1 00:00:00,060 --> 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 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,600 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,600 --> 00:00:17,275 at ocw.mit.edu. 8 00:00:20,592 --> 00:00:22,550 BARTON ZWIEBACH: [INAUDIBLE] of today's lecture 9 00:00:22,550 --> 00:00:28,220 is coherent states of the harmonic oscillator. 10 00:00:28,220 --> 00:00:33,860 So let me begin by telling you about some things 11 00:00:33,860 --> 00:00:40,730 we've learned in the last lecture, and here they are. 12 00:00:40,730 --> 00:00:46,080 We learned how to calculate the so-called Heisenberg operators. 13 00:00:46,080 --> 00:00:48,480 Remember, if you have a Schrodinger operator, 14 00:00:48,480 --> 00:00:52,710 you subject it to this transformation 15 00:00:52,710 --> 00:00:55,070 with a unitary operator. 16 00:00:55,070 --> 00:00:57,880 That creates time evolution and that gives you 17 00:00:57,880 --> 00:00:59,980 the Heisenberg operator. 18 00:00:59,980 --> 00:01:04,319 We learned things about Heisenberg expectation values. 19 00:01:04,319 --> 00:01:09,710 If the Hamiltonian is time independent, h, time 20 00:01:09,710 --> 00:01:14,550 independent, the formula is quite simple 21 00:01:14,550 --> 00:01:18,180 and gives you the Heisenberg operator at the later time. 22 00:01:18,180 --> 00:01:21,650 So we did this. 23 00:01:21,650 --> 00:01:23,970 We found, in fact, Heisenberg operators 24 00:01:23,970 --> 00:01:26,760 satisfy equations of motion. 25 00:01:26,760 --> 00:01:29,960 And we calculated the Heisenberg operators 26 00:01:29,960 --> 00:01:32,260 for the harmonic oscillator. 27 00:01:32,260 --> 00:01:35,430 That was our main achievement last time, 28 00:01:35,430 --> 00:01:40,410 a formula for the time development of the x 29 00:01:40,410 --> 00:01:44,830 and p operators in the Heisenberg picture. 30 00:01:44,830 --> 00:01:50,680 And that really contains all the information of the dynamics, 31 00:01:50,680 --> 00:01:56,000 as you will see today, when we will be using this stuff. 32 00:01:56,000 --> 00:02:02,242 Now, I suggested that you read-- and you may do it later. 33 00:02:02,242 --> 00:02:03,700 There's no need that you've done it 34 00:02:03,700 --> 00:02:07,560 for today-- the information on the time 35 00:02:07,560 --> 00:02:11,980 development of the creation and annihilation operators. 36 00:02:11,980 --> 00:02:18,340 You see, the a and a dagger are different inverses of x and p, 37 00:02:18,340 --> 00:02:19,880 are linear combinations. 38 00:02:19,880 --> 00:02:23,450 So the a and the a dagger operators 39 00:02:23,450 --> 00:02:26,776 also can be further Schrodinger operators 40 00:02:26,776 --> 00:02:29,530 that have no time dependence. 41 00:02:29,530 --> 00:02:33,150 And suddenly, if you go to the Heisenberg picture, 42 00:02:33,150 --> 00:02:35,290 the creation and annihilation operators 43 00:02:35,290 --> 00:02:39,240 become time dependent operators. 44 00:02:39,240 --> 00:02:41,700 So that's in the notes. 45 00:02:41,700 --> 00:02:44,070 You can read about it. 46 00:02:44,070 --> 00:02:49,070 So we define the time dependent operator, a hat, 47 00:02:49,070 --> 00:02:53,470 to be the Heisenberg version of a hat. 48 00:02:53,470 --> 00:02:56,100 And you're supposed to do a calculation 49 00:02:56,100 --> 00:03:03,770 and try it or read it, and the answer is very nice, 50 00:03:03,770 --> 00:03:07,030 simply a phase dependence. 51 00:03:07,030 --> 00:03:12,450 The a is a at time equals 0, the Schrodinger 1 times 52 00:03:12,450 --> 00:03:15,550 e to the minus i omega t. 53 00:03:15,550 --> 00:03:21,470 Then a dagger is just what you would expect, 54 00:03:21,470 --> 00:03:27,330 the dagger of this, which the face has an opposite sign and a 55 00:03:27,330 --> 00:03:29,660 becomes a dagger. 56 00:03:29,660 --> 00:03:35,160 Finally, if you substitute this a and a 57 00:03:35,160 --> 00:03:37,920 daggers in this formula. 58 00:03:37,920 --> 00:03:40,640 For example, you could say x Heisenberg 59 00:03:40,640 --> 00:03:44,450 is a Heisenberg plus a dagger Heisenberg. 60 00:03:44,450 --> 00:03:47,640 And you substitute those Heisenberg values there, 61 00:03:47,640 --> 00:03:49,780 you will obtain this. 62 00:03:49,780 --> 00:03:51,200 Same for the momentum. 63 00:03:51,200 --> 00:03:54,280 If you put Heisenberg, Heisenberg, Heisenberg, 64 00:03:54,280 --> 00:03:56,880 remember, if you have an equality of Schrodinger 65 00:03:56,880 --> 00:03:59,230 operators, it also holds when you 66 00:03:59,230 --> 00:04:03,270 put Heisenberg in every operator. 67 00:04:03,270 --> 00:04:07,180 And therefore, if you put the Heisenberg a, a, 68 00:04:07,180 --> 00:04:10,940 and use those values, you will recover this equation. 69 00:04:10,940 --> 00:04:17,640 So in a sense, these equations are equivalent to these ones. 70 00:04:21,040 --> 00:04:25,370 And that's basically our situation. 71 00:04:25,370 --> 00:04:28,550 This is what we've learned so far, 72 00:04:28,550 --> 00:04:35,480 and our goal today is to apply this to understand 73 00:04:35,480 --> 00:04:38,660 coherent states of the harmonic oscillator. 74 00:04:38,660 --> 00:04:40,570 Now, why do we want to understand 75 00:04:40,570 --> 00:04:44,400 coherent states of the harmonic oscillator? 76 00:04:44,400 --> 00:04:47,550 You want to understand coherent states because the energy 77 00:04:47,550 --> 00:04:53,150 eigenstates are extraordinarily quantum. 78 00:04:53,150 --> 00:04:56,720 The energy eigenstates of the harmonic oscillator 79 00:04:56,720 --> 00:05:00,600 don't look at all-- and you've seen the expectation 80 00:05:00,600 --> 00:05:02,120 value of the position. 81 00:05:02,120 --> 00:05:04,350 It's time independent. 82 00:05:04,350 --> 00:05:05,710 It just doesn't change. 83 00:05:05,710 --> 00:05:11,030 Expectation value of any operator in a stationary state 84 00:05:11,030 --> 00:05:12,400 is a constant. 85 00:05:12,400 --> 00:05:13,730 It just doesn't change. 86 00:05:13,730 --> 00:05:19,295 So you have any eigenstate, any energy eigenstate 87 00:05:19,295 --> 00:05:21,680 of the harmonic oscillator, you ask, 88 00:05:21,680 --> 00:05:24,660 what is the position of this particle doing? 89 00:05:24,660 --> 00:05:25,230 Nothing. 90 00:05:25,230 --> 00:05:27,680 What is the momentum of this particle doing? 91 00:05:27,680 --> 00:05:29,520 Nothing. 92 00:05:29,520 --> 00:05:33,230 So nevertheless, of course, it's an interesting state, 93 00:05:33,230 --> 00:05:38,770 but we want to construct quantum mechanical states that 94 00:05:38,770 --> 00:05:41,630 behave a little like the classical states we're 95 00:05:41,630 --> 00:05:43,830 accustomed to. 96 00:05:43,830 --> 00:05:48,100 And that's what coherent states do. 97 00:05:48,100 --> 00:05:51,260 We'll have an application of coherent states 98 00:05:51,260 --> 00:05:54,710 to light, photons, coherent photons. 99 00:05:54,710 --> 00:05:55,580 What are they? 100 00:05:55,580 --> 00:05:59,880 We'll see it later this week. 101 00:05:59,880 --> 00:06:03,200 So that's the reason we want to understand coherent states, 102 00:06:03,200 --> 00:06:07,640 because we want some states that in some ways 103 00:06:07,640 --> 00:06:12,980 behave classically, or close to classically. 104 00:06:12,980 --> 00:06:15,640 So they have many applications, these states, 105 00:06:15,640 --> 00:06:20,350 and you will see some of them in this lecture. 106 00:06:20,350 --> 00:06:24,430 I'm going to try to keep this blackboard there, untouched, 107 00:06:24,430 --> 00:06:26,336 so that we can refer to these equations. 108 00:06:40,860 --> 00:06:47,540 So our first step is considering translation operators. 109 00:06:47,540 --> 00:06:51,740 So let's consider the unitary translation operator. 110 00:06:51,740 --> 00:06:59,600 So translation operators. 111 00:06:59,600 --> 00:07:06,580 So this translation operator that I will write as T sub x0 112 00:07:06,580 --> 00:07:10,590 will be defined to be the exponential of e 113 00:07:10,590 --> 00:07:18,090 to the minus i p hat x0 over h bar. 114 00:07:18,090 --> 00:07:20,565 You have seen such operators before. 115 00:07:23,630 --> 00:07:27,380 We've seen a lot of them in the homework. 116 00:07:27,380 --> 00:07:30,770 So first of all, why is it unitary? 117 00:07:37,360 --> 00:07:43,655 well, it's unitary because x0 is supposed to be a real number. 118 00:07:48,040 --> 00:07:49,650 p is Hermitian. 119 00:07:49,650 --> 00:07:52,565 Therefore, this with the i is anti-Hermitian, 120 00:07:52,565 --> 00:07:55,580 and an exponential of anti-Hermitian operator 121 00:07:55,580 --> 00:07:57,940 is unitary. 122 00:07:57,940 --> 00:08:01,710 Now, it has, actually, a very simple property. 123 00:08:01,710 --> 00:08:09,585 The multiplication of two of those operators is what? 124 00:08:09,585 --> 00:08:14,130 Well, you have an exponential, e to the minus ipx0, 125 00:08:14,130 --> 00:08:17,790 and an exponential followed, e to the minus ipy0. 126 00:08:20,740 --> 00:08:24,120 Now, if you're well trained in 805, 127 00:08:24,120 --> 00:08:27,850 you should get a little nervous for a second 128 00:08:27,850 --> 00:08:30,610 because you don't know, can I treat it easily? 129 00:08:30,610 --> 00:08:33,690 And then you relax and say, yes, these two operators, 130 00:08:33,690 --> 00:08:38,929 whatsoever the numbers here, this with another one with a y0 131 00:08:38,929 --> 00:08:40,360 would commute. 132 00:08:40,360 --> 00:08:44,260 Therefore, they can be put together in the exponential, 133 00:08:44,260 --> 00:08:47,440 and this is T of x0 plus y0. 134 00:08:54,840 --> 00:09:00,000 No combo Baker-Hausdorff needed here. 135 00:09:00,000 --> 00:09:03,140 It's just straightforward. 136 00:09:03,140 --> 00:09:11,440 So what is Tx0 dagger? 137 00:09:11,440 --> 00:09:14,610 T x0 dagger, if you take the dagger, 138 00:09:14,610 --> 00:09:19,580 you change this i for a minus i, so it's 139 00:09:19,580 --> 00:09:23,920 exactly the same as changing the sign of x0. 140 00:09:23,920 --> 00:09:29,630 So this is T of minus x0. 141 00:09:29,630 --> 00:09:35,430 And by this identity, T of minus x0 with a T of x0 142 00:09:35,430 --> 00:09:39,830 would be T of 0, which is the unit operator. 143 00:09:39,830 --> 00:09:47,800 So T of minus x0 is the inverse of T of x0, 144 00:09:47,800 --> 00:09:53,082 confirming that the operator is unitary. 145 00:09:53,082 --> 00:09:55,600 The inverse is the dagger. 146 00:09:59,580 --> 00:10:03,040 So I used here that this is the inverse 147 00:10:03,040 --> 00:10:08,860 because T minus x0 times T x0 is T of 0 is equal to 1. 148 00:10:08,860 --> 00:10:13,980 So I could mention here, T of 0 is equal to the unit operator. 149 00:10:18,030 --> 00:10:21,170 So these are our translation operators, 150 00:10:21,170 --> 00:10:23,570 but you don't get the intuition of what 151 00:10:23,570 --> 00:10:27,210 they do unless you compute a little more. 152 00:10:27,210 --> 00:10:30,700 And a little more than you should compute is this. 153 00:10:30,700 --> 00:10:36,960 What is T x0 dagger x T x0? 154 00:10:43,420 --> 00:10:49,610 And what is T x0 dagger p T x0? 155 00:10:54,210 --> 00:10:58,300 Now, why do we ask for these particular things? 156 00:10:58,300 --> 00:11:02,410 Why don't I ask, what is x hat multiplied by T x0? 157 00:11:02,410 --> 00:11:04,180 Why do I ask this? 158 00:11:04,180 --> 00:11:07,660 It is because an operator acting on an operator 159 00:11:07,660 --> 00:11:08,820 always does this. 160 00:11:08,820 --> 00:11:12,490 If you say an operator is acting on another operator, 161 00:11:12,490 --> 00:11:14,580 the first operator that is acting, 162 00:11:14,580 --> 00:11:16,720 you put it here with its inverse. 163 00:11:16,720 --> 00:11:20,420 It happens to be unitary, so you put the dagger, 164 00:11:20,420 --> 00:11:23,750 and you put the operator here. 165 00:11:23,750 --> 00:11:26,580 And this is the right thing to do. 166 00:11:26,580 --> 00:11:30,200 It has a simple answer and a simple interpretation, 167 00:11:30,200 --> 00:11:31,190 as we'll see now. 168 00:11:31,190 --> 00:11:35,910 So what is T, this commutator, supposed to be? 169 00:11:35,910 --> 00:11:40,220 Well, you can probably imagine what this is. 170 00:11:40,220 --> 00:11:42,220 You've calculated it in homework, 171 00:11:42,220 --> 00:11:44,960 so I will not do it again. 172 00:11:44,960 --> 00:11:48,470 This is x plus x0. 173 00:11:48,470 --> 00:11:54,080 So you get the operator, x, plus x0 times the unit operator. 174 00:11:54,080 --> 00:11:57,490 That was done before. 175 00:11:57,490 --> 00:12:03,730 And here, you get just p. 176 00:12:03,730 --> 00:12:04,360 Why? 177 00:12:04,360 --> 00:12:08,690 Because p hat is the only operator 178 00:12:08,690 --> 00:12:13,880 that exists in this translation thing, so p commutes with p. 179 00:12:13,880 --> 00:12:19,110 So these two operators commute and the T tagger hits the T, 180 00:12:19,110 --> 00:12:21,960 and it's equal to 1, so that's a simple thing. 181 00:12:24,780 --> 00:12:30,400 So why is this reasonable? 182 00:12:30,400 --> 00:12:34,160 It's because of the following situation. 183 00:12:34,160 --> 00:12:40,840 If you have a state, psi, you can ask, for example, 184 00:12:40,840 --> 00:12:46,120 what is the expectation value of x in the state psi? 185 00:12:50,930 --> 00:12:54,970 And if this state represents a particle that 186 00:12:54,970 --> 00:13:01,610 is sitting somewhere here, roughly, the expectation value 187 00:13:01,610 --> 00:13:05,860 of x is basically that vector that 188 00:13:05,860 --> 00:13:09,190 tells you where the particle is. 189 00:13:09,190 --> 00:13:16,260 So you could ask, then, what is the expectation 190 00:13:16,260 --> 00:13:21,560 value of x in the state T x0 psi? 191 00:13:28,500 --> 00:13:33,690 So you want to know, what does T x0 really do? 192 00:13:33,690 --> 00:13:38,330 Here, it seems to say something, takes the operator 193 00:13:38,330 --> 00:13:41,440 and displaces it, but that seems abstract. 194 00:13:41,440 --> 00:13:46,530 If you ask this question, this seems more physical. 195 00:13:46,530 --> 00:13:50,470 You had a state, you act with an operator, it's another state. 196 00:13:50,470 --> 00:13:52,350 How does it look? 197 00:13:52,350 --> 00:13:58,140 Well, this expectation value would be the expectation value 198 00:13:58,140 --> 00:14:07,470 of x on T x0 psi, and the [? brau ?] 199 00:14:07,470 --> 00:14:11,240 would be psi T x0 dagger. 200 00:14:14,330 --> 00:14:17,280 So actually, that expectation value 201 00:14:17,280 --> 00:14:22,220 builds precisely this combination, 202 00:14:22,220 --> 00:14:24,660 and that's why it's meaningful. 203 00:14:24,660 --> 00:14:38,790 And since you know what this is, this is psi x plus x0 psi. 204 00:14:38,790 --> 00:14:41,910 This is equal to the expectation value 205 00:14:41,910 --> 00:14:50,150 of x in the original state plus x0 times 1. 206 00:14:53,510 --> 00:15:00,790 So the expectation value of x in the new state, the x0 psi, 207 00:15:00,790 --> 00:15:05,920 is the expectation value of x in the old state plus x0. 208 00:15:05,920 --> 00:15:12,050 So indeed, if this is x, you could do this for vectors, 209 00:15:12,050 --> 00:15:15,100 and here is x0. 210 00:15:15,100 --> 00:15:22,680 Well, the expectation value of x in the new state, the T x0 211 00:15:22,680 --> 00:15:28,800 operator, took the state and moved it by a displacement x0 212 00:15:28,800 --> 00:15:33,840 so that the new expectation value is the old one plus x0. 213 00:15:37,450 --> 00:15:42,320 So that's physically why these things are relevant. 214 00:15:42,320 --> 00:15:49,420 A couple of other things you've shown in the homework, 215 00:15:49,420 --> 00:15:56,900 and you could retry doing them, is that T x0 on the x state, 216 00:15:56,900 --> 00:16:04,050 by this intuition, should be the x plus x0 state. 217 00:16:04,050 --> 00:16:07,810 It moves the state to the right. 218 00:16:07,810 --> 00:16:18,745 And if psi has a wave function, psi of x, T x0 of psi 219 00:16:18,745 --> 00:16:24,180 has a wave function, psi of x minus x0, 220 00:16:24,180 --> 00:16:27,860 since you know that psi of x minus x0 221 00:16:27,860 --> 00:16:33,300 is the wave function translated by x0 to the right. 222 00:16:33,300 --> 00:16:35,480 The sign is always the opposite one. 223 00:16:35,480 --> 00:16:37,930 When you write psi of x minus x0, 224 00:16:37,930 --> 00:16:42,600 the function has been moved to the right x0. 225 00:16:42,600 --> 00:16:48,410 So this is our whole discussion and reminder 226 00:16:48,410 --> 00:16:52,480 of what the translation operators are. 227 00:16:52,480 --> 00:16:55,170 So we've got our translation operator. 228 00:16:55,170 --> 00:16:57,670 Let's see how we can use it. 229 00:16:57,670 --> 00:17:01,880 And we'll use it to define the coherent states. 230 00:17:01,880 --> 00:17:05,715 So here comes the definition of what the coherent state is. 231 00:17:08,250 --> 00:17:12,069 It's a beginning definition, or a working definition, 232 00:17:12,069 --> 00:17:16,250 until we understand it enough that we can generalize it. 233 00:17:16,250 --> 00:17:18,400 By the time we finish the lecture, 234 00:17:18,400 --> 00:17:23,329 this definition will be generalized in a very nice way, 235 00:17:23,329 --> 00:17:25,829 in a very elegant way. 236 00:17:25,829 --> 00:17:28,975 So coherent states. 237 00:17:36,110 --> 00:17:37,480 So here it goes. 238 00:17:37,480 --> 00:17:43,010 I'm going to take the vacuum state of the harmonic 239 00:17:43,010 --> 00:17:46,840 oscillator, the ground state of the harmonic oscillator, 240 00:17:46,840 --> 00:17:54,430 and simply displace it with a translation operator by x0. 241 00:17:54,430 --> 00:18:04,260 So this is going to be e to the minus i p hat x0 over h bar 0. 242 00:18:04,260 --> 00:18:06,680 And I want a name for this state, 243 00:18:06,680 --> 00:18:09,190 and that's the worst part of it. 244 00:18:09,190 --> 00:18:11,590 There's no great name for it. 245 00:18:15,160 --> 00:18:19,000 I don't know if any notation is very good. 246 00:18:19,000 --> 00:18:23,760 If it's very good, it's cumbersome, 247 00:18:23,760 --> 00:18:25,890 so I'll write it like this. 248 00:18:25,890 --> 00:18:27,230 A little misleading. 249 00:18:27,230 --> 00:18:30,510 I'll put a tilde over the state. 250 00:18:30,510 --> 00:18:32,490 You could say it's a tilde over the x, 251 00:18:32,490 --> 00:18:34,430 but it really, morally speaking, is 252 00:18:34,430 --> 00:18:36,480 a tilde over the whole state. 253 00:18:36,480 --> 00:18:40,440 It means that this thing, you should 254 00:18:40,440 --> 00:18:44,820 read there's an x0 here used for the translation operator that 255 00:18:44,820 --> 00:18:46,480 appears here. 256 00:18:46,480 --> 00:18:50,770 So that's the state, x tilde 0. 257 00:18:50,770 --> 00:18:54,100 Intuitively, you know what it is. 258 00:18:54,100 --> 00:18:57,450 You have the harmonic oscillator potential. 259 00:18:57,450 --> 00:18:58,910 Here is x. 260 00:18:58,910 --> 00:19:02,640 The ground state is some wave function like that. 261 00:19:02,640 --> 00:19:07,670 This state has been moved to position x0, 262 00:19:07,670 --> 00:19:11,880 and presumably some sort of wave function like that, 263 00:19:11,880 --> 00:19:15,670 because this translates the wave function. 264 00:19:15,670 --> 00:19:19,900 So the ground state moves it up there to the right. 265 00:19:19,900 --> 00:19:21,260 That's what it is. 266 00:19:21,260 --> 00:19:23,240 That's a coherent state. 267 00:19:23,240 --> 00:19:27,800 And there's no time dependence here so far, 268 00:19:27,800 --> 00:19:31,020 so this is the state at some instant of time. 269 00:19:31,020 --> 00:19:34,580 The coherent state, maybe call it at time equals zero. 270 00:19:34,580 --> 00:19:38,850 Let's leave time frozen for a little while 271 00:19:38,850 --> 00:19:41,640 until we understand what this state does. 272 00:19:41,640 --> 00:19:45,190 Then we'll put the time back. 273 00:19:45,190 --> 00:19:48,475 So a few remarks on this. 274 00:19:51,220 --> 00:19:58,085 x0 x0 is how much? 275 00:20:04,350 --> 00:20:07,670 Now, don't think these are position eigenstates. 276 00:20:07,670 --> 00:20:09,380 That's a possible mistake. 277 00:20:09,380 --> 00:20:13,550 That's not a position eigenstate. 278 00:20:13,550 --> 00:20:16,780 This is a coherent state. 279 00:20:16,780 --> 00:20:18,680 If these would be position eigenstate, 280 00:20:18,680 --> 00:20:21,220 you say delta of this minus that, 281 00:20:21,220 --> 00:20:24,890 but it's nothing to do with that. 282 00:20:24,890 --> 00:20:27,840 Can you tell without doing any computation what 283 00:20:27,840 --> 00:20:29,330 is this number? 284 00:20:29,330 --> 00:20:30,890 How much should be? 285 00:20:30,890 --> 00:20:31,650 Yes? 286 00:20:31,650 --> 00:20:32,587 AUDIENCE: 1. 287 00:20:32,587 --> 00:20:33,920 BARTON ZWIEBACH: It should be 1. 288 00:20:33,920 --> 00:20:34,440 Why? 289 00:20:34,440 --> 00:20:38,630 Because it's a unitary operator acting on this thing, 290 00:20:38,630 --> 00:20:40,160 so it preserve length. 291 00:20:40,160 --> 00:20:44,100 So this should be equal to 0 0, should be 1. 292 00:20:44,100 --> 00:20:46,680 Very good. 293 00:20:46,680 --> 00:20:48,550 No need to do the computation. 294 00:20:48,550 --> 00:20:50,710 It's just 1. 295 00:20:50,710 --> 00:21:02,260 Psi associated to this state is the ground state wave function 296 00:21:02,260 --> 00:21:04,380 at x minus x0. 297 00:21:04,380 --> 00:21:14,435 Where this refers to the wave function, x0 is psi 0 of x. 298 00:21:17,440 --> 00:21:20,400 So this is what I was saying here. 299 00:21:20,400 --> 00:21:22,830 The wave function has been translated 300 00:21:22,830 --> 00:21:27,670 to x0, the remark over there. 301 00:21:30,180 --> 00:21:33,450 So these are our coherent states and we 302 00:21:33,450 --> 00:21:39,240 want to understand the first few basic things about them 303 00:21:39,240 --> 00:21:42,435 so we can do the following simple computations. 304 00:21:51,130 --> 00:21:53,540 So if I have to do the following, 305 00:21:53,540 --> 00:21:55,500 if I have to compute the expectation 306 00:21:55,500 --> 00:22:04,850 value of any operator, A, on a coherent state, 307 00:22:04,850 --> 00:22:09,180 I use the fact that I want to go back to the vacuum, 308 00:22:09,180 --> 00:22:15,820 so I put T x0 dagger A T x0 0. 309 00:22:19,160 --> 00:22:25,360 Because that way, I trace back to what the vacuum is doing. 310 00:22:25,360 --> 00:22:29,840 It's much easier to do that than to try to calculate something 311 00:22:29,840 --> 00:22:31,250 from scratch. 312 00:22:31,250 --> 00:22:46,830 So for example, we have here that x0 x x0, well, you 313 00:22:46,830 --> 00:22:53,090 would replace it by T x T, T dagger x T, which you know 314 00:22:53,090 --> 00:22:56,140 is x hat plus x0. 315 00:22:56,140 --> 00:23:03,960 We calculated it a few seconds ago, top blackboard. 316 00:23:03,960 --> 00:23:07,570 And therefore, you got what is the expectation 317 00:23:07,570 --> 00:23:10,800 value of x on the ground state? 318 00:23:10,800 --> 00:23:13,210 x0, very good. 319 00:23:13,210 --> 00:23:18,560 And therefore, we just got x0, which is what you would expect. 320 00:23:18,560 --> 00:23:23,100 The expectation value of x on the coherent state is x0. 321 00:23:23,100 --> 00:23:24,270 You're there. 322 00:23:24,270 --> 00:23:26,410 You've been displaced. 323 00:23:26,410 --> 00:23:37,080 How about the momentum, x0 p hat x0? 324 00:23:37,080 --> 00:23:43,430 Well, p acted by the translation operator is unchanged. 325 00:23:43,430 --> 00:23:49,570 Therefore, we got 0 p 0, and again that 0, so this state 326 00:23:49,570 --> 00:23:53,330 still has no momentum. 327 00:23:53,330 --> 00:23:57,860 It represents a T equals 0, a state that is over here. 328 00:23:57,860 --> 00:24:02,780 And just by looking at it, it's just sitting there, 329 00:24:02,780 --> 00:24:05,140 has no momentum whatsoever. 330 00:24:08,440 --> 00:24:11,310 Another question that is interesting, 331 00:24:11,310 --> 00:24:16,201 what is the expectation value of the Hamiltonian 332 00:24:16,201 --> 00:24:18,010 on the coherent state? 333 00:24:21,080 --> 00:24:28,640 Well, this should be, now you imagine in your head, 334 00:24:28,640 --> 00:24:37,350 T dagger H T. Now, H is p squared over 2m, 335 00:24:37,350 --> 00:24:41,800 and that p squared over 2m gets unchanged. 336 00:24:41,800 --> 00:24:49,520 p squared over 2m is not changed because T dagger and T does 337 00:24:49,520 --> 00:24:53,330 nothing to it, T dagger from the left, T. 338 00:24:53,330 --> 00:24:54,120 Nevertheless. 339 00:24:54,120 --> 00:25:01,240 the Hamiltonian has a 1/2 m omega squared x hat, 340 00:25:01,240 --> 00:25:06,960 and x hat is changed by becoming x hat plus x0. 341 00:25:16,120 --> 00:25:20,840 Well, we don't want to compute too hard, do 342 00:25:20,840 --> 00:25:22,280 too much effort here. 343 00:25:22,280 --> 00:25:26,920 So first, we realize that here's the p squared over 2m 344 00:25:26,920 --> 00:25:31,070 and here's the m omega squared x hat squared, so that's 345 00:25:31,070 --> 00:25:32,980 the whole Hamiltonian. 346 00:25:32,980 --> 00:25:44,500 So we got 0 H 0 plus the extra terms that come here. 347 00:25:44,500 --> 00:25:46,720 But what terms come here? 348 00:25:46,720 --> 00:25:54,250 There's a product of an x0 and an x between 0 and 0. 349 00:25:54,250 --> 00:26:01,220 x0 is a number, so you have an x between 0 and 0, and that's 0. 350 00:26:01,220 --> 00:26:05,810 So the cross product here won't contribute to the expectation 351 00:26:05,810 --> 00:26:10,240 value, so the last term that is there is 1/2 m 352 00:26:10,240 --> 00:26:16,310 is a number omega squared, x0 squared. 353 00:26:16,310 --> 00:26:18,200 And what is the expectation value 354 00:26:18,200 --> 00:26:19,890 of the Hamiltonian on the vacuum? 355 00:26:19,890 --> 00:26:28,450 It's h omega over 2 plus 1/2 m omega squared, x0 squared. 356 00:26:28,450 --> 00:26:35,000 And you start seeing classical behavior. 357 00:26:35,000 --> 00:26:38,400 The expectation value of the energy at this point 358 00:26:38,400 --> 00:26:44,840 is a little quantum thing plus the whole cost 359 00:26:44,840 --> 00:26:49,080 of stretching something all the way to x0. 360 00:26:49,080 --> 00:26:57,690 1/2 of k squared, k for the oscillator, x0 squared. 361 00:26:57,690 --> 00:27:02,500 So the energy of this thing is quite reasonably approximated, 362 00:27:02,500 --> 00:27:07,250 if x0 is large enough, by the second term, 363 00:27:07,250 --> 00:27:09,870 and this is the cost of energy of having 364 00:27:09,870 --> 00:27:13,450 a particle of the potential. 365 00:27:13,450 --> 00:27:16,280 So it's behaving in a reasonable way. 366 00:27:19,100 --> 00:27:26,910 You can do a couple more little exercises that I'll 367 00:27:26,910 --> 00:27:35,613 put here as things for you to check. 368 00:27:35,613 --> 00:27:36,113 Exercise. 369 00:27:38,660 --> 00:27:44,770 x0 tilde x squared x0 tilde. 370 00:27:47,480 --> 00:27:48,330 Just calculate. 371 00:27:48,330 --> 00:27:51,250 It's just useful to have. 372 00:27:51,250 --> 00:27:56,650 x0 squared plus h bar over 2m omega. 373 00:27:56,650 --> 00:28:09,290 And x0 tilde p squared x0 tilde is mh omega over 2. 374 00:28:09,290 --> 00:28:21,535 And finally, x0 tilde xp plus px x0 tilde is equal to 0. 375 00:28:29,740 --> 00:28:31,520 Any questions? 376 00:28:31,520 --> 00:28:34,580 These are exercises for you to practice 377 00:28:34,580 --> 00:28:36,072 a little these expectation values. 378 00:28:38,700 --> 00:28:40,700 Questions on what we've done so far? 379 00:28:40,700 --> 00:28:42,254 Yes? 380 00:28:42,254 --> 00:28:44,739 AUDIENCE: You said these coherent states is 381 00:28:44,739 --> 00:28:47,224 most significant only in the ground state, 382 00:28:47,224 --> 00:28:53,964 or is it also important to use them for [INAUDIBLE]? 383 00:28:56,770 --> 00:29:01,770 BARTON ZWIEBACH: Well, we've defined the coherent state 384 00:29:01,770 --> 00:29:05,110 by taking the ground state and moving it, 385 00:29:05,110 --> 00:29:07,420 and these are particularly interesting. 386 00:29:07,420 --> 00:29:10,800 You could try to figure out what would happen if you would take 387 00:29:10,800 --> 00:29:13,320 an excited state and you move it. 388 00:29:13,320 --> 00:29:16,410 Things are a little more complicated. 389 00:29:16,410 --> 00:29:18,360 PROFESSOR: And in a sense, they can all 390 00:29:18,360 --> 00:29:21,790 be understood in terms of what we do to the ground state. 391 00:29:21,790 --> 00:29:25,190 So we will not focus on them too much. 392 00:29:25,190 --> 00:29:27,860 In a sense, you will see when we generalize 393 00:29:27,860 --> 00:29:33,430 this how what we're doing is very special, in at least 394 00:29:33,430 --> 00:29:35,660 one simple way. 395 00:29:35,660 --> 00:29:41,040 So we'll always focus on translating the grounds. 396 00:29:41,040 --> 00:29:43,560 Other questions? 397 00:29:43,560 --> 00:29:44,690 Yes. 398 00:29:44,690 --> 00:29:47,180 AUDIENCE: Where does the term [INAUDIBLE] arise 399 00:29:47,180 --> 00:29:50,120 and why does it persevere [INAUDIBLE] 400 00:29:50,120 --> 00:29:54,940 PROFESSOR: OK, here is the thing of the coherent state. 401 00:29:54,940 --> 00:30:00,000 Is this an energy eigenstate at this moment? 402 00:30:00,000 --> 00:30:01,030 What do you think? 403 00:30:01,030 --> 00:30:06,090 Is this an energy eigenstate-- this state over here? 404 00:30:06,090 --> 00:30:09,250 No, it won't be an energy eigenstate. 405 00:30:09,250 --> 00:30:11,050 There's something funny about it. 406 00:30:11,050 --> 00:30:15,010 Energy eigenstates are always diffuse things. 407 00:30:15,010 --> 00:30:18,050 They never look like that. 408 00:30:18,050 --> 00:30:19,970 So this is not an energy eigenstate, 409 00:30:19,970 --> 00:30:22,820 and you've done things with non-energy eigenstates. 410 00:30:22,820 --> 00:30:25,430 They change shape. 411 00:30:25,430 --> 00:30:28,950 As they evolve, they change shape. 412 00:30:28,950 --> 00:30:32,160 What we will see very soon is that this state, 413 00:30:32,160 --> 00:30:37,170 if we let it go, it will start moving back and forth 414 00:30:37,170 --> 00:30:39,630 without changing shape. 415 00:30:39,630 --> 00:30:43,450 It's going to do an amazing thing. 416 00:30:43,450 --> 00:30:46,120 Energy eigenstates-- you're super-close to energy 417 00:30:46,120 --> 00:30:46,870 eigenstates. 418 00:30:46,870 --> 00:30:50,710 You get something that changes in time and the shape changes, 419 00:30:50,710 --> 00:30:53,210 and you've even done problems like that. 420 00:30:53,210 --> 00:30:58,620 But this state is so exceptional that even 421 00:30:58,620 --> 00:31:01,815 as we let it go in time, it's going to change, 422 00:31:01,815 --> 00:31:05,740 but the shape is not going to spread out. 423 00:31:05,740 --> 00:31:08,440 Do you remember when you considered a pulse 424 00:31:08,440 --> 00:31:13,270 in a free particle, how it disappears and stretches away? 425 00:31:13,270 --> 00:31:15,460 Well, in the harmonic oscillator, 426 00:31:15,460 --> 00:31:19,340 this has been so well prepared that this thing, as time goes 427 00:31:19,340 --> 00:31:23,660 by, will just move and oscillate like a particle. 428 00:31:23,660 --> 00:31:26,350 And it does so coherently. 429 00:31:26,350 --> 00:31:28,430 It doesn't change shape. 430 00:31:28,430 --> 00:31:32,780 When we talk about light, coherent light 431 00:31:32,780 --> 00:31:34,610 is what you get from lasers. 432 00:31:34,610 --> 00:31:36,500 And so if you want understand lasers, 433 00:31:36,500 --> 00:31:38,230 you have to understand coherent states. 434 00:31:41,020 --> 00:31:46,140 OK, so this brings us there to time evolution. 435 00:31:46,140 --> 00:31:48,376 So let's do time evolution. 436 00:32:05,480 --> 00:32:06,760 So what will happen? 437 00:32:06,760 --> 00:32:16,440 We'll have a state x0 goes to x0 comma t. 438 00:32:16,440 --> 00:32:17,880 So that's the notation. 439 00:32:17,880 --> 00:32:23,530 That's what we'll mean by the state at a later time. 440 00:32:23,530 --> 00:32:26,510 And how are we going to explore this? 441 00:32:26,510 --> 00:32:30,290 Well, we're all set with our Heisenberg operator, there. 442 00:32:30,290 --> 00:32:33,410 We'll take expectation values of things 443 00:32:33,410 --> 00:32:36,420 to figure out how things look. 444 00:32:36,420 --> 00:32:39,300 So what do we have here? 445 00:32:39,300 --> 00:32:46,550 We'll ask for X0 t, and we'll put the Schrodinger operator 446 00:32:46,550 --> 00:32:50,910 in between here-- X0 t, and this is 447 00:32:50,910 --> 00:32:54,760 what we'll call the expectation value of A 448 00:32:54,760 --> 00:32:58,230 as time goes by in the X0 0 state. 449 00:33:01,510 --> 00:33:03,120 This is what we call this. 450 00:33:03,120 --> 00:33:06,720 But then, we have the time evolution. 451 00:33:06,720 --> 00:33:14,320 So this is equal to the original state, Heisenberg operator 452 00:33:14,320 --> 00:33:17,785 of A-- original state. 453 00:33:21,650 --> 00:33:26,460 And if you wish, you could then put the t operator-- 454 00:33:26,460 --> 00:33:30,280 as we have in the top blackboard to the right-- 455 00:33:30,280 --> 00:33:32,130 and reduce it even more. 456 00:33:32,130 --> 00:33:37,930 But we've computed a lot of this coherent state expectation 457 00:33:37,930 --> 00:33:41,510 value, so let's leave it like that. 458 00:33:41,510 --> 00:33:44,810 So you could, if you wish, say this 459 00:33:44,810 --> 00:33:56,970 is equal to 0-- T X0 dagger A T X0 0. 460 00:33:56,970 --> 00:34:01,080 So you can ultimately reduce the expectation values 461 00:34:01,080 --> 00:34:02,810 of things on the vacuum. 462 00:34:05,830 --> 00:34:07,680 So OK, we're all set. 463 00:34:07,680 --> 00:34:09,125 Let's try to do one. 464 00:34:17,210 --> 00:34:19,960 And the reason this is a nice calculation 465 00:34:19,960 --> 00:34:21,820 is that the time evolution of this state 466 00:34:21,820 --> 00:34:23,370 is a little complicated. 467 00:34:23,370 --> 00:34:26,540 We'll figure it out later, but it's easier 468 00:34:26,540 --> 00:34:28,449 to work with the time evolved state. 469 00:34:28,449 --> 00:34:31,820 So here it goes-- what is the expectation 470 00:34:31,820 --> 00:34:40,120 value of X as a function of time on the X0 state? 471 00:34:40,120 --> 00:34:45,620 Well, it says here take the X0 state, 472 00:34:45,620 --> 00:34:49,739 and take the Heisenberg value of X. 473 00:34:49,739 --> 00:34:56,960 So we have it up there-- X hat cosine omega t plus b hat 474 00:34:56,960 --> 00:35:03,260 over M omega sine omega t X0. 475 00:35:03,260 --> 00:35:06,630 Forget about time evolution of the coherent states. 476 00:35:06,630 --> 00:35:09,240 We evolved the operator. 477 00:35:09,240 --> 00:35:13,510 On the other hand, we have that the expectation value of p 478 00:35:13,510 --> 00:35:19,090 is 0 in the coherent state, and the expectation value of X 479 00:35:19,090 --> 00:35:20,980 is X0. 480 00:35:20,980 --> 00:35:27,760 So end of story-- calculation over-- X0 cosine of omega t. 481 00:35:27,760 --> 00:35:30,630 That's expectation value in time. 482 00:35:30,630 --> 00:35:32,806 This thing is oscillating classically. 483 00:35:36,110 --> 00:35:39,300 That's nice as can be. 484 00:35:39,300 --> 00:35:46,720 So classical behavior again, of a quantum state. 485 00:35:46,720 --> 00:35:55,690 How about expectation value of p X0 of t? 486 00:35:55,690 --> 00:35:58,535 If it's oscillating, it better be moving, 487 00:35:58,535 --> 00:36:00,720 and it better have some momentum. 488 00:36:00,720 --> 00:36:04,430 So let's put the momentum operator here, 489 00:36:04,430 --> 00:36:07,110 the Heisenberg one. 490 00:36:07,110 --> 00:36:14,500 So we'll have p hat cosine omega t minus m omega x 491 00:36:14,500 --> 00:36:22,160 hat sine omega t X0 tilde. 492 00:36:22,160 --> 00:36:27,080 And this is 0, but X has X0 there, 493 00:36:27,080 --> 00:36:35,510 so minus m omega X0 sine of omega t, which 494 00:36:35,510 --> 00:36:46,110 is equal to m d dt of the expectation value of X. 495 00:36:46,110 --> 00:36:50,430 Here it is-- expectation value of X. m d dt of that 496 00:36:50,430 --> 00:36:53,900 is minus m omega X0 sine omega t. 497 00:36:53,900 --> 00:36:57,220 That's what it should be. 498 00:36:57,220 --> 00:37:00,760 And this thing is really oscillating classically-- 499 00:37:00,760 --> 00:37:04,330 not only the exposition, but the momentum is doing that. 500 00:37:07,130 --> 00:37:09,750 Now, the other thing that we can compute-- 501 00:37:09,750 --> 00:37:15,650 and we want to compute-- is the key thing. 502 00:37:15,650 --> 00:37:17,110 You have this state. 503 00:37:17,110 --> 00:37:20,420 We said it's coherent evolution. 504 00:37:20,420 --> 00:37:23,800 So the ground state is this state 505 00:37:23,800 --> 00:37:27,190 that is a minimum uncertainty packet. 506 00:37:27,190 --> 00:37:31,800 It has a delta X uncertainty mix and a delta p. 507 00:37:31,800 --> 00:37:37,280 Their product saturates the uncertainty in equality. 508 00:37:37,280 --> 00:37:41,710 And when we move the state X0, well, the delta X 509 00:37:41,710 --> 00:37:42,650 will be the same. 510 00:37:42,650 --> 00:37:45,970 The delta p will be the same, and it's that. 511 00:37:45,970 --> 00:37:49,730 But now as it starts to move, we want 512 00:37:49,730 --> 00:37:52,630 to see if the shape is kept the same. 513 00:37:52,630 --> 00:37:56,100 Maybe it fattens up, and shrinks down, 514 00:37:56,100 --> 00:37:58,770 and does things in the middle. 515 00:37:58,770 --> 00:38:02,200 So the issue of coherency of this state 516 00:38:02,200 --> 00:38:07,270 is the issue whether the uncertainties remain the same. 517 00:38:07,270 --> 00:38:09,670 If the uncertainties remain the same, 518 00:38:09,670 --> 00:38:13,070 and they are saturated-- the product is saturating 519 00:38:13,070 --> 00:38:17,700 the inequality, you know that the shape has to be Gaussian, 520 00:38:17,700 --> 00:38:22,210 and it must be the same shape that is running around. 521 00:38:22,210 --> 00:38:28,570 So what we need to compute is the uncertainty in X, 522 00:38:28,570 --> 00:38:29,800 for example. 523 00:38:29,800 --> 00:38:42,130 So how do delta X of t and delta p of p behave? 524 00:38:45,610 --> 00:38:48,220 That's our question. 525 00:38:48,220 --> 00:38:50,305 And let's see how they do. 526 00:38:54,210 --> 00:38:57,060 Well, we have this computation-- actually, 527 00:38:57,060 --> 00:38:59,640 if you don't have the Heisenberg picture, 528 00:38:59,640 --> 00:39:02,930 it's kind of a nightmare. 529 00:39:02,930 --> 00:39:07,450 With the Heisenberg picture, it's a lot easier. 530 00:39:07,450 --> 00:39:12,250 Delta x squared of t would be the expectation 531 00:39:12,250 --> 00:39:21,120 value of X0 t of X squared, X0 t, 532 00:39:21,120 --> 00:39:30,790 minus the expectation value of X0 t X, X0 t squared. 533 00:39:35,200 --> 00:39:39,396 I wrote what the definition of the uncertainty squared is. 534 00:39:39,396 --> 00:39:43,450 It's the expectation value of the operator squared, 535 00:39:43,450 --> 00:39:46,330 minus the square of the expectation 536 00:39:46,330 --> 00:39:49,250 value of the operator. 537 00:39:49,250 --> 00:39:52,390 And of course, everything is going to turn Heisenberg 538 00:39:52,390 --> 00:39:57,490 immediately, so this thing-- maybe I can go one more 539 00:39:57,490 --> 00:40:08,070 line here-- would be X0 X Heisenberg squared of t, 540 00:40:08,070 --> 00:40:16,170 X0 minus-- this is simple-- this we've calculated. 541 00:40:16,170 --> 00:40:20,010 it's that expectation value at the top 542 00:40:20,010 --> 00:40:23,530 is the expectation value of X in time. 543 00:40:23,530 --> 00:40:24,250 It's that. 544 00:40:24,250 --> 00:40:29,650 So this is minus X0 squared cosine squared of omega t. 545 00:40:37,950 --> 00:40:39,680 So what do we have to do? 546 00:40:39,680 --> 00:40:42,670 We have to focus on this term. 547 00:40:45,860 --> 00:40:50,960 So this term is equal to X0. 548 00:40:50,960 --> 00:40:53,430 And you have X Heisenberg squared, 549 00:40:53,430 --> 00:40:57,590 so let's do it-- X squared cosine squared 550 00:40:57,590 --> 00:41:04,550 omega t plus p hat squared over m 551 00:41:04,550 --> 00:41:12,810 squared w squared sine squared omega t plus 1 552 00:41:12,810 --> 00:41:28,240 over mw cosine omega t sine omega t X p plus pX X0 tilde. 553 00:41:32,290 --> 00:41:36,500 That shows that term, and I just squared that thing, 554 00:41:36,500 --> 00:41:41,260 but that I suggested a few exercises here. 555 00:41:41,260 --> 00:41:44,340 This is 0. 556 00:41:44,340 --> 00:41:48,770 In fact, it's 0 in the ground state as well, so this is 0. 557 00:41:51,850 --> 00:41:57,410 X squared gives you the top equation-- 558 00:41:57,410 --> 00:42:05,420 X0 squared plus h bar over 2 m omega cosine 559 00:42:05,420 --> 00:42:21,040 squared omega t-- plus p squared over m squared w squared, 560 00:42:21,040 --> 00:42:25,730 so p squared is m h omega over 2. 561 00:42:25,730 --> 00:42:28,590 And then you have m squared omega 562 00:42:28,590 --> 00:42:31,860 squared sine squared omega t. 563 00:42:35,590 --> 00:42:38,470 And that's this whole term. 564 00:42:38,470 --> 00:42:41,020 And the thing that we're supposed to do 565 00:42:41,020 --> 00:42:45,370 is subtract this here. 566 00:42:45,370 --> 00:42:51,210 You see that the X0 squared cosine squared of omega t 567 00:42:51,210 --> 00:42:52,130 cancels here. 568 00:42:56,790 --> 00:42:58,810 So what do we get? 569 00:42:58,810 --> 00:43:03,680 h bar over 2 mw cosine squared omega t. 570 00:43:03,680 --> 00:43:13,980 But this thing is also h bar over 2 mw sine squared omega t. 571 00:43:13,980 --> 00:43:16,890 So this whole thing, all the times 572 00:43:16,890 --> 00:43:20,840 have disappeared-- delta X squared-- 573 00:43:20,840 --> 00:43:24,410 the time dependence here has disappeared with that, 574 00:43:24,410 --> 00:43:28,090 and the cosine squared with sine squared have combined, 575 00:43:28,090 --> 00:43:33,050 and you get h bar over 2 m omega, which 576 00:43:33,050 --> 00:43:38,500 was-- this is, I'm sorry, of t. 577 00:43:38,500 --> 00:43:41,000 We work very hard to put the t there. 578 00:43:41,000 --> 00:43:42,970 We should leave it. 579 00:43:42,970 --> 00:43:48,080 The uncertainty as a function of time has not changed. 580 00:43:48,080 --> 00:43:52,160 It is the original uncertainty of the ground state. 581 00:43:52,160 --> 00:43:55,030 So this is moving in a nice way. 582 00:43:55,030 --> 00:44:00,790 You're supposed to compute now as well the uncertainty in p. 583 00:44:00,790 --> 00:44:04,050 I leave that as an exercise-- delta 584 00:44:04,050 --> 00:44:14,180 p squared of t equal m h bar omega over 2. 585 00:44:14,180 --> 00:44:15,290 So this is an exercise. 586 00:44:19,340 --> 00:44:22,560 Practice with coherent states. 587 00:44:22,560 --> 00:44:24,520 It's worth doing it, I think. 588 00:44:24,520 --> 00:44:27,170 Actually there's going to be a problem in the homework 589 00:44:27,170 --> 00:44:31,570 set, in which you're going to ask to do most of these things, 590 00:44:31,570 --> 00:44:33,970 including things I'm doing here on the blackboard. 591 00:44:33,970 --> 00:44:37,350 So you will practice this. 592 00:44:37,350 --> 00:44:46,230 So between these two, delta p, delta X-- delta X of t, 593 00:44:46,230 --> 00:44:52,760 delta p of t is, in fact, equal to h bar over 2. 594 00:44:52,760 --> 00:44:57,190 And this is a minimum uncertainty thing. 595 00:44:57,190 --> 00:44:58,750 And it behaves quite nicely. 596 00:45:02,320 --> 00:45:05,950 All right, so the name coherent now should make sense. 597 00:45:05,950 --> 00:45:09,480 You've produced a quantum state that 598 00:45:09,480 --> 00:45:13,790 has about the energy of a state that you're familiar with, 599 00:45:13,790 --> 00:45:18,310 and it moves classically, and it doesn't change shape 600 00:45:18,310 --> 00:45:20,575 as it moves, so it moves coherent. 601 00:45:23,520 --> 00:45:26,870 So our next task, therefore, will 602 00:45:26,870 --> 00:45:30,390 be to understand this in the energy basis. 603 00:45:30,390 --> 00:45:34,990 Because in the energy basis, it looks like a miracle. 604 00:45:34,990 --> 00:45:36,980 You've suddenly managed to produce 605 00:45:36,980 --> 00:45:41,020 a set of states of different energies, 606 00:45:41,020 --> 00:45:44,150 created the superposition, and suddenly, it 607 00:45:44,150 --> 00:45:45,620 moves in a nice way. 608 00:45:45,620 --> 00:45:48,450 Why does that happen? 609 00:45:48,450 --> 00:45:51,550 So we need to understand the energy basis. 610 00:45:51,550 --> 00:45:53,560 And as we do that, we'll understand 611 00:45:53,560 --> 00:45:58,090 how to generalize the coherent states completely. 612 00:45:58,090 --> 00:46:02,690 So let's go on with that, and let's 613 00:46:02,690 --> 00:46:04,860 explore this in the energy basis. 614 00:46:22,690 --> 00:46:23,985 So what do we have? 615 00:46:28,140 --> 00:46:33,940 We have the coherent state-- no need to put the time yet-- 616 00:46:33,940 --> 00:46:42,000 is e to the exponential of minus i p hat X0 over h bar. 617 00:46:46,370 --> 00:46:48,940 There is, as you've seen already, 618 00:46:48,940 --> 00:46:53,040 at length scale in the harmonic oscillator-- famous length 619 00:46:53,040 --> 00:46:58,580 scale, and we'll have an abbreviation for it. 620 00:46:58,580 --> 00:47:06,060 It's the length scale d0 squared h bar over m omega. 621 00:47:06,060 --> 00:47:11,850 You can use the parameters of the harmonic oscillator-- h bar 622 00:47:11,850 --> 00:47:14,820 and m and omega-- to produce a length scale. 623 00:47:14,820 --> 00:47:18,110 And that length scale is d0. 624 00:47:18,110 --> 00:47:24,930 It's essentially the uncertainty in the position in the ground 625 00:47:24,930 --> 00:47:28,770 state, up to the square root of 2. 626 00:47:28,770 --> 00:47:32,930 It's the way-- you want to construct a length-- there 627 00:47:32,930 --> 00:47:36,060 it is-- the only way you can construct a length. 628 00:47:36,060 --> 00:47:38,480 So I'm going to use that notation. 629 00:47:38,480 --> 00:47:49,330 So let me put what the p is into that formula, and simplify it. 630 00:47:49,330 --> 00:47:53,830 So this is on the vacuum-- I'm sorry, I stopped half the way. 631 00:47:53,830 --> 00:48:02,590 So this is the exponential of X0 over square root of 2d a dagger 632 00:48:02,590 --> 00:48:07,325 minus a on the vacuum. 633 00:48:13,590 --> 00:48:18,800 Plug in the p, get the h bars, and you 634 00:48:18,800 --> 00:48:22,710 will see that d enters in that way. 635 00:48:22,710 --> 00:48:27,740 It's the way it has to enter, because this exponential should 636 00:48:27,740 --> 00:48:32,850 have no units, and therefore X0 over d0 has no units, 637 00:48:32,850 --> 00:48:36,820 and the a's and the a daggers have no units. 638 00:48:36,820 --> 00:48:40,640 So it couldn't be any way different like that. 639 00:48:40,640 --> 00:48:45,960 The i also shouldn't be there, because this operator-- 640 00:48:45,960 --> 00:48:49,550 the i was there to make this anti-Hermitian. 641 00:48:49,550 --> 00:48:55,942 But this, with this real, is already anti-Hermitian. 642 00:48:55,942 --> 00:48:57,150 You see, you take the dagger. 643 00:48:57,150 --> 00:48:59,450 It becomes minus itself. 644 00:48:59,450 --> 00:49:01,250 So this is anti-Hermitian. 645 00:49:01,250 --> 00:49:05,280 No need for an i-- in fact, an i would be wrong, 646 00:49:05,280 --> 00:49:07,660 so there's no i. 647 00:49:07,660 --> 00:49:08,630 And that's this. 648 00:49:11,940 --> 00:49:17,780 Now, we want to figure out how this looks in the energy basis, 649 00:49:17,780 --> 00:49:23,630 so what are we going to do? 650 00:49:23,630 --> 00:49:25,370 We're going to have to do something 651 00:49:25,370 --> 00:49:27,610 with that exponential. 652 00:49:27,610 --> 00:49:29,710 We're going to have to reorder it. 653 00:49:29,710 --> 00:49:31,520 This is a job for Baker-Campbell-Hausdorff. 654 00:49:37,570 --> 00:49:38,660 Which one? 655 00:49:38,660 --> 00:49:45,600 Well, this one-- e to the X plus Y is equal to e to X, 656 00:49:45,600 --> 00:49:50,670 e to the Y, e to the minus 1/2-- I 657 00:49:50,670 --> 00:49:59,510 don't know this by heart-- XY, and it stops there. 658 00:49:59,510 --> 00:50:06,460 If and only if X commutator with Y commutes with X, 659 00:50:06,460 --> 00:50:09,100 and commutes with Y. 660 00:50:09,100 --> 00:50:13,700 There was a problem in the test that there 661 00:50:13,700 --> 00:50:17,690 was an operator with Xp plus pX acting on X, 662 00:50:17,690 --> 00:50:22,630 and after you commuted, you get X again, 663 00:50:22,630 --> 00:50:25,830 and you have to keep including terms. 664 00:50:25,830 --> 00:50:40,410 So this stops here if XY commutes with X and Y. 665 00:50:40,410 --> 00:50:42,330 And why do I want this? 666 00:50:42,330 --> 00:50:46,990 Because I actually want to split the creation 667 00:50:46,990 --> 00:50:49,150 and the manipulation operators. 668 00:50:49,150 --> 00:50:53,630 I want them in separate exponentials. 669 00:50:53,630 --> 00:50:59,650 We have that energy eigenstates are creation operators 670 00:50:59,650 --> 00:51:05,080 on the vacuum, but here I have creation minus destruction. 671 00:51:05,080 --> 00:51:07,500 So if I expand the exponential, I'm 672 00:51:07,500 --> 00:51:10,030 going to get lots of creation and destruction, 673 00:51:10,030 --> 00:51:14,510 and I'm going to spend hours trying to sort it out. 674 00:51:14,510 --> 00:51:19,460 If you did expand it, I bet you won't see it through 675 00:51:19,460 --> 00:51:22,920 so easily-- probably will take you forever, 676 00:51:22,920 --> 00:51:26,440 and it might not work out. 677 00:51:26,440 --> 00:51:29,520 So expanding an exponential is something 678 00:51:29,520 --> 00:51:33,480 that we should be reluctant to do. 679 00:51:33,480 --> 00:51:36,500 On the other hand, this is a nice option, 680 00:51:36,500 --> 00:51:38,810 because then you think of this as e 681 00:51:38,810 --> 00:51:47,310 to the X0 over square root of 2d0 a dagger minus X0 682 00:51:47,310 --> 00:51:49,525 over square root of 2d0 a. 683 00:51:55,520 --> 00:52:03,820 And I chose this analogy X with this, and Y with this. 684 00:52:03,820 --> 00:52:08,300 It's like Y is this thing minus that, and X is that. 685 00:52:08,300 --> 00:52:10,360 You could have done it the other way around, 686 00:52:10,360 --> 00:52:13,510 but then you would run into trouble again. 687 00:52:13,510 --> 00:52:14,440 Why? 688 00:52:14,440 --> 00:52:21,040 Because I want a Y factor that has the manipulators 689 00:52:21,040 --> 00:52:26,480 to be to the right of the X factor that has the creation. 690 00:52:26,480 --> 00:52:27,240 Why? 691 00:52:27,240 --> 00:52:33,740 Because if I have annihilators closer to the vacuum, 692 00:52:33,740 --> 00:52:36,100 that's good. 693 00:52:36,100 --> 00:52:40,110 Annihilators closer to the vacuum is what you really want, 694 00:52:40,110 --> 00:52:42,840 because if you have creators close to the vacuum, 695 00:52:42,840 --> 00:52:45,290 they create state, but then you have the manipulators, 696 00:52:45,290 --> 00:52:47,640 and you have to start working them out. 697 00:52:47,640 --> 00:52:49,590 On the other hand, if the annihilators 698 00:52:49,590 --> 00:52:52,880 are close to the vacuum, they just kill the vacuum 699 00:52:52,880 --> 00:52:55,020 and you can forget them. 700 00:52:55,020 --> 00:53:00,130 So it's very important that you identify X with this, 701 00:53:00,130 --> 00:53:03,840 and Y with this whole thing. 702 00:53:07,170 --> 00:53:18,830 So that this is e to the X0 over square root of 2D0 a dagger, e 703 00:53:18,830 --> 00:53:26,470 to the minus X0 over square root of 2d0 a. 704 00:53:26,470 --> 00:53:29,860 And now you're supposed to do the commutator of these two 705 00:53:29,860 --> 00:53:30,600 things. 706 00:53:30,600 --> 00:53:32,560 And the commutator is the commutator 707 00:53:32,560 --> 00:53:39,140 of an a dagger with an a, and that is 1. 708 00:53:39,140 --> 00:53:44,860 So this commutator is a number-- crucial, 709 00:53:44,860 --> 00:53:46,850 because if it wasn't a number, if it 710 00:53:46,850 --> 00:53:49,680 would be an a or an a dagger, it would not 711 00:53:49,680 --> 00:53:54,320 commute with a, X, and Y, and you have to include more terms. 712 00:53:54,320 --> 00:53:57,190 So the fact that this commutator is a number 713 00:53:57,190 --> 00:53:59,680 allows you to use this formula. 714 00:53:59,680 --> 00:54:05,960 So now we'll put this factor here, minus 1/2. 715 00:54:05,960 --> 00:54:08,370 Then you have X with Y, and that's 716 00:54:08,370 --> 00:54:17,740 X0 minus X0 over square root of 2d squared, 717 00:54:17,740 --> 00:54:20,610 minus-- and this factor squared-- an a dagger 718 00:54:20,610 --> 00:54:22,620 with a, which is minus 1. 719 00:54:25,340 --> 00:54:26,860 So that's that whole operator. 720 00:54:31,800 --> 00:54:36,860 So let's write it. 721 00:54:36,860 --> 00:54:40,550 The coherent state, therefore, X0 tilde, 722 00:54:40,550 --> 00:54:47,610 is equal to e to the X0 over square root of 2d0 a hat 723 00:54:47,610 --> 00:54:55,390 dagger, e to the minus X0 over square root of 2d0 a, 724 00:54:55,390 --> 00:55:04,130 and this factor that seems to be e to the minus 1/4 X0 725 00:55:04,130 --> 00:55:07,420 squared over d0 squared. 726 00:55:07,420 --> 00:55:12,340 And here is this nice vacuum. 727 00:55:12,340 --> 00:55:14,770 Yes, factor is right. 728 00:55:14,770 --> 00:55:17,680 So what is this? 729 00:55:17,680 --> 00:55:22,540 Well, this is a number, so I can pull it to the left. 730 00:55:22,540 --> 00:55:27,280 And here is the exponential of the annihilator operator. 731 00:55:27,280 --> 00:55:29,480 Now expand the exponential. 732 00:55:29,480 --> 00:55:30,680 It's 1. 733 00:55:30,680 --> 00:55:34,900 That survives, but the first term has an a-- kills it. 734 00:55:34,900 --> 00:55:38,120 The second term has an a squared-- kills it. 735 00:55:38,120 --> 00:55:40,260 Everything kills it. 736 00:55:40,260 --> 00:55:44,640 This thing acting on the vacuum is just 1. 737 00:55:44,640 --> 00:55:47,380 That's why this is simple. 738 00:55:47,380 --> 00:55:48,920 So what have we got? 739 00:55:48,920 --> 00:55:56,290 In the state X0 tilde is e to the minus 1/4 X0 squared 740 00:55:56,290 --> 00:56:04,660 over d0 squared times e to the X0 over square root of 2d0 741 00:56:04,660 --> 00:56:07,115 a dagger on the vacuum. 742 00:56:17,640 --> 00:56:22,700 Well this is nice-- not quite energy eigenstates, 743 00:56:22,700 --> 00:56:25,480 but we're almost there. 744 00:56:25,480 --> 00:56:27,050 What is this? 745 00:56:27,050 --> 00:56:33,780 e to the minus 1/4, X0 squared over d0 squared. 746 00:56:33,780 --> 00:56:35,210 And now expand. 747 00:56:35,210 --> 00:56:39,850 This is the sum from n equals 1 to infinity, 748 00:56:39,850 --> 00:56:43,600 1 over n factorial. 749 00:56:43,600 --> 00:56:47,948 X0 over square root of 2d0 to the n, 750 00:56:47,948 --> 00:56:52,530 a hat dagger to the n on the vacuum. 751 00:56:58,220 --> 00:57:02,680 And what was the nth energy eigenstate? 752 00:57:02,680 --> 00:57:04,000 You probably remember. 753 00:57:04,000 --> 00:57:07,130 The nth energy eigenstate is a dagger 754 00:57:07,130 --> 00:57:12,370 to the n on the vacuum over square root of n factorial. 755 00:57:12,370 --> 00:57:16,320 So we've got a little more than the square root 756 00:57:16,320 --> 00:57:17,560 of 2 n factorial. 757 00:57:17,560 --> 00:57:22,480 So maybe I'll do it here. 758 00:57:22,480 --> 00:57:31,450 We get e to the minus 1/4 X0 squared over d0 squared, 759 00:57:31,450 --> 00:57:42,300 sum from n equals 1 to infinity, 1 760 00:57:42,300 --> 00:57:51,410 over square root of n factorial, X0 over the square root of 2 d0 761 00:57:51,410 --> 00:57:55,610 to the n times the nth energy eigenstate. 762 00:57:58,730 --> 00:58:07,180 It's a little messy, but not so bad. 763 00:58:07,180 --> 00:58:10,080 I think actually I won't need that anymore. 764 00:58:10,080 --> 00:58:11,445 Well no, I may. 765 00:58:11,445 --> 00:58:11,945 I will. 766 00:58:14,670 --> 00:58:22,650 So let's write it maybe again. 767 00:58:22,650 --> 00:58:25,630 Well, it's OK. 768 00:58:25,630 --> 00:58:30,540 Let's write it as follows-- Cn n. 769 00:58:37,144 --> 00:58:42,950 OK, so I got some cn's and n's. 770 00:58:42,950 --> 00:58:46,600 So this is a very precise superposition 771 00:58:46,600 --> 00:58:51,040 of energy eigenstates, very delicate superposition 772 00:58:51,040 --> 00:58:54,290 of energy eigenstates. 773 00:58:54,290 --> 00:59:00,610 Let me write it in the following way-- cn squared. 774 00:59:03,440 --> 00:59:07,200 Why would I care about cn squared? 775 00:59:07,200 --> 00:59:13,940 cn squared is the probability to find the coherent state 776 00:59:13,940 --> 00:59:17,020 in the nth energy eigenstate. 777 00:59:17,020 --> 00:59:21,540 The amplitude to have it in the nth energy eigenstate is cn. 778 00:59:21,540 --> 00:59:26,020 So that probability to find it in the nth energy eigenstate 779 00:59:26,020 --> 00:59:34,080 is cn squared, is the probability 4x tilde 0 780 00:59:34,080 --> 00:59:39,680 2b in the nth energy eigenstate. 781 00:59:39,680 --> 00:59:40,800 That is what? 782 00:59:40,800 --> 00:59:50,220 Exponential of minus 1/2 x0 squared over d0 squared. 783 00:59:50,220 --> 00:59:51,770 And I have to square that. 784 00:59:51,770 --> 00:59:56,322 So I have 1 over n factorial. 785 00:59:56,322 --> 00:59:58,980 I have to square that coefficient there. 786 00:59:58,980 --> 01:00:03,970 So it's x0 squared over 2d0 squared-- that's 787 01:00:03,970 --> 01:00:06,700 nice, it's the same one here-- to the n. 788 01:00:12,670 --> 01:00:17,210 So it's easier to think of this if you 789 01:00:17,210 --> 01:00:28,410 invent a new letter, lambda, to be x0 squared over 2d0 squared. 790 01:00:30,940 --> 01:00:39,850 Then, cn squared is equal to e to the minus lambda lambda 791 01:00:39,850 --> 01:00:42,170 to the n over n factorial. 792 01:01:06,400 --> 01:01:07,680 Yes. 793 01:01:07,680 --> 01:01:10,039 AUDIENCE: Is that something that we 794 01:01:10,039 --> 01:01:13,532 should expect to be true for any time of [INAUDIBLE], 795 01:01:13,532 --> 01:01:15,528 or is that something that we [INAUDIBLE]? 796 01:01:18,720 --> 01:01:24,120 PROFESSOR: Well, let me say it this way. 797 01:01:24,120 --> 01:01:26,400 In a second, it will become clear 798 01:01:26,400 --> 01:01:29,280 that this was almost necessary. 799 01:01:29,280 --> 01:01:35,050 I actually don't know very deeply why this is true. 800 01:01:35,050 --> 01:01:38,570 And I'm always a little puzzled and uncomfortable at this point 801 01:01:38,570 --> 01:01:40,840 in 805. 802 01:01:40,840 --> 01:01:43,925 So what is really strange about this 803 01:01:43,925 --> 01:01:47,245 is that this is the so-called Poisson distribution. 804 01:01:49,990 --> 01:01:53,290 So there's something about this energy eigenstate 805 01:01:53,290 --> 01:01:58,270 that their Poisson distributed in a coherent state. 806 01:01:58,270 --> 01:02:04,150 So these are probabilities, as I claimed, to find an n. 807 01:02:04,150 --> 01:02:09,750 And indeed, let's check the sum of the cn squareds from n 808 01:02:09,750 --> 01:02:12,410 equals 1 to infinity. 809 01:02:12,410 --> 01:02:13,500 Let's see what it is. 810 01:02:13,500 --> 01:02:17,734 And you will see, you cannot tinker with this. 811 01:02:17,734 --> 01:02:21,530 This is e to the minus lambda the sum from n 812 01:02:21,530 --> 01:02:26,850 equals 1 to infinity lambda to the n over n factorial. 813 01:02:26,850 --> 01:02:29,810 And that sum-- it's not from n equals 1. 814 01:02:29,810 --> 01:02:32,370 It's 0 to infinity, I'm sorry. 815 01:02:32,370 --> 01:02:34,750 Did I write once anywhere? 816 01:02:34,750 --> 01:02:37,940 Yeah, it should be 0. 817 01:02:37,940 --> 01:02:39,425 OK, this is 0. 818 01:02:42,250 --> 01:02:49,290 There is the ground state, so from n equals 0 to infinity. 819 01:02:49,290 --> 01:02:52,090 And this is e to the minus lambda e 820 01:02:52,090 --> 01:02:54,030 to the lambda, which is 1. 821 01:02:58,140 --> 01:03:01,390 So yes, this is Poisson distributed. 822 01:03:04,170 --> 01:03:07,930 It's some sort of distribution like that. 823 01:03:07,930 --> 01:03:18,470 So if you have the n's, the cn's, Poisson distributions 824 01:03:18,470 --> 01:03:22,485 have to do with if you have a radioactive material. 825 01:03:22,485 --> 01:03:24,030 It has a lifetime. 826 01:03:24,030 --> 01:03:26,870 And you say, how many events should I-- 827 01:03:26,870 --> 01:03:28,690 the lifetime is five years. 828 01:03:28,690 --> 01:03:32,850 How many events should you expect to happen in a week? 829 01:03:32,850 --> 01:03:34,130 These are Poisson distributed. 830 01:03:37,100 --> 01:03:40,740 So it's a Poisson distribution. 831 01:03:40,740 --> 01:03:43,930 It's a very nice thing. 832 01:03:43,930 --> 01:03:49,160 So let me just make one more remark about it. 833 01:03:49,160 --> 01:03:56,530 And it's quite something. 834 01:04:03,040 --> 01:04:08,380 So one question that you could ask 835 01:04:08,380 --> 01:04:13,130 is, what is the most probable n? 836 01:04:13,130 --> 01:04:15,030 That's a good question. 837 01:04:15,030 --> 01:04:17,026 You have a coherent state. 838 01:04:17,026 --> 01:04:18,650 So it's going to have the superposition 839 01:04:18,650 --> 01:04:20,570 of the vacuum, the first. 840 01:04:20,570 --> 01:04:27,300 What is the most probable n, so the expectation value of n? 841 01:04:27,300 --> 01:04:30,960 Now, I'm thinking of it probabilistically. 842 01:04:30,960 --> 01:04:33,916 So I'm thinking this is a probability distribution. 843 01:04:36,860 --> 01:04:40,020 Then, I will show it for you that this is really 844 01:04:40,020 --> 01:04:41,390 computing what you want. 845 01:04:41,390 --> 01:04:46,270 But probabilistically, what is the expectation value of n? 846 01:04:46,270 --> 01:04:49,995 You should sum n times the probability that you get n. 847 01:04:53,590 --> 01:05:02,160 So this is sum over ne to the minus lambda lambda 848 01:05:02,160 --> 01:05:04,570 to the n over n factorial. 849 01:05:04,570 --> 01:05:07,152 So you got an n there. 850 01:05:07,152 --> 01:05:11,410 And the way you get an n there-- well, the e to the minus lambda 851 01:05:11,410 --> 01:05:14,100 goes out. 852 01:05:14,100 --> 01:05:18,230 And the n can be reproduced by doing lambda d d 853 01:05:18,230 --> 01:05:20,670 lambda on the sum. 854 01:05:23,590 --> 01:05:27,630 Because lambda d d lambda on this sum brings down this n, 855 01:05:27,630 --> 01:05:31,380 puts back the lambda so it gives you the thing you had, 856 01:05:31,380 --> 01:05:32,700 and that's what it is. 857 01:05:32,700 --> 01:05:38,485 So here, you get e to the minus lambda lambda 858 01:05:38,485 --> 01:05:44,280 d d lambda of e to the lambda. 859 01:05:44,280 --> 01:05:47,730 And that is lambda, OK? 860 01:05:51,220 --> 01:05:59,765 So the expectation value of the most sort of not the peak, 861 01:05:59,765 --> 01:06:05,590 but the expected value of n in this distribution, the level 862 01:06:05,590 --> 01:06:08,980 that you're going to be excited is basically lambda. 863 01:06:08,980 --> 01:06:16,140 So if x0 is 1,000 times bigger than d0, 864 01:06:16,140 --> 01:06:22,290 you've moved this thing 1,000 times the quantum uncertainty. 865 01:06:22,290 --> 01:06:26,010 Then, you're occupying most strongly the levels 866 01:06:26,010 --> 01:06:29,160 at 1 million. 867 01:06:29,160 --> 01:06:37,590 You get x0 over d0 controls which n is the most likely. 868 01:06:37,590 --> 01:06:41,140 Indeed, look, this n-- suppose you 869 01:06:41,140 --> 01:06:48,375 would compute x0 tilde n hat x0 tilde. 870 01:06:53,120 --> 01:06:56,290 This is what you would think is an occupation number. 871 01:06:56,290 --> 01:06:59,110 This sounds a little hand wavy. 872 01:06:59,110 --> 01:07:02,890 But this is the number operator, the expected value 873 01:07:02,890 --> 01:07:06,030 of the number operator, in the coherent state. 874 01:07:06,030 --> 01:07:12,410 But this is-- you have that the coherent state is this. 875 01:07:12,410 --> 01:07:15,640 So let's substitute that in there. 876 01:07:15,640 --> 01:07:19,860 So you get two sums over n and over m. 877 01:07:19,860 --> 01:07:32,500 And you would have cm star m N n cn. 878 01:07:32,500 --> 01:07:37,030 I've substituted x0 and x0 dagger. 879 01:07:37,030 --> 01:07:40,140 The c's in fact are real. 880 01:07:40,140 --> 01:07:44,240 And then, the number operator on here is little n. 881 01:07:44,240 --> 01:07:46,380 And then, you get the Kronecker delta. 882 01:07:46,380 --> 01:07:53,690 So this is sum over n and m cmcn-- it's real. 883 01:07:53,690 --> 01:07:57,880 And then, you get n delta m, n. 884 01:07:57,880 --> 01:08:03,800 So this is in fact the sum over ncn squared. 885 01:08:03,800 --> 01:08:06,780 So what we wrote here, this is really 886 01:08:06,780 --> 01:08:09,195 the expectation value of the number operator. 887 01:08:12,260 --> 01:08:17,040 And one can do more calculations here. 888 01:08:17,040 --> 01:08:20,229 A calculation that is particularly interesting 889 01:08:20,229 --> 01:08:24,109 to discover what these states look like 890 01:08:24,109 --> 01:08:30,069 is the uncertainty in the energy. 891 01:08:30,069 --> 01:08:36,060 So that's another sort of relevant measure. 892 01:08:36,060 --> 01:08:39,180 How big is the uncertainty in the energy? 893 01:08:41,800 --> 01:08:48,200 What are, basically, the delta E associated 894 01:08:48,200 --> 01:08:49,510 to the coherent state? 895 01:08:49,510 --> 01:08:52,720 How does it look like? 896 01:08:52,720 --> 01:08:53,795 Is it very sharp? 897 01:08:56,460 --> 01:08:59,490 So it's a good question. 898 01:08:59,490 --> 01:09:02,220 And it's in the notes. 899 01:09:02,220 --> 01:09:05,399 I leave it for you to try to calculate it. 900 01:09:05,399 --> 01:09:14,080 Delta E in the coherent state x0, how much is it? 901 01:09:14,080 --> 01:09:18,029 And it turns out to be the following-- h omega 902 01:09:18,029 --> 01:09:20,384 x0 over square root of 2d. 903 01:09:24,689 --> 01:09:28,160 So actually, maybe this is a little surprising. 904 01:09:28,160 --> 01:09:37,529 But delta E over h omega is equal to x0 over d. 905 01:09:40,040 --> 01:09:46,689 So actually, the energy uncertainty 906 01:09:46,689 --> 01:09:50,180 for a classical look in coherent state-- I'm sorry, 907 01:09:50,180 --> 01:09:52,189 I'm missing a square root of 2 there. 908 01:09:56,480 --> 01:10:00,020 So what is a classical looking coherent state? 909 01:10:00,020 --> 01:10:05,580 It's a state in which x0 is much bigger than the quantum d. 910 01:10:05,580 --> 01:10:09,450 So x0 is much bigger than d. 911 01:10:09,450 --> 01:10:17,530 So in that case, this is a large number for a classical state-- 912 01:10:17,530 --> 01:10:21,480 "classical" state. 913 01:10:21,480 --> 01:10:26,480 But in that case, the uncertainty in delta E 914 01:10:26,480 --> 01:10:31,280 is really big compared to the spacing 915 01:10:31,280 --> 01:10:33,540 of the harmonic oscillator. 916 01:10:33,540 --> 01:10:37,150 So you have-- here is the ground state. 917 01:10:37,150 --> 01:10:39,270 Here is h omega. 918 01:10:39,270 --> 01:10:43,040 Here is the coherent state, maybe. 919 01:10:43,040 --> 01:10:48,740 And you have a lot of energy levels that are excited. 920 01:10:48,740 --> 01:10:52,730 So if x0 over this is 1,000, well, 921 01:10:52,730 --> 01:10:57,740 at least 1,000 energy levels are excited. 922 01:10:57,740 --> 01:11:01,810 But you shouldn't fear that too much. 923 01:11:01,810 --> 01:11:07,960 Because at the same time, the expectation value of E 924 01:11:07,960 --> 01:11:11,980 over delta E-- the expectation of E 925 01:11:11,980 --> 01:11:15,120 is something we calculated at the beginning of the lecture. 926 01:11:15,120 --> 01:11:17,520 You have the oscillator displaced. 927 01:11:17,520 --> 01:11:23,320 So this is roughly 1/2 m omega squared x0 squared. 928 01:11:23,320 --> 01:11:25,310 Throw away the ground state energy. 929 01:11:25,310 --> 01:11:27,570 That's supposed to be very little. 930 01:11:27,570 --> 01:11:33,920 Delta E is h bar omega x0 over square root of 2. 931 01:11:33,920 --> 01:11:36,265 And this is, again, the same ratio. 932 01:11:39,980 --> 01:11:46,620 So yes, this state is very funny. 933 01:11:46,620 --> 01:11:52,990 It contains an uncertainty that measured in harmonic oscillator 934 01:11:52,990 --> 01:11:56,080 levels contains many levels. 935 01:11:56,080 --> 01:12:01,280 But still, the uncertainty is much smaller 936 01:12:01,280 --> 01:12:05,490 by the same amount than the average energy. 937 01:12:05,490 --> 01:12:10,510 So this state is a state of some almost definite energy, 938 01:12:10,510 --> 01:12:12,830 the uncertainty being much smaller. 939 01:12:12,830 --> 01:12:15,020 But even though it's much smaller, 940 01:12:15,020 --> 01:12:19,500 it still contains a lot of levels of the oscillator. 941 01:12:22,490 --> 01:12:28,510 So that I think gives you a reasonable picture of this. 942 01:12:28,510 --> 01:12:32,010 So you're ready for a generalization. 943 01:12:32,010 --> 01:12:37,890 This is a time to generalize the coherent states 944 01:12:37,890 --> 01:12:40,220 and produce the set of coherent states 945 01:12:40,220 --> 01:12:45,073 that are most useful eventually, and most flexible. 946 01:12:48,050 --> 01:12:50,560 And we do them as follows. 947 01:12:50,560 --> 01:12:59,290 We basically are inspired by this formula 948 01:12:59,290 --> 01:13:01,020 to write the following operator. 949 01:13:05,500 --> 01:13:07,760 And here, we change notation. 950 01:13:07,760 --> 01:13:09,500 This x0 was here. 951 01:13:09,500 --> 01:13:11,420 But now we'll introduce what is called 952 01:13:11,420 --> 01:13:16,190 the alpha coherent state. 953 01:13:16,190 --> 01:13:19,860 Most general coherent state is going 954 01:13:19,860 --> 01:13:26,070 to be obtained by acted with a unitary operator on the vacuum. 955 01:13:26,070 --> 01:13:30,126 So far so good-- D of alpha unitary. 956 01:13:34,060 --> 01:13:39,070 But now generalize what you have there. 957 01:13:39,070 --> 01:13:42,630 Here, you put a minus a dagger minus a, 958 01:13:42,630 --> 01:13:46,030 because that was anti-Hermitian, and you put the real constant. 959 01:13:46,030 --> 01:13:52,580 Now this alpha will belong to the complex numbers. 960 01:13:52,580 --> 01:13:56,430 Quantum mechanics is all about complex numbers. 961 01:13:56,430 --> 01:14:00,570 You've got complex vector spaces, complex numbers. 962 01:14:00,570 --> 01:14:02,040 It's all over the place. 963 01:14:02,040 --> 01:14:04,660 So how do we do this? 964 01:14:04,660 --> 01:14:10,630 We do this exponential of alpha a dagger. 965 01:14:10,630 --> 01:14:13,360 And I want it to be anti-Hermitian. 966 01:14:13,360 --> 01:14:24,580 So I should put minus alpha star a on the vacuum. 967 01:14:24,580 --> 01:14:29,180 This thing for alpha equals-- this real number 968 01:14:29,180 --> 01:14:31,060 reduces to that. 969 01:14:31,060 --> 01:14:34,720 But now, with alpha complex, it's 970 01:14:34,720 --> 01:14:38,210 a little more complicated operator. 971 01:14:38,210 --> 01:14:39,460 And it's more general. 972 01:14:39,460 --> 01:14:42,770 But it's still unitary. 973 01:14:42,770 --> 01:14:44,840 And it preserves a norm. 974 01:14:44,840 --> 01:14:49,280 And its most of what you want from these states. 975 01:14:49,280 --> 01:14:52,660 So the first thing you do to figure out 976 01:14:52,660 --> 01:14:58,740 what this operator does is to calculate something 977 01:14:58,740 --> 01:15:03,410 that maybe you would not expect it to be simple. 978 01:15:03,410 --> 01:15:06,740 But it's worth doing. 979 01:15:06,740 --> 01:15:11,330 What is a acting on the alpha state? 980 01:15:14,560 --> 01:15:19,590 Well, I would have to do a acting 981 01:15:19,590 --> 01:15:26,060 on this exponential of alpha a dagger minus alpha 982 01:15:26,060 --> 01:15:29,480 star a on the vacuum. 983 01:15:34,470 --> 01:15:39,350 Now, a kills the vacuum. 984 01:15:39,350 --> 01:15:43,460 So maybe you're accustomed already to the next step. 985 01:15:43,460 --> 01:15:47,470 I can replace that product by a commutator. 986 01:15:47,470 --> 01:15:50,640 Because the other ordering is 0. 987 01:15:50,640 --> 01:15:54,050 So this is equal to the commutator. 988 01:15:54,050 --> 01:16:00,060 Because the other term when a is on the other side is 0 anyway. 989 01:16:00,060 --> 01:16:02,900 And now I have to compute the commutator 990 01:16:02,900 --> 01:16:04,535 of a with an exponential. 991 01:16:08,000 --> 01:16:11,160 Again, it's a little scary. 992 01:16:11,160 --> 01:16:16,586 But A with an exponential e to the B-- 993 01:16:16,586 --> 01:16:19,990 it's in the formula sheet, this Campbell-Baker-Hausdorff 994 01:16:19,990 --> 01:16:37,080 again-- is A, B e to the B if A, B commutes with B. 995 01:16:37,080 --> 01:16:47,310 Well, this is A. This is B. A with B 996 01:16:47,310 --> 01:16:53,920 is-- A with B, the exponent-- just alpha times 1 997 01:16:53,920 --> 01:16:54,670 because of this. 998 01:16:54,670 --> 01:16:55,590 So it's a number. 999 01:16:55,590 --> 01:16:57,770 So this is safe. 1000 01:16:57,770 --> 01:17:01,306 So you get alpha times the same exponential. 1001 01:17:04,270 --> 01:17:07,180 But the same exponential means the state 1002 01:17:07,180 --> 01:17:14,120 alpha-- a little quick, isn't it? 1003 01:17:14,120 --> 01:17:20,190 OK, A with B, this factor, was alpha. 1004 01:17:20,190 --> 01:17:26,160 And e to the B anyway on the vacuum is the state alpha. 1005 01:17:26,160 --> 01:17:28,910 So there you go. 1006 01:17:28,910 --> 01:17:30,665 You have achieved the impossible. 1007 01:17:33,510 --> 01:17:37,375 You've diagonalized a non-Hermitian operator. 1008 01:17:40,000 --> 01:17:45,080 This is not Hermitian, and you found its eigenvalues. 1009 01:17:45,080 --> 01:17:48,850 How could that happen? 1010 01:17:48,850 --> 01:17:52,340 Well, it can happen. 1011 01:17:52,340 --> 01:17:54,350 But then, all of the theorems that you 1012 01:17:54,350 --> 01:17:58,290 like about Hermitian operators don't hold. 1013 01:17:58,290 --> 01:17:59,810 So it's a fluke. 1014 01:17:59,810 --> 01:18:01,350 This can be done. 1015 01:18:01,350 --> 01:18:05,860 But then, states that correspond to different eigenvalues 1016 01:18:05,860 --> 01:18:08,770 will not be orthogonal, and they will not 1017 01:18:08,770 --> 01:18:11,680 form a complete set of states, and nothing 1018 01:18:11,680 --> 01:18:16,220 will be quite as you may think it would be. 1019 01:18:16,220 --> 01:18:19,450 But still, it's quite remarkable that this can be done. 1020 01:18:22,020 --> 01:18:26,610 So this characterizes the coherent state in a nice way. 1021 01:18:26,610 --> 01:18:29,600 They're eigenstates of the destruction operator. 1022 01:18:29,600 --> 01:18:34,310 And they're the most general exponentials 1023 01:18:34,310 --> 01:18:40,500 of creation and annihilation operators acting on the vacuum. 1024 01:18:40,500 --> 01:18:49,000 Now, we knew that when alpha is real, it has to do with x0. 1025 01:18:49,000 --> 01:18:53,520 So we've put a complex alpha. 1026 01:18:53,520 --> 01:18:55,440 What will it do? 1027 01:18:55,440 --> 01:18:59,650 A complex alpha, what it does is gives 1028 01:18:59,650 --> 01:19:03,570 the original coherent state some momentum. 1029 01:19:03,570 --> 01:19:06,760 Remember, the original state that we had was an x0. 1030 01:19:06,760 --> 01:19:08,490 And how did it move? 1031 01:19:08,490 --> 01:19:11,100 x0 cosine of omega t. 1032 01:19:11,100 --> 01:19:14,550 So at time equals 0, it had 0 momentum. 1033 01:19:14,550 --> 01:19:19,150 This creates a coherent state at x0, 1034 01:19:19,150 --> 01:19:21,980 and it gives it a momentum controlled 1035 01:19:21,980 --> 01:19:25,170 by the imaginary part of this thing. 1036 01:19:25,170 --> 01:19:28,250 In fact, we can do this as follows. 1037 01:19:28,250 --> 01:19:30,460 You can ask, what is the expectation 1038 01:19:30,460 --> 01:19:33,960 value of x in this state? 1039 01:19:33,960 --> 01:19:37,660 Well, x is written here. 1040 01:19:37,660 --> 01:19:49,041 It's d over square root of 2 alpha a plus a dagger alpha. 1041 01:19:49,041 --> 01:19:53,160 And look, these are easy to compute. 1042 01:19:53,160 --> 01:19:57,110 a gives an alpha, gives you alpha. 1043 01:19:57,110 --> 01:20:00,380 a dagger and alpha, you don't know what it is. 1044 01:20:00,380 --> 01:20:05,620 But a dagger on [? brau ?] alpha is alpha star. 1045 01:20:05,620 --> 01:20:07,940 So this one you know on the right. 1046 01:20:07,940 --> 01:20:10,980 This one you know on the left. 1047 01:20:10,980 --> 01:20:16,070 It gives you the over square root of 2 alpha plus 1048 01:20:16,070 --> 01:20:17,880 alpha star. 1049 01:20:17,880 --> 01:20:23,670 So it's square root of 2d real of alpha. 1050 01:20:23,670 --> 01:20:28,535 So the real part of alpha is the expectation value of x. 1051 01:20:31,400 --> 01:20:33,475 So I'll go here. 1052 01:20:39,420 --> 01:20:43,840 I'm almost done-- waiting for the punch line. 1053 01:20:47,560 --> 01:20:52,740 Similarly, you can calculate the expectation value 1054 01:20:52,740 --> 01:20:55,111 of the momentum. 1055 01:20:55,111 --> 01:21:00,010 It will be alpha p alpha. 1056 01:21:00,010 --> 01:21:02,620 And p is a minus a dagger. 1057 01:21:02,620 --> 01:21:07,230 So you're going to get alpha star minus alpha, so 1058 01:21:07,230 --> 01:21:08,770 the imaginary part. 1059 01:21:08,770 --> 01:21:15,310 So p is actually square root of 2 h 1060 01:21:15,310 --> 01:21:20,570 bar over d imaginary part of alpha. 1061 01:21:20,570 --> 01:21:22,250 So the physics is clear. 1062 01:21:22,250 --> 01:21:24,920 Maybe the formulas are a little messy. 1063 01:21:24,920 --> 01:21:31,005 But when you have an alpha, the real part of alpha 1064 01:21:31,005 --> 01:21:34,590 is telling you where you're positioning the coherent state. 1065 01:21:34,590 --> 01:21:37,020 The imaginary part of alpha is telling 1066 01:21:37,020 --> 01:21:39,450 what kick are you giving to it. 1067 01:21:39,450 --> 01:21:43,690 And you now have produced the most general coherent state. 1068 01:21:43,690 --> 01:21:49,490 So how do we describe that geometrically? 1069 01:21:49,490 --> 01:21:51,655 We imagine it, the alpha plane. 1070 01:21:55,050 --> 01:21:57,970 And here it is. 1071 01:21:57,970 --> 01:22:02,810 The alpha plane, here is the vector, the complex number 1072 01:22:02,810 --> 01:22:06,530 alpha that you've chosen maybe for your state, 1073 01:22:06,530 --> 01:22:09,970 some particular complex value alpha. 1074 01:22:09,970 --> 01:22:13,100 On the x-axis, the real part of alpha 1075 01:22:13,100 --> 01:22:20,020 is the expectation value of x over square root of 2d. 1076 01:22:20,020 --> 01:22:22,970 The real part of alpha is the expectation value 1077 01:22:22,970 --> 01:22:25,620 of x over square root of 2d. 1078 01:22:25,620 --> 01:22:29,670 And the imaginary part of alpha is the expectation value 1079 01:22:29,670 --> 01:22:35,950 of p over square root of 2 h bar d. 1080 01:22:39,100 --> 01:22:43,120 And there it is, your state at time equals 0. 1081 01:22:45,730 --> 01:22:49,880 What is it going to do a little later? 1082 01:22:49,880 --> 01:22:52,080 Well, that will be the last thing 1083 01:22:52,080 --> 01:22:54,610 I want to calculate for you. 1084 01:22:54,610 --> 01:22:56,675 It's a nice answer. 1085 01:22:56,675 --> 01:22:58,820 And you should see it. 1086 01:22:58,820 --> 01:23:01,555 It's going to take me two minutes. 1087 01:23:04,290 --> 01:23:05,990 And what is it? 1088 01:23:05,990 --> 01:23:12,180 Well, alpha at time t, you have to evolve the state-- e 1089 01:23:12,180 --> 01:23:18,580 to the minus iHt over h bar on the state, which 1090 01:23:18,580 --> 01:23:27,970 is e to the alpha a dagger minus alpha star a e to the output 1091 01:23:27,970 --> 01:23:37,090 and into the iHt over h bar, and an e to the minus iHt over h 1092 01:23:37,090 --> 01:23:38,640 bar on the vacuum. 1093 01:23:38,640 --> 01:23:40,420 So I put the one here. 1094 01:23:40,420 --> 01:23:42,830 I evolve with this. 1095 01:23:42,830 --> 01:23:47,210 But I take the state and put this and that. 1096 01:23:47,210 --> 01:23:49,265 This is simple. 1097 01:23:49,265 --> 01:23:54,980 It's e to the minus iH omega over 2. 1098 01:23:54,980 --> 01:23:56,980 That's the energy of the ground state. 1099 01:23:59,500 --> 01:24:01,850 But what is this part? 1100 01:24:05,980 --> 01:24:11,080 It's pretty much the Heisenberg operator. 1101 01:24:11,080 --> 01:24:13,020 But the sign came out wrong. 1102 01:24:16,000 --> 01:24:18,530 Well, it didn't come out wrong. 1103 01:24:18,530 --> 01:24:21,250 It's what it is. 1104 01:24:21,250 --> 01:24:24,290 It just means that what I have to put here 1105 01:24:24,290 --> 01:24:29,000 is the Heisenberg operator at minus t. 1106 01:24:29,000 --> 01:24:31,250 Because I have t for minus t. 1107 01:24:31,250 --> 01:24:36,010 So this is e to the alpha a Heisenberg at 1108 01:24:36,010 --> 01:24:40,536 minus t dagger minus alpha star. 1109 01:24:43,040 --> 01:24:45,910 I'm sorry, I have too many parentheses here. 1110 01:24:48,690 --> 01:24:52,880 That's it, much better-- minus alpha 1111 01:24:52,880 --> 01:25:00,050 star a Heisenberg of minus t acting on this thing. 1112 01:25:00,050 --> 01:25:01,400 And what is this? 1113 01:25:01,400 --> 01:25:04,800 Well, we have the formula for the Heisenberg states here. 1114 01:25:04,800 --> 01:25:11,840 So you've got e to the alpha H a dagger of minus t 1115 01:25:11,840 --> 01:25:20,390 would be e to the minus I omega t a dagger. 1116 01:25:20,390 --> 01:25:23,950 And here, you have minus alpha star e 1117 01:25:23,950 --> 01:25:30,210 to the i omega t a on e to the minus 1118 01:25:30,210 --> 01:25:35,110 i omega h bar over 2 times the vacuum. 1119 01:25:35,110 --> 01:25:36,430 And look what has happened. 1120 01:25:39,130 --> 01:25:44,860 Alpha has become alpha times e to the minus i omega t. 1121 01:25:44,860 --> 01:25:47,200 Because the star is here. 1122 01:25:47,200 --> 01:25:50,200 It's minus the star one. 1123 01:25:50,200 --> 01:25:52,460 So the only thing that has changed 1124 01:25:52,460 --> 01:25:57,590 is that this state, alpha at time t, 1125 01:25:57,590 --> 01:26:03,590 is e to the minus ih bar omega over 2. 1126 01:26:03,590 --> 01:26:05,290 I'm sorry, I'm missing a t here. 1127 01:26:07,810 --> 01:26:09,160 AUDIENCE: [INAUDIBLE] 1128 01:26:09,160 --> 01:26:18,800 PROFESSOR: Yeah, I dropped-- yeah, minus i omega t over 2, 1129 01:26:18,800 --> 01:26:29,370 minus i omega t over 2, minus i omega t over 2, 1130 01:26:29,370 --> 01:26:35,880 times the coherent state, the time independent coherent state 1131 01:26:35,880 --> 01:26:40,090 of value e to the minus i omega t alpha. 1132 01:26:40,090 --> 01:26:41,745 That's a new complex number. 1133 01:26:45,390 --> 01:26:48,290 That's what has happened. 1134 01:26:48,290 --> 01:26:51,940 The number alpha has become e to the minus i omega t. 1135 01:26:51,940 --> 01:26:55,410 Now, this is a face for the whole state multiplicative. 1136 01:26:55,410 --> 01:26:56,900 It's irrelevant. 1137 01:26:56,900 --> 01:26:59,040 So what has this alpha done? 1138 01:26:59,040 --> 01:27:03,390 It has been rotated by e to the minus i omega t. 1139 01:27:03,390 --> 01:27:09,880 So this at the time t is the state alpha t. 1140 01:27:09,880 --> 01:27:11,220 Here is the state alpha. 1141 01:27:11,220 --> 01:27:15,500 And it has rotated by omega t. 1142 01:27:15,500 --> 01:27:18,650 So the coherent state can be visualized 1143 01:27:18,650 --> 01:27:22,870 as a complex number in this complex plane. 1144 01:27:22,870 --> 01:27:26,110 This real part is the expectation value of x at time 1145 01:27:26,110 --> 01:27:28,760 equals 0 whose imaginary part is the expectation 1146 01:27:28,760 --> 01:27:31,720 value of the momentum at time equals 0. 1147 01:27:31,720 --> 01:27:33,700 And how does it evolve? 1148 01:27:33,700 --> 01:27:37,480 This state just rotates with frequency omega 1149 01:27:37,480 --> 01:27:40,510 all along and forever. 1150 01:27:40,510 --> 01:27:43,120 All right, that's it for today. 1151 01:27:43,120 --> 01:27:44,670 See you on Wednesday. 1152 01:27:44,670 --> 01:27:46,470 [APPLAUSE] 1153 01:27:46,470 --> 01:27:48,431 PROFESSOR: Thank you, thank you.