1 00:00:00,060 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,810 under a Creative Commons license. 3 00:00:03,810 --> 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 --> 00:00:18,050 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:18,050 --> 00:00:18,730 at ocw.mit.edu. 8 00:00:21,790 --> 00:00:23,720 PROFESSOR: What we have to do today 9 00:00:23,720 --> 00:00:35,080 is study a very important example of a two- state system. 10 00:00:35,080 --> 00:00:39,090 That will be the ammonia molecule, 11 00:00:39,090 --> 00:00:43,555 and will lead us to understand how masers work. 12 00:00:46,180 --> 00:00:49,660 Masers are what for microwaves, the same thing 13 00:00:49,660 --> 00:00:54,450 as lasers are for light-- it's just a different frequency. 14 00:00:54,450 --> 00:00:59,560 Masers is microwaves and lasers is for light. 15 00:00:59,560 --> 00:01:00,710 It's the same thing. 16 00:01:00,710 --> 00:01:06,020 So it's a very nice application of two-state systems. 17 00:01:06,020 --> 00:01:10,560 And then we'll discuss over the last part of the lecture 18 00:01:10,560 --> 00:01:13,230 some aspects of nuclear magnetic resonance. 19 00:01:13,230 --> 00:01:15,600 I don't think I'll get to the end of it, 20 00:01:15,600 --> 00:01:18,770 because it's quite a bit of material. 21 00:01:18,770 --> 00:01:23,040 But we'll try to see what we can do. 22 00:01:23,040 --> 00:01:25,410 So let me remind you of the last thing 23 00:01:25,410 --> 00:01:27,050 we were doing last time, that this 24 00:01:27,050 --> 00:01:30,220 is going to be in the backdrop of what we do today. 25 00:01:30,220 --> 00:01:35,130 We spoke about Hamiltonians for a two-state system that 26 00:01:35,130 --> 00:01:40,910 were the most general two by two Hermitian matrix specified 27 00:01:40,910 --> 00:01:44,500 by four real numbers-- g0 and the three components 28 00:01:44,500 --> 00:01:48,510 of the vector g multiplied by the Pauli matrices. 29 00:01:48,510 --> 00:01:50,460 This is Hermitian. 30 00:01:50,460 --> 00:01:57,890 This can be written in this way, in which we've identified 31 00:01:57,890 --> 00:02:05,700 Hamiltonians for spins, in the sense that g dot sigma-- 32 00:02:05,700 --> 00:02:09,169 really, sigma is proportional to S, 33 00:02:09,169 --> 00:02:14,130 so this is equal to omega dot S, where 34 00:02:14,130 --> 00:02:19,430 omega-- Larmor-- is 2g over h bar. 35 00:02:19,430 --> 00:02:24,215 And we explained last time that if you have a term omega l 36 00:02:24,215 --> 00:02:32,130 dot S, spins will rotate with angular velocity omega l 37 00:02:32,130 --> 00:02:35,420 vector, which means they rotate around the axis defined 38 00:02:35,420 --> 00:02:38,990 by the vector omega l, with an angular velocity 39 00:02:38,990 --> 00:02:42,240 equal to the magnitude of the vector omega l. 40 00:02:42,240 --> 00:02:45,490 So that's Larmor precession. 41 00:02:45,490 --> 00:02:50,390 This Larmor precession in the case of a magnetic field 42 00:02:50,390 --> 00:02:55,040 is given by minus lambda times the magnetic field, gamma times 43 00:02:55,040 --> 00:02:59,000 the magnetic field, where gamma is that constant that relates 44 00:02:59,000 --> 00:03:01,660 the magnetic moment of the particle 45 00:03:01,660 --> 00:03:06,250 to the spin angular momentum of the particle. 46 00:03:06,250 --> 00:03:10,330 Then we got, moreover, that the energy 47 00:03:10,330 --> 00:03:14,840 levels of this Hamiltonian-- this is a two-state systems, 48 00:03:14,840 --> 00:03:16,930 so it's a two-dimensional vector space 49 00:03:16,930 --> 00:03:19,780 that can be at most two energy eigenstates. 50 00:03:19,780 --> 00:03:23,090 That's the simple thing about two-state systems. 51 00:03:23,090 --> 00:03:25,760 These two energy eigenstates have 52 00:03:25,760 --> 00:03:30,100 the energies equal to g0 plus/minus g. 53 00:03:30,100 --> 00:03:36,280 And the plus corresponds to the spin state n plus, 54 00:03:36,280 --> 00:03:39,760 and the minus corresponds to the spin state n minus. 55 00:03:39,760 --> 00:03:43,600 And you don't have to talk spin states when 56 00:03:43,600 --> 00:03:47,030 you write this spin states over here. 57 00:03:47,030 --> 00:03:50,840 The plus, you should think of spinning in the plus direction, 58 00:03:50,840 --> 00:03:55,840 but the thing that we call plus is the first basis vector. 59 00:03:55,840 --> 00:03:59,680 And the thing that we call minus is the second basis vector 60 00:03:59,680 --> 00:04:01,340 of this state space. 61 00:04:01,340 --> 00:04:04,240 Therefore, if you've given a matrix, 62 00:04:04,240 --> 00:04:07,460 Hamiltonian has nothing to do with spins. 63 00:04:07,460 --> 00:04:11,100 You still have the notion that the first basis 64 00:04:11,100 --> 00:04:16,560 vector, whatever it is-- an iron moving in this direction-- 65 00:04:16,560 --> 00:04:20,920 is the mathematical analog of a spin up. 66 00:04:20,920 --> 00:04:24,720 And the second basis vector-- whatever else it may be-- 67 00:04:24,720 --> 00:04:27,210 is the analog of the spin down. 68 00:04:27,210 --> 00:04:32,430 So this will be important for what we do now, 69 00:04:32,430 --> 00:04:37,270 as we begin the study of the ammonia molecule, 70 00:04:37,270 --> 00:04:38,170 and its states. 71 00:04:38,170 --> 00:04:43,900 So having reviewed the key ideas from the last part 72 00:04:43,900 --> 00:04:46,905 of last lecture, are there any questions? 73 00:04:57,870 --> 00:05:08,486 So let me begin with this ammonia molecule-- double M, 74 00:05:08,486 --> 00:05:12,370 M-O-N-I-A, is NH3. 75 00:05:16,525 --> 00:05:19,340 It's used as a fertilizer. 76 00:05:19,340 --> 00:05:25,820 It's a gas, has strong odor, no color, fertilizers, 77 00:05:25,820 --> 00:05:28,090 cleaning products, pharmaceuticals, 78 00:05:28,090 --> 00:05:30,100 all kinds of things. 79 00:05:30,100 --> 00:05:37,390 It has the shape of a flattened tetrahedron with a nitrogen 80 00:05:37,390 --> 00:05:43,050 atom at one corner, say, and the base the three hydrogen atoms. 81 00:05:43,050 --> 00:05:48,350 If it would not be a flattened tetrahedron, this angle 82 00:05:48,350 --> 00:05:52,610 over here-- if it would be an equilateral, regular 83 00:05:52,610 --> 00:05:54,830 tetrahedron, this angle over there 84 00:05:54,830 --> 00:05:57,110 would be 60 degrees, because this every phase 85 00:05:57,110 --> 00:06:00,900 would be an equilateral triangle. 86 00:06:00,900 --> 00:06:03,520 But if it's a flattened tetrahedron, 87 00:06:03,520 --> 00:06:08,860 if it will be totally flat-- then n would be at the base. 88 00:06:08,860 --> 00:06:12,380 The angle in between these two edges 89 00:06:12,380 --> 00:06:16,910 would be 120 degrees, because they have to add up to 360. 90 00:06:16,910 --> 00:06:22,960 Well, this has 108 degrees, so it's pretty flat, 91 00:06:22,960 --> 00:06:24,460 this tetrahedron. 92 00:06:24,460 --> 00:06:29,060 And you can imagine, actually, this is a two-state system, 93 00:06:29,060 --> 00:06:34,000 because just as the nitrogen can be like this-- 94 00:06:34,000 --> 00:06:40,530 can be up with respect to the base of the hydrogens, 95 00:06:40,530 --> 00:06:42,690 it could also be down. 96 00:06:42,690 --> 00:06:46,940 And so you can imagine this molecule rotating, 97 00:06:46,940 --> 00:06:51,100 and suddenly the up nitrogen goes down, 98 00:06:51,100 --> 00:06:54,660 and this keeps rotating-- a possible transition 99 00:06:54,660 --> 00:07:01,170 of the system, in which, well, I don't 100 00:07:01,170 --> 00:07:02,920 know if I'm drawing it well. 101 00:07:02,920 --> 00:07:04,480 I don't think so. 102 00:07:04,480 --> 00:07:13,120 But roughly, it could be like this-- the nitrogen is down. 103 00:07:13,120 --> 00:07:18,390 And this, in principle, would be like two 104 00:07:18,390 --> 00:07:21,060 possible configurations of this system. 105 00:07:21,060 --> 00:07:23,570 There's a barrier. 106 00:07:23,570 --> 00:07:26,900 This is in equilibrium, so if you try to push it, 107 00:07:26,900 --> 00:07:27,810 it's not easy. 108 00:07:27,810 --> 00:07:30,010 But if you did manage to push it, 109 00:07:30,010 --> 00:07:32,140 it would be stable in the other direction. 110 00:07:32,140 --> 00:07:40,965 It is as if you had a potential for the nitrogen-- a V of z-- 111 00:07:40,965 --> 00:07:48,220 the direction is z-- in which it can be here, up or down. 112 00:07:48,220 --> 00:07:51,330 And it's stable in either one, but there's 113 00:07:51,330 --> 00:07:54,920 a big barrier in between. 114 00:07:54,920 --> 00:07:58,160 So that's the story of this nitrogen atom. 115 00:07:58,160 --> 00:08:02,240 And we're going to try to describe 116 00:08:02,240 --> 00:08:04,140 this as a two-state system. 117 00:08:04,140 --> 00:08:07,200 So I need some notation. 118 00:08:07,200 --> 00:08:09,410 Well, I'm going to have the first basis 119 00:08:09,410 --> 00:08:18,830 state to be called up for N up, and the second basis state 120 00:08:18,830 --> 00:08:22,800 is going to be called down for nitrogen down. 121 00:08:34,809 --> 00:08:38,390 And now, I'm going to try to write 122 00:08:38,390 --> 00:08:40,960 the Hamiltonian for this system. 123 00:08:40,960 --> 00:08:47,470 Well, you know what sort of happens here. 124 00:08:47,470 --> 00:08:51,000 Your intuition in quantum mechanics with wave functions 125 00:08:51,000 --> 00:08:52,380 should be similar. 126 00:08:52,380 --> 00:08:56,220 Look-- this is not the two-state system, 127 00:08:56,220 --> 00:08:59,110 because there may be many energy eigenstates, 128 00:08:59,110 --> 00:09:02,930 but you know that the ground state looks like a wave 129 00:09:02,930 --> 00:09:04,365 function, just like this. 130 00:09:09,120 --> 00:09:14,050 And the first excited state could 131 00:09:14,050 --> 00:09:25,340 look like a wave function that is like-- oops-- this. 132 00:09:25,340 --> 00:09:28,720 Pretty much the same thing, but you flip one, 133 00:09:28,720 --> 00:09:31,200 and if the barrier is sufficiently high, 134 00:09:31,200 --> 00:09:35,860 these two energy levels are not that different. 135 00:09:35,860 --> 00:09:38,460 So the question is, how do we model this? 136 00:09:38,460 --> 00:09:45,065 There may be an energy E0 for the system, 137 00:09:45,065 --> 00:09:48,500 a ground state energy maybe, and a little bit of a higher 138 00:09:48,500 --> 00:09:49,760 energy. 139 00:09:49,760 --> 00:09:52,885 So we're going to write the Hamiltonian. 140 00:09:55,860 --> 00:10:00,740 And I'm going to put E0 for the moment. 141 00:10:03,440 --> 00:10:09,270 And my first basis state is 1 and up. 142 00:10:09,270 --> 00:10:12,570 This would be the 1 0, and here would 143 00:10:12,570 --> 00:10:16,060 be the 0 1, the second basis state. 144 00:10:16,060 --> 00:10:19,640 And this is saying that 1 0, the N up, 145 00:10:19,640 --> 00:10:23,760 is an energy eigenstate of energy E0. 146 00:10:23,760 --> 00:10:30,290 And down is an energy eigenstate of energy in up as well. 147 00:10:30,290 --> 00:10:32,140 But that can't be the story. 148 00:10:32,140 --> 00:10:35,660 There cannot be two degenerate energy eigenstates. 149 00:10:35,660 --> 00:10:39,810 Your intuition tells you that this is impossible. 150 00:10:39,810 --> 00:10:43,050 One dimensional potential wouldn't say that. 151 00:10:43,050 --> 00:10:46,750 So there must be something else happening. 152 00:10:46,750 --> 00:10:49,720 This cannot be the whole Hamiltonian that describes 153 00:10:49,720 --> 00:10:51,830 the physics of the problem. 154 00:10:51,830 --> 00:10:54,420 So what we're going to do is try to tinker 155 00:10:54,420 --> 00:10:57,330 with this Hamiltonian, a simple tinkering that 156 00:10:57,330 --> 00:11:01,080 is going to give us the physics that we expect. 157 00:11:01,080 --> 00:11:05,680 So I'm going to put a constant delta here. 158 00:11:05,680 --> 00:11:10,160 This should be Hermitian, so I should put the delta as well, 159 00:11:10,160 --> 00:11:13,170 another constant there. 160 00:11:13,170 --> 00:11:16,720 For convenience, however, I'd rather 161 00:11:16,720 --> 00:11:19,300 put the minus sign there. 162 00:11:19,300 --> 00:11:25,410 I will define delta to be positive for definiteness, 163 00:11:25,410 --> 00:11:33,950 and for convenience, however, I will put here minus delta. 164 00:11:33,950 --> 00:11:39,840 Now, you could say look, you say that, but maybe 165 00:11:39,840 --> 00:11:41,050 it's not for convenience. 166 00:11:41,050 --> 00:11:43,500 Maybe it changes the physics. 167 00:11:43,500 --> 00:11:46,130 Well, it cannot change the physics, 168 00:11:46,130 --> 00:11:49,810 because these things are the matrix elements 169 00:11:49,810 --> 00:11:56,200 of the Hamiltonian-- the 1 2, and the 2 1 matrix elements. 170 00:11:56,200 --> 00:12:00,370 And I could decide to change what they call the first basis 171 00:12:00,370 --> 00:12:04,810 vector, to call it minus the first basis vector. 172 00:12:04,810 --> 00:12:07,810 This would change the sign of this, change the sign of that, 173 00:12:07,810 --> 00:12:10,000 without changing those signs. 174 00:12:10,000 --> 00:12:14,460 So this sign is a matter of a basis. 175 00:12:14,460 --> 00:12:18,490 So we certainly have not made any assumption 176 00:12:18,490 --> 00:12:23,960 by putting that to be a minus sign over there. 177 00:12:23,960 --> 00:12:27,080 Now, once you have this Hamiltonian, 178 00:12:27,080 --> 00:12:29,510 this delta is going to be some energy. 179 00:12:29,510 --> 00:12:33,260 And that's going to be what mimics the physics here, 180 00:12:33,260 --> 00:12:36,720 because these states are not going to be any more energy 181 00:12:36,720 --> 00:12:39,050 eigenstates. 182 00:12:39,050 --> 00:12:41,700 The matrix is not diagonal anymore. 183 00:12:41,700 --> 00:12:44,650 So the 1 0 vector, and the 0 1 vector 184 00:12:44,650 --> 00:12:48,630 are not any more energy eigenstates. 185 00:12:48,630 --> 00:12:51,420 Moreover, it's interesting to try 186 00:12:51,420 --> 00:12:55,600 to figure out what it has to do with our previous system. 187 00:12:55,600 --> 00:13:03,090 So this is E0 times 1 minus delta times sigma 1. 188 00:13:07,120 --> 00:13:09,520 And that's a good thing to know. 189 00:13:13,130 --> 00:13:18,350 So in this case, comparing to this g 190 00:13:18,350 --> 00:13:23,180 is the vector in the x direction, 191 00:13:23,180 --> 00:13:26,590 because its g multiplying by sigma 1. 192 00:13:26,590 --> 00:13:31,410 And it has magnitude delta. 193 00:13:31,410 --> 00:13:35,560 So we notice-- and we're going to make a picture later-- 194 00:13:35,560 --> 00:13:43,360 is that g, in this case so far, is equal to delta times 195 00:13:43,360 --> 00:13:46,000 the unit vector in the x direction, 196 00:13:46,000 --> 00:13:48,885 minus delta times the unit vector in the x direction. 197 00:13:55,570 --> 00:13:58,390 So g is equal to delta. 198 00:14:01,760 --> 00:14:03,900 So, OK, we've written those. 199 00:14:03,900 --> 00:14:07,440 Let's then figure out what are the ground 200 00:14:07,440 --> 00:14:10,650 states and the excited states. 201 00:14:10,650 --> 00:14:15,510 And this is a two by two matrix, and a simple one, at that. 202 00:14:15,510 --> 00:14:19,070 So you could just do it, or better 203 00:14:19,070 --> 00:14:21,020 to figure out what we're doing. 204 00:14:21,020 --> 00:14:24,090 We'll use our formulas before. 205 00:14:24,090 --> 00:14:25,940 Yes, George. 206 00:14:25,940 --> 00:14:27,860 AUDIENCE: So why is it that we mandate 207 00:14:27,860 --> 00:14:29,300 that delta has to be real? 208 00:14:29,300 --> 00:14:31,700 I mean, that's not the most general form. 209 00:14:31,700 --> 00:14:35,190 PROFESSOR: That's right, it's not the most general form. 210 00:14:35,190 --> 00:14:38,380 So at this moment, we're trying to do 211 00:14:38,380 --> 00:14:42,860 what any reasonable physicist does. 212 00:14:42,860 --> 00:14:46,850 Without delta, it doesn't match the physics. 213 00:14:46,850 --> 00:14:50,330 So let's try the simplest thing that can work, 214 00:14:50,330 --> 00:14:53,380 and a delta real-- we'll see if it works. 215 00:14:53,380 --> 00:14:57,890 And if it works, we'll worry later about different things. 216 00:14:57,890 --> 00:15:01,290 So we'll put the simplest thing at this moment, 217 00:15:01,290 --> 00:15:06,400 but indeed, we could put more complicated things. 218 00:15:06,400 --> 00:15:13,660 So given this, in fact, we know what it the energy eigenstates 219 00:15:13,660 --> 00:15:17,180 should be, or we more or less can 220 00:15:17,180 --> 00:15:19,910 guess what the energy eigenstates should be. 221 00:15:19,910 --> 00:15:26,930 Let me tell you the energies are g0 plus/minus g, 222 00:15:26,930 --> 00:15:36,340 so you're going to get E0 plus delta and e0 minus 223 00:15:36,340 --> 00:15:46,180 delta as the energies-- E excited, and E ground. 224 00:15:46,180 --> 00:15:49,490 And the gap between these two energies-- 225 00:15:49,490 --> 00:15:55,220 the gap between these two energy levels is 2 delta. 226 00:15:59,740 --> 00:16:00,840 So there we go. 227 00:16:00,840 --> 00:16:03,540 We've already produced something good. 228 00:16:03,540 --> 00:16:06,130 We have two energy eigenstates. 229 00:16:06,130 --> 00:16:08,420 There should be a small energy difference, 230 00:16:08,420 --> 00:16:10,820 and that gap is 2 delta. 231 00:16:10,820 --> 00:16:13,950 Now, what are those states? 232 00:16:13,950 --> 00:16:18,750 Well, it's not too hard to see that the eigenstate that 233 00:16:18,750 --> 00:16:23,980 has this energy is the excited state, is 234 00:16:23,980 --> 00:16:30,500 1 over square root of 2, 1 minus 1. 235 00:16:30,500 --> 00:16:33,640 If you add with this matrix on this, 236 00:16:33,640 --> 00:16:37,690 that's the energy eigenstate. 237 00:16:37,690 --> 00:16:39,860 And the energy eigenstate for this one 238 00:16:39,860 --> 00:16:45,460 is 1 over square root of 2, 1 1. 239 00:16:45,460 --> 00:16:47,170 Let's write them. 240 00:16:47,170 --> 00:16:48,900 These are the eigenvectors. 241 00:16:48,900 --> 00:16:53,440 Let's write them as 1 over square root of 2, 242 00:16:53,440 --> 00:17:02,310 nitrogen up minus nitrogen down, and 1 over square root of 2 243 00:17:02,310 --> 00:17:06,430 nitrogen up plus nitrogen down. 244 00:17:13,670 --> 00:17:19,839 So I want to, even though it's not complicated to do this, 245 00:17:19,839 --> 00:17:24,859 and we have called these states that way, so it's all clear. 246 00:17:24,859 --> 00:17:32,260 I want you to see how that comes from our spin way of thinking. 247 00:17:32,260 --> 00:17:35,000 So you know there's this molecule, 248 00:17:35,000 --> 00:17:38,060 and for this molecule, only one direction matters. 249 00:17:38,060 --> 00:17:40,580 We could have called it x, if we wanted. 250 00:17:40,580 --> 00:17:43,930 In fact, maybe x would have been a better name. 251 00:17:43,930 --> 00:17:46,080 On the other hand, for spin states, 252 00:17:46,080 --> 00:17:49,680 there are three dimensions-- x, y, and z. 253 00:17:49,680 --> 00:17:54,870 So we have to think in an abstract way. 254 00:17:54,870 --> 00:17:57,780 So where is this vector G? 255 00:17:57,780 --> 00:18:00,870 We said G is minus delta Ex. 256 00:18:00,870 --> 00:18:02,970 So this is the x-axis. 257 00:18:02,970 --> 00:18:04,340 This is the y-axis. 258 00:18:04,340 --> 00:18:07,440 This is the z-axis. 259 00:18:07,440 --> 00:18:09,210 G is here. 260 00:18:09,210 --> 00:18:12,830 The vector G goes back over here, 261 00:18:12,830 --> 00:18:17,000 is minus delta x hat vector. 262 00:18:22,370 --> 00:18:29,650 Now, what if you have g in that way? 263 00:18:29,650 --> 00:18:37,210 You know that the excited state is 264 00:18:37,210 --> 00:18:39,000 one of these states over here. 265 00:18:39,000 --> 00:18:47,530 Let's see-- this N plus is the excited state, 266 00:18:47,530 --> 00:18:51,640 and N minus is the lower state. 267 00:18:51,640 --> 00:18:54,950 So the excited state should point 268 00:18:54,950 --> 00:19:01,120 in the direction of g vector, because N corresponds 269 00:19:01,120 --> 00:19:03,170 to the direction of the g vector. 270 00:19:03,170 --> 00:19:13,110 G is positive here, this little g is positive. 271 00:19:13,110 --> 00:19:17,590 So g is in there. 272 00:19:17,590 --> 00:19:25,720 N is in there as well, because g and n are parallel. 273 00:19:25,720 --> 00:19:30,800 And the excited state should correspond 274 00:19:30,800 --> 00:19:35,970 to N, a vector in the plus n direction. 275 00:19:35,970 --> 00:19:39,585 So the excited state should be here. 276 00:19:43,750 --> 00:19:45,870 It's a spin state in that direction. 277 00:19:48,730 --> 00:19:50,400 That's what that formula says. 278 00:19:50,400 --> 00:19:56,050 And the ground state should be a spin state in the minus N 279 00:19:56,050 --> 00:20:00,500 direction, so this must be the ground state. 280 00:20:00,500 --> 00:20:05,150 So this I call the excited state, 281 00:20:05,150 --> 00:20:09,410 and this the ground state. 282 00:20:09,410 --> 00:20:15,850 And indeed, remember now that what is your translation. 283 00:20:15,850 --> 00:20:20,260 1 and 2-- the 1 and 2 states are like the plus and minus 284 00:20:20,260 --> 00:20:21,450 of spins. 285 00:20:21,450 --> 00:20:29,460 So in terms of spin language, this excited state 286 00:20:29,460 --> 00:20:32,180 is the plus minus the minus. 287 00:20:35,490 --> 00:20:48,190 And this is the plus plus the minus, because the up is plus, 288 00:20:48,190 --> 00:20:51,810 the down is minus. 289 00:20:51,810 --> 00:20:56,380 So indeed, this state-- you probably remember it. 290 00:20:56,380 --> 00:20:59,290 This is a spin along the x direction. 291 00:20:59,290 --> 00:21:02,710 So the ground state must be like a spin along the x direction. 292 00:21:02,710 --> 00:21:04,870 That's here. 293 00:21:04,870 --> 00:21:10,370 The excited state is a spin, the orthogonal one in the minus x 294 00:21:10,370 --> 00:21:13,530 direction, so it must be a state orthogonal to this one, 295 00:21:13,530 --> 00:21:18,310 as to this, and it points in the other direction. 296 00:21:18,310 --> 00:21:23,950 So those are our spins. 297 00:21:23,950 --> 00:21:28,800 And we had that the gap delta-- the gap is 2 omega. 298 00:21:28,800 --> 00:21:34,100 It's an energy, so it's what we called 299 00:21:34,100 --> 00:21:40,950 h omega naught a photon-- the transition energy. 300 00:21:40,950 --> 00:21:45,380 I could give this energy in eVs, but I actually don't have it. 301 00:21:45,380 --> 00:21:49,880 I have the wavelength of the frequency of the associated 302 00:21:49,880 --> 00:21:50,500 photon. 303 00:21:50,500 --> 00:21:53,720 So this corresponds to a frequency 304 00:21:53,720 --> 00:22:03,430 nu of 23.87 gigahertz, and a lambda of about 305 00:22:03,430 --> 00:22:09,155 1.26 centimeters-- more or less half an inch. 306 00:22:13,940 --> 00:22:20,980 So that is the transition difference 307 00:22:20,980 --> 00:22:22,730 between these two levels. 308 00:22:22,730 --> 00:22:24,510 So this is something people knew-- 309 00:22:24,510 --> 00:22:29,280 there's two levels above this molecule, 310 00:22:29,280 --> 00:22:32,280 and they correspond to the result 311 00:22:32,280 --> 00:22:35,920 of this perturbation that splits them. 312 00:22:35,920 --> 00:22:39,970 So in a sense, we've got a nice model-- perfectly reasonable 313 00:22:39,970 --> 00:22:43,830 model, without introducing much complexity-- of what 314 00:22:43,830 --> 00:22:46,750 this thing is doing. 315 00:22:46,750 --> 00:22:50,850 So let's do a little exercise. 316 00:22:50,850 --> 00:22:56,410 Let's see how does the N up state evolve in time? 317 00:23:12,650 --> 00:23:19,000 So we have psi at time equals 0 be in the state up. 318 00:23:19,000 --> 00:23:20,350 What is it later? 319 00:23:23,640 --> 00:23:27,210 There are many ways to do it-- many, many ways to do it. 320 00:23:27,210 --> 00:23:33,340 The quickest, in principle, is to think about spins, 321 00:23:33,340 --> 00:23:36,420 even if just a little painful. 322 00:23:36,420 --> 00:23:41,340 But let's think about spins. 323 00:23:41,340 --> 00:23:52,210 Omega l is going to be around the direction of g. 324 00:23:52,210 --> 00:23:55,645 So think of the state of the spin. 325 00:23:59,590 --> 00:24:03,465 The N up state, the up state, is here. 326 00:24:10,440 --> 00:24:18,340 And then it's going to precess with angular frequency 327 00:24:18,340 --> 00:24:20,650 vector in the direction of g. 328 00:24:20,650 --> 00:24:23,980 So it's going to precess in the direction of g. 329 00:24:23,980 --> 00:24:28,090 So you can imagine now this vector precessing. 330 00:24:28,090 --> 00:24:31,310 And it's going to go-- since it's essentially the minus x 331 00:24:31,310 --> 00:24:34,520 direction-- precession in time is going to flip it 332 00:24:34,520 --> 00:24:39,040 to the y-axis, and then make it rotate in the z-y axis. 333 00:24:39,040 --> 00:24:41,430 That's all what it's going to do. 334 00:24:41,430 --> 00:24:45,330 So you have a picture of what it's going to do. 335 00:24:45,330 --> 00:24:49,950 We might as well calculate a little, although the picture is 336 00:24:49,950 --> 00:24:53,660 complete, and the frequency is known, and everything. 337 00:24:53,660 --> 00:24:58,300 But what you do here, of course, is 338 00:24:58,300 --> 00:25:02,210 you try to write it in terms of energy eigenstates. 339 00:25:02,210 --> 00:25:07,970 And the up state is the 1 over square root 340 00:25:07,970 --> 00:25:10,035 times the sum of e plus g. 341 00:25:14,130 --> 00:25:17,710 And you know the energies of those two states, 342 00:25:17,710 --> 00:25:19,580 so you know how they evolve in time. 343 00:25:22,540 --> 00:25:23,970 It will be in the notes. 344 00:25:23,970 --> 00:25:26,220 You can do this. 345 00:25:26,220 --> 00:25:36,130 After you now evolve, with e to the minus i ht over h bar, 346 00:25:36,130 --> 00:25:44,630 you then go back from e to g, to up and down, 347 00:25:44,630 --> 00:25:47,690 because that's sort of the intuition that we want to 348 00:25:47,690 --> 00:25:49,850 Have. 349 00:25:49,850 --> 00:25:54,570 So it's not difficult. 350 00:25:54,570 --> 00:25:57,000 None of these steps are difficult. 351 00:25:57,000 --> 00:26:00,190 e and g are written in terms of up and down. 352 00:26:00,190 --> 00:26:03,060 So what does one get? 353 00:26:03,060 --> 00:26:12,130 One gets psi of t is equal e to the minus i, et over h bar 354 00:26:12,130 --> 00:26:18,830 times cosine of t delta over h bar, 355 00:26:18,830 --> 00:26:28,670 times the state up plus i sine of t delta over h bar, 356 00:26:28,670 --> 00:26:29,625 state down. 357 00:26:40,120 --> 00:26:44,970 This is the time evolution, so the probabilities, for example, 358 00:26:44,970 --> 00:26:53,360 to be up is the square of this one-- cosine squared 359 00:26:53,360 --> 00:26:58,570 of t delta over h bar. 360 00:26:58,570 --> 00:27:02,380 And the probability to be down is sine 361 00:27:02,380 --> 00:27:04,340 squared of the same thing. 362 00:27:08,220 --> 00:27:13,190 So this poor nitrogen molecule, if it 363 00:27:13,190 --> 00:27:16,860 happens to have the nitrogen up, is 364 00:27:16,860 --> 00:27:19,120 going to start rotating like crazy, 365 00:27:19,120 --> 00:27:21,420 even if you don't do anything. 366 00:27:21,420 --> 00:27:26,740 It's just sitting there, and it's rotating up and down, 367 00:27:26,740 --> 00:27:34,780 with a speed doing this thing 23 billion times a second. 368 00:27:34,780 --> 00:27:37,250 Molecule's up and down, because it's not 369 00:27:37,250 --> 00:27:40,860 in a stationary eigenstate. 370 00:27:40,860 --> 00:27:45,940 Now, here, actually, you may think 371 00:27:45,940 --> 00:27:52,000 that something is a little funny, because you would say, 372 00:27:52,000 --> 00:27:57,850 well, the frequency of rotation is like delta over h bar, 373 00:27:57,850 --> 00:28:03,440 but the Larmor frequency is supposed to be 2g over h bar, 374 00:28:03,440 --> 00:28:08,960 so it would correspond to a Larmor frequency of 2 delta 375 00:28:08,960 --> 00:28:14,196 over h bar, which is exactly the frequency of the photons. 376 00:28:18,940 --> 00:28:21,140 But there's no contradiction here. 377 00:28:21,140 --> 00:28:25,940 This is, in fact, rotating at that speed, 378 00:28:25,940 --> 00:28:27,080 at twice that speed. 379 00:28:27,080 --> 00:28:29,560 Because if you remember, for a spin state, 380 00:28:29,560 --> 00:28:35,550 this was the cosine of theta over 2. 381 00:28:35,550 --> 00:28:39,270 Therefore, as it changes, that's the way theta over 2 382 00:28:39,270 --> 00:28:40,480 is changing. 383 00:28:40,480 --> 00:28:44,080 But theta, which is the angle of this physical rotation, 384 00:28:44,080 --> 00:28:46,770 changes twice as fast. 385 00:28:46,770 --> 00:28:50,405 So it's, again, those 1/2s of spin states that 386 00:28:50,405 --> 00:28:52,910 are very confusing sometimes. 387 00:28:52,910 --> 00:28:55,040 But there's no contradiction. 388 00:28:55,040 --> 00:29:00,870 The sort of Larmor frequency of the equivalent spin problem 389 00:29:00,870 --> 00:29:03,800 is exactly the same as the frequency 390 00:29:03,800 --> 00:29:07,160 of the original problem. 391 00:29:07,160 --> 00:29:16,240 So now we want to make this into something more practical. 392 00:29:16,240 --> 00:29:20,920 And for that, what we explore is the fact 393 00:29:20,920 --> 00:29:25,255 that this molecule has an electric dipole moment. 394 00:29:29,210 --> 00:29:39,050 So the molecule as we pictured it there, 395 00:29:39,050 --> 00:29:42,210 as it happens, the electrons of the hydrogen 396 00:29:42,210 --> 00:29:44,600 sort of cluster near the nitrogen. 397 00:29:44,600 --> 00:29:47,620 So this up region is kind of negative. 398 00:29:47,620 --> 00:29:50,190 The bottom region is kind of positive, 399 00:29:50,190 --> 00:29:55,285 and there is an electric dipole moment pointing down. 400 00:30:00,760 --> 00:30:04,960 So this is a pretty important property 401 00:30:04,960 --> 00:30:07,720 of this molecule, this dipole moment. 402 00:30:07,720 --> 00:30:11,380 And electric dipoles we usually call p. 403 00:30:11,380 --> 00:30:14,800 But for some reason-- maybe I should change the notes, 404 00:30:14,800 --> 00:30:18,440 at some stage or maybe this is discussed very nicely 405 00:30:18,440 --> 00:30:20,680 in Feynman's lectures on physics. 406 00:30:20,680 --> 00:30:24,000 He uses mu for this, like for magnetic dipole. 407 00:30:24,000 --> 00:30:28,890 So I will actually use mu as well, now. 408 00:30:28,890 --> 00:30:33,300 So this thing has an electric dipole, 409 00:30:33,300 --> 00:30:40,970 and therefore the energy is the electric dipole 410 00:30:40,970 --> 00:30:42,830 dotted with the electric field. 411 00:30:46,720 --> 00:30:50,210 And that electric field is an external electric field. 412 00:30:50,210 --> 00:30:52,810 You have this little dipole, which is this molecule, 413 00:30:52,810 --> 00:30:54,940 and you put it inside an electric field, 414 00:30:54,940 --> 00:30:58,010 and there's a contribution to the energy, 415 00:30:58,010 --> 00:31:03,220 just because the dipole is sitting on an electric field. 416 00:31:03,220 --> 00:31:08,560 And that means our Hamiltonian is now changed. 417 00:31:08,560 --> 00:31:14,460 So I will consider the case in which we 418 00:31:14,460 --> 00:31:19,150 have an electric field in the z direction-- 419 00:31:19,150 --> 00:31:22,760 a positive electric field in the z direction-- 420 00:31:22,760 --> 00:31:33,620 so that E is equal to E times z. 421 00:31:33,620 --> 00:31:40,680 And mu would be equal to minus mu times z, 422 00:31:40,680 --> 00:31:42,240 because it points down. 423 00:31:42,240 --> 00:31:47,580 We've assumed that the dipole is down. 424 00:31:47,580 --> 00:31:52,870 And the dipole is down for the case of spin 425 00:31:52,870 --> 00:31:54,740 in the z direction. 426 00:31:58,430 --> 00:32:04,420 So look what we get here-- this energy contribution is 427 00:32:04,420 --> 00:32:08,400 essentially mu E. And it's the energy that 428 00:32:08,400 --> 00:32:15,370 is acquired by the state in which the nitrogen is up. 429 00:32:15,370 --> 00:32:19,030 This is for nitrogen up. 430 00:32:19,030 --> 00:32:21,930 So what we've discovered-- if we want 431 00:32:21,930 --> 00:32:24,890 to model this in the Hamiltonian is that we can take 432 00:32:24,890 --> 00:32:33,150 the Hamiltonian that we have-- E0 minus delta, 433 00:32:33,150 --> 00:32:43,400 e0 minus delta, and the energy of the state up with nitrogen 434 00:32:43,400 --> 00:32:47,950 up, is this one-- mu E. So we add 435 00:32:47,950 --> 00:32:52,030 mu E. And the one with the spin down 436 00:32:52,030 --> 00:32:56,470 will be the opposite, so it will be minus mu E. 437 00:32:56,470 --> 00:33:02,710 And this is our reasonable expectation for the Hamiltonian 438 00:33:02,710 --> 00:33:09,920 of this molecule inside an electric field. 439 00:33:09,920 --> 00:33:15,410 So this is the NH3 in E field. 440 00:33:20,160 --> 00:33:24,630 So again, we can wonder what kind of thing happens here. 441 00:33:24,630 --> 00:33:29,370 And the best thing is to first say this is E0 1 442 00:33:29,370 --> 00:33:31,730 minus delta sigma 1. 443 00:33:31,730 --> 00:33:38,250 And then you see, oh, it's mu E sigma 3. 444 00:33:38,250 --> 00:33:40,570 So getting to that, and I realize 445 00:33:40,570 --> 00:33:43,265 this is a little more painful than the other one. 446 00:33:43,265 --> 00:33:45,840 And we don't have to do it, because we've 447 00:33:45,840 --> 00:33:48,430 solved the general problem. 448 00:33:48,430 --> 00:33:51,630 And the energies, this time, are going 449 00:33:51,630 --> 00:33:58,340 to be E of the excited one, and E of the lower one-- ground 450 00:33:58,340 --> 00:34:00,200 state. 451 00:34:00,200 --> 00:34:03,850 It's going to be E0 plus g. 452 00:34:03,850 --> 00:34:10,310 And g was the magnitude of the vector g. 453 00:34:10,310 --> 00:34:14,250 So it's the magnitude of the vector g 454 00:34:14,250 --> 00:34:24,810 that now has components minus delta 0 and mu E. 455 00:34:24,810 --> 00:34:29,570 So here we get plus square root of delta 456 00:34:29,570 --> 00:34:34,080 squared plus mu E squared. 457 00:34:34,080 --> 00:34:42,590 And here is 0 minus square root of delta squared plus mu 458 00:34:42,590 --> 00:34:43,455 E squared. 459 00:34:50,520 --> 00:34:51,802 So there we go. 460 00:34:51,802 --> 00:34:55,090 If we know how the energies behave, 461 00:34:55,090 --> 00:34:57,030 even if we have some electric field-- 462 00:34:57,030 --> 00:35:02,550 and typically delta is such, and mu 463 00:35:02,550 --> 00:35:06,230 is such that, for most electric fields 464 00:35:06,230 --> 00:35:09,220 that you ever have in the lab-- this 465 00:35:09,220 --> 00:35:12,330 is very small compared to that. 466 00:35:12,330 --> 00:35:14,740 The dipole moment is sufficiently small 467 00:35:14,740 --> 00:35:17,710 that the energies that you get from here 468 00:35:17,710 --> 00:35:21,230 are pale compared to the difference of energies 469 00:35:21,230 --> 00:35:22,580 over there. 470 00:35:22,580 --> 00:35:24,890 So you can approximate. 471 00:35:24,890 --> 00:35:34,235 This is E0 plus delta plus 1/2 mu E squared over delta. 472 00:35:37,520 --> 00:35:40,240 This is for mu E small. 473 00:35:40,240 --> 00:35:49,410 Here E0 minus delta minus 1/2 mu E over delta squared. 474 00:35:49,410 --> 00:35:54,870 And this is when mu E is much smaller than delta. 475 00:35:54,870 --> 00:35:57,280 Now, the only reason I point this out 476 00:35:57,280 --> 00:36:02,380 is because it does provide a technological opportunity 477 00:36:02,380 --> 00:36:10,720 to separate a beam of particles into excited- 478 00:36:10,720 --> 00:36:12,580 and the ground-state level. 479 00:36:12,580 --> 00:36:15,660 Sort of like Stern-Gerlach experiment, 480 00:36:15,660 --> 00:36:23,820 you put, now, this beam that has this ammonia molecules. 481 00:36:23,820 --> 00:36:26,290 And you put them inside an electric field 482 00:36:26,290 --> 00:36:28,520 that has a gradient. 483 00:36:28,520 --> 00:36:32,170 In a gradient, this state is going 484 00:36:32,170 --> 00:36:35,520 to try to go to minimize its energy. 485 00:36:35,520 --> 00:36:38,680 So it's going to go to the regions of the electric field 486 00:36:38,680 --> 00:36:41,530 where the electric field is small. 487 00:36:41,530 --> 00:36:45,290 This particle minimizes its energy 488 00:36:45,290 --> 00:36:48,370 when it goes to the regions of the electric field 489 00:36:48,370 --> 00:36:51,000 when the electric field is big. 490 00:36:51,000 --> 00:36:54,570 So it's like putting it in a Stern-Gerlach experiment. 491 00:36:54,570 --> 00:36:57,890 You have your beam, and you separate them. 492 00:36:57,890 --> 00:36:59,470 You have your beam and you manage 493 00:36:59,470 --> 00:37:05,170 to separate the things that can be in an excited state, 494 00:37:05,170 --> 00:37:07,400 and the things that are in the ground state. 495 00:37:11,480 --> 00:37:17,420 And now what you do is insert these excited states 496 00:37:17,420 --> 00:37:20,315 into a resonant cavity. 497 00:37:25,170 --> 00:37:29,480 Have a little hole here, and a little hole here, 498 00:37:29,480 --> 00:37:35,605 and E comes in, and something comes out. 499 00:37:41,090 --> 00:37:45,930 So we're getting now to the design of the maser. 500 00:37:45,930 --> 00:37:50,050 The idea that we're trying to do is 501 00:37:50,050 --> 00:37:59,670 that we try to make a cavity tuned 502 00:37:59,670 --> 00:38:06,400 to 23.7 gigahertz-- the frequency associated 503 00:38:06,400 --> 00:38:07,295 with a gap. 504 00:38:10,510 --> 00:38:19,610 And we just insert those Es over there, these excited states 505 00:38:19,610 --> 00:38:26,030 over there, and hope that by the time they go out, 506 00:38:26,030 --> 00:38:27,010 they become a g. 507 00:38:30,460 --> 00:38:34,940 Because if they go from E-- say there was an electric field 508 00:38:34,940 --> 00:38:38,370 here to separate them, and then E's over here. 509 00:38:38,370 --> 00:38:40,630 This is the excited state. 510 00:38:40,630 --> 00:38:42,830 There's no more electric fields over here. 511 00:38:42,830 --> 00:38:46,590 It just comes into the cavity as an excited state. 512 00:38:46,590 --> 00:38:51,140 The excited state has energy E0 plus delta. 513 00:38:51,140 --> 00:38:57,590 And then, if it manages to go out of the cavity as the ground 514 00:38:57,590 --> 00:39:02,250 state, then it would have energy E0 minus delta. 515 00:39:02,250 --> 00:39:05,390 It must've lost energy to delta. 516 00:39:05,390 --> 00:39:08,360 That can go nicely into the electromagnetic field 517 00:39:08,360 --> 00:39:13,400 and become one photon-- a one photon state 518 00:39:13,400 --> 00:39:16,510 in the cavity of the right frequency, 519 00:39:16,510 --> 00:39:20,130 because the cavity is tuned to that. 520 00:39:20,130 --> 00:39:22,980 The only difficulty with this assumption 521 00:39:22,980 --> 00:39:28,420 is that E is an energy eigenstate. 522 00:39:28,420 --> 00:39:31,030 So energy eigenstates are lazy. 523 00:39:31,030 --> 00:39:32,930 They're stationary states. 524 00:39:32,930 --> 00:39:34,980 They don't like to change. 525 00:39:34,980 --> 00:39:38,630 So there's no reason why it should go out as g. 526 00:39:38,630 --> 00:39:41,650 It's excited state. 527 00:39:41,650 --> 00:39:44,810 It's perfectly happy to remain excited forever. 528 00:39:44,810 --> 00:39:51,210 So what must happen somehow is that there's 529 00:39:51,210 --> 00:39:56,010 an electric field here in the cavity, 530 00:39:56,010 --> 00:39:59,850 and that stimulates this thing to make the transition, 531 00:39:59,850 --> 00:40:03,370 because once there's an electric field, 532 00:40:03,370 --> 00:40:07,800 E is not anymore an energy eigenstate. 533 00:40:07,800 --> 00:40:10,400 The E of the original system is not anymore 534 00:40:10,400 --> 00:40:13,070 an energy eigenstate, and nor is this. 535 00:40:13,070 --> 00:40:16,030 So then it's going to change in time. 536 00:40:16,030 --> 00:40:18,670 So the problem is a delicate one in which 537 00:40:18,670 --> 00:40:23,960 we want to somehow have an electric field here 538 00:40:23,960 --> 00:40:26,580 that is self-consistent with the idea 539 00:40:26,580 --> 00:40:32,800 that this excited state goes out as the ground state. 540 00:40:32,800 --> 00:40:37,530 And that's why it's microwave amplification 541 00:40:37,530 --> 00:40:40,310 by stimulated emission of radiation, 542 00:40:40,310 --> 00:40:43,660 because you're going to amplify a signal here. 543 00:40:43,660 --> 00:40:46,840 It's a microwave, 1 centimeter wavelength-- 544 00:40:46,840 --> 00:40:48,610 that's a microwave. 545 00:40:48,610 --> 00:40:52,090 And the stimulation is the fact that this wouldn't do it 546 00:40:52,090 --> 00:40:56,200 unless there's some electric field already. 547 00:40:56,200 --> 00:40:58,880 So you could say, well, so how does it get started? 548 00:40:58,880 --> 00:41:02,010 There's no electric field to begin with. 549 00:41:02,010 --> 00:41:04,880 Well, you know quantum mechanics, 550 00:41:04,880 --> 00:41:08,540 and you know that in general, there are little fluctuations, 551 00:41:08,540 --> 00:41:13,960 and there's energies-- small photons, one 552 00:41:13,960 --> 00:41:16,920 or two photons that suddenly appear because of anything. 553 00:41:16,920 --> 00:41:19,770 Any motion of charges in here produces 554 00:41:19,770 --> 00:41:21,570 an electromagnetic wave. 555 00:41:21,570 --> 00:41:25,390 So at the beginning, yes-- there's no many photons here. 556 00:41:25,390 --> 00:41:30,290 But somehow, by having it resonate at that frequency, 557 00:41:30,290 --> 00:41:32,310 it's very easy to get those photons. 558 00:41:32,310 --> 00:41:35,540 And a few appear, and a few molecules start to turn in, 559 00:41:35,540 --> 00:41:40,020 and then very soon this is full with energy, in which there's 560 00:41:40,020 --> 00:41:44,860 a consistent configuration of some electric field oscillating 561 00:41:44,860 --> 00:41:50,360 and producing precisely the right transitions here. 562 00:41:50,360 --> 00:41:56,030 So I want to use the next 50 minutes to describe that math. 563 00:41:56,030 --> 00:41:57,550 How do we do this? 564 00:41:57,550 --> 00:42:00,800 Because it just shows really the sort 565 00:42:00,800 --> 00:42:03,620 of hard part of the problem. 566 00:42:03,620 --> 00:42:06,780 How do you get consistently a field, 567 00:42:06,780 --> 00:42:09,310 and the radiation going on? 568 00:42:09,310 --> 00:42:17,230 So maybe I should call this E prime and g prime. 569 00:42:17,230 --> 00:42:18,670 They shouldn't be confused. 570 00:42:18,670 --> 00:42:23,750 E and g are these states that we had before. 571 00:42:23,750 --> 00:42:26,840 And E prime and g prime, we never wrote those states, 572 00:42:26,840 --> 00:42:31,280 but they are deformed states due to an electric field. 573 00:42:31,280 --> 00:42:35,140 OK, so what do we have to do? 574 00:42:35,140 --> 00:42:38,190 Well here is E and g. 575 00:42:38,190 --> 00:42:42,650 And we had a Hamiltonian. 576 00:42:42,650 --> 00:42:45,330 There's going to be an electric field here, 577 00:42:45,330 --> 00:42:49,500 so this Hamiltonian is the relevant one. 578 00:42:49,500 --> 00:42:52,180 The only problem with this Hamiltonian 579 00:42:52,180 --> 00:42:56,130 is that this is going to be a time-dependent field, something 580 00:42:56,130 --> 00:42:58,920 that we're a little scared of-- Hamiltonians with time 581 00:42:58,920 --> 00:43:03,640 dependence-- for good reason, because they're hard. 582 00:43:03,640 --> 00:43:05,800 But anyway, let's try to see. 583 00:43:05,800 --> 00:43:08,960 Today's all about Hamiltonians with time dependence. 584 00:43:08,960 --> 00:43:11,220 So there's going to be a time-dependent 585 00:43:11,220 --> 00:43:12,550 is going to be the wave here. 586 00:43:12,550 --> 00:43:15,330 So that's the relevant Hamiltonian. 587 00:43:15,330 --> 00:43:20,580 But it's the Hamiltonian in the 1 2 basis, in that up nitrogen, 588 00:43:20,580 --> 00:43:23,030 down nitrogen basis. 589 00:43:23,030 --> 00:43:27,270 I want that Hamiltonian in the Eg basis. 590 00:43:27,270 --> 00:43:28,230 It's better. 591 00:43:28,230 --> 00:43:29,620 It's more useful. 592 00:43:29,620 --> 00:43:38,907 So let's try to see how that looks-- Hamiltonian in the Eg 593 00:43:38,907 --> 00:43:39,406 basis. 594 00:43:45,500 --> 00:43:57,860 H prime in E equal 1 g equal 2 primes, maybe put basis. 595 00:44:00,760 --> 00:44:07,290 So here is the Hamiltonian in this basis. 596 00:44:07,290 --> 00:44:11,480 In the 1 2 basis, I have to pass to the other basis, the Eg 597 00:44:11,480 --> 00:44:13,050 basis. 598 00:44:13,050 --> 00:44:14,890 So it's not complicated. 599 00:44:14,890 --> 00:44:21,830 It takes a little work, but it's nothing all that difficult. 600 00:44:21,830 --> 00:44:30,000 For example, in the 1 prime h 1 prime, which would be the 1 1 601 00:44:30,000 --> 00:44:34,380 element of this matrix, I'm supposed 602 00:44:34,380 --> 00:44:37,603 to put here 1 prime is EhE. 603 00:44:41,570 --> 00:44:45,000 And now I'm supposed to say, OK, what is this? 604 00:44:45,000 --> 00:44:48,360 Well, E is 1 over square root of 2. 605 00:44:53,580 --> 00:44:57,390 E was 1 over square root of 2, 1 minus 1. 606 00:45:00,030 --> 00:45:09,450 H is the original H, so it's E0 plus mu E minus delta minus 607 00:45:09,450 --> 00:45:21,220 delta E0 minus mu E. And E is 1 1, 1 minus 1, again. 608 00:45:21,220 --> 00:45:23,420 And there's also the square root of 2. 609 00:45:23,420 --> 00:45:27,150 So at the end, this is a 1/2. 610 00:45:27,150 --> 00:45:28,880 So this is the kind of thing you have 611 00:45:28,880 --> 00:45:30,820 to do to pass to this basis. 612 00:45:30,820 --> 00:45:34,590 So I think I'll do that in the notes. 613 00:45:34,590 --> 00:45:37,270 And this calculation is simple. 614 00:45:37,270 --> 00:45:39,715 In this case, it gives E0 plus delta. 615 00:45:43,100 --> 00:45:47,383 And in retrospect, that sort of is pretty reasonable. 616 00:45:51,620 --> 00:45:56,700 This is E0 plus delta, and this is E0 minus delta. 617 00:45:56,700 --> 00:45:59,270 And if you didn't have an electric field-- 618 00:45:59,270 --> 00:46:02,960 indeed in this basis, the first state 619 00:46:02,960 --> 00:46:05,750 is the excited state who has this energy. 620 00:46:05,750 --> 00:46:09,490 The second state is the ground state, and has this energy. 621 00:46:09,490 --> 00:46:12,920 And that makes sense if there's no mu E. 622 00:46:12,920 --> 00:46:18,420 Well, the mu E still shows up, and it shows up here. 623 00:46:24,410 --> 00:46:27,380 So that is the Hamiltonian in this basis, 624 00:46:27,380 --> 00:46:30,120 and the general state in this basis 625 00:46:30,120 --> 00:46:35,080 is the amplitude to be excited, and the amplitude 626 00:46:35,080 --> 00:46:37,930 to be in the ground state. 627 00:46:37,930 --> 00:46:42,240 This is the general psi of t. 628 00:46:45,070 --> 00:46:50,550 So your Schrodinger equation in this mu basis-- not mu basis, 629 00:46:50,550 --> 00:46:59,080 in the Eg basis, you see it's-- Eg basis is the basis of energy 630 00:46:59,080 --> 00:47:01,750 eigenstates if you don't have electric field. 631 00:47:01,750 --> 00:47:03,910 But once you have an electric field, 632 00:47:03,910 --> 00:47:07,220 it's not anymore energy eigenstates, and much worse 633 00:47:07,220 --> 00:47:09,540 if you have a time-dependent electric field. 634 00:47:13,940 --> 00:47:22,570 So the Schrodinger equation is i h bar d dt of CE Cg 635 00:47:22,570 --> 00:47:26,440 is equal to this mu matrix. 636 00:47:26,440 --> 00:47:30,810 And now, E0 is totally irrelevant for everything. 637 00:47:30,810 --> 00:47:33,400 It's a constant of the unit matrix. 638 00:47:33,400 --> 00:47:35,150 Let's put E0 to 0. 639 00:47:35,150 --> 00:47:37,580 There's no need to keep it. 640 00:47:37,580 --> 00:47:39,870 E0 equal zero. 641 00:47:39,870 --> 00:47:51,803 So we have delta mu E minus delta times CECg. 642 00:47:56,550 --> 00:48:02,270 I d dt of psi, h psi-- the Schrodinger equation. 643 00:48:02,270 --> 00:48:05,150 Now, the real difficulty that we have 644 00:48:05,150 --> 00:48:09,000 is that E is a function of time. 645 00:48:09,000 --> 00:48:14,210 So this is not all that trivial. 646 00:48:14,210 --> 00:48:21,900 So what you do to solve this is simplify the equation by saying 647 00:48:21,900 --> 00:48:23,960 what would the solution be if you 648 00:48:23,960 --> 00:48:26,440 didn't have a function of time? 649 00:48:26,440 --> 00:48:35,920 Then you would have-- if you didn't have E, CE Cg of time 650 00:48:35,920 --> 00:48:43,250 would be E to the minus i ht. 651 00:48:43,250 --> 00:48:48,170 So this would be i delta t-- the energy of this, 652 00:48:48,170 --> 00:48:51,330 if there's no electric field, the Hamiltonian 653 00:48:51,330 --> 00:48:52,960 will be delta minus delta. 654 00:48:52,960 --> 00:48:57,650 So here, I have my psi ht over h bar. 655 00:48:57,650 --> 00:49:00,640 And for the lower state, you would have E 656 00:49:00,640 --> 00:49:06,460 to the plus i delta t over h bar as solutions, 657 00:49:06,460 --> 00:49:10,930 if you didn't have this. 658 00:49:10,930 --> 00:49:12,680 This would be the solutions. 659 00:49:12,680 --> 00:49:15,070 But we want now better. 660 00:49:15,070 --> 00:49:17,060 So what we're going to say is well, 661 00:49:17,060 --> 00:49:19,810 that's a solution if I neglect this. 662 00:49:19,810 --> 00:49:22,040 So this cannot be the real solution. 663 00:49:22,040 --> 00:49:31,860 So I'll put here beta sub E of t, and a beta sub g of t. 664 00:49:31,860 --> 00:49:38,550 And sure-- if no electric field, betas are 1. 665 00:49:38,550 --> 00:49:40,200 They're not necessary. 666 00:49:40,200 --> 00:49:42,230 But if there is an electric field, 667 00:49:42,230 --> 00:49:44,080 the betas are going to be complicated, 668 00:49:44,080 --> 00:49:45,320 so we need them there. 669 00:49:49,670 --> 00:49:52,180 So this is like an [INAUDIBLE]. 670 00:49:52,180 --> 00:49:59,230 Now you could plug this back, and calculate what you get. 671 00:49:59,230 --> 00:50:01,380 And it should not be too surprising 672 00:50:01,380 --> 00:50:07,260 that you're going to get in here something in which these deltas 673 00:50:07,260 --> 00:50:11,330 are going to disappear, because this thing takes care of that. 674 00:50:11,330 --> 00:50:13,850 So there's a little bit of algebra 675 00:50:13,850 --> 00:50:17,760 here, maybe two, three lines of algebra. 676 00:50:17,760 --> 00:50:20,440 And let me give you what you get-- 677 00:50:20,440 --> 00:50:27,030 I h bar d dt of beta E beta g. 678 00:50:27,030 --> 00:50:31,100 Now the equation is really for this quantities-- 679 00:50:31,100 --> 00:50:37,100 and it's 0 E to the i omega 0t times mu 680 00:50:37,100 --> 00:50:42,220 E, E to the minus i omega 0t times 681 00:50:42,220 --> 00:50:52,710 mu E0, beta E, beta g, where omega 0 is the Larmor 682 00:50:52,710 --> 00:50:57,485 frequency, or 2 delta over h bar. 683 00:51:01,500 --> 00:51:06,290 So some calculation involved, but I 684 00:51:06,290 --> 00:51:09,370 hope you don't lose track of the physics here. 685 00:51:09,370 --> 00:51:11,290 The physics is that the amplitude 686 00:51:11,290 --> 00:51:15,030 to be in E and the amplitudes to be in g 687 00:51:15,030 --> 00:51:23,000 have now been replaced by beta and beta g, which are expected 688 00:51:23,000 --> 00:51:25,600 to be simpler things to calculate. 689 00:51:25,600 --> 00:51:28,476 And in fact, since the probability to be in E 690 00:51:28,476 --> 00:51:31,820 is the norm of to this thing squared, 691 00:51:31,820 --> 00:51:36,940 beta is as good as C to know how likely is the particle 692 00:51:36,940 --> 00:51:39,910 to be in E, or how likely is the particle 693 00:51:39,910 --> 00:51:41,750 to be in the ground state. 694 00:51:41,750 --> 00:51:45,370 You could use beta, because they differ by a phase. 695 00:51:45,370 --> 00:51:49,110 So betas have still the physical significance 696 00:51:49,110 --> 00:51:53,310 of the amplitude to be in E or g. 697 00:51:53,310 --> 00:51:55,780 And we're still having an electric field 698 00:51:55,780 --> 00:51:57,950 that is time dependent here. 699 00:51:57,950 --> 00:52:03,540 So it's time to put the time dependence, 700 00:52:03,540 --> 00:52:08,810 and what are we going to do? 701 00:52:08,810 --> 00:52:13,833 Well we're going to find a sort of self-consistent operation 702 00:52:13,833 --> 00:52:21,070 of this device, in which we will have an electromagnetic field, 703 00:52:21,070 --> 00:52:29,150 E. E of t will be 2 E-- letter E, 704 00:52:29,150 --> 00:52:33,460 that now is going to be a constant with an E0, cosine 705 00:52:33,460 --> 00:52:38,240 omega 0 t-- again the Larmor frequency or the photon 706 00:52:38,240 --> 00:52:42,710 frequency that is emitted by the possible transition. 707 00:52:42,710 --> 00:52:46,510 So we will consider the case when the cavity has already 708 00:52:46,510 --> 00:52:49,620 that electric field that is going precisely 709 00:52:49,620 --> 00:52:55,530 at the right speed to do things well. 710 00:52:55,530 --> 00:52:59,200 So this Et, with the 2 conveniently put here, 711 00:52:59,200 --> 00:53:06,990 is equal to E0 E to the i omega naught t plus e to the minus i 712 00:53:06,990 --> 00:53:08,460 omega naught t. 713 00:53:12,420 --> 00:53:18,170 So when you multiply these things, what do you get? 714 00:53:18,170 --> 00:53:21,650 Let me do one of them for you to see. 715 00:53:21,650 --> 00:53:27,690 You get this i-- the top one-- h I'm 716 00:53:27,690 --> 00:53:29,230 going to put to the other side. 717 00:53:29,230 --> 00:53:34,980 Beta E is going to be beta E dot, 718 00:53:34,980 --> 00:53:40,710 is going to couple with this to beta g. 719 00:53:40,710 --> 00:53:45,790 So that thing is going to be this electric field times mu, 720 00:53:45,790 --> 00:53:51,530 so mu e0, the h bar from the left-hand side, 721 00:53:51,530 --> 00:53:55,630 and you have E to the i omega naught t multiplying this, 722 00:53:55,630 --> 00:54:04,520 so it's 1 plus E to the 2i omega naught t times beta g. 723 00:54:04,520 --> 00:54:05,833 So that's the first equation. 724 00:54:08,950 --> 00:54:13,090 Not all that easy, for sure. 725 00:54:13,090 --> 00:54:20,520 Second equation-- i beta g dot is 726 00:54:20,520 --> 00:54:27,020 equal to mu E0 over h bar, 1 plus 727 00:54:27,020 --> 00:54:35,620 e to the minus 2i omega naught t beta E of t. 728 00:54:41,490 --> 00:54:46,400 And now you have to think maybe a little like engineers, 729 00:54:46,400 --> 00:54:50,860 or physicists, or mathematicians-- 730 00:54:50,860 --> 00:54:53,050 whatever you prefer. 731 00:54:53,050 --> 00:54:55,520 But you want to convince yourself 732 00:54:55,520 --> 00:54:58,410 that what you want to do is true. 733 00:54:58,410 --> 00:55:00,050 And what do you want to do? 734 00:55:00,050 --> 00:55:04,290 I like to forget about these curves, basically. 735 00:55:04,290 --> 00:55:08,720 That's what we want to do. 736 00:55:08,720 --> 00:55:11,510 Why would that be true? 737 00:55:11,510 --> 00:55:14,900 Well, here's a reason why it's true. 738 00:55:14,900 --> 00:55:19,610 This is a number that sort of goes between 1 and minus 1, 739 00:55:19,610 --> 00:55:21,470 with some phase. 740 00:55:21,470 --> 00:55:27,590 Mu E0, however, over h bar, is a very small number. 741 00:55:27,590 --> 00:55:32,010 Mu E0 we're thinking of-- we're saying compared 742 00:55:32,010 --> 00:55:40,250 to the natural scales of the problem, this energy, mu E0, 743 00:55:40,250 --> 00:55:43,370 is much smaller than delta. 744 00:55:43,370 --> 00:55:50,770 And delta is the thing that is related to omega naught, which 745 00:55:50,770 --> 00:55:56,460 h bar omega naught is equal to 2 delta. 746 00:55:56,460 --> 00:56:05,120 So essentially, mu E0 is an energy 747 00:56:05,120 --> 00:56:09,290 which is very, very slow compared to h omega naught. 748 00:56:09,290 --> 00:56:15,110 Now, being very small, whatever this is, 749 00:56:15,110 --> 00:56:18,390 this time derivative is going to be very small. 750 00:56:18,390 --> 00:56:22,410 So beta E and beta g are going to be 751 00:56:22,410 --> 00:56:27,500 very small-time derivatives-- going to move slowly. 752 00:56:27,500 --> 00:56:33,460 If they move slowly over the time that this oscillates, 753 00:56:33,460 --> 00:56:35,595 this hasn't changed a lot. 754 00:56:35,595 --> 00:56:39,440 And therefore, the average of this function over that time 755 00:56:39,440 --> 00:56:42,705 is 0, and it doesn't contribute to the differential equation. 756 00:56:46,860 --> 00:56:48,340 It's actually a good argument. 757 00:56:54,140 --> 00:56:56,300 You can try to convince yourselves, 758 00:56:56,300 --> 00:56:59,870 or maybe I'll try better in some way, or something. 759 00:56:59,870 --> 00:57:01,350 But it is right. 760 00:57:01,350 --> 00:57:05,190 I've tried it, actually, out with computers 761 00:57:05,190 --> 00:57:07,940 and with Mathematica, and things like that. 762 00:57:07,940 --> 00:57:10,580 And it's really absolutely true-- 763 00:57:10,580 --> 00:57:13,450 that if you think of differential equations 764 00:57:13,450 --> 00:57:16,370 as integrals, that you integrate with this 765 00:57:16,370 --> 00:57:19,920 in the right-hand side, you can see that if this-- really, 766 00:57:19,920 --> 00:57:21,570 the time derivative is controlled 767 00:57:21,570 --> 00:57:25,950 by this, that corresponds to a frequency much smaller 768 00:57:25,950 --> 00:57:27,280 than omega naught. 769 00:57:27,280 --> 00:57:29,330 These ones don't matter. 770 00:57:29,330 --> 00:57:36,420 So it's really interesting, and still not trivial. 771 00:57:36,420 --> 00:57:45,090 But this is a case where we end up ignoring part of the thing. 772 00:57:45,090 --> 00:57:54,430 So what do we get, then? i beta E dot is equal mu E0 over h bar 773 00:57:54,430 --> 00:57:57,400 beta g. 774 00:57:57,400 --> 00:58:08,470 And the second equation-- i beta g dot equal mu E0 over h bar 775 00:58:08,470 --> 00:58:13,880 beta E. Therefore, if you multiply by another i 776 00:58:13,880 --> 00:58:20,840 here, and differentiate i-- i beta double dot of E 777 00:58:20,840 --> 00:58:24,670 is equal mu E0 over h bar. 778 00:58:24,670 --> 00:58:29,170 The i extra that we borrowed, the dot here. 779 00:58:29,170 --> 00:58:32,090 You can use the second equation. 780 00:58:32,090 --> 00:58:42,410 So you get mu E0 over h-bar squared beta E. Therefore, 781 00:58:42,410 --> 00:58:53,150 beta E double dot is equal to minus mu E0 over h-bar beta E. 782 00:58:53,150 --> 00:58:56,580 And you, see your you're rotating 783 00:58:56,580 --> 00:59:09,170 with the a of-- put the h-bar-- with a frequency 784 00:59:09,170 --> 00:59:12,220 that i much smaller than omega 0. 785 00:59:12,220 --> 00:59:16,740 So this as a frequency, this is like omega 0. 786 00:59:19,310 --> 00:59:23,280 So indeed, the rate of change of this thing 787 00:59:23,280 --> 00:59:25,660 goes with the frequency that's much smaller. 788 00:59:25,660 --> 00:59:28,760 And it's all right, actually. 789 00:59:28,760 --> 00:59:33,230 So, we've had that, and then we can write the solution, 790 00:59:33,230 --> 00:59:35,460 finally. 791 00:59:35,460 --> 00:59:37,510 So, what is it? 792 00:59:37,510 --> 00:59:47,180 Beta E of T is cosine mu E0 t over h-bar. 793 00:59:49,730 --> 00:59:57,350 And the probability to be in the E state 794 00:59:57,350 --> 01:00:02,460 is the square of that amplitude, so the probability 795 01:00:02,460 --> 01:00:07,870 to be in the E state at time t is the square of that. 796 01:00:07,870 --> 01:00:15,690 So it's cos squared mu E 0 t over h-bar. 797 01:00:15,690 --> 01:00:18,640 Again, I'm sorry, this beta is here, 798 01:00:18,640 --> 01:00:23,450 and the probability to be in the excited state 799 01:00:23,450 --> 01:00:26,620 is this square of ce, but the square of ce 800 01:00:26,620 --> 01:00:32,190 is the square of beta E, so I just square this. 801 01:00:32,190 --> 01:00:33,080 So there you go. 802 01:00:36,010 --> 01:00:46,070 We have this thing, and we now have an understanding 803 01:00:46,070 --> 01:00:49,950 of how this goes as time goes by. 804 01:00:49,950 --> 01:00:58,030 In fact, this mu e 0 T or h-bar under bar goes by. 805 01:00:58,030 --> 01:00:59,900 This starts to as a cosine squared, 806 01:00:59,900 --> 01:01:02,680 and then it goes like this. 807 01:01:02,680 --> 01:01:06,370 And this is the place when this is equal to pi over 2. 808 01:01:10,500 --> 01:01:14,490 So, what do we need? 809 01:01:14,490 --> 01:01:19,350 We need the place where this happens-- 810 01:01:19,350 --> 01:01:28,490 we can call that time T-- be such that mu E 0 T over h-bar 811 01:01:28,490 --> 01:01:35,360 be equal to pi over 2, or 3 pi over 2. 812 01:01:35,360 --> 01:01:41,210 For those values of time, the probability to be excited 813 01:01:41,210 --> 01:01:45,380 is zero, and therefore you must be in the ground state. 814 01:01:45,380 --> 01:01:51,830 These two probabilities squared added go to 1. 815 01:01:51,830 --> 01:01:57,460 So you're either E or G. So if you have zero probability to be 816 01:01:57,460 --> 01:02:06,580 in E, you will be in the state G. 817 01:02:06,580 --> 01:02:11,250 So, the whole issue is basically at this moment 818 01:02:11,250 --> 01:02:15,900 that you must give the right speed, 819 01:02:15,900 --> 01:02:19,830 or for a given the speed of the molecules, 820 01:02:19,830 --> 01:02:23,340 there will be a time that it takes to traverse. 821 01:02:23,340 --> 01:02:28,140 That time is related to the steady state 822 01:02:28,140 --> 01:02:33,990 value of this electric field by this relation. 823 01:02:33,990 --> 01:02:37,960 So you need the right velocity-- the molecules have 824 01:02:37,960 --> 01:02:39,950 to be at the right temperature, that's 825 01:02:39,950 --> 01:02:44,450 when they'll have a velocity-- so that they travel in such 826 01:02:44,450 --> 01:02:46,980 a way that is consistent with this. 827 01:02:46,980 --> 01:02:54,370 As they do that, each of these particles that goes from E to G 828 01:02:54,370 --> 01:02:57,400 gives out one photon, one quantum 829 01:02:57,400 --> 01:03:00,090 of the electromagnetic field, and helps 830 01:03:00,090 --> 01:03:03,580 build the time dependent electric field 831 01:03:03,580 --> 01:03:06,990 that we started at the beginning. 832 01:03:06,990 --> 01:03:12,730 Now if you, at some point for some speed of the molecules, 833 01:03:12,730 --> 01:03:15,790 you saturate this and you build some electric field, 834 01:03:15,790 --> 01:03:21,600 and then you have your cavity operating at the nominal way. 835 01:03:21,600 --> 01:03:24,410 And then, of course, you want to use this for something, 836 01:03:24,410 --> 01:03:29,260 so you shine, let it go out, and shine those microwaves 837 01:03:29,260 --> 01:03:30,660 or do something. 838 01:03:30,660 --> 01:03:32,560 And if you want to recharge it, you 839 01:03:32,560 --> 01:03:36,870 keep adding nitrogen molecules. 840 01:03:36,870 --> 01:03:40,280 Now, this was a great discovery, actually. 841 01:03:40,280 --> 01:03:43,590 Charles Townes, Gordon, and Zeiger, 842 01:03:43,590 --> 01:03:46,750 built this ammonia maser in 1953. 843 01:03:46,750 --> 01:03:50,110 They got the Nobel Prize. 844 01:03:50,110 --> 01:03:55,450 Nobel Prize, Charles Townes got it in 1964. 845 01:03:55,450 --> 01:03:57,500 And he emphasized that this masers 846 01:03:57,500 --> 01:04:00,830 do the most perfect amplification 847 01:04:00,830 --> 01:04:04,280 consistent with the uncertainty principle. 848 01:04:04,280 --> 01:04:08,040 This is a coherent state of flight that is built here, 849 01:04:08,040 --> 01:04:11,560 and it's much better than any vacuum amplifier or anything 850 01:04:11,560 --> 01:04:14,730 like that, because the thing that this giving out 851 01:04:14,730 --> 01:04:18,980 those photons is a molecule that this uncharged. 852 01:04:18,980 --> 01:04:21,680 So it doesn't disturb the electromagnetic field 853 01:04:21,680 --> 01:04:24,290 in the cavity as it goes through. 854 01:04:24,290 --> 01:04:28,170 Many times you use an electron, for example, 855 01:04:28,170 --> 01:04:30,000 to give out some energy. 856 01:04:30,000 --> 01:04:31,800 But the electron itself is charged, 857 01:04:31,800 --> 01:04:34,640 so it produces additional electromagnetic fields, 858 01:04:34,640 --> 01:04:36,630 shot noise, all kinds of noise. 859 01:04:36,630 --> 01:04:40,610 This is absolutely quiet device in which 860 01:04:40,610 --> 01:04:46,990 is this turns from one state to another smoothly, stimulated 861 01:04:46,990 --> 01:04:50,080 by the electric field, because if there was no electric field 862 01:04:50,080 --> 01:04:54,460 would be an energy eigenstate and then it gives out photon 863 01:04:54,460 --> 01:04:57,170 after photon. 864 01:04:57,170 --> 01:05:04,660 So the uncertainties actually, in this noble lecture, 865 01:05:04,660 --> 01:05:07,830 which is fun to read-- in fact, you 866 01:05:07,830 --> 01:05:12,730 remember delta N delta phi supposed 867 01:05:12,730 --> 01:05:15,400 to be greater than 1/2. 868 01:05:15,400 --> 01:05:19,150 Well, for coherent states as we more or less discussed, 869 01:05:19,150 --> 01:05:22,480 this thing is saturated. 870 01:05:22,480 --> 01:05:23,980 And what do you have now? 871 01:05:23,980 --> 01:05:26,900 You have a coherent state of light. 872 01:05:26,900 --> 01:05:29,430 You may remember that the expectation 873 01:05:29,430 --> 01:05:34,850 value of N in a coherent state alpha was alpha squared. 874 01:05:34,850 --> 01:05:40,430 And you call this the number n of photons. 875 01:05:40,430 --> 01:05:47,100 And the uncertainty in N was, in fact, alpha. 876 01:05:47,100 --> 01:05:49,070 So, it's square root of n. 877 01:05:52,580 --> 01:06:01,830 So for this thing, we have the situation 878 01:06:01,830 --> 01:06:07,720 in which we are working with a coherent state that 879 01:06:07,720 --> 01:06:08,750 saturates this. 880 01:06:08,750 --> 01:06:17,900 So delta n times delta phi is about one half. 881 01:06:17,900 --> 01:06:21,810 And delta n is square root of the number of photons. 882 01:06:21,810 --> 01:06:26,310 So delta phi is about to 1 over 2 square root 883 01:06:26,310 --> 01:06:27,686 of the number of photons. 884 01:06:32,940 --> 01:06:35,160 And you can imagine the cavity easily 885 01:06:35,160 --> 01:06:39,680 can have 10 to the 12 photons, and to 15 photons. 886 01:06:39,680 --> 01:06:42,020 Something fairly big. 887 01:06:42,020 --> 01:06:45,580 And you get an uncertainty phase. 888 01:06:45,580 --> 01:06:48,340 The thing is coherent. 889 01:06:48,340 --> 01:06:50,800 All the pieces of that electromagnetic wave, 890 01:06:50,800 --> 01:06:55,700 the phases are very coherent up to an incredibly great 891 01:06:55,700 --> 01:06:58,550 accuracy. 892 01:06:58,550 --> 01:07:03,410 So it's a great discovery and the beginning of many things 893 01:07:03,410 --> 01:07:07,480 that were done here, in fact, by Professor Kleppner and others 894 01:07:07,480 --> 01:07:10,480 in the '60s with other type of lasers 895 01:07:10,480 --> 01:07:14,130 and masers and this whole thing. 896 01:07:14,130 --> 01:07:16,910 So, that's pretty much what's I wanted 897 01:07:16,910 --> 01:07:19,580 to say about the ammonia and these things, 898 01:07:19,580 --> 01:07:21,550 and we're going to use the last 15 899 01:07:21,550 --> 01:07:25,570 minutes to begin something else, NMR. 900 01:07:25,570 --> 01:07:30,040 And, so are there any questions so this Point? 901 01:07:30,040 --> 01:07:31,946 Yes? 902 01:07:31,946 --> 01:07:32,862 AUDIENCE: [INAUDIBLE]. 903 01:07:36,180 --> 01:07:37,910 If it started changing between states, 904 01:07:37,910 --> 01:07:41,389 I sort of imagine a photon being omitted and absorbed, 905 01:07:41,389 --> 01:07:44,868 what exactly is happening then? [INAUDIBLE]? 906 01:07:44,868 --> 01:07:47,850 I just happened to catch it in the right moment. 907 01:07:47,850 --> 01:07:50,310 PROFESSOR: Well, in this case, it's basically 908 01:07:50,310 --> 01:07:55,310 omitted all the time, because we've tuned the cavity in such 909 01:07:55,310 --> 01:07:59,930 a way that if it comes as E, goes out as G. 910 01:07:59,930 --> 01:08:03,540 So the whole process by the time that it entered 911 01:08:03,540 --> 01:08:07,050 and it went out, it has to have omitted one photon. 912 01:08:10,092 --> 01:08:11,675 AUDIENCE: In the midst of the process, 913 01:08:11,675 --> 01:08:13,830 how does it get back somehow, because it's 914 01:08:13,830 --> 01:08:15,920 sort of oscillating between states. 915 01:08:15,920 --> 01:08:17,800 PROFESSOR: It doesn't get it back. 916 01:08:17,800 --> 01:08:21,660 If the cavity would be badly designed in such a way 917 01:08:21,660 --> 01:08:26,710 that it may be twice as long, it would out still come as E. 918 01:08:26,710 --> 01:08:29,090 So it would make the one transition 919 01:08:29,090 --> 01:08:31,319 and then absorb another thing, it 920 01:08:31,319 --> 01:08:33,684 would just not generate anything. 921 01:08:33,684 --> 01:08:34,600 AUDIENCE: [INAUDIBLE]. 922 01:08:34,600 --> 01:08:35,266 PROFESSOR: Yeah. 923 01:08:35,266 --> 01:08:36,279 AUDIENCE: Interesting. 924 01:08:36,279 --> 01:08:40,990 PROFESSOR: And you know, a more complete discussion 925 01:08:40,990 --> 01:08:46,410 of this, of course, if you really want to do everything, 926 01:08:46,410 --> 01:08:49,689 you would have to treat the photon states. 927 01:08:49,689 --> 01:08:53,970 Here, we treated this as a wave coupled 928 01:08:53,970 --> 01:08:57,620 to the quantum system of the molecule. 929 01:08:57,620 --> 01:09:01,810 You could treat the photons themselves as quanta 930 01:09:01,810 --> 01:09:04,790 and do quantum mechanics of the photon field on that. 931 01:09:04,790 --> 01:09:08,210 And that would be a wave that you could calculate things more 932 01:09:08,210 --> 01:09:09,529 completely . 933 01:09:09,529 --> 01:09:12,960 But this is not what we do now. 934 01:09:12,960 --> 01:09:14,520 Any other questions? 935 01:09:14,520 --> 01:09:15,170 Yes? 936 01:09:15,170 --> 01:09:18,104 AUDIENCE: So physically, are the nitrogen molecules all 937 01:09:18,104 --> 01:09:21,038 fixed in the same orientation going in to this device? 938 01:09:21,038 --> 01:09:22,029 How does that happen? 939 01:09:22,029 --> 01:09:23,184 PROFESSOR: Are the what? 940 01:09:23,184 --> 01:09:25,600 AUDIENCE: The molecules all fixed in the same orientation? 941 01:09:25,600 --> 01:09:26,250 PROFESSOR: Yes. 942 01:09:26,250 --> 01:09:30,229 Essentially it's not quite like an orientation. 943 01:09:30,229 --> 01:09:31,530 It's an energy eigenstate. 944 01:09:34,040 --> 01:09:37,430 So basically what you have to do is 945 01:09:37,430 --> 01:09:41,270 have this beam of ammonia molecules 946 01:09:41,270 --> 01:09:44,399 and do this beam splitter that we talked about 947 01:09:44,399 --> 01:09:45,399 with the electric field. 948 01:09:45,399 --> 01:09:49,399 And you split the beam into some that are all excited, 949 01:09:49,399 --> 01:09:51,170 and some that are ground. 950 01:09:51,170 --> 01:09:52,670 And that's it. 951 01:09:52,670 --> 01:09:57,063 You need that everything that enters here is excited state. 952 01:10:00,440 --> 01:10:01,140 OK. 953 01:10:01,140 --> 01:10:05,280 So another time dependent problem 954 01:10:05,280 --> 01:10:09,890 that we're going to discuss today and continue next time 955 01:10:09,890 --> 01:10:15,920 is the NMR problem, nuclear magnetic resonance. 956 01:10:15,920 --> 01:10:18,425 So this is a pretty interesting problem. 957 01:10:21,480 --> 01:10:23,295 Our nuclear magnetic resonance. 958 01:10:23,295 --> 01:10:28,550 And it all begins by having a magnetic field that 959 01:10:28,550 --> 01:10:33,640 has a big component here, B0, I think. 960 01:10:33,640 --> 01:10:35,120 It's a good name for it. 961 01:10:42,330 --> 01:10:45,770 B0 Z, indeed. 962 01:10:45,770 --> 01:10:51,960 And then you're going to have some magnetic field that 963 01:10:51,960 --> 01:10:58,720 is going to be rotating here in time, in the xy plane. 964 01:10:58,720 --> 01:11:01,570 We'll assume that time equals 0, is here, 965 01:11:01,570 --> 01:11:05,860 and then it's rotating with some angular frequency omega. 966 01:11:05,860 --> 01:11:11,680 So the total magnetic field is B0 Z hat 967 01:11:11,680 --> 01:11:18,580 plus some number, B 1-- I'll write it here-- 968 01:11:18,580 --> 01:11:29,560 plus B 1 cosine omega t times x hat minus sine omega t times 969 01:11:29,560 --> 01:11:31,936 y hat. 970 01:11:31,936 --> 01:11:39,170 So, indeed, in the xy plane, seen like this, 971 01:11:39,170 --> 01:11:43,280 you have it here and it's rotating with angular frequency 972 01:11:43,280 --> 01:11:44,820 omega this way, clockwise. 973 01:11:48,250 --> 01:11:49,050 All right. 974 01:11:49,050 --> 01:11:51,540 So we have this magnetic field. 975 01:11:51,540 --> 01:11:53,500 And of course we're going to try to figure out 976 01:11:53,500 --> 01:11:56,670 what spins do in it. 977 01:11:56,670 --> 01:12:00,170 And the magnetic field is time dependent. 978 01:12:00,170 --> 01:12:06,710 So we're in risk of getting a time dependent Hamiltonian. 979 01:12:06,710 --> 01:12:08,590 So what is the Hamiltonian? 980 01:12:08,590 --> 01:12:11,520 This possibly time dependent, it's 981 01:12:11,520 --> 01:12:17,520 supposed to be minus gamma B times the spin. 982 01:12:22,800 --> 01:12:26,160 So what is that? 983 01:12:26,160 --> 01:12:38,410 It's minus gamma B0 Sz-- The z component matches 984 01:12:38,410 --> 01:12:48,040 with the z component of the spin-- plus B1, Sx cosine omega 985 01:12:48,040 --> 01:12:52,520 t minus Sy sine omega t. 986 01:12:57,000 --> 01:13:04,050 And, well, it's as bad as you could imagine, pretty much. 987 01:13:04,050 --> 01:13:07,460 This Hamiltonian is time dependent. 988 01:13:07,460 --> 01:13:11,170 And there's some good news if even if it's time dependent, 989 01:13:11,170 --> 01:13:13,210 they commute at different times. 990 01:13:13,210 --> 01:13:15,620 The time evolution is EC. 991 01:13:15,620 --> 01:13:18,590 But no, they don't commute at different times. 992 01:13:18,590 --> 01:13:25,290 Time equal 0, for example, you have Sz and Sx, 993 01:13:25,290 --> 01:13:28,800 but at a later time, you will have Sz and Sy, 994 01:13:28,800 --> 01:13:31,840 and they just don't commute. 995 01:13:31,840 --> 01:13:38,610 So we have no cookbook recipe to solve this problem. 996 01:13:38,610 --> 01:13:42,560 We have to figure it out. 997 01:13:42,560 --> 01:13:46,490 And there are several ways to figure it out. 998 01:13:46,490 --> 01:13:48,590 I had a way to figure it out that I explained 999 01:13:48,590 --> 01:13:53,760 in previous years, but today I suddenly thought 1000 01:13:53,760 --> 01:13:58,670 I could explain it in a different way that is maybe 1001 01:13:58,670 --> 01:14:02,300 a little harder conceptually, but explains 1002 01:14:02,300 --> 01:14:04,950 more of what's going on. 1003 01:14:04,950 --> 01:14:09,070 So this is what I want to try to do now. 1004 01:14:09,070 --> 01:14:13,130 And basically what we're going to try to do 1005 01:14:13,130 --> 01:14:19,560 is get the main intuition going for this problem. 1006 01:14:19,560 --> 01:14:22,480 I have the Schrodinger equation for this problem that 1007 01:14:22,480 --> 01:14:26,810 is very complicated with a time dependent Hamiltonian. 1008 01:14:26,810 --> 01:14:31,210 So if you have a wave function-- and now 1009 01:14:31,210 --> 01:14:33,740 I'm going to be a little rough for a little while. 1010 01:14:33,740 --> 01:14:37,365 A h-bar D T H psi. 1011 01:14:40,290 --> 01:14:42,300 And it-- oops. 1012 01:14:42,300 --> 01:14:44,700 What is it? 1013 01:14:44,700 --> 01:14:52,515 Let's see which lights-- Lights, general. 1014 01:14:52,515 --> 01:14:53,015 OK. 1015 01:14:56,300 --> 01:15:01,620 So if you have a problem like this, you might say, 1016 01:15:01,620 --> 01:15:06,390 well, I would like to do something non-trivial to this, 1017 01:15:06,390 --> 01:15:10,770 so I want to maybe change the Hamiltonian 1018 01:15:10,770 --> 01:15:13,760 and do the same physical problem. 1019 01:15:13,760 --> 01:15:18,090 Now that's not easy, because if it's the same physical problem, 1020 01:15:18,090 --> 01:15:21,400 how can you change the Hamiltonian all that much? 1021 01:15:21,400 --> 01:15:23,940 So one thing you could do is you can 1022 01:15:23,940 --> 01:15:30,910 try to change the states by doing a unitary transformation, 1023 01:15:30,910 --> 01:15:35,910 and then hope that the unitary transformation acts 1024 01:15:35,910 --> 01:15:37,960 on this Hamiltonian. 1025 01:15:37,960 --> 01:15:39,525 So this will be the new states. 1026 01:15:42,390 --> 01:15:45,650 And you would hope that this unitary transformation 1027 01:15:45,650 --> 01:15:48,590 would somehow, working with these new states, 1028 01:15:48,590 --> 01:15:51,510 would simplify the Hamiltonian. 1029 01:15:51,510 --> 01:15:53,950 But unitary transformations in general 1030 01:15:53,950 --> 01:15:56,800 is just like a change of bases. 1031 01:15:56,800 --> 01:16:00,560 It's not going to do all that much, 1032 01:16:00,560 --> 01:16:04,940 unless the unitary transformation has 1033 01:16:04,940 --> 01:16:08,360 time dependence, in which case it doesn't just 1034 01:16:08,360 --> 01:16:13,420 rotate the Hamiltonian but messes up this term. 1035 01:16:13,420 --> 01:16:18,420 But this is really your only possibility of fixing things, 1036 01:16:18,420 --> 01:16:24,050 is to try to do a time independent unitary 1037 01:16:24,050 --> 01:16:27,450 transformation that somehow you started with a spin 1038 01:16:27,450 --> 01:16:31,770 Hamiltonian, and you're going to do a unitary transformation, 1039 01:16:31,770 --> 01:16:37,280 and perhaps it's going to be a time independent one. 1040 01:16:37,280 --> 01:16:41,230 But this U is going to depend on time. 1041 01:16:41,230 --> 01:16:43,660 And this is not the whole Hamiltonian, 1042 01:16:43,660 --> 01:16:46,180 because there's a problem with this part. 1043 01:16:46,180 --> 01:16:50,240 So the idea is sort of vague, but you're 1044 01:16:50,240 --> 01:16:54,210 getting a little of the gist of what we try to do. 1045 01:16:54,210 --> 01:17:01,860 So the first thing I tried to do is something maybe intuitive 1046 01:17:01,860 --> 01:17:05,860 about though this system. 1047 01:17:05,860 --> 01:17:18,870 I could say that suppose I have a system here, 1048 01:17:18,870 --> 01:17:21,870 and the Hamiltonian is 0. 1049 01:17:21,870 --> 01:17:22,780 Nothing. 1050 01:17:22,780 --> 01:17:26,060 Nothing happens in that system. 1051 01:17:26,060 --> 01:17:29,500 Now the thing that this curious about this, 1052 01:17:29,500 --> 01:17:32,390 that this magnetic field is rotating. 1053 01:17:32,390 --> 01:17:36,910 So let's try to imagine how would physics 1054 01:17:36,910 --> 01:17:43,450 look if you have the xy axis, and you have the xy axis 1055 01:17:43,450 --> 01:17:49,400 rotating with angular velocity omega. 1056 01:17:49,400 --> 01:17:53,820 So the xy in the plane is rotating with angular velocity 1057 01:17:53,820 --> 01:17:54,610 omega. 1058 01:17:54,610 --> 01:17:56,730 What would happen? 1059 01:17:56,730 --> 01:18:01,890 So there's an Hs that is originally 0. 1060 01:18:01,890 --> 01:18:06,480 You say it's 0, and any spin state stays there. 1061 01:18:06,480 --> 01:18:07,970 There's no time evolution. 1062 01:18:07,970 --> 01:18:09,940 H is equal to 0. 1063 01:18:09,940 --> 01:18:14,210 Nevertheless, if you jump into this rotating frame, 1064 01:18:14,210 --> 01:18:17,700 all the spin states that were static, for you 1065 01:18:17,700 --> 01:18:19,890 they're rotating. 1066 01:18:19,890 --> 01:18:23,350 And in fact, for you, they are processing around the z-axis 1067 01:18:23,350 --> 01:18:26,370 if you're in the rotating frame. 1068 01:18:26,370 --> 01:18:29,180 Therefore, in the rotating frame, 1069 01:18:29,180 --> 01:18:33,360 you must have some Hamiltonian, even though there's 1070 01:18:33,360 --> 01:18:36,850 no Hamiltonian in the static frame. 1071 01:18:36,850 --> 01:18:40,960 Because in the rotating frame, you feel things spinning. 1072 01:18:40,960 --> 01:18:44,130 So in the rotating frame, if there's 1073 01:18:44,130 --> 01:18:47,120 a static spin along the x direction, 1074 01:18:47,120 --> 01:18:52,380 you now see it spinning around the z-axis with frequency 1075 01:18:52,380 --> 01:18:56,450 omega, in fact, going the plus z direction. 1076 01:18:56,450 --> 01:18:59,430 So in the rotating frame you can ask, 1077 01:18:59,430 --> 01:19:03,310 what is the new Hamiltonian in the rotating frame? 1078 01:19:09,920 --> 01:19:12,450 I mean, the rotating frame, the Hamiltonian 1079 01:19:12,450 --> 01:19:18,070 should be such that it rotates the spins with angular velocity 1080 01:19:18,070 --> 01:19:20,650 omega around the z-axis. 1081 01:19:20,650 --> 01:19:25,470 You may recall that this is done by all the unitary operator 1082 01:19:25,470 --> 01:19:33,040 E to the minus i omega t Sz hat over h-bar. 1083 01:19:33,040 --> 01:19:38,740 This is the operator that rotates, spins, 1084 01:19:38,740 --> 01:19:42,460 with omega t around the z direction, which 1085 01:19:42,460 --> 01:19:46,150 is precisely what you want this Hamiltonian to do. 1086 01:19:46,150 --> 01:19:52,490 So the Hamiltonian must be e to the minus i ht. 1087 01:19:52,490 --> 01:19:57,920 So this rotating Hamiltonian must be omega Sz hat. 1088 01:20:00,520 --> 01:20:03,360 So the Hamiltonian in the new frame 1089 01:20:03,360 --> 01:20:07,470 is that because that Hamiltonian produces the rotation of states 1090 01:20:07,470 --> 01:20:11,270 that you see because your friend that is not rotating 1091 01:20:11,270 --> 01:20:13,710 is telling you that no state is moving. 1092 01:20:13,710 --> 01:20:15,430 H is 0. 1093 01:20:15,430 --> 01:20:17,610 So the intuition is that you've passed 1094 01:20:17,610 --> 01:20:19,620 from the original [INAUDIBLE] to this one, 1095 01:20:19,620 --> 01:20:22,440 and you have to add 6 this. 1096 01:20:22,440 --> 01:20:24,240 But what we'll see now is that you 1097 01:20:24,240 --> 01:20:26,000 have to add the little-- you have 1098 01:20:26,000 --> 01:20:29,705 to do other things a little if this is not equal to 0. 1099 01:20:34,360 --> 01:20:37,360 Here is what we want to do. 1100 01:20:37,360 --> 01:20:40,530 Now there's also a couple of ways of doing this, 1101 01:20:40,530 --> 01:20:44,090 but let me try a little calculation 1102 01:20:44,090 --> 01:20:46,710 to do this second part. 1103 01:20:46,710 --> 01:20:52,150 So, we're going to take, therefore, to have, 1104 01:20:52,150 --> 01:20:57,040 for example of psi R a rotating wave function 1105 01:20:57,040 --> 01:21:01,760 that is going to be given by a unitary operator 1106 01:21:01,760 --> 01:21:05,640 times the physical wave function you want to solve. 1107 01:21:05,640 --> 01:21:09,340 This is what you want, and this is 1108 01:21:09,340 --> 01:21:14,220 what you hope has a simpler dynamics, psi R. 1109 01:21:14,220 --> 01:21:17,400 So, let's try to figure out what is the Schrodinger 1110 01:21:17,400 --> 01:21:20,770 equation for psi R if you know the Schrodinger 1111 01:21:20,770 --> 01:21:22,430 equation for psi. 1112 01:21:22,430 --> 01:21:30,600 So, I dt I h-bar dt psi is equal to Hs psi, 1113 01:21:30,600 --> 01:21:34,810 let's see what is the Schrodinger equation satisfied 1114 01:21:34,810 --> 01:21:36,220 by this one. 1115 01:21:36,220 --> 01:21:47,590 So I h-bar dt of psi R is I h-bar dt of U psi. 1116 01:21:47,590 --> 01:21:50,450 So let's differentiate this. 1117 01:21:50,450 --> 01:21:56,860 This is I h-bar dt of U, and I will have the psi. 1118 01:21:56,860 --> 01:22:02,480 But then I will put them a U dagger, there and another U 1119 01:22:02,480 --> 01:22:08,980 acting on psi so that it gives me psi R. 1120 01:22:08,980 --> 01:22:13,530 So the first derivative is acting just on the U. I acted, 1121 01:22:13,530 --> 01:22:18,560 and then I put U dagger U and recreated the psi R. 1122 01:22:18,560 --> 01:22:21,310 The other terms is in which it acts 1123 01:22:21,310 --> 01:22:25,770 on psi, so I get plus I h-bar. 1124 01:22:25,770 --> 01:22:28,520 The U goes away, and now we get dt 1125 01:22:28,520 --> 01:22:36,580 of psi, which is I h-bar dt of psi is Hs I psi, 1126 01:22:36,580 --> 01:22:38,060 so I must delete this. 1127 01:22:42,160 --> 01:22:45,040 So the second term, when I acted here, 1128 01:22:45,040 --> 01:22:46,280 I act with the whole thing. 1129 01:22:46,280 --> 01:22:50,070 I ddt, I h-bar ddt on psi. 1130 01:22:50,070 --> 01:22:54,240 The U is out, and I put Hs psi. 1131 01:22:54,240 --> 01:22:59,120 And here, I can, of course, put a U dagger U 1132 01:22:59,120 --> 01:23:04,590 and put back the psi R, so I will do that as well. 1133 01:23:04,590 --> 01:23:08,690 U dagger psi R. 1134 01:23:08,690 --> 01:23:16,280 So actually you have now a the Schrodinger equation which 1135 01:23:16,280 --> 01:23:22,870 is of the form I h-bar dt of psi R 1136 01:23:22,870 --> 01:23:31,710 is equal to U Hs U dagger plus I h-bar 1137 01:23:31,710 --> 01:23:43,250 dt of U U dagger of psi R. Now that's it. 1138 01:23:43,250 --> 01:23:45,530 This is the new Hamiltonian. 1139 01:23:45,530 --> 01:23:49,940 This is the rotating Hamiltonian, essentially, 1140 01:23:49,940 --> 01:23:53,770 that we're trying to figure out that is going to be simpler, 1141 01:23:53,770 --> 01:23:55,140 we hope. 1142 01:23:55,140 --> 01:23:59,350 And there we go here. 1143 01:23:59,350 --> 01:24:02,860 We have this similarity transformation 1144 01:24:02,860 --> 01:24:07,340 of the Hamiltonian and we have this extra term. 1145 01:24:07,340 --> 01:24:11,620 Now suppose the original Hamiltonian had been 0. 1146 01:24:11,620 --> 01:24:15,130 We want the new Hamiltonian, given our argument, 1147 01:24:15,130 --> 01:24:19,770 to be a this for the rotating systems. 1148 01:24:19,770 --> 01:24:26,440 So I will say that this U is such 1149 01:24:26,440 --> 01:24:34,170 that you get I h-bar dt U U dagger being precisely 1150 01:24:34,170 --> 01:24:37,780 omega Sz. 1151 01:24:37,780 --> 01:24:48,235 And for that the U is nothing else but e to the minus I omega 1152 01:24:48,235 --> 01:24:57,840 tSz over h-bar, which is in fact that thing that we had there. 1153 01:24:57,840 --> 01:25:01,536 So that's what U is. 1154 01:25:01,536 --> 01:25:09,030 And in that way, this whole term becomes just omega Sz hat. 1155 01:25:09,030 --> 01:25:13,670 So we're almost, in a sense, done with this thing, 1156 01:25:13,670 --> 01:25:18,850 because we have made some good progress. 1157 01:25:18,850 --> 01:25:22,320 Except that we still don't know if everything has worked out 1158 01:25:22,320 --> 01:25:23,660 or not. 1159 01:25:23,660 --> 01:25:27,700 I'll continue here. 1160 01:25:27,700 --> 01:25:33,460 And just one minute to close up the discussion 1161 01:25:33,460 --> 01:25:36,180 by saying what has happened. 1162 01:25:36,180 --> 01:25:43,830 So, what has happened is that we have found that there's 1163 01:25:43,830 --> 01:25:57,870 a new Hamiltonian, HR equal U Hs U plus I h-bar dtU U dagger. 1164 01:25:57,870 --> 01:26:03,870 And then U is given by e to the minus I omega tSz 1165 01:26:03,870 --> 01:26:07,190 hat over h-bar. 1166 01:26:07,190 --> 01:26:20,000 And psi-- as we said, psi of t is 1167 01:26:20,000 --> 01:26:30,440 equal to e to the I omega tSz hat over h-bar psi R of t. 1168 01:26:30,440 --> 01:26:37,880 So this came because we said that psi R was going 1169 01:26:37,880 --> 01:26:43,420 to be U psi, and therefore I took U and the inverse 1170 01:26:43,420 --> 01:26:44,230 and you get this. 1171 01:26:44,230 --> 01:26:49,120 So look what the problem has turned into. 1172 01:26:49,120 --> 01:26:53,760 It has turned into a problem for psi R 1173 01:26:53,760 --> 01:26:59,000 with a Schrodinger equation that has a Hamilton HR in which 1174 01:26:59,000 --> 01:27:01,370 in this piece is very simple. 1175 01:27:01,370 --> 01:27:04,498 It's omega Sz hat. 1176 01:27:04,498 --> 01:27:08,800 And now the crucial point is whether-- here I 1177 01:27:08,800 --> 01:27:17,550 have U dagger-- whether this thing is time independent. 1178 01:27:17,550 --> 01:27:20,160 And this should be time independent 1179 01:27:20,160 --> 01:27:23,810 if we got our physics right, and that's exactly where we'll 1180 01:27:23,810 --> 01:27:29,010 take on the next time, and prove that it's time independent 1181 01:27:29,010 --> 01:27:31,960 and then we can solve the complete problem. 1182 01:27:31,960 --> 01:27:36,680 All right, there will be notes posted on this soon. 1183 01:27:36,680 --> 01:27:38,830 Good luck finishing your homework, 1184 01:27:38,830 --> 01:27:42,400 and I think Monday is a holiday. 1185 01:27:42,400 --> 01:27:43,230 Is that right? 1186 01:27:43,230 --> 01:27:45,020 Well, no class Monday. 1187 01:27:45,020 --> 01:27:47,270 We'll see you on Wednesday.