050 So I will write it all together here. 313 00:22:28,050 --> 00:22:35,410 Therefore, Ex of z t-- and now I've put a hat here. 314 00:22:35,410 --> 00:22:42,430 And z t, the t is the t of a Heisenberg operator now. 315 00:22:42,430 --> 00:22:55,260 Is equal to E naught e to the minus i omega t a plus 316 00:22:55,260 --> 00:23:04,420 e to the i omega t a hat dagger sine of kz, 317 00:23:04,420 --> 00:23:12,590 where this constant E zero is h bar omega over epsilon 0 V. 318 00:23:12,590 --> 00:23:16,860 It's just a mnemonic for some constant at this moment. 319 00:23:16,860 --> 00:23:20,030 So we plugged-in here, there's all these factors. 320 00:23:20,030 --> 00:23:23,210 There's the omega and there's the q. 321 00:23:25,910 --> 00:23:29,880 So all these factors together give you this. 322 00:23:29,880 --> 00:23:32,690 The factor and sine of kz. 323 00:23:32,690 --> 00:23:41,650 And this is the electromagnetic field operator. 324 00:23:41,650 --> 00:23:45,150 The electric field is not anymore an electric field. 325 00:23:45,150 --> 00:23:45,985 It's an operator. 326 00:23:50,420 --> 00:23:53,980 So if we want to get an intuition 327 00:23:53,980 --> 00:23:57,910 about this electric field operator, 328 00:23:57,910 --> 00:24:01,690 let's try to find its expectation value. 329 00:24:01,690 --> 00:24:02,820 It's an operator. 330 00:24:02,820 --> 00:24:07,160 The closest thing we can have an intuition about an operator 331 00:24:07,160 --> 00:24:08,840 is its expectation value. 332 00:24:08,840 --> 00:24:10,240 So very good. 333 00:24:10,240 --> 00:24:13,540 Let's take a photon state and energy 334 00:24:13,540 --> 00:24:19,860 eigenstate of the harmonic oscillator of occupation number 335 00:24:19,860 --> 00:24:22,720 n. 336 00:24:22,720 --> 00:24:29,700 And we have a state now of n photons, an energy eigenstate. 337 00:24:29,700 --> 00:24:39,260 In fact, with energy n h omega plus this 1/2 h bar omega. 338 00:24:39,260 --> 00:24:42,380 And let's figure out what is the expectation 339 00:24:42,380 --> 00:24:47,310 value of Ex in that state n. 340 00:24:47,310 --> 00:24:49,100 So we go to this formula. 341 00:24:49,100 --> 00:24:56,335 And we say, OK, it's E naught e to the minus i omega t n-- 342 00:24:56,335 --> 00:24:57,960 and we're all very curious. 343 00:24:57,960 --> 00:25:00,750 We want to see how the electromagnetic field 344 00:25:00,750 --> 00:25:05,830 of the n-th state of the harmonic oscillator, n 345 00:25:05,830 --> 00:25:10,290 photons in an energy eigenstate, how does that wave look? 346 00:25:10,290 --> 00:25:12,020 Let's see. 347 00:25:12,020 --> 00:25:27,020 n a hat n plus e to the i omega t n a dagger n sine kz. 348 00:25:31,120 --> 00:25:33,350 So this is a field operator. 349 00:25:33,350 --> 00:25:35,370 So we put it in a state and we want 350 00:25:35,370 --> 00:25:40,056 to know how does the field look in that state. 351 00:25:40,056 --> 00:25:42,622 And how much do we get? 352 00:25:42,622 --> 00:25:44,625 STUDENT: [INAUDIBLE]. 353 00:25:44,625 --> 00:25:45,166 PROFESSOR: 0. 354 00:25:47,880 --> 00:25:50,450 OK, that seems a little strange. 355 00:25:50,450 --> 00:25:55,310 Because indeed, the matrix element 356 00:25:55,310 --> 00:25:58,150 of a in an energy eigenstate is 0. 357 00:25:58,150 --> 00:26:01,410 This reduces, makes n minus 1 [INAUDIBLE] to this. 358 00:26:01,410 --> 00:26:03,190 So this is 0. 359 00:26:03,190 --> 00:26:04,990 And this is n plus 1 n. 360 00:26:04,990 --> 00:26:05,760 This is 0. 361 00:26:05,760 --> 00:26:11,470 So actually no great illumination nation 362 00:26:11,470 --> 00:26:13,550 has happened. 363 00:26:13,550 --> 00:26:15,750 We got 0. 364 00:26:15,750 --> 00:26:22,780 So actually this is not too strange. 365 00:26:22,780 --> 00:26:26,590 Energy eigenstates are very unintuitive. 366 00:26:26,590 --> 00:26:29,285 The energy eigenstate of a harmonic oscillator, 367 00:26:29,285 --> 00:26:32,930 the n-th state is some sort of wave that is like that. 368 00:26:32,930 --> 00:26:36,290 Nothing changes in time in that wave. 369 00:26:36,290 --> 00:26:39,900 Nothing all that interesting happens. 370 00:26:39,900 --> 00:26:43,830 So the fact that this electromagnetic field operator 371 00:26:43,830 --> 00:26:49,540 has zero expectation value on this n photon state 372 00:26:49,540 --> 00:26:52,940 is maybe not too surprising. 373 00:26:52,940 --> 00:26:59,430 So let's take a more thoughtful state. 374 00:26:59,430 --> 00:27:03,380 We've said so many times that coherent states 375 00:27:03,380 --> 00:27:06,120 act like classical states. 376 00:27:06,120 --> 00:27:12,570 So let's put a coherent state of photons into this state. 377 00:27:12,570 --> 00:27:14,350 So let's see. 378 00:27:14,350 --> 00:27:18,100 Now the state will be an alpha state, 379 00:27:18,100 --> 00:27:21,010 which is a coherent state. 380 00:27:21,010 --> 00:27:27,800 And therefore, the expectation value of Ex on the alpha state 381 00:27:27,800 --> 00:27:35,210 will be equal to E not e to the minus i 382 00:27:35,210 --> 00:27:42,740 omega t alpha a alpha plus e to the plus 383 00:27:42,740 --> 00:27:55,040 i omega t alpha a hat dagger alpha sine of kz. 384 00:27:55,040 --> 00:27:58,780 Well, we're in better shape now. 385 00:27:58,780 --> 00:28:02,900 a on alpha is the number alpha, as we 386 00:28:02,900 --> 00:28:04,330 reviewed at the beginning. 387 00:28:04,330 --> 00:28:06,940 And then alpha with alpha is 1. 388 00:28:06,940 --> 00:28:10,440 Remember, it's a unitary transform of the vacuum. 389 00:28:10,440 --> 00:28:13,460 Therefore, this whole thing is alpha. 390 00:28:13,460 --> 00:28:20,070 So this is E not alpha being a number e to the minus i omega 391 00:28:20,070 --> 00:28:33,200 t plus here is alpha star e to the i omega t sine of kz. 392 00:28:33,200 --> 00:28:36,520 And now we're very happy. 393 00:28:36,520 --> 00:28:42,470 The coherent state is the state for which the expectation value 394 00:28:42,470 --> 00:28:45,640 of the electromagnetic field is precisely 395 00:28:45,640 --> 00:28:50,540 the kind of waves you've seen all your life. 396 00:28:50,540 --> 00:28:54,340 This wave, travelling waves, stationary waves. 397 00:28:54,340 --> 00:28:59,950 All those only appeared because only on coherent states 398 00:28:59,950 --> 00:29:04,020 a and a dagger have expectation values. 399 00:29:04,020 --> 00:29:09,870 So what we really call a classical wave resonating 400 00:29:09,870 --> 00:29:16,090 in a cavity is a coherent state of the electromagnetic field 401 00:29:16,090 --> 00:29:18,410 in this sense. 402 00:29:18,410 --> 00:29:21,860 The state of photons form a coherent state. 403 00:29:21,860 --> 00:29:24,880 They're not an energy ion state. 404 00:29:24,880 --> 00:29:28,370 They're not positioned for anything. 405 00:29:28,370 --> 00:29:31,230 They're not the number eigen states either, 406 00:29:31,230 --> 00:29:33,620 because they're not energy eigen states. 407 00:29:33,620 --> 00:29:35,570 They have uncertainties. 408 00:29:35,570 --> 00:29:39,900 But they have a nice, classical picture. 409 00:29:39,900 --> 00:29:45,630 The expectation value of the operator is a real wave. 410 00:29:45,630 --> 00:29:50,840 So any time in 802 or in 8022, you have a classical wave 411 00:29:50,840 --> 00:29:55,170 to analyze, the quantum description of that wave 412 00:29:55,170 --> 00:29:58,680 is a coherent state of the electromagnetic field. 413 00:29:58,680 --> 00:30:02,230 Lasers are coherent states of the electromagnetic field. 414 00:30:02,230 --> 00:30:05,330 They have these uncertainties that we discussed 415 00:30:05,330 --> 00:30:09,280 last time with number and phase that are very strong. 416 00:30:09,280 --> 00:30:14,380 If the number goes large, then certainty on the face 417 00:30:14,380 --> 00:30:15,890 is extremely small. 418 00:30:15,890 --> 00:30:18,240 So there we go. 419 00:30:18,240 --> 00:30:19,590 This is a coherent state. 420 00:30:19,590 --> 00:30:24,650 We can do a little more on that, write it more explicitly. 421 00:30:24,650 --> 00:30:29,620 This is epsilon 2 E not, the real part 422 00:30:29,620 --> 00:30:36,905 of alpha e to the minus i omega t sine of kz. 423 00:30:39,750 --> 00:30:42,100 And if we write, for example, alpha 424 00:30:42,100 --> 00:30:51,760 to be length of alpha e to the i theta, 425 00:30:51,760 --> 00:30:57,910 then this would be 2 E not. 426 00:30:57,910 --> 00:31:02,080 Length of alpha would go out, and the i theta 427 00:31:02,080 --> 00:31:05,370 to minus i omega t would give you 428 00:31:05,370 --> 00:31:12,910 cosine of omega t minus theta sine of kz. 429 00:31:12,910 --> 00:31:19,700 And this is something like a standing wave. 430 00:31:19,700 --> 00:31:26,170 It just changes in time and with a fixed spatial distribution. 431 00:31:26,170 --> 00:31:31,800 So it's a classical wave, and nevertheless, it 432 00:31:31,800 --> 00:31:35,755 has a good description classically, 433 00:31:35,755 --> 00:31:38,850 a good description quantum mechanically. 434 00:31:38,850 --> 00:31:40,820 It's a coherent state. 435 00:31:40,820 --> 00:31:47,220 And its energy is the expectation value 436 00:31:47,220 --> 00:31:49,035 of the Hamiltonian. 437 00:31:49,035 --> 00:31:52,960 The expectation value of the energy-- 438 00:31:52,960 --> 00:32:00,570 let me write this expectation value-- of H is H omega. 439 00:32:00,570 --> 00:32:05,840 Expectation value of N plus 1/2. 440 00:32:05,840 --> 00:32:12,080 And in a coherent state, the expectation value of N 441 00:32:12,080 --> 00:32:16,995 is alpha squared. 442 00:32:22,560 --> 00:32:27,350 So you have this of the coherent state 443 00:32:27,350 --> 00:32:30,130 alpha has alpha squared photons. 444 00:32:37,050 --> 00:32:39,490 And that's because it's the number operator, 445 00:32:39,490 --> 00:32:44,620 and that's pretty much the end of our story for photon states. 446 00:32:44,620 --> 00:32:46,380 There's more that one could do. 447 00:32:46,380 --> 00:32:51,160 One could do basically all kinds of things put together 448 00:32:51,160 --> 00:32:52,050 different modes. 449 00:32:52,050 --> 00:32:54,420 We considered here one mode. 450 00:32:54,420 --> 00:32:56,130 You could consider electric fields 451 00:32:56,130 --> 00:32:59,470 have super procession of modes and discuss 452 00:32:59,470 --> 00:33:02,640 commutation relations for the field operators, 453 00:33:02,640 --> 00:33:04,410 and all kinds of things. 454 00:33:04,410 --> 00:33:07,330 But that's really a quantum field theory course. 455 00:33:07,330 --> 00:33:11,220 At this moment, the main story I wanted to get across 456 00:33:11,220 --> 00:33:16,900 is that naturally, the harmonic oscillator has entered here, 457 00:33:16,900 --> 00:33:19,030 but in a very funny way. 458 00:33:19,030 --> 00:33:23,370 q and p were not positioned in momentum, 459 00:33:23,370 --> 00:33:27,330 were basically electric field and magnetic field. 460 00:33:27,330 --> 00:33:29,270 And there's an uncertainty between 461 00:33:29,270 --> 00:33:31,030 electric and magnetic fields. 462 00:33:33,550 --> 00:33:36,770 And the result of all this is that at the end of the day, 463 00:33:36,770 --> 00:33:41,210 you have a description by a harmonic oscillator 464 00:33:41,210 --> 00:33:43,570 and with energy levels that correspond 465 00:33:43,570 --> 00:33:47,320 to different amount of photons in the field. 466 00:33:47,320 --> 00:33:50,140 Finally, the classical things, if you 467 00:33:50,140 --> 00:33:52,600 want to recover classical waves, you 468 00:33:52,600 --> 00:33:55,670 must consider coherence states. 469 00:33:55,670 --> 00:33:57,810 These are the states that were classical. 470 00:33:57,810 --> 00:34:01,710 When you looked at the harmonic oscillator doing motion 471 00:34:01,710 --> 00:34:03,680 and for electromagnetic field, they 472 00:34:03,680 --> 00:34:06,690 give you the classical wave picture 473 00:34:06,690 --> 00:34:10,000 of an electric and magnetic field oscillating 474 00:34:10,000 --> 00:34:12,520 in position and time. 475 00:34:12,520 --> 00:34:16,190 So are there any questions? 476 00:34:16,190 --> 00:34:17,358 Yes. 477 00:34:17,358 --> 00:34:20,690 AUDIENCE: If we're associating h bar with [INAUDIBLE], what 478 00:34:20,690 --> 00:34:24,389 object would you associate the zero point energy with? 479 00:34:24,389 --> 00:34:26,520 PROFESSOR: Well, it's a zero point energy 480 00:34:26,520 --> 00:34:29,610 of this quantum of vibration. 481 00:34:29,610 --> 00:34:36,739 So just like an electromagnetic field, 482 00:34:36,739 --> 00:34:40,080 basically, if this is like q and p, 483 00:34:40,080 --> 00:34:42,900 there's a minimum energy state in which 484 00:34:42,900 --> 00:34:45,790 you're in the ground state of the harmonic oscillator. 485 00:34:45,790 --> 00:34:51,100 But E and B cannot be zero, like delta x and delta p cannot be 486 00:34:51,100 --> 00:34:51,600 zero. 487 00:34:51,600 --> 00:34:54,969 So every mode of the electromagnetic field 488 00:34:54,969 --> 00:34:56,540 has a zero point energy. 489 00:34:56,540 --> 00:34:57,790 You cannot reduce it. 490 00:34:57,790 --> 00:35:01,450 So the vacuum of the electromagnetic field 491 00:35:01,450 --> 00:35:04,720 has a lot of zero point energies, 492 00:35:04,720 --> 00:35:07,050 one for every mode of radiation. 493 00:35:07,050 --> 00:35:11,750 Now, that zero point energies don't get you in trouble 494 00:35:11,750 --> 00:35:14,780 unless you're trying to do gravity. 495 00:35:14,780 --> 00:35:19,210 Gravity's the universal force and universal interaction 496 00:35:19,210 --> 00:35:22,300 that notes every bit of energy. 497 00:35:22,300 --> 00:35:24,410 So your zero point energies are quite 498 00:35:24,410 --> 00:35:26,840 important if you consider gravity. 499 00:35:26,840 --> 00:35:32,390 And you would have encountered here the first complication 500 00:35:32,390 --> 00:35:34,880 associated with quantum field theory. 501 00:35:34,880 --> 00:35:37,350 Every mode of the electromagnetic field-- 502 00:35:37,350 --> 00:35:40,360 a frequency one, a frequency 1.1, a frequency 503 00:35:40,360 --> 00:35:45,090 1.2-- every one of them has a ground state energy 504 00:35:45,090 --> 00:35:47,260 of 1/2H bar omega. 505 00:35:47,260 --> 00:35:50,470 If you add them all up, you get infinity. 506 00:35:50,470 --> 00:35:55,120 So you get an infinity of ground state energies. 507 00:35:55,120 --> 00:35:58,480 And people have learned how to work with this infinities. 508 00:35:58,480 --> 00:36:01,220 That infinity is not physical. 509 00:36:01,220 --> 00:36:04,850 But, if you suitably treat it, you 510 00:36:04,850 --> 00:36:06,970 can figure out all kinds of things. 511 00:36:06,970 --> 00:36:09,090 And there's several people, I think 512 00:36:09,090 --> 00:36:11,040 even some undergraduates, working 513 00:36:11,040 --> 00:36:14,950 on this with Professor Kardar and Professor Jaffe, 514 00:36:14,950 --> 00:36:18,540 called Casimir energies, in which the zero point 515 00:36:18,540 --> 00:36:21,320 energies of the electromagnetic field 516 00:36:21,320 --> 00:36:23,550 are treated in a more careful way, 517 00:36:23,550 --> 00:36:26,850 and the infinities are seen to be irrelevant, 518 00:36:26,850 --> 00:36:29,100 but there are some physical dependence 519 00:36:29,100 --> 00:36:31,710 on the parameters that keeps there. 520 00:36:31,710 --> 00:36:35,180 So you see the origin of this is because every mode 521 00:36:35,180 --> 00:36:38,730 of the electromagnetic field has a zero point energy, just 522 00:36:38,730 --> 00:36:41,170 like any quantum oscillator. 523 00:36:41,170 --> 00:36:42,576 Yes. 524 00:36:42,576 --> 00:36:43,492 AUDIENCE: [INAUDIBLE]. 525 00:36:49,044 --> 00:36:49,960 PROFESSOR: Absolutely. 526 00:36:49,960 --> 00:36:50,876 AUDIENCE: [INAUDIBLE]. 527 00:36:59,380 --> 00:37:03,540 PROFESSOR: Well, uncountable things, 528 00:37:03,540 --> 00:37:05,800 we already have seen some. 529 00:37:05,800 --> 00:37:08,110 Maybe they didn't look that sophisticated, 530 00:37:08,110 --> 00:37:13,470 but we had position states that were uncountable. 531 00:37:13,470 --> 00:37:17,270 So the electromagnetic field, yes, it has uncountable things. 532 00:37:17,270 --> 00:37:20,465 And there's nothing wrong about it. 533 00:37:20,465 --> 00:37:22,006 You just have to work with integrals. 534 00:37:24,787 --> 00:37:25,703 AUDIENCE: [INAUDIBLE]. 535 00:37:34,760 --> 00:37:37,290 PROFESSOR: Well, no, no. 536 00:37:37,290 --> 00:37:42,450 They're not really normalized because just like these states, 537 00:37:42,450 --> 00:37:44,760 the position states are not normalized, 538 00:37:44,760 --> 00:37:47,200 they're delta function normalized and things 539 00:37:47,200 --> 00:37:48,200 like that. 540 00:37:48,200 --> 00:37:53,040 So look, if you want to avoid conceptual troubles with that, 541 00:37:53,040 --> 00:37:55,760 people and many physicists and textbooks 542 00:37:55,760 --> 00:38:03,120 on quantum field theory begin with space, a big, big, box. 543 00:38:03,120 --> 00:38:06,680 And then you see that it works for any size box, 544 00:38:06,680 --> 00:38:11,000 and then you say, well, it will work if the box is infinite. 545 00:38:11,000 --> 00:38:13,540 And we just proceed. 546 00:38:13,540 --> 00:38:14,240 All right. 547 00:38:14,240 --> 00:38:18,910 So I'll move on now to the second part of the lecture 548 00:38:18,910 --> 00:38:23,400 that deals with two-state systems and spin 549 00:38:23,400 --> 00:38:27,160 states and goes back and puts together a few of the things 550 00:38:27,160 --> 00:38:27,980 we've been doing. 551 00:38:33,357 --> 00:38:34,190 AUDIENCE: Professor? 552 00:38:34,190 --> 00:38:35,076 PROFESSOR: Yes. 553 00:38:35,076 --> 00:38:35,520 AUDIENCE: Could you close the sun shade? 554 00:38:35,520 --> 00:38:37,170 I can't really see the board. 555 00:38:37,170 --> 00:38:39,560 PROFESSOR: OK, sure. 556 00:38:42,285 --> 00:38:42,785 Board. 557 00:38:50,980 --> 00:38:55,400 I think maybe we need all the way? 558 00:38:55,400 --> 00:38:57,060 No, that won't make a difference. 559 00:38:57,060 --> 00:38:59,385 It's the other shades, I think. 560 00:39:03,990 --> 00:39:05,050 I'll leave it like that. 561 00:39:08,930 --> 00:39:11,960 Maybe I should use another board for the people 562 00:39:11,960 --> 00:39:15,000 that watch these movies. 563 00:39:15,000 --> 00:39:16,380 That may be better. 564 00:39:16,380 --> 00:39:18,330 So let's do this board. 565 00:39:18,330 --> 00:39:22,025 OK, so here's what we want to understand. 566 00:39:28,470 --> 00:39:29,515 Two-state systems. 567 00:39:34,440 --> 00:39:36,500 It's probably going to be about this, 568 00:39:36,500 --> 00:39:39,290 and two more lectures on that. 569 00:39:39,290 --> 00:39:45,410 And what we want to understand first is spin procession. 570 00:39:45,410 --> 00:39:47,550 You say, well, spin procession looks 571 00:39:47,550 --> 00:39:52,020 like a very particular kind of problem. 572 00:39:52,020 --> 00:39:55,040 When you have spins, you have magnetic fields. 573 00:39:55,040 --> 00:39:57,900 But at the end of the day, what we will see 574 00:39:57,900 --> 00:40:02,820 is that spin process-- you can view any two-state system 575 00:40:02,820 --> 00:40:05,980 as a system in which you've put a spin in a magnetic field. 576 00:40:05,980 --> 00:40:11,200 Even though you may be talking about electrons 577 00:40:11,200 --> 00:40:16,580 shared between two atoms, or ammonia molecule, or anything 578 00:40:16,580 --> 00:40:17,240 like that. 579 00:40:17,240 --> 00:40:20,750 Mathematically, you go back always to spins. 580 00:40:20,750 --> 00:40:25,330 Because spins are things we have become familiar already. 581 00:40:25,330 --> 00:40:27,570 So we exploit that to the maximum. 582 00:40:27,570 --> 00:40:34,780 So we do the one thing we haven't done rigorously so far, 583 00:40:34,780 --> 00:40:38,600 and then we'll explore this analogy to some point. 584 00:40:38,600 --> 00:40:42,350 So what was our discussion of spin? 585 00:40:42,350 --> 00:40:46,040 So two-state systems, and we'll begin with spin procession. 586 00:40:52,350 --> 00:40:57,150 So the idea of spin procession all arises, as you remember, 587 00:40:57,150 --> 00:41:02,920 because if you have a charged particle that has some spin, 588 00:41:02,920 --> 00:41:07,310 there's a relation between the particle's magnetic moment 589 00:41:07,310 --> 00:41:11,320 and the spin, or the angular momentum, of that particle, 590 00:41:11,320 --> 00:41:14,000 of that little ball of material. 591 00:41:14,000 --> 00:41:17,110 And we made an argument that this was just 592 00:41:17,110 --> 00:41:22,320 q over 2m times the angular momentum. 593 00:41:22,320 --> 00:41:23,940 And this will be angular momentum. 594 00:41:28,180 --> 00:41:30,720 This was classical. 595 00:41:30,720 --> 00:41:36,040 Nevertheless, the fact that we claim 596 00:41:36,040 --> 00:41:40,170 is true quantum mechanically is that in fact this idea is 597 00:41:40,170 --> 00:41:45,380 roughly right, except that there's two modifications. 598 00:41:45,380 --> 00:41:50,480 The true magnetic moment that enters into the Hamiltonian 599 00:41:50,480 --> 00:41:54,910 under the particle has is not quite the same 600 00:41:54,910 --> 00:41:57,350 as suggested by the classical argument, 601 00:41:57,350 --> 00:41:59,783 but it's modified by a g factor. 602 00:42:03,440 --> 00:42:07,670 And that modification is important. 603 00:42:07,670 --> 00:42:14,770 And this S is not just a plain classical angular momentum 604 00:42:14,770 --> 00:42:18,130 of a rotated ball with some mass and some radius, 605 00:42:18,130 --> 00:42:22,565 but it's a spin angular momentum and intrinsic angular momentum. 606 00:42:28,460 --> 00:42:31,690 A rather abstract thing that in fact 607 00:42:31,690 --> 00:42:35,340 should be best viewed as an operator, 608 00:42:35,340 --> 00:42:38,280 and that's the way we've thought about it. 609 00:42:38,280 --> 00:42:42,920 The magnetic [INAUDIBLE] now becomes an operator, 610 00:42:42,920 --> 00:42:47,370 because it's proportional to the spin operator. 611 00:42:47,370 --> 00:42:48,620 So it's an operator. 612 00:42:51,200 --> 00:42:59,320 And different values of g apply for different particles. 613 00:42:59,320 --> 00:43:04,900 And we saw that g equals 2 applies for the electron. 614 00:43:04,900 --> 00:43:07,620 That's a famous value, in fact predicted 615 00:43:07,620 --> 00:43:11,220 by Dirac's equation, relativistic equation, 616 00:43:11,220 --> 00:43:17,800 for the electron, and observed to great accuracy of course 617 00:43:17,800 --> 00:43:19,780 as well. 618 00:43:19,780 --> 00:43:23,350 And for other particles, like the proton or the neutron, 619 00:43:23,350 --> 00:43:26,240 the quantity g has different values. 620 00:43:26,240 --> 00:43:31,440 You might be surprised that the neutron has a dipole moment. 621 00:43:31,440 --> 00:43:34,190 Because you would say a neutron is an uncharged particle, 622 00:43:34,190 --> 00:43:37,980 so a charge rotating doesn't do anything. 623 00:43:37,980 --> 00:43:40,120 Nevertheless, a neutron is uncharged, 624 00:43:40,120 --> 00:43:44,060 but it has three quarks, two with some charge, one 625 00:43:44,060 --> 00:43:46,890 with an opposite charge to the other two. 626 00:43:46,890 --> 00:43:51,130 And if they distribute cleverly, say the negative ones are 627 00:43:51,130 --> 00:43:53,910 farther away from the center, and in the center 628 00:43:53,910 --> 00:44:01,090 is the positive one, this could have angular magnetic moment. 629 00:44:01,090 --> 00:44:03,960 And in fact, it does have magnetic moment. 630 00:44:03,960 --> 00:44:08,910 The neutron has a significant magnetic moment. 631 00:44:08,910 --> 00:44:12,040 So at the end of the day, we're going 632 00:44:12,040 --> 00:44:19,780 to write this as mu equal gamma S. 633 00:44:19,780 --> 00:44:24,516 And this constant gamma is going to summarize everything, g, q, 634 00:44:24,516 --> 00:44:27,940 m, all these things. 635 00:44:27,940 --> 00:44:30,560 And this will be a good notation. 636 00:44:30,560 --> 00:44:34,410 Gamma S is brief and simple. 637 00:44:34,410 --> 00:44:37,370 And this constant, we're going to use it. 638 00:44:37,370 --> 00:44:49,180 So the Hamiltonian minus mu dot B is a quantum Hamiltonian 639 00:44:49,180 --> 00:44:51,490 because mu is an operator. 640 00:44:51,490 --> 00:44:54,760 B, at this moment, even though we were just 641 00:44:54,760 --> 00:44:57,090 talking about photon states, this 642 00:44:57,090 --> 00:45:00,890 will be a static magnetic field typically. 643 00:45:00,890 --> 00:45:03,560 Can be a to time dependent, but it will not 644 00:45:03,560 --> 00:45:07,610 be sufficiently important if it has time dependence, 645 00:45:07,610 --> 00:45:11,030 and we have to quantize it and to think of it as a quantum 646 00:45:11,030 --> 00:45:11,740 field. 647 00:45:11,740 --> 00:45:15,040 But in some problems of radiation 648 00:45:15,040 --> 00:45:19,280 of electromagnetic fields by the motion of spins, 649 00:45:19,280 --> 00:45:22,580 you would have to quantize the electromagnetic field. 650 00:45:22,580 --> 00:45:24,290 But this is not the case now. 651 00:45:24,290 --> 00:45:31,080 So this is minus gamma S dot B. And we typically 652 00:45:31,080 --> 00:45:38,730 like to write it as minus gamma B dot S. 653 00:45:38,730 --> 00:45:47,050 And that means very explicitly minus gamma BxSx operator 654 00:45:47,050 --> 00:45:52,781 plus BySy operator plus BzSz operator. 655 00:45:57,020 --> 00:46:03,540 So let me remind you of a simple situation when 656 00:46:03,540 --> 00:46:06,640 you had a magnetic field in the z direction. 657 00:46:09,280 --> 00:46:17,370 B along z if B is B times z hat. 658 00:46:20,380 --> 00:46:26,351 Then H is minus gamma. 659 00:46:26,351 --> 00:46:26,850 BSz. 660 00:46:34,210 --> 00:46:38,370 And the unitary operator that generates time evolution 661 00:46:38,370 --> 00:46:49,210 of states, the unitary operator u of t0 is exponential minus i. 662 00:46:49,210 --> 00:46:51,440 I'll call it H sub s for spin. 663 00:46:57,640 --> 00:47:01,080 H sub s t over H bar. 664 00:47:03,660 --> 00:47:10,790 And I'll put it like this exponential of minus i minus 665 00:47:10,790 --> 00:47:18,080 gamma B t Sz over H bar. 666 00:47:18,080 --> 00:47:22,850 So I substituted what Hs is, moved 667 00:47:22,850 --> 00:47:29,020 the t sort of inside the parentheses minus gamma B Sz. 668 00:47:29,020 --> 00:47:30,890 I put the z out and put this here. 669 00:47:35,360 --> 00:47:38,930 So far so good? 670 00:47:38,930 --> 00:47:42,300 This is our time development operator. 671 00:47:42,300 --> 00:47:46,500 Now, I want you to recall one property that you only 672 00:47:46,500 --> 00:47:50,440 justified by checking it in the homework. 673 00:47:50,440 --> 00:47:54,770 But in the next few lectures, we will just 674 00:47:54,770 --> 00:47:57,540 make sure you understand this why it's true in general. 675 00:47:57,540 --> 00:48:04,630 But we talked in the homework about an operator Rn sub alpha, 676 00:48:04,630 --> 00:48:14,670 which was exponential of minus i alpha Sn over H bar. 677 00:48:19,200 --> 00:48:26,320 Where n was a unit vector, and Sn 678 00:48:26,320 --> 00:48:34,905 is defined as n dot S. So nxSx, nySy, nzSz. 679 00:48:39,420 --> 00:48:46,720 So this operator that you considered 680 00:48:46,720 --> 00:48:48,730 was called the rotation operator, 681 00:48:48,730 --> 00:48:52,740 and it did perform rotation of spin states. 682 00:48:52,740 --> 00:48:59,560 In fact, what it did was rotate any spin state by an angle, 683 00:48:59,560 --> 00:49:03,550 alpha, around the nth direction. 684 00:49:03,550 --> 00:49:08,430 So if you had the n direction here, 685 00:49:08,430 --> 00:49:23,000 and you had any spin state in some arbitrary direction, 686 00:49:23,000 --> 00:49:27,920 it would rotate it by an angle alpha around this. 687 00:49:27,920 --> 00:49:30,595 So you have this, it would rotate it 688 00:49:30,595 --> 00:49:37,390 to another point over here with an angle alpha in between. 689 00:49:37,390 --> 00:49:43,800 So in words, it rotates by an angle alpha, 690 00:49:43,800 --> 00:49:47,830 rotates spin states. 691 00:49:51,280 --> 00:49:53,370 And when you think of a spin state, 692 00:49:53,370 --> 00:49:58,520 you must think of some n vector, n prime vector. 693 00:49:58,520 --> 00:50:02,240 So maybe n prime here would be a good notation. 694 00:50:02,240 --> 00:50:05,760 So you have a spin state in the n prime direction. 695 00:50:05,760 --> 00:50:11,090 Remember your spin states were of the form n plus minus. 696 00:50:11,090 --> 00:50:14,150 Well, the state that points in the direction n 697 00:50:14,150 --> 00:50:18,900 is n plus, so some n prime direction. 698 00:50:18,900 --> 00:50:26,510 This operator rotates those states by an angle alpha. 699 00:50:26,510 --> 00:50:30,820 Now, it probably is a little vague in your mind, that idea, 700 00:50:30,820 --> 00:50:33,450 because you checked it several weeks ago. 701 00:50:33,450 --> 00:50:36,450 And you only checked it by taking some particular states 702 00:50:36,450 --> 00:50:37,500 and rotating them. 703 00:50:37,500 --> 00:50:42,380 So we will have to elaborate on this, and we will. 704 00:50:42,380 --> 00:50:44,890 So this will become clear that this 705 00:50:44,890 --> 00:50:55,500 rotates any spin state by an angle alpha and rotates 706 00:50:55,500 --> 00:51:02,900 spin states using an axis, with respect 707 00:51:02,900 --> 00:51:06,800 to the axis defined by n over here. 708 00:51:06,800 --> 00:51:11,810 So that's interpretation of this state, of this operator. 709 00:51:11,810 --> 00:51:13,260 That's what it does. 710 00:51:13,260 --> 00:51:15,823 And now I want you to look at this operator. 711 00:51:18,550 --> 00:51:21,480 Well, it's similar. 712 00:51:21,480 --> 00:51:25,820 In fact, this plays the role of alpha, 713 00:51:25,820 --> 00:51:28,105 and this plays the role of Sn. 714 00:51:30,890 --> 00:51:33,950 So this is the spin in the z direction, 715 00:51:33,950 --> 00:51:37,820 and this operator must rotate states 716 00:51:37,820 --> 00:51:43,120 by this angle alpha, which is gamma Bt. 717 00:51:43,120 --> 00:51:49,840 If what we said is right, that's what this operator must do. 718 00:51:49,840 --> 00:51:53,010 Even though I think you've done this calculation as part 719 00:51:53,010 --> 00:51:56,730 of tests, problems, or other problems, 720 00:51:56,730 --> 00:52:00,510 practice problems, not quite homework., 721 00:52:00,510 --> 00:52:03,580 I want to do this calculation again. 722 00:52:03,580 --> 00:52:10,490 So let's take an arbitrary spin state, xyz. 723 00:52:13,190 --> 00:52:20,000 Now, don't confuse the arbitrary spin states with the n here. 724 00:52:20,000 --> 00:52:23,310 The n is here the axis around which 725 00:52:23,310 --> 00:52:25,540 this Hamiltonian rotates states. 726 00:52:25,540 --> 00:52:27,010 But there's no states here. 727 00:52:27,010 --> 00:52:28,909 This is a rotation operator. 728 00:52:28,909 --> 00:52:30,450 I'm sorry, I called it a Hamiltonian. 729 00:52:30,450 --> 00:52:31,770 It's not precise. 730 00:52:31,770 --> 00:52:34,090 This is a unitary operator. 731 00:52:34,090 --> 00:52:36,200 It rotates states. 732 00:52:36,200 --> 00:52:40,500 And this is the direction, the axis, of rotation. 733 00:52:40,500 --> 00:52:42,790 Your spin state is another object. 734 00:52:42,790 --> 00:52:45,420 It's a spin that lives in some direction. 735 00:52:45,420 --> 00:52:48,300 So here, we're having the magnetic field 736 00:52:48,300 --> 00:52:49,450 in the z direction. 737 00:52:49,450 --> 00:52:51,460 So the magnetic field is here. 738 00:52:54,710 --> 00:53:00,080 And we'll put a spin state over here, an n, a spin state that 739 00:53:00,080 --> 00:53:04,815 has some value of phi and some value of theta. 740 00:53:07,360 --> 00:53:10,410 And that's the spin state at time equals zero. 741 00:53:10,410 --> 00:53:18,430 So psi 0 is the spin state this that with your formula sheet, 742 00:53:18,430 --> 00:53:24,930 this cosine theta over 2 plus plus sine theta over 2 e 743 00:53:24,930 --> 00:53:29,360 to the i phi, I think with a plus, yes. 744 00:53:29,360 --> 00:53:38,800 I'll call it phi not, and maybe theta 0, y 0, and minus. 745 00:53:38,800 --> 00:53:43,140 So this is a state, a spin state pointing 746 00:53:43,140 --> 00:53:45,610 in this direction, the direction n. 747 00:53:45,610 --> 00:53:49,810 That was the general formula for a spin state. 748 00:53:49,810 --> 00:53:55,520 Now we are going to apply the operator, the time evolution 749 00:53:55,520 --> 00:53:56,800 operator. 750 00:53:56,800 --> 00:54:01,660 But let's do a preliminary calculation. 751 00:54:01,660 --> 00:54:15,190 HS on plus is minus gamma B Sz on plus minus gamma B H 752 00:54:15,190 --> 00:54:22,510 bar over 2 plus, and Hs minus is equal to minus 753 00:54:22,510 --> 00:54:31,200 gamma BSz on minus equal plus gamma B H bar over 2 minus. 754 00:54:35,840 --> 00:54:42,360 So we want to add with this operator on this state. 755 00:54:42,360 --> 00:54:47,610 So here we have it, the state that any time 756 00:54:47,610 --> 00:54:54,740 is going to be E to the minus iHst over H bar times 757 00:54:54,740 --> 00:54:59,380 this state over here acting on psi is 0. 758 00:55:03,810 --> 00:55:07,050 So let's do it. 759 00:55:07,050 --> 00:55:12,370 Well, on the first term is cosine theta 0 over 2. 760 00:55:12,370 --> 00:55:16,560 And you have this exponent acting on plus. 761 00:55:16,560 --> 00:55:21,360 But the exponent has Hs that's acting on plus is this. 762 00:55:21,360 --> 00:55:24,810 So you can just put that thing on the exponent. 763 00:55:24,810 --> 00:55:30,470 So you put e to the minus i, and Hs on plus 764 00:55:30,470 --> 00:55:38,220 is this, minus gamma B H bar over 2. 765 00:55:38,220 --> 00:55:44,820 Then you have the p and the H bar and the plus. 766 00:55:48,360 --> 00:55:54,730 And continue here. 767 00:56:01,740 --> 00:56:10,610 So we just need to do the second term, plus sine theta over 2, 768 00:56:10,610 --> 00:56:12,440 e to the minus i. 769 00:56:12,440 --> 00:56:15,000 And now the same thing, but with a plus sign. 770 00:56:15,000 --> 00:56:24,940 Plus gamma B H bar over 2, t over H bar on the minus state. 771 00:56:24,940 --> 00:56:29,800 So just in case I got you confused 772 00:56:29,800 --> 00:56:34,320 and the small type is a problem here, 773 00:56:34,320 --> 00:56:38,790 this operator active on initial state just acts on plus, 774 00:56:38,790 --> 00:56:39,770 then acts on minus. 775 00:56:39,770 --> 00:56:43,940 On plus, the operator is an eigen state. 776 00:56:43,940 --> 00:56:46,550 So you can just put the number in the exponential. 777 00:56:46,550 --> 00:56:51,270 So you put the plus eigen value, the minus eigen value. 778 00:56:51,270 --> 00:56:54,890 So what do we get? 779 00:56:54,890 --> 00:57:01,486 Psi t is equal, cosine theta not over 2, 780 00:57:01,486 --> 00:57:15,290 e to the i gamma B t over 2 plus sine theta not over 2, e 781 00:57:15,290 --> 00:57:23,270 to the minus i gamma B t over 2 minus. 782 00:57:23,270 --> 00:57:25,250 Now, this state this is not quite-- 783 00:57:25,250 --> 00:57:27,730 I hope I got my signs right. 784 00:57:27,730 --> 00:57:28,230 Yes. 785 00:57:30,900 --> 00:57:34,520 This state is not quite in readable form. 786 00:57:34,520 --> 00:57:37,750 To compare it with a general end state, 787 00:57:37,750 --> 00:57:39,430 you need null phase here. 788 00:57:39,430 --> 00:57:44,100 So we must factor this phase out. 789 00:57:44,100 --> 00:57:47,400 e to the i gamma B t over 2. 790 00:57:47,400 --> 00:57:49,140 And it's an irrelevant phase. 791 00:57:49,140 --> 00:57:54,370 So then you have cosine theta not over 2 plus sine theta not 792 00:57:54,370 --> 00:57:56,010 over 2. 793 00:57:56,010 --> 00:58:00,105 I'm sorry, I forgot to have the e to the i phi not here. 794 00:58:12,990 --> 00:58:14,750 I didn't copy it. 795 00:58:14,750 --> 00:58:17,840 So here, what do we have? 796 00:58:17,840 --> 00:58:27,050 e to the i phi not minus gamma B t minus. 797 00:58:27,050 --> 00:58:30,520 Look, when you factor this one out, 798 00:58:30,520 --> 00:58:33,150 you get minus the same thing here. 799 00:58:33,150 --> 00:58:36,160 So this becomes a minus 1. 800 00:58:36,160 --> 00:58:40,040 And then you put the two faces together, and you got that. 801 00:58:40,040 --> 00:58:41,770 So now you look at this state, and you 802 00:58:41,770 --> 00:58:44,305 say, oh, I know what this is. 803 00:58:44,305 --> 00:58:49,440 This is a spin state that has theta 804 00:58:49,440 --> 00:58:54,130 as a function of time, just theta not. 805 00:58:54,130 --> 00:58:58,460 But the angle, phi, as a function of time 806 00:58:58,460 --> 00:59:01,780 is phi not minus gamma B t. 807 00:59:05,480 --> 00:59:12,080 So this spin will precess and will go like this. 808 00:59:16,260 --> 00:59:26,750 Phi not minus gamma B t is the phi as a function of time. 809 00:59:26,750 --> 00:59:30,420 So have the magnetic field. 810 00:59:30,420 --> 00:59:34,790 You have a procession of the spin over here. 811 00:59:34,790 --> 00:59:37,450 So this is spin procession. 812 00:59:37,450 --> 00:59:45,320 And indeed, this is exactly what we're claiming here. 813 00:59:45,320 --> 00:59:49,030 If this rotates states by an angle alpha, 814 00:59:49,030 --> 00:59:53,650 this operator, this Hamiltonian that we've discussed here, 815 00:59:53,650 --> 00:59:59,540 must rotate states by this angle alpha, which is minus gamma Bt, 816 00:59:59,540 --> 01:00:01,370 along the z-axis. 817 01:00:01,370 --> 01:00:05,990 So you have the z-axis, and you rotate by minus gamma Bt. 818 01:00:05,990 --> 01:00:12,190 The sine is the reason the phi decreases in time and goes 819 01:00:12,190 --> 01:00:17,900 in this direction, as opposed to going in the other direction. 820 01:00:17,900 --> 01:00:22,340 So this is a basic confirmation of what the spin is doing. 821 01:00:22,340 --> 01:00:26,540 And I want to give you the general result 822 01:00:26,540 --> 01:00:29,920 so that you can really use it more clearly. 823 01:00:29,920 --> 01:00:33,030 So I think the lights are gone, so we 824 01:00:33,030 --> 01:00:35,420 can go to this blackboard. 825 01:00:35,420 --> 01:00:40,270 First of all, classical picture. 826 01:00:40,270 --> 01:00:43,310 What is it about spin procession? 827 01:00:43,310 --> 01:00:47,870 Is it a quantum phenomenon or a classical phenomenon, or both? 828 01:00:47,870 --> 01:00:49,250 Well, it's really both. 829 01:00:49,250 --> 01:00:51,980 And this idea of procession, you can get it 830 01:00:51,980 --> 01:00:54,280 from the classical picture as well. 831 01:00:54,280 --> 01:00:55,540 So what do you have? 832 01:00:55,540 --> 01:01:03,760 If you have a mu in a B field, you get a torque. 833 01:01:09,360 --> 01:01:12,890 And that you can easily convince yourself. 834 01:01:12,890 --> 01:01:15,820 I'm sure you've done the computation in 802. 835 01:01:15,820 --> 01:01:18,440 You have a little square wire not 836 01:01:18,440 --> 01:01:20,240 aligned with the magnetic field. 837 01:01:20,240 --> 01:01:23,110 You calculate the force on one side, the force on the other. 838 01:01:23,110 --> 01:01:25,070 You see that there is a torque. 839 01:01:25,070 --> 01:01:32,295 And the torque is given by mu cross B. That's enm. 840 01:01:34,810 --> 01:01:42,170 On the other hand, the rate of change of angular momentum 841 01:01:42,170 --> 01:01:43,060 is the torque. 842 01:01:45,830 --> 01:01:57,430 So this is mu cross B. But mu is gamma S, so this 843 01:01:57,430 --> 01:02:15,520 is gamma S cross B. And this is minus gamma B cross S. 844 01:02:15,520 --> 01:02:16,380 OK. 845 01:02:16,380 --> 01:02:26,140 This equation, which I rewrite it here, ds/dt equals 846 01:02:26,140 --> 01:02:34,650 minus gamma B cross S, is a particular case 847 01:02:34,650 --> 01:02:42,460 of a very famous equation in classical mechanics, 848 01:02:42,460 --> 01:02:46,940 and this equation for a rotating vector. 849 01:02:46,940 --> 01:03:05,515 If you have a vector, dx/dt is omega cross x. 850 01:03:09,680 --> 01:03:13,790 This is the equation satisfied by a vector 851 01:03:13,790 --> 01:03:17,670 x that is rotating with angular frequency 852 01:03:17,670 --> 01:03:23,710 omega around the axis defined by the vector omega. 853 01:03:23,710 --> 01:03:25,430 A famous equation. 854 01:03:25,430 --> 01:03:32,500 OK, so you have here omega vector is omega n. 855 01:03:32,500 --> 01:03:35,430 So here is the direction of n, the unit vector. 856 01:03:35,430 --> 01:03:36,630 Here's omega. 857 01:03:36,630 --> 01:03:40,310 And you have a vector x over here. 858 01:03:40,310 --> 01:03:43,560 Then this vector, the solution of this equation, 859 01:03:43,560 --> 01:03:50,270 is a vector that is rotating around omega with the angular 860 01:03:50,270 --> 01:03:51,860 velocity magnitude of omega. 861 01:03:54,720 --> 01:03:57,130 In the notes, I just give a little hint 862 01:03:57,130 --> 01:03:58,640 of how you derive that. 863 01:03:58,640 --> 01:04:01,860 But truly speaking, you guys should 864 01:04:01,860 --> 01:04:04,870 be able to just scribble a few notes 865 01:04:04,870 --> 01:04:08,290 if you don't know the situation by heart, and convince yourself 866 01:04:08,290 --> 01:04:10,320 this is true. 867 01:04:10,320 --> 01:04:16,690 So this equation is of that form in which 868 01:04:16,690 --> 01:04:23,750 the omega x is played by S. Omega is minus gamma B. 869 01:04:23,750 --> 01:04:30,010 So this defines what is called the Larmor frequency, which 870 01:04:30,010 --> 01:04:37,455 is minus gamma B, is the Larmor frequency. 871 01:04:45,560 --> 01:04:48,600 Now, this Larmor frequency is precisely 872 01:04:48,600 --> 01:04:54,520 that one because was minus gamma B. 873 01:04:54,520 --> 01:04:57,470 And here you have minus gamma B times t. 874 01:04:57,470 --> 01:05:00,570 Omega times t is the angle. 875 01:05:00,570 --> 01:05:05,960 So in fact, this is rotating with a Larmor frequency. 876 01:05:05,960 --> 01:05:06,930 And there you go. 877 01:05:06,930 --> 01:05:10,420 In the same blackboard, you have a classical mechanics 878 01:05:10,420 --> 01:05:12,810 derivation of the Larmor frequency 879 01:05:12,810 --> 01:05:14,900 and a quantum mechanical derivation 880 01:05:14,900 --> 01:05:16,730 of the Larmor frequency. 881 01:05:16,730 --> 01:05:20,700 Again, at the end of the day, this is no coincidence. 882 01:05:20,700 --> 01:05:24,380 We've made dynamical classical variables 883 01:05:24,380 --> 01:05:30,700 into quantum operators, and we haven't changed the physics. 884 01:05:30,700 --> 01:05:33,650 Mu dot B is a classical energy. 885 01:05:33,650 --> 01:05:38,270 Well, it became Hamiltonian, and it's doing the right thing. 886 01:05:38,270 --> 01:05:42,460 So we can now use the Larmor frequency 887 01:05:42,460 --> 01:05:46,110 to rewrite the Hamiltonian, of course. 888 01:05:46,110 --> 01:05:46,755 It's here. 889 01:05:50,300 --> 01:05:54,680 So a little bit of emphasis is worth it. 890 01:05:54,680 --> 01:06:03,750 Hs is minus mu dot B, and it's minus gamma B dot S, 891 01:06:03,750 --> 01:06:13,450 and it's finally equal to omega L dot S. 892 01:06:13,450 --> 01:06:17,550 So if somebody gives you a Hamiltonian 893 01:06:17,550 --> 01:06:21,070 that at the end of the day, you can write it 894 01:06:21,070 --> 01:06:28,270 as some vector dot S, you already know that for spins, 895 01:06:28,270 --> 01:06:33,710 that is the Larmor frequency of rotation. 896 01:06:33,710 --> 01:06:36,890 It's a very simple thing. 897 01:06:36,890 --> 01:06:40,140 Hs, something times S, well that's 898 01:06:40,140 --> 01:06:45,340 precisely the rotation frequency for the spin states. 899 01:06:45,340 --> 01:06:47,200 They will all rotate that way. 900 01:06:51,360 --> 01:06:55,000 So we can say that the spin states in this Hamiltonian 901 01:06:55,000 --> 01:06:59,510 rotate with omega L frequency. 902 01:06:59,510 --> 01:07:02,180 So that's good. 903 01:07:02,180 --> 01:07:06,980 That's a general discussion of precession in a magnetic field. 904 01:07:06,980 --> 01:07:12,600 But I want to go one more step in generalization. 905 01:07:12,600 --> 01:07:15,820 It's a simple step, but let's just take it. 906 01:07:15,820 --> 01:07:19,740 So that you see even more generally 907 01:07:19,740 --> 01:07:25,045 why any system can be thought of as a spin system. 908 01:07:29,640 --> 01:07:31,310 And this is quite practical. 909 01:07:31,310 --> 01:07:34,150 In fact, it's probably the best way 910 01:07:34,150 --> 01:07:39,030 to imagine physically, the effects of any Hamiltonian. 911 01:07:39,030 --> 01:07:44,320 So let's consider time-independent Hamiltonians 912 01:07:44,320 --> 01:07:50,070 the most general Hamiltonian for a two-state system. 913 01:07:50,070 --> 01:07:51,250 How can it be? 914 01:07:51,250 --> 01:07:54,210 Well, a two-state system, remember 915 01:07:54,210 --> 01:07:56,620 two-state system is a word. 916 01:07:56,620 --> 01:08:03,450 It really means a system with two basis states. 917 01:08:03,450 --> 01:08:06,150 Once you have two basis states, a plus and the minus 918 01:08:06,150 --> 01:08:08,810 have infinitely many states, of course. 919 01:08:08,810 --> 01:08:12,700 But two-state system is two basis states. 920 01:08:12,700 --> 01:08:15,330 And therefore, in the Hamiltonian, 921 01:08:15,330 --> 01:08:19,930 in this two basis states, is a 2 by 2 matrix. 922 01:08:19,930 --> 01:08:22,350 And it's a 2 by 2 Hermitian matrix. 923 01:08:22,350 --> 01:08:25,710 So there's not too much it can be. 924 01:08:25,710 --> 01:08:28,800 In fact, you can have a constant that I 925 01:08:28,800 --> 01:08:38,000 will call maybe not the base notation, g not and g not. 926 01:08:38,000 --> 01:08:39,569 And that's Hermitian. 927 01:08:39,569 --> 01:08:42,960 It's real constant. 928 01:08:42,960 --> 01:08:47,620 You can put a g3 and a minus g3. 929 01:08:47,620 --> 01:08:50,380 That's still Hermitian. 930 01:08:50,380 --> 01:08:51,790 And that's reasonable. 931 01:08:51,790 --> 01:08:54,590 There's no reason why this number should be equal to this. 932 01:08:54,590 --> 01:08:59,090 So there are two numbers here that are arbitrary, real. 933 01:08:59,090 --> 01:09:01,460 And therefore, you can put them wherever you want. 934 01:09:01,460 --> 01:09:08,260 And I decided to call one g not plus g3 and one g not minus g3. 935 01:09:08,260 --> 01:09:12,760 Here, I can put again an arbitrary complex number, 936 01:09:12,760 --> 01:09:15,310 as long as I put here the complex conjugate. 937 01:09:15,310 --> 01:09:25,399 So I will call this g1 minus ig2, and this g1 plus ig2. 938 01:09:25,399 --> 01:09:28,620 And that's the most general 2 by 2 Hamiltonian. 939 01:09:28,620 --> 01:09:32,100 Tonya If those would be time-dependent functions, 940 01:09:32,100 --> 01:09:36,680 this is the most general Hamiltonian ever for a 2 941 01:09:36,680 --> 01:09:37,430 by 2 system. 942 01:09:37,430 --> 01:09:39,490 It doesn't get more complicated. 943 01:09:39,490 --> 01:09:42,399 That's a great advantage of this. 944 01:09:42,399 --> 01:09:44,630 But I've written it in a way that you 945 01:09:44,630 --> 01:09:47,109 can recognize something. 946 01:09:47,109 --> 01:09:49,910 You can recognize that this is g not 947 01:09:49,910 --> 01:09:56,420 times the identity plus g1 times sigma 1 plus g2 times sigma 948 01:09:56,420 --> 01:10:01,390 2 plus g3 times sigma 3. 949 01:10:01,390 --> 01:10:05,680 And this is because the Pauli matrices are, together 950 01:10:05,680 --> 01:10:12,310 with the unit matrix, a basis for all Hermitian 2 951 01:10:12,310 --> 01:10:14,500 by 2 matrices. 952 01:10:14,500 --> 01:10:17,980 So the Pauli matrices are Hermitian. 953 01:10:17,980 --> 01:10:19,920 The unit matrix is Hermitian. 954 01:10:19,920 --> 01:10:22,590 The most general 2 by 2 Hermitian matrix 955 01:10:22,590 --> 01:10:25,890 is a number times the one matrix, then number times 956 01:10:25,890 --> 01:10:29,470 the first part, then number, second, number, third. 957 01:10:29,470 --> 01:10:31,840 OK. 958 01:10:31,840 --> 01:10:35,890 So at this moment, we've got the most general Hamiltonian. 959 01:10:35,890 --> 01:10:42,540 And I will write it as g not times 1 plus g vector 960 01:10:42,540 --> 01:10:51,340 dot sigma, where g vector is g1, g2, g3. 961 01:10:55,910 --> 01:11:03,050 If we write the g vector as length 962 01:11:03,050 --> 01:11:07,840 of g, which is just the letter g, 963 01:11:07,840 --> 01:11:11,520 shouldn't be confused because we have g not, g1, g2, g3, 964 01:11:11,520 --> 01:11:14,430 but we haven't had a g without an index. 965 01:11:14,430 --> 01:11:18,760 So g without an index is going to be the magnitude of g 966 01:11:18,760 --> 01:11:23,640 vector, and n is going to be a particular vector. 967 01:11:23,640 --> 01:11:27,860 So look, you're talking about the most general Hamiltonian, 968 01:11:27,860 --> 01:11:31,280 and you're saying it's most easily understood 969 01:11:31,280 --> 01:11:35,200 as g not multiplying the identity, and that g 970 01:11:35,200 --> 01:11:37,525 vector multiplying the sigma vector. 971 01:11:40,220 --> 01:11:44,870 So on the other hand, g is this. 972 01:11:44,870 --> 01:11:51,040 So this is also g not 1 plus g times n dot sigma. 973 01:11:57,120 --> 01:11:59,945 But let's continue here. 974 01:12:07,320 --> 01:12:09,930 We know how to solve this problem. 975 01:12:09,930 --> 01:12:12,330 And you can say, well, all right. 976 01:12:12,330 --> 01:12:16,740 I have to diagonalize this matrix, find the eigen vectors, 977 01:12:16,740 --> 01:12:20,612 find the eigenvalues, and all that. 978 01:12:20,612 --> 01:12:23,470 But you've done all that work. 979 01:12:23,470 --> 01:12:25,830 It's already been done. 980 01:12:25,830 --> 01:12:28,120 What were the eigen states? 981 01:12:28,120 --> 01:12:33,620 Well, n dot sigma, the eigen states 982 01:12:33,620 --> 01:12:39,200 were the end states, the spin states, n plus minus. 983 01:12:39,200 --> 01:12:44,090 And they were plus minus n comma plus minus. 984 01:12:47,630 --> 01:12:53,720 Remember that S is H over 2 sigma. 985 01:12:53,720 --> 01:13:01,840 So this corresponds to n dot S on n plus minus equal plus 986 01:13:01,840 --> 01:13:06,080 minus H bar over 2 n plus minus, which 987 01:13:06,080 --> 01:13:09,560 might be the form in which you remember it better. 988 01:13:09,560 --> 01:13:14,180 But the sigma matrices, n dot sigma 989 01:13:14,180 --> 01:13:17,660 is diagonalized precisely by this thing. 990 01:13:17,660 --> 01:13:22,810 So in fact, you never have to diagonalize this matrix. 991 01:13:22,810 --> 01:13:25,760 It's already been done for you. 992 01:13:25,760 --> 01:13:30,060 And these are the eigen states of this Hamiltonian. 993 01:13:30,060 --> 01:13:37,650 And what is the value of the energy on n plus minus? 994 01:13:37,650 --> 01:13:44,570 Well, energy on n plus minus is g not times 1 plus g n 995 01:13:44,570 --> 01:13:48,985 dot sigma on the n plus minus. 996 01:13:51,920 --> 01:13:55,790 And g not times 1 here on this state 997 01:13:55,790 --> 01:14:03,830 is g not plus g n dot sigma, the thing is plus minus. 998 01:14:03,830 --> 01:14:11,250 So plus minus g, n plus minus. 999 01:14:11,250 --> 01:14:15,680 So in fact, you have the energies, 1000 01:14:15,680 --> 01:14:17,960 and you have the eigen vectors. 1001 01:14:17,960 --> 01:14:31,076 So the eigen states are n plus with energy equal g not plus g 1002 01:14:31,076 --> 01:14:38,670 and n minus with energy equal g not minus g. 1003 01:14:44,580 --> 01:14:48,220 So what we did by inventing the Pauli matrices 1004 01:14:48,220 --> 01:14:50,530 and inventing spin states and all 1005 01:14:50,530 --> 01:14:55,230 that was solve for you the most general 2 1006 01:14:55,230 --> 01:14:58,930 by 2 Hamiltonian, Hermitian Hamiltonian. 1007 01:14:58,930 --> 01:15:01,440 If you have a 2 by 2 Hermitian matrix, 1008 01:15:01,440 --> 01:15:04,320 you don't have to diagonalize it by hand. 1009 01:15:04,320 --> 01:15:07,760 You know the answers are this state. 1010 01:15:10,510 --> 01:15:12,470 And how do you build those states? 1011 01:15:12,470 --> 01:15:16,470 Well, you know what n is because you know the g's. 1012 01:15:16,470 --> 01:15:20,210 If you know the three g's, you know what the vector g is. 1013 01:15:20,210 --> 01:15:22,000 You know what the vector n is. 1014 01:15:22,000 --> 01:15:24,640 You know what this g is as well. 1015 01:15:24,640 --> 01:15:26,630 And therefore, with a vector n, you 1016 01:15:26,630 --> 01:15:30,170 construct this state, as you know already very well. 1017 01:15:30,170 --> 01:15:34,060 And given that you know g and g not, 1018 01:15:34,060 --> 01:15:41,040 well the energies are this, at the splitting 1019 01:15:41,040 --> 01:15:44,800 is 2g between those states. 1020 01:15:44,800 --> 01:15:46,560 This is the ground state. 1021 01:15:46,560 --> 01:15:50,180 This is the excited state. 1022 01:15:50,180 --> 01:15:53,370 Splitting two g's, so you look at the Hamiltonian, 1023 01:15:53,370 --> 01:15:57,080 and you say, what's the splitting between the two eigen 1024 01:15:57,080 --> 01:15:59,020 states of this. 1025 01:15:59,020 --> 01:16:05,620 You just take this numbers, compute g, and multiply by 2. 1026 01:16:05,620 --> 01:16:09,250 Now, last thing that you would want 1027 01:16:09,250 --> 01:16:13,720 to do with this Hamiltonian is time evolution. 1028 01:16:13,720 --> 01:16:18,380 So what do we say about time evolution? 1029 01:16:18,380 --> 01:16:25,700 Well, we have here H is equal to this. 1030 01:16:31,640 --> 01:16:48,530 And we also had omega L dot S. So omega L dot S in here 1031 01:16:48,530 --> 01:16:51,950 should be identified with this. 1032 01:16:51,950 --> 01:17:00,020 So sigma and S, as you remember, S is equal H bar over 2 sigma. 1033 01:17:00,020 --> 01:17:07,150 So this term can be written as g vector sigma. 1034 01:17:07,150 --> 01:17:09,790 In fact, this is better from here. 1035 01:17:09,790 --> 01:17:14,790 g vector sigma, and sigma is H bar over 2S. 1036 01:17:19,430 --> 01:17:21,640 I got a 2 over H bar. 1037 01:17:25,540 --> 01:17:28,190 2 over H bar S. 1038 01:17:28,190 --> 01:17:37,685 So from here, g dot sigma is 2g over H bar S. And remember, 1039 01:17:37,685 --> 01:17:44,370 a Hamiltonian for a spin system, whatever's 1040 01:17:44,370 --> 01:17:47,830 multiplying the vector that is multiplying S 1041 01:17:47,830 --> 01:17:52,970 is omega L. So in this system-- I 1042 01:17:52,970 --> 01:18:00,850 will write it like that-- omega L is 2g over H bar. 1043 01:18:04,400 --> 01:18:07,920 And this is a great physical help. 1044 01:18:07,920 --> 01:18:12,080 Because now that you have this, I 1045 01:18:12,080 --> 01:18:14,530 should remark this part of the Hamiltonian 1046 01:18:14,530 --> 01:18:16,840 is the one that does procession. 1047 01:18:16,840 --> 01:18:21,640 A part proportional to the identity cannot do procession, 1048 01:18:21,640 --> 01:18:27,000 is just a constant term that produces a constant phase, 1049 01:18:27,000 --> 01:18:28,490 just produces a pure phase. 1050 01:18:28,490 --> 01:18:31,060 That's a change, an overall phase 1051 01:18:31,060 --> 01:18:33,310 that doesn't change the state. 1052 01:18:33,310 --> 01:18:39,230 You would have an extra factor of e to the minus i times 1053 01:18:39,230 --> 01:18:44,260 that constant, g not t over H bar, 1054 01:18:44,260 --> 01:18:46,280 multiplying all the states. 1055 01:18:46,280 --> 01:18:49,970 Doesn't change the action on plus or minus state. 1056 01:18:49,970 --> 01:18:52,120 It's an overall phase. 1057 01:18:52,120 --> 01:18:56,960 This term in the Hamiltonian is almost never very important. 1058 01:18:56,960 --> 01:19:00,130 It doesn't do anything to the physical states, 1059 01:19:00,130 --> 01:19:02,550 just gives them pure phases. 1060 01:19:02,550 --> 01:19:05,200 And this term is the thing that matters. 1061 01:19:05,200 --> 01:19:08,590 So now with this Hamiltonian, because g 1062 01:19:08,590 --> 01:19:11,570 dot sigma is the form of the Hamiltonian, 1063 01:19:11,570 --> 01:19:15,350 and we've identified this physical phenomenon of Larmor 1064 01:19:15,350 --> 01:19:21,170 frequency, if you know your vector g for any Hamiltonian, 1065 01:19:21,170 --> 01:19:25,160 this might be the [INAUDIBLE] for ammonia molecule, 1066 01:19:25,160 --> 01:19:28,830 then you know how the states evolve in time. 1067 01:19:28,830 --> 01:19:31,840 Because you represent the state. 1068 01:19:31,840 --> 01:19:35,370 You have one state and a second state. 1069 01:19:35,370 --> 01:19:37,950 You think of the one state as the plus 1070 01:19:37,950 --> 01:19:40,420 of a spin, the minus of a spin. 1071 01:19:40,420 --> 01:19:43,890 And then you know that this is processing with this Larmor 1072 01:19:43,890 --> 01:19:45,840 frequency. 1073 01:19:45,840 --> 01:19:50,050 So it may sound a little abstract at this moment, 1074 01:19:50,050 --> 01:19:54,130 but this gives you the way to evolve any arbitrary state 1075 01:19:54,130 --> 01:19:55,290 intuitively. 1076 01:19:55,290 --> 01:19:58,530 You know the vector V where it points. 1077 01:19:58,530 --> 01:20:02,500 You know where your state points in the configuration space. 1078 01:20:02,500 --> 01:20:06,240 And you have a physical picture of what it does in time. 1079 01:20:06,240 --> 01:20:08,350 It's always precessing. 1080 01:20:08,350 --> 01:20:12,930 Therefore, the dynamics of a two-state system in time 1081 01:20:12,930 --> 01:20:16,410 is always procession, and that's what we have to learn. 1082 01:20:16,410 --> 01:20:21,580 So next time will be ammonia molecule, and then NMR.