1 00:00:00,060 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,800 under a Creative Commons license. 3 00:00:03,800 --> 00:00:06,530 Your support will help MIT OpenCourseWare continue 4 00:00:06,530 --> 00:00:10,130 to offer high quality educational resources for free. 5 00:00:10,130 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,116 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,116 --> 00:00:17,565 at ocw.mit.edu. 8 00:00:23,062 --> 00:00:26,099 PROFESSOR: Any questions from last lecture 9 00:00:26,099 --> 00:00:27,140 from before spring break? 10 00:00:32,369 --> 00:00:36,800 No questions, nothing? 11 00:00:36,800 --> 00:00:38,090 Nothing at all? 12 00:00:38,090 --> 00:00:39,610 We you all still on vacation? 13 00:00:39,610 --> 00:00:41,750 Thank you. 14 00:00:41,750 --> 00:00:42,556 Yeah. 15 00:00:42,556 --> 00:00:45,185 AUDIENCE: Could you talk about coherence states? 16 00:00:45,185 --> 00:00:45,890 PROFESSOR: Ah. 17 00:00:45,890 --> 00:00:47,690 OK, well, that's a great question. 18 00:00:47,690 --> 00:00:51,640 So coherence states-- actually, how many people 19 00:00:51,640 --> 00:00:54,420 have looked at the optional problems? 20 00:00:54,420 --> 00:00:55,170 Nice, OK, good. 21 00:00:58,550 --> 00:01:00,270 Good, so coherence states were a topic 22 00:01:00,270 --> 00:01:01,750 that we touched on the problem sets 23 00:01:01,750 --> 00:01:07,570 and on the optional problems, and the optional problems 24 00:01:07,570 --> 00:01:20,052 are mostly on the harmonic oscillator and nice problems 25 00:01:20,052 --> 00:01:22,510 revealing some of the structure of the harmonic oscillator, 26 00:01:22,510 --> 00:01:25,114 and it generalizes quite boldly. 27 00:01:25,114 --> 00:01:27,030 But here's the basic idea of coherence states. 28 00:01:27,030 --> 00:01:29,040 Let me just talk you through the basic ideas 29 00:01:29,040 --> 00:01:31,510 rather than do any calculations. 30 00:01:31,510 --> 00:01:34,987 So what's the ground state of a harmonic oscillator? 31 00:01:34,987 --> 00:01:36,570 What does its wave function look like? 32 00:01:36,570 --> 00:01:37,256 AUDIENCE: Gaussian. 33 00:01:37,256 --> 00:01:38,560 PROFESSOR: It's a Gaussian, exactly. 34 00:01:38,560 --> 00:01:40,101 It's minimum uncertainty wave packet. 35 00:01:40,101 --> 00:01:41,460 How does it evolve in time? 36 00:01:41,460 --> 00:01:42,460 AUDIENCE: Phase. 37 00:01:42,460 --> 00:01:43,520 PROFESSOR: Yeah, phase. 38 00:01:43,520 --> 00:01:44,960 It's an energy eigenstate, it's the ground state, 39 00:01:44,960 --> 00:01:46,760 so it just evolves in time with the phase. 40 00:01:46,760 --> 00:01:49,030 So if we look at the wave function for the ground 41 00:01:49,030 --> 00:01:51,850 state, phi naught, it's something like e to the minus x 42 00:01:51,850 --> 00:01:56,472 squared over 2a squared with some normalization coefficient, 43 00:01:56,472 --> 00:01:57,930 which I'm not going to worry about. 44 00:02:01,830 --> 00:02:05,074 So this is a minimal uncertainty wave packet. 45 00:02:05,074 --> 00:02:06,990 Its position distribution is time independent, 46 00:02:06,990 --> 00:02:07,980 because it's a stationary statement. 47 00:02:07,980 --> 00:02:10,188 It's momentum distribution, which is also a Gaussian, 48 00:02:10,188 --> 00:02:13,570 the Fourier transform, is time independent. 49 00:02:13,570 --> 00:02:16,420 And so this thing up to its rotation by the overall phase 50 00:02:16,420 --> 00:02:19,510 just sits there and remains a Gaussian. 51 00:02:19,510 --> 00:02:21,340 Now, here's a question. 52 00:02:21,340 --> 00:02:25,610 Suppose I take my harmonic oscillator potential, 53 00:02:25,610 --> 00:02:31,020 and I take my Gaussian, but I displace it a little bit. 54 00:02:31,020 --> 00:02:34,530 It's the same ground state, it's the same state 55 00:02:34,530 --> 00:02:37,730 but I've just displaced it over a little. 56 00:02:37,730 --> 00:02:40,370 What do you expect to happen? 57 00:02:40,370 --> 00:02:43,470 How will this state evolve in time? 58 00:02:43,470 --> 00:02:45,980 So, we know how to solve that problem. 59 00:02:45,980 --> 00:02:49,580 We take this state, it's a known wave function at time 0. 60 00:02:49,580 --> 00:02:53,040 We expand it in the basis of energy eigenstates, 61 00:02:53,040 --> 00:02:55,910 each energy eigenstate evolves in time with a phase, 62 00:02:55,910 --> 00:02:58,220 so we put in that phase and redo the sum, 63 00:02:58,220 --> 00:03:00,262 and recover the time evolution of the full state. 64 00:03:00,262 --> 00:03:02,636 And we've done this a number of time on the problem sets. 65 00:03:02,636 --> 00:03:03,162 Yeah. 66 00:03:03,162 --> 00:03:10,220 AUDIENCE: Would the Gaussian, the displaced Gaussian 67 00:03:10,220 --> 00:03:15,338 evolve the same way, keep its width, 68 00:03:15,338 --> 00:03:17,310 if it had any other initial width 69 00:03:17,310 --> 00:03:19,581 other than the one of the ground state? 70 00:03:19,581 --> 00:03:21,080 PROFESSOR: Let me come back to that, 71 00:03:21,080 --> 00:03:23,121 because it's a little more of a precise question. 72 00:03:25,460 --> 00:03:28,340 So we know how to solve this problem practically, 73 00:03:28,340 --> 00:03:30,560 algorithmically. 74 00:03:30,560 --> 00:03:31,666 But here's a nice fact. 75 00:03:31,666 --> 00:03:33,540 I'm not going to derive any equations, that's 76 00:03:33,540 --> 00:03:35,650 part of the point of the optional problems, 77 00:03:35,650 --> 00:03:37,560 but here's a nice fact about this state. 78 00:03:37,560 --> 00:03:39,601 So it's clearly a minimum uncertainty wave packet 79 00:03:39,601 --> 00:03:41,750 because at time 0, because it's just 80 00:03:41,750 --> 00:03:45,620 the same Gaussian just translated over a little bit. 81 00:03:45,620 --> 00:03:47,480 So what we'd expect, naively, from solving 82 00:03:47,480 --> 00:03:49,530 is we expand this in Fourier modes 83 00:03:49,530 --> 00:03:53,420 and then we let this the system evolve in time. 84 00:03:53,420 --> 00:03:55,740 We let each individual Fourier mode-- 85 00:03:55,740 --> 00:03:58,130 or, sorry, Fourier mode-- we let each individual energy 86 00:03:58,130 --> 00:04:01,080 eigenstate evolve in time, pick up a phase, and now what we get 87 00:04:01,080 --> 00:04:02,440 is a superposition. 88 00:04:02,440 --> 00:04:05,280 Instead of sum over n at time 0, we'd 89 00:04:05,280 --> 00:04:08,770 have sum over n cn phi n of x. 90 00:04:08,770 --> 00:04:13,422 Now, as a function of time, we get e minus I omega nt, 91 00:04:13,422 --> 00:04:15,630 and these phases are going to change the interference 92 00:04:15,630 --> 00:04:17,647 from summing up all these energy eigenstates, 93 00:04:17,647 --> 00:04:19,230 and so the system will change in time. 94 00:04:19,230 --> 00:04:22,019 Because the way that the various terms in the superposition 95 00:04:22,019 --> 00:04:23,625 interfere will change in time. 96 00:04:23,625 --> 00:04:25,750 So very naively, what you might expect, if you just 97 00:04:25,750 --> 00:04:27,190 took a random function-- for example, 98 00:04:27,190 --> 00:04:28,980 if I took a harmonic oscillator potential 99 00:04:28,980 --> 00:04:32,800 and I took some stupid function that did something like this. 100 00:04:32,800 --> 00:04:34,590 What would you expect it to do over time? 101 00:04:34,590 --> 00:04:37,020 Well, due to all the complicated interference effects, 102 00:04:37,020 --> 00:04:40,190 you'd expect this to turn into, basically, some schmutz. 103 00:04:40,190 --> 00:04:43,586 Just some crazy interference pattern. 104 00:04:43,586 --> 00:04:45,460 The thing that's nice about a coherent state, 105 00:04:45,460 --> 00:04:47,000 and this is where it gets its name, 106 00:04:47,000 --> 00:04:51,930 is the way that all of these interference effects 107 00:04:51,930 --> 00:04:56,950 conspire together to evolve the state is to leave it 108 00:04:56,950 --> 00:05:02,610 a Gaussian that does nothing but translates in time. 109 00:05:02,610 --> 00:05:05,600 A coherent state is coherent because it remains coherently 110 00:05:05,600 --> 00:05:07,800 a Gaussian as it moves along. 111 00:05:07,800 --> 00:05:09,610 It oscillates back and forth. 112 00:05:09,610 --> 00:05:14,200 And in fact, the peak, the center of this Gaussian wave 113 00:05:14,200 --> 00:05:17,040 packet, oscillates with precisely the frequency 114 00:05:17,040 --> 00:05:19,070 of the trap. 115 00:05:19,070 --> 00:05:22,930 It behaves just as a classical particle would have had you 116 00:05:22,930 --> 00:05:25,560 displaced it from the center of the harmonic oscillator trap, 117 00:05:25,560 --> 00:05:26,850 it oscillates back and forth. 118 00:05:30,507 --> 00:05:32,590 Now, on the other hand, you know that its momentum 119 00:05:32,590 --> 00:05:35,276 is changing in time, right. 120 00:05:35,276 --> 00:05:36,650 And at any given point, as you've 121 00:05:36,650 --> 00:05:39,680 shown on the problem set that's due tomorrow-- 122 00:05:39,680 --> 00:05:42,610 or you will have shown-- at any given moment in time, 123 00:05:42,610 --> 00:05:44,790 the momentum can be understood as the overall phase. 124 00:05:44,790 --> 00:05:46,630 You could just change that momentum by the overall phase, 125 00:05:46,630 --> 00:05:48,741 so it's the spatial rate of change of the phase, e 126 00:05:48,741 --> 00:05:49,960 to the ikx. 127 00:05:49,960 --> 00:05:52,890 So you know that the way the phase depends on position 128 00:05:52,890 --> 00:05:54,400 is changing over time. 129 00:05:54,400 --> 00:05:57,440 So it can't be quite so simple that the wave function, rather 130 00:05:57,440 --> 00:05:59,370 than the probability distribution, 131 00:05:59,370 --> 00:06:00,850 is remaining a Gaussian over time. 132 00:06:00,850 --> 00:06:03,010 It's not, it's got all sorts of complicated phases. 133 00:06:03,010 --> 00:06:06,060 But the upshot is that the probability distribution 134 00:06:06,060 --> 00:06:10,180 oscillates back and forth perfectly coherently. 135 00:06:10,180 --> 00:06:14,010 So that seems a little magical, and there's a nice way 136 00:06:14,010 --> 00:06:16,150 to understand how that magic arises, 137 00:06:16,150 --> 00:06:18,240 and it's to understand the following. 138 00:06:18,240 --> 00:06:27,380 If I take a state, phi 0, but I translate it x minus x0. 139 00:06:27,380 --> 00:06:28,790 OK. 140 00:06:28,790 --> 00:06:31,514 So we know that this state, phi 0 of x, 141 00:06:31,514 --> 00:06:32,930 was the ground state, what does it 142 00:06:32,930 --> 00:06:35,887 mean to be the ground state of the harmonic oscillator? 143 00:06:35,887 --> 00:06:37,470 How do I check if I'm the ground state 144 00:06:37,470 --> 00:06:40,700 of the harmonic oscillator? 145 00:06:40,700 --> 00:06:43,221 Look, I walk up to you, I'm like, hey, 146 00:06:43,221 --> 00:06:45,220 I'm the ground state of the harmonic oscillator. 147 00:06:45,220 --> 00:06:45,880 You're suspicious. 148 00:06:45,880 --> 00:06:46,530 What do you do? 149 00:06:46,530 --> 00:06:47,470 AUDIENCE: Annihilate. 150 00:06:47,470 --> 00:06:48,050 PROFESSOR: Annihilate. 151 00:06:48,050 --> 00:06:48,620 Exactly. 152 00:06:48,620 --> 00:06:50,870 You act with the annihilation operator. 153 00:06:50,870 --> 00:06:51,810 Curse you. 154 00:06:51,810 --> 00:06:55,129 So you act with the annihilation operator and you get 0, right. 155 00:06:55,129 --> 00:06:57,670 What happens if I act with the annihilation operator on this? 156 00:06:57,670 --> 00:06:59,720 Is this the ground state? 157 00:06:59,720 --> 00:07:01,381 No, it's been displaced by x0. 158 00:07:01,381 --> 00:07:03,880 And meanwhile, I told you that it oscillates back and forth. 159 00:07:03,880 --> 00:07:06,296 So what happens if you act with the annihilation operator? 160 00:07:06,296 --> 00:07:07,260 Should you get 0? 161 00:07:07,260 --> 00:07:08,510 No, what are you going to get? 162 00:07:08,510 --> 00:07:09,240 AUDIENCE: Something weird. 163 00:07:09,240 --> 00:07:10,824 PROFESSOR: Some random schmutz, right? 164 00:07:10,824 --> 00:07:12,989 If you just take a and you act on some stupid state, 165 00:07:12,989 --> 00:07:14,670 you'll just get some other stupid state. 166 00:07:14,670 --> 00:07:18,397 Except for Gaussians that have been displaced, 167 00:07:18,397 --> 00:07:19,980 you get a constant times the same wave 168 00:07:19,980 --> 00:07:23,700 function, phi 0 of x minus x0. 169 00:07:23,700 --> 00:07:24,810 Aha. 170 00:07:24,810 --> 00:07:26,630 It turns out that these displaced Gaussians 171 00:07:26,630 --> 00:07:29,140 are eigenstates of the annihilation operator. 172 00:07:29,140 --> 00:07:31,610 What does that mean? 173 00:07:31,610 --> 00:07:34,200 Well, it means they're coherent states. 174 00:07:34,200 --> 00:07:36,070 And so the optional problems are a working 175 00:07:36,070 --> 00:07:39,870 through of the study of the eigenfunctions 176 00:07:39,870 --> 00:07:45,710 of the annihilation operator, the coherent states. 177 00:07:45,710 --> 00:07:46,810 Is that cool? 178 00:07:46,810 --> 00:07:48,505 It's a state, it's a superposition. 179 00:07:48,505 --> 00:07:49,880 You can think about it like this. 180 00:07:49,880 --> 00:07:52,036 It's a superposition of the energy eigenstate. 181 00:07:52,036 --> 00:07:54,160 Any state is a superposition of energy eigenstates, 182 00:07:54,160 --> 00:07:56,076 and a coherent state is just some particularly 183 00:07:56,076 --> 00:07:58,810 special superposition of energy eigenstates. 184 00:07:58,810 --> 00:08:00,710 So you can think about it literally as c0 phi 185 00:08:00,710 --> 00:08:03,620 0 plus c1 phi 1 plus c2 phi 2, blah blah blah. 186 00:08:03,620 --> 00:08:05,160 What does the annihilation operator 187 00:08:05,160 --> 00:08:06,960 do, what does the lowering operator do? 188 00:08:06,960 --> 00:08:08,501 It takes any state and then lowers it 189 00:08:08,501 --> 00:08:09,490 with some coefficient. 190 00:08:09,490 --> 00:08:12,190 So a coherent state, an eigenstate of the annihilation 191 00:08:12,190 --> 00:08:14,550 operator, must be a state such that if you 192 00:08:14,550 --> 00:08:18,914 take c 0 phi 0 plus c1 phi 1 plus c2 phi 2 193 00:08:18,914 --> 00:08:21,080 and you hit it with the annihilation operator, which 194 00:08:21,080 --> 00:08:27,690 will give you see c1 phi 0 plus c2 phi 1 divided by root 2 plus 195 00:08:27,690 --> 00:08:31,554 dot, dot, dot-- or times root 2 plus dot, dot, dot. 196 00:08:31,554 --> 00:08:32,929 It gives you the same state back. 197 00:08:32,929 --> 00:08:35,570 So that gave you a relation between the coefficients c0 198 00:08:35,570 --> 00:08:38,110 and c1, c1 and c2, c2 and c3. 199 00:08:38,110 --> 00:08:42,280 They've all got to be suitable multiples of each other. 200 00:08:42,280 --> 00:08:45,550 And what you can show-- and this is something you work through, 201 00:08:45,550 --> 00:08:48,055 not quite in this order, on the optional problems-- what 202 00:08:48,055 --> 00:08:50,190 you can show is that doing so it gives you 203 00:08:50,190 --> 00:08:52,240 a translated Gaussian. 204 00:08:52,240 --> 00:08:53,710 OK. 205 00:08:53,710 --> 00:08:55,990 So physically, that's what a coherent state is. 206 00:08:55,990 --> 00:08:58,520 Another way to say what a coherent state is, 207 00:08:58,520 --> 00:09:00,850 it's as close to a classical object 208 00:09:00,850 --> 00:09:02,610 as you're going to get by building 209 00:09:02,610 --> 00:09:04,460 a quantum mechanical wave function. 210 00:09:04,460 --> 00:09:07,450 It's something that behaves just like a classical particle 211 00:09:07,450 --> 00:09:10,890 in that potential would have behaved, 212 00:09:10,890 --> 00:09:12,460 in the harmonic oscillator potential. 213 00:09:12,460 --> 00:09:13,710 Did that answer your question? 214 00:09:13,710 --> 00:09:15,420 OK. 215 00:09:15,420 --> 00:09:18,080 Anything else? 216 00:09:18,080 --> 00:09:20,110 So, Matt works with these for a living. 217 00:09:20,110 --> 00:09:22,708 Matt, do you want to add anything? 218 00:09:22,708 --> 00:09:23,666 All right. 219 00:09:27,020 --> 00:09:28,440 I like these. 220 00:09:28,440 --> 00:09:31,190 They show up all over the place. 221 00:09:31,190 --> 00:09:34,465 This is like the spherical cow of wave functions 222 00:09:34,465 --> 00:09:35,340 because it's awesome. 223 00:09:35,340 --> 00:09:37,550 It's an eigenfunction, it's an annihilation operator, 224 00:09:37,550 --> 00:09:39,285 it behaves like a classical particle. 225 00:09:39,285 --> 00:09:42,790 AUDIENCE: Is there an easy way to define coherent states 226 00:09:42,790 --> 00:09:45,690 in potentials where you don't have nice operators like a? 227 00:09:45,690 --> 00:09:46,630 PROFESSOR: No, I mean, that's usually 228 00:09:46,630 --> 00:09:48,070 what we mean by a coherent state. 229 00:09:48,070 --> 00:09:52,350 So the term coherent state is often used interchangeably 230 00:09:52,350 --> 00:09:54,041 to mean many different things, which 231 00:09:54,041 --> 00:09:56,290 in the case of the harmonic oscillator, are identical. 232 00:09:56,290 --> 00:09:59,520 One is a state which is a Gaussian. 233 00:09:59,520 --> 00:10:00,020 OK. 234 00:10:00,020 --> 00:10:02,680 So in the harmonic oscillator system, 235 00:10:02,680 --> 00:10:04,930 that's a particularly nice state because it oscillates 236 00:10:04,930 --> 00:10:09,984 particularly nicely and it maintains its probability 237 00:10:09,984 --> 00:10:12,400 distribution as a function of time roughly by translating, 238 00:10:12,400 --> 00:10:13,316 possibly, some phases. 239 00:10:15,834 --> 00:10:17,750 But people often use the phrase coherent state 240 00:10:17,750 --> 00:10:19,690 even when you're not harmonic. 241 00:10:19,690 --> 00:10:22,480 And it's useful to keep in mind that we're often not harmonic, 242 00:10:22,480 --> 00:10:25,590 nothing's truly harmonic. 243 00:10:25,590 --> 00:10:27,470 You had another question a moment ago though, 244 00:10:27,470 --> 00:10:29,550 and your question was about the width. 245 00:10:29,550 --> 00:10:30,050 Yeah. 246 00:10:30,050 --> 00:10:32,890 AUDIENCE: It just translates without changing its width, 247 00:10:32,890 --> 00:10:35,020 with or without changing its shape at all. 248 00:10:35,020 --> 00:10:36,811 If the Gaussian were of a different width, 249 00:10:36,811 --> 00:10:37,810 would that still happen? 250 00:10:37,810 --> 00:10:40,210 PROFESSOR: Yeah, it does, although the details of how 251 00:10:40,210 --> 00:10:41,630 it does so are slightly different. 252 00:10:41,630 --> 00:10:43,710 That's called a squeeze state. 253 00:10:43,710 --> 00:10:45,460 The basic idea of a squeeze state is this. 254 00:10:45,460 --> 00:10:47,790 Suppose I have a harmonic oscillator-- 255 00:10:47,790 --> 00:10:50,030 and this is actually one way people 256 00:10:50,030 --> 00:10:51,640 build squeeze states in labs-- so 257 00:10:51,640 --> 00:10:52,413 suppose I have a harmonic oscillator 258 00:10:52,413 --> 00:10:54,130 and I put the system in a ground state. 259 00:10:54,130 --> 00:10:55,296 So there's its ground state. 260 00:10:55,296 --> 00:10:59,290 And its width is correlated with the frequency of the potential. 261 00:11:01,814 --> 00:11:03,230 Now, suppose I take this potential 262 00:11:03,230 --> 00:11:07,230 at some moment in time and I control the potential, 263 00:11:07,230 --> 00:11:09,790 the potential is created by some laser field, for example. 264 00:11:09,790 --> 00:11:14,440 And some annoying grad student walks over to the control panel 265 00:11:14,440 --> 00:11:17,840 and doubles the power of the laser. 266 00:11:17,840 --> 00:11:20,227 All of a sudden, I've squeezed the potential. 267 00:11:20,227 --> 00:11:22,560 But my system is already in a state which is a Gaussian, 268 00:11:22,560 --> 00:11:24,530 it's just the wrong Gaussian. 269 00:11:24,530 --> 00:11:25,620 So what does this guy do? 270 00:11:25,620 --> 00:11:26,453 Well, this is funny. 271 00:11:26,453 --> 00:11:29,964 The true ground state, once we've squeezed, 272 00:11:29,964 --> 00:11:31,630 the true ground state would be something 273 00:11:31,630 --> 00:11:34,701 that's much narrower in position space. 274 00:11:34,701 --> 00:11:36,200 I didn't draw that very well, but it 275 00:11:36,200 --> 00:11:38,554 would be much narrower in position space, 276 00:11:38,554 --> 00:11:39,970 and thus its distribution would be 277 00:11:39,970 --> 00:11:42,000 much broader in momentum space. 278 00:11:42,000 --> 00:11:44,240 So the state that we put the system, 279 00:11:44,240 --> 00:11:48,940 or we've left the system in, has too much position uncertainty 280 00:11:48,940 --> 00:11:51,070 and too little momentum uncertainty. 281 00:11:51,070 --> 00:11:53,840 It's been squeezed compared to the normal state 282 00:11:53,840 --> 00:11:56,200 in the delta x delta p plane. 283 00:11:56,200 --> 00:11:58,442 It's uncertainty relation is still extreme, 284 00:11:58,442 --> 00:12:00,650 it still saturates the uncertainty bound because it's 285 00:12:00,650 --> 00:12:04,050 a Gaussian, but it doesn't have the specific delta x and delta 286 00:12:04,050 --> 00:12:06,340 p associated with the true ground 287 00:12:06,340 --> 00:12:08,190 state of the squeezed potential. 288 00:12:08,190 --> 00:12:10,980 OK, so now you can ask what does this guy do in time. 289 00:12:10,980 --> 00:12:12,910 And that's one of the optional problems, too, 290 00:12:12,910 --> 00:12:15,410 and it has many of the nice properties of coherent states, 291 00:12:15,410 --> 00:12:18,541 it's periodic, it evolves much like a classical particle, 292 00:12:18,541 --> 00:12:20,040 but it's uncertainties are different 293 00:12:20,040 --> 00:12:22,240 and they change shape. 294 00:12:22,240 --> 00:12:24,770 It's an interesting story, that's the squeezed state. 295 00:12:24,770 --> 00:12:25,632 Yeah. 296 00:12:25,632 --> 00:12:29,040 AUDIENCE: Do other potentials have 297 00:12:29,040 --> 00:12:31,962 coherent states that are Gaussian? 298 00:12:31,962 --> 00:12:33,260 PROFESSOR: It sort of depends. 299 00:12:33,260 --> 00:12:37,170 So other potentials have states that behave classically. 300 00:12:37,170 --> 00:12:37,750 Yeah. 301 00:12:37,750 --> 00:12:40,550 There are generally-- so systems that aren't the harmonic 302 00:12:40,550 --> 00:12:42,950 oscillator do have very special states that 303 00:12:42,950 --> 00:12:46,070 behave like a classical particle. 304 00:12:46,070 --> 00:12:49,380 But they're not as simple as annihilations 305 00:12:49,380 --> 00:12:52,260 by the annihilation operator. 306 00:12:52,260 --> 00:12:53,718 Come to my office and I'll tell you 307 00:12:53,718 --> 00:12:55,530 about analogous toys for something 308 00:12:55,530 --> 00:12:59,210 called supersymmetric quantum models, 309 00:12:59,210 --> 00:13:01,030 where there's a nice story there. 310 00:13:01,030 --> 00:13:02,150 OK. 311 00:13:02,150 --> 00:13:04,460 I'm going to cut off coherent states for the moment 312 00:13:04,460 --> 00:13:07,950 and move on to where we are now. 313 00:13:07,950 --> 00:13:13,770 OK, so last time we talked about scattering 314 00:13:13,770 --> 00:13:16,950 of a particle, a quantum particle of mass m 315 00:13:16,950 --> 00:13:19,322 against a barrier. 316 00:13:19,322 --> 00:13:20,780 And we made a classical prediction, 317 00:13:20,780 --> 00:13:22,180 which I didn't quite phrase this way. 318 00:13:22,180 --> 00:13:23,596 But we made a classical prediction 319 00:13:23,596 --> 00:13:27,640 that if you took this particle of mass m 320 00:13:27,640 --> 00:13:29,880 and threw it against a barrier of height v0, 321 00:13:29,880 --> 00:13:32,180 that the probability that it will transmit 322 00:13:32,180 --> 00:13:37,135 across this barrier to infinity is basically 0. 323 00:13:37,135 --> 00:13:38,635 So this is the classical prediction. 324 00:13:44,600 --> 00:13:47,390 It will not transmit to infinity until the energy goes 325 00:13:47,390 --> 00:13:48,927 above the potential. 326 00:13:48,927 --> 00:13:51,510 And when the energy is above the potential, it will slow down, 327 00:13:51,510 --> 00:13:53,720 but it will transmit 100% of the time. 328 00:13:59,871 --> 00:14:01,370 So this is our classical prediction. 329 00:14:04,340 --> 00:14:06,840 And so we sought to solve this problem, we did. 330 00:14:06,840 --> 00:14:08,260 It was pretty straightforward. 331 00:14:08,260 --> 00:14:11,555 And the energy eigenstate took the form-- well, 332 00:14:11,555 --> 00:14:12,930 we know how to solve it out here, 333 00:14:12,930 --> 00:14:13,940 we know how to solve it out here, 334 00:14:13,940 --> 00:14:15,900 because these are just constant potentials 335 00:14:15,900 --> 00:14:18,420 so it's just plain waves. 336 00:14:18,420 --> 00:14:20,130 Let's take the case of the energy 337 00:14:20,130 --> 00:14:22,920 is less than the potential. 338 00:14:22,920 --> 00:14:23,760 So this guy. 339 00:14:23,760 --> 00:14:26,027 Then out here, it's in a classically disallowed state, 340 00:14:26,027 --> 00:14:27,360 so it's got to be a exponential. 341 00:14:27,360 --> 00:14:29,026 If we want it to be normalized, well, we 342 00:14:29,026 --> 00:14:30,575 need it to be a decaying exponential. 343 00:14:30,575 --> 00:14:32,200 Out here it's oscillatory, because it's 344 00:14:32,200 --> 00:14:33,610 a classically allowed region. 345 00:14:33,610 --> 00:14:36,610 And so the general form of the wave function of the energy 346 00:14:36,610 --> 00:14:40,320 eigenstate is a superposition of a wave moving this way 347 00:14:40,320 --> 00:14:42,960 with positive momentum, a wave with negative momentum, 348 00:14:42,960 --> 00:14:45,700 a contribution with minus k, and out here it 349 00:14:45,700 --> 00:14:47,370 had to be the decaying exponential. 350 00:14:47,370 --> 00:14:50,210 And then by matching, requiring that the wave function was 351 00:14:50,210 --> 00:14:53,480 oscillatory and then exponentially decaying out 352 00:14:53,480 --> 00:14:57,119 here, and requiring that it was continuous and differentiably 353 00:14:57,119 --> 00:14:59,160 continuous-- that its derivative was continuous-- 354 00:14:59,160 --> 00:15:00,510 we found matching conditions between 355 00:15:00,510 --> 00:15:01,926 the various coefficients, and this 356 00:15:01,926 --> 00:15:03,440 was the solution in general. 357 00:15:03,440 --> 00:15:03,940 OK. 358 00:15:07,260 --> 00:15:12,090 Now, in particular, this term, which 359 00:15:12,090 --> 00:15:17,830 corresponds to the component of the wave function moving 360 00:15:17,830 --> 00:15:22,740 towards the barrier on the left hand side, has amplitude 1. 361 00:15:22,740 --> 00:15:29,370 This term, which corresponds to a wave on the left 362 00:15:29,370 --> 00:15:32,360 with negative momentum, so moving to the left, 363 00:15:32,360 --> 00:15:35,390 has amplitude k minus i alpha over k plus i alpha, 364 00:15:35,390 --> 00:15:39,470 where h bar squared k squared upon 2m 365 00:15:39,470 --> 00:15:44,100 is equal to-- that's just the energy, e-- and h bar squared 366 00:15:44,100 --> 00:15:47,600 alpha squared over 2m-- that's on this side-- is 367 00:15:47,600 --> 00:15:48,590 equal to v0 minus e. 368 00:15:51,940 --> 00:15:54,640 But this, notably, is a pure phase. 369 00:15:54,640 --> 00:15:58,484 And we understood that, so we'll call this parameter r, 370 00:15:58,484 --> 00:15:59,900 because this is the reflected wave 371 00:15:59,900 --> 00:16:02,100 and this is an amplitude rather than a probability, 372 00:16:02,100 --> 00:16:03,183 so we'll call it little r. 373 00:16:06,200 --> 00:16:09,450 And we notice that the probability 374 00:16:09,450 --> 00:16:12,285 that we get out-- good. 375 00:16:12,285 --> 00:16:13,910 So if we want to ask the question, what 376 00:16:13,910 --> 00:16:16,368 is the transmission probability, the probability that I get 377 00:16:16,368 --> 00:16:19,210 from far out here to far out here, what is that probability? 378 00:16:22,600 --> 00:16:26,364 So if I consider a state that starts out as a localized wave 379 00:16:26,364 --> 00:16:28,530 packet way out here and I send it in but with energy 380 00:16:28,530 --> 00:16:30,250 below the barrier, what's the probability 381 00:16:30,250 --> 00:16:32,460 that I'll get arbitrarily far out here, 382 00:16:32,460 --> 00:16:35,880 that I will subsequently find the particle very far out here? 383 00:16:35,880 --> 00:16:36,866 0, right, exactly. 384 00:16:36,866 --> 00:16:38,990 And you can see that because here's the probability 385 00:16:38,990 --> 00:16:40,910 amplitude, the norm squared is the probability 386 00:16:40,910 --> 00:16:43,610 density as a function of the position. 387 00:16:43,610 --> 00:16:46,470 And it goes like a constant times e to the minus alpha x. 388 00:16:46,470 --> 00:16:48,770 For large x, this exponential kills us. 389 00:16:48,770 --> 00:16:52,540 And we made that more precise by talking about the current. 390 00:16:52,540 --> 00:16:54,330 We said look, the current that gets 391 00:16:54,330 --> 00:16:58,980 transmitted is equal to-- that's a funny way to write things. 392 00:16:58,980 --> 00:17:00,840 H bar-- sorry. 393 00:17:00,840 --> 00:17:02,520 The current that gets transmitted, 394 00:17:02,520 --> 00:17:10,650 which equal to h bar upon 2mi psi complex conjugate dx 395 00:17:10,650 --> 00:17:17,936 psi minus psi dx psi complex conjugate on the right. 396 00:17:17,936 --> 00:17:21,359 So we'll put right, right, right, right. 397 00:17:21,359 --> 00:17:23,859 This is just equal to 0, and the easiest way 398 00:17:23,859 --> 00:17:25,359 to see that is something you already 399 00:17:25,359 --> 00:17:27,839 showed on a problem set. 400 00:17:27,839 --> 00:17:29,630 The current vanishes when the wave function 401 00:17:29,630 --> 00:17:31,650 can be made real up to a phase. 402 00:17:31,650 --> 00:17:35,960 And since this is some number, but in particular it's 403 00:17:35,960 --> 00:17:37,870 an overall constant complex number, 404 00:17:37,870 --> 00:17:39,310 and this is a real wave function, 405 00:17:39,310 --> 00:17:41,140 we know that this is 0. 406 00:17:41,140 --> 00:17:43,640 So the transmitted current out here is 0. 407 00:17:43,640 --> 00:17:45,950 The flux of particles moving out to the right is 0. 408 00:17:48,876 --> 00:17:50,250 So nothing gets out to the right. 409 00:17:50,250 --> 00:17:53,040 So that was for the energy less than the potential, 410 00:17:53,040 --> 00:17:56,440 and we've now re-derived this result, 411 00:17:56,440 --> 00:17:57,580 the classical prediction. 412 00:17:57,580 --> 00:18:01,305 So the classical prediction worked pretty well. 413 00:18:01,305 --> 00:18:04,100 Now, importantly, in general, we also 414 00:18:04,100 --> 00:18:06,537 wanted to define the transmission probability 415 00:18:06,537 --> 00:18:08,120 a little more carefully, and I'm going 416 00:18:08,120 --> 00:18:09,850 to define the transmission probability 417 00:18:09,850 --> 00:18:12,270 as the current of transmitted particles 418 00:18:12,270 --> 00:18:15,580 on the right divided by the current of probability that 419 00:18:15,580 --> 00:18:18,460 was incident. 420 00:18:18,460 --> 00:18:21,450 And similarly, we can define a probability 421 00:18:21,450 --> 00:18:23,640 that the particle reflects, which 422 00:18:23,640 --> 00:18:27,320 is the ratio of the reflected current, the current 423 00:18:27,320 --> 00:18:30,650 of the reflected beam, to the incident current. 424 00:18:36,370 --> 00:18:38,600 So this is going to be the transmission probability 425 00:18:38,600 --> 00:18:39,891 and the reflection probability. 426 00:18:42,721 --> 00:18:43,220 Yeah. 427 00:18:43,220 --> 00:18:45,460 AUDIENCE: Is there a square on t? 428 00:18:45,460 --> 00:18:46,580 PROFESSOR: Thank you. 429 00:18:50,700 --> 00:18:53,340 OK. 430 00:18:53,340 --> 00:18:57,150 So let's do a slightly different example. 431 00:18:57,150 --> 00:18:59,750 In this example, I want to study the same system, the wall, 432 00:18:59,750 --> 00:19:01,705 but I want to consider a ball incident 433 00:19:01,705 --> 00:19:03,182 from the left-- a ball. 434 00:19:03,182 --> 00:19:05,390 An object, a quantum particle incident from the left, 435 00:19:05,390 --> 00:19:08,499 with energy greater than v. Greater than v0. 436 00:19:08,499 --> 00:19:10,790 So again, classically, what do you expect in this case. 437 00:19:10,790 --> 00:19:13,640 You send in the particle, it loses a little bit of energy 438 00:19:13,640 --> 00:19:15,570 going up the potential barrier, but it's still 439 00:19:15,570 --> 00:19:18,070 got a positive kinetic energy, and so it just keeps rolling. 440 00:19:21,580 --> 00:19:23,680 Just like my car making it up the driveway, 441 00:19:23,680 --> 00:19:24,950 just barely makes it. 442 00:19:27,690 --> 00:19:28,400 OK. 443 00:19:28,400 --> 00:19:32,350 So here again, what's the form of the wave function. 444 00:19:32,350 --> 00:19:36,600 Phi e can be put in the form on the left. 445 00:19:36,600 --> 00:19:41,970 It's going to be e to the ikx plus b to the minus ikx, 446 00:19:41,970 --> 00:19:44,120 and that's over here on the left. 447 00:19:44,120 --> 00:19:47,286 And on the right it's going in the form ce 448 00:19:47,286 --> 00:19:53,510 to the ik-- in fact, let me call this k1x and k2x-- or sorry, 449 00:19:53,510 --> 00:19:55,680 k1x, which is on the left. 450 00:19:55,680 --> 00:20:03,077 H bar squared k1 squared upon 2m is equal to e. 451 00:20:03,077 --> 00:20:05,660 And on the right, we're going to have k2x because it's, again, 452 00:20:05,660 --> 00:20:07,076 a classically allowed region, it's 453 00:20:07,076 --> 00:20:09,720 going to be oscillatory but with a different momentum. 454 00:20:09,720 --> 00:20:15,370 e to the ik2x plus de to the minus ik2x 455 00:20:15,370 --> 00:20:22,890 where h bar squared k2 squared upon 2m 456 00:20:22,890 --> 00:20:26,110 is equal to e minus v0, which is a positive number, 457 00:20:26,110 --> 00:20:26,874 so that's good. 458 00:20:26,874 --> 00:20:28,040 So here's our wave function. 459 00:20:32,194 --> 00:20:33,610 Now, before we do anything else, I 460 00:20:33,610 --> 00:20:37,210 want to just interpret this quickly. 461 00:20:37,210 --> 00:20:41,954 So again, this is like a wave that has positive momentum, 462 00:20:41,954 --> 00:20:44,370 and I'm going to say that it's like a contribution, a term 463 00:20:44,370 --> 00:20:46,572 in the superposition, that is moving to the right, 464 00:20:46,572 --> 00:20:48,905 and the way to think of it as moving to the right, well, 465 00:20:48,905 --> 00:20:50,020 first off, it's got positive momentum. 466 00:20:50,020 --> 00:20:52,277 But more to the point, this is an energy eigenstate. 467 00:20:52,277 --> 00:20:53,860 So how does this state evolve in time? 468 00:20:57,767 --> 00:20:59,350 It's an energy eigenstate with energy, 469 00:20:59,350 --> 00:21:00,520 how does it evolve in time. 470 00:21:00,520 --> 00:21:01,500 AUDIENCE: [INAUDIBLE] 471 00:21:01,500 --> 00:21:03,480 PROFESSOR: It rotates by an overall phase, exactly. 472 00:21:03,480 --> 00:21:05,271 So we just get this times an e to the minus 473 00:21:05,271 --> 00:21:08,039 i omega t, where h bar omega is equal to e, the energy. 474 00:21:08,039 --> 00:21:09,580 And that's true on the left or right, 475 00:21:09,580 --> 00:21:12,050 because the energy is just a constant. 476 00:21:12,050 --> 00:21:16,830 So this term goes as e to the ikx minus i 477 00:21:16,830 --> 00:21:20,050 omega t, or kx minus omega t. 478 00:21:20,050 --> 00:21:22,660 And this is a point of constant phase 479 00:21:22,660 --> 00:21:23,871 in this move to the right. 480 00:21:23,871 --> 00:21:26,120 As t increases-- in order for the phase to be constant 481 00:21:26,120 --> 00:21:28,570 or for the phase, for example, to be 0-- as t increases, 482 00:21:28,570 --> 00:21:29,920 x must increase. 483 00:21:29,920 --> 00:21:32,680 By assumption here, k is a positive number. 484 00:21:32,680 --> 00:21:36,380 On the other hand, this guy-- that was a k1-- the second term 485 00:21:36,380 --> 00:21:39,790 is of the form e to the minus ik1x, 486 00:21:39,790 --> 00:21:44,800 and when I add the omega t, minus i omega t, 487 00:21:44,800 --> 00:21:46,244 this becomes kx plus omega t. 488 00:21:46,244 --> 00:21:48,160 So point of constant phase, for example, phase 489 00:21:48,160 --> 00:21:51,450 equals 0, in order for this to stay 0 then, as t increases, 490 00:21:51,450 --> 00:21:54,449 x must become negative. x must decrease. 491 00:21:54,449 --> 00:21:55,990 So that's why we say this corresponds 492 00:21:55,990 --> 00:21:59,170 to a component of the wave function, 493 00:21:59,170 --> 00:22:03,030 a term in the superposition which is moving to the left. 494 00:22:03,030 --> 00:22:05,660 And this is one which is moving to the right. 495 00:22:05,660 --> 00:22:06,950 Sorry, my right, your left. 496 00:22:06,950 --> 00:22:08,950 Sorry, this is moving to your right 497 00:22:08,950 --> 00:22:10,415 and this is moving to your left. 498 00:22:10,415 --> 00:22:11,900 OK? 499 00:22:11,900 --> 00:22:15,150 It's that little z2, it's harder than it seems. 500 00:22:15,150 --> 00:22:17,740 OK. 501 00:22:17,740 --> 00:22:20,110 In this set up, we could imagine two different kinds 502 00:22:20,110 --> 00:22:21,560 of scattering experiments. 503 00:22:21,560 --> 00:22:23,310 We can imagine a scattering experiment 504 00:22:23,310 --> 00:22:26,179 where we send in a particle from the left and ask what happens. 505 00:22:26,179 --> 00:22:27,970 So if you send in a particle from the left, 506 00:22:27,970 --> 00:22:30,414 what are the logical possibilities? 507 00:22:30,414 --> 00:22:31,580 AUDIENCE: It can go through. 508 00:22:31,580 --> 00:22:33,410 PROFESSOR: It could go through, or? 509 00:22:33,410 --> 00:22:34,390 AUDIENCE: Reflect back. 510 00:22:34,390 --> 00:22:34,980 PROFESSOR: Reflect back. 511 00:22:34,980 --> 00:22:37,354 Are you ever going to get a particle spontaneously coming 512 00:22:37,354 --> 00:22:38,620 from infinity out here? 513 00:22:38,620 --> 00:22:39,120 Not so much. 514 00:22:39,120 --> 00:22:41,161 So if you're sending a particle in from the left, 515 00:22:41,161 --> 00:22:44,030 what can you say about these coefficients? 516 00:22:44,030 --> 00:22:46,472 d equals 0 and a is not 0, right. 517 00:22:46,472 --> 00:22:48,680 Because d corresponds to a particle on the right hand 518 00:22:48,680 --> 00:22:50,590 side coming in this way. 519 00:22:50,590 --> 00:22:52,480 So that's a different scattering experiment. 520 00:22:52,480 --> 00:22:58,570 So in particular, coming in from the left 521 00:22:58,570 --> 00:23:01,790 means d equals 0 and a not equal to 0. 522 00:23:04,580 --> 00:23:12,250 Coming in from the right means a equals 0 and d 523 00:23:12,250 --> 00:23:14,741 not equal to 0 by exactly the same token. 524 00:23:14,741 --> 00:23:15,240 Cool? 525 00:23:19,270 --> 00:23:21,750 So I want to emphasize this. 526 00:23:21,750 --> 00:23:24,480 a and d, when you think of this as a scattering process, a 527 00:23:24,480 --> 00:23:26,640 and d are in guys. 528 00:23:26,640 --> 00:23:30,850 In, in. 529 00:23:30,850 --> 00:23:33,810 And c and b are out. 530 00:23:33,810 --> 00:23:36,090 This term corresponds to moving away from the barrier, 531 00:23:36,090 --> 00:23:38,339 this term corresponds to moving away from the barrier, 532 00:23:38,339 --> 00:23:39,960 just on the left or on the right. 533 00:23:39,960 --> 00:23:40,770 OK? 534 00:23:40,770 --> 00:23:44,849 Out and out. 535 00:23:44,849 --> 00:23:45,890 Everyone happy with that? 536 00:23:45,890 --> 00:23:47,127 Yeah. 537 00:23:47,127 --> 00:23:52,020 AUDIENCE: According to that graph, if d is-- no, that one. 538 00:23:52,020 --> 00:23:52,520 Yeah. 539 00:23:52,520 --> 00:23:56,570 If d is bigger than [INAUDIBLE] then the transplants will be 1. 540 00:23:56,570 --> 00:23:59,814 Doesn't that mean that d is also 0? 541 00:23:59,814 --> 00:24:06,180 PROFESSOR: So this is the classical prediction. 542 00:24:06,180 --> 00:24:09,680 I've written it, classical prediction. 543 00:24:09,680 --> 00:24:10,590 So let me rephrase. 544 00:24:10,590 --> 00:24:12,580 The question was basically, look, 545 00:24:12,580 --> 00:24:14,180 that classical prediction implies 546 00:24:14,180 --> 00:24:17,700 that b must be 0 on the other scattering process, 547 00:24:17,700 --> 00:24:20,040 and that doesn't sound right. 548 00:24:20,040 --> 00:24:20,765 So is that true? 549 00:24:20,765 --> 00:24:23,015 No, it's not true, and we'll see it again in a second. 550 00:24:23,015 --> 00:24:25,760 So very good intuition. 551 00:24:25,760 --> 00:24:26,510 That was good. 552 00:24:26,510 --> 00:24:31,060 OK, good. 553 00:24:31,060 --> 00:24:34,470 So in general, if we have a general wave function, 554 00:24:34,470 --> 00:24:37,920 general superposition of the two states with energy e with 555 00:24:37,920 --> 00:24:40,060 a,b,c, and d all non-0, that's fine. 556 00:24:40,060 --> 00:24:42,010 That just corresponds to sending some stuff 557 00:24:42,010 --> 00:24:44,120 in from the right and some stuff in-- sorry, 558 00:24:44,120 --> 00:24:47,170 some stuff in from the left and some stuff in from the right. 559 00:24:47,170 --> 00:24:48,606 Yeah? 560 00:24:48,606 --> 00:24:50,230 But it's, of course, going to be easier 561 00:24:50,230 --> 00:24:52,240 if we can just do a simpler experiment. 562 00:24:52,240 --> 00:24:55,670 If we send in stuff from the left or send in 563 00:24:55,670 --> 00:24:56,560 stuff from the right. 564 00:24:56,560 --> 00:24:58,887 And if we solve those problems independently, 565 00:24:58,887 --> 00:25:00,470 we can then just superpose the results 566 00:25:00,470 --> 00:25:01,900 to get the general solution. 567 00:25:01,900 --> 00:25:02,410 Yeah. 568 00:25:02,410 --> 00:25:04,310 So it will suffice to always either set 569 00:25:04,310 --> 00:25:06,522 d to 0 or a to 0, corresponding to sending things 570 00:25:06,522 --> 00:25:08,730 in from the left or sending things in from the right. 571 00:25:08,730 --> 00:25:09,955 And then we can just take a general superposition 572 00:25:09,955 --> 00:25:11,424 to get the general answer. 573 00:25:11,424 --> 00:25:12,465 Everyone happy with that? 574 00:25:15,290 --> 00:25:18,550 So let's do that in this set up. 575 00:25:18,550 --> 00:25:20,460 So here's our wave function. 576 00:25:20,460 --> 00:25:23,900 And you guys are now adept at solving the energy eigenvalue 577 00:25:23,900 --> 00:25:26,564 equation, it's just the matching conditions for a,b,c, and d. 578 00:25:26,564 --> 00:25:27,980 I'm not going to solve it for you, 579 00:25:27,980 --> 00:25:30,040 I'm just going to write down the results. 580 00:25:30,040 --> 00:25:33,140 So the results-- in fact, I'll use a new border. 581 00:25:38,730 --> 00:25:43,120 The results for this guy, now for e greater than v, 582 00:25:43,120 --> 00:25:49,270 are that-- doo doo doo, what just happened. 583 00:25:49,270 --> 00:25:50,019 Right, OK, good. 584 00:25:50,019 --> 00:25:52,060 So let's look at the case d equal 0 corresponding 585 00:25:52,060 --> 00:25:53,832 to coming in from the left. 586 00:25:53,832 --> 00:25:55,290 So in the case of in from the left, 587 00:25:55,290 --> 00:26:03,620 c is equal to 2k1 over k1 plus k2, 588 00:26:03,620 --> 00:26:08,800 and b is equal to k1 minus k2 over k1 plus k2. 589 00:26:13,110 --> 00:26:17,490 And this tells us, running through the definition of j, 590 00:26:17,490 --> 00:26:22,130 the currents, and t and r, gives us that, at the end of the day, 591 00:26:22,130 --> 00:26:27,780 the reflection coefficient is equal to k1 minus k2 upon k1 592 00:26:27,780 --> 00:26:32,790 plus k2 [INAUDIBLE] squared, whereas the transmission 593 00:26:32,790 --> 00:26:40,090 amplitude or probability is equal to 4k1 k2 over k1 594 00:26:40,090 --> 00:26:40,900 plus k2 squared. 595 00:26:45,450 --> 00:26:47,870 And a good exercise for yourself is to re-derive this, 596 00:26:47,870 --> 00:26:50,540 you're going to have to do that on the problem set as a warm 597 00:26:50,540 --> 00:26:53,259 up for a problem. 598 00:26:53,259 --> 00:26:54,800 So at this point we've got an answer, 599 00:26:54,800 --> 00:26:57,870 but it's not terribly satisfying because k1 and k2, what's 600 00:26:57,870 --> 00:27:02,630 the-- so let's put this in terms of a more 601 00:27:02,630 --> 00:27:03,980 easily interpretable form. 602 00:27:03,980 --> 00:27:07,290 k1 and k2 are nothing other than code for the energy 603 00:27:07,290 --> 00:27:08,996 and the energy minus the potential. 604 00:27:08,996 --> 00:27:11,370 Right, so we must be able to rewrite this purely in terms 605 00:27:11,370 --> 00:27:14,700 of the energy in the potential e and v. In fact, 606 00:27:14,700 --> 00:27:16,920 if we go through and divide both top and bottom 607 00:27:16,920 --> 00:27:20,480 by k1 squared for both r and t, this 608 00:27:20,480 --> 00:27:23,320 has a nice expression in terms of the ratio 609 00:27:23,320 --> 00:27:25,030 of the potential to the energy. 610 00:27:25,030 --> 00:27:29,210 And the nice expression is 1 minus the square root of 1 611 00:27:29,210 --> 00:27:36,410 minus v0 over e over 1 plus the square root 612 00:27:36,410 --> 00:27:42,970 of 1 minus v0 over e norm squared. 613 00:27:42,970 --> 00:27:44,730 And suddenly, the transmission amplitude 614 00:27:44,730 --> 00:27:54,410 has the form 4 root 1 minus v0 over e over 1 615 00:27:54,410 --> 00:27:59,855 plus the square root of 1 minus v0 over e squared. 616 00:28:02,780 --> 00:28:04,660 I don't remember which version of the notes 617 00:28:04,660 --> 00:28:06,720 I posted for this year's course. 618 00:28:06,720 --> 00:28:10,280 In the version from 2011 there is a typo 619 00:28:10,280 --> 00:28:13,730 that had here e over v, and in the version from 2012, that 620 00:28:13,730 --> 00:28:17,790 was corrected in the notes to being v0 over e. 621 00:28:17,790 --> 00:28:21,820 So I'll check, but let me just warn you about that. 622 00:28:21,820 --> 00:28:23,940 On one or two pages of the notes, at some point, 623 00:28:23,940 --> 00:28:25,790 these guys were inverted with respect to each other. 624 00:28:25,790 --> 00:28:27,414 But the reason to write this out is now 625 00:28:27,414 --> 00:28:29,220 we can plot the following. 626 00:28:29,220 --> 00:28:32,010 We can now plot the quantum version of this plot. 627 00:28:32,010 --> 00:28:34,630 We can plot the actual quantum transmission 628 00:28:34,630 --> 00:28:39,070 as a function of energy, and the classical prediction 629 00:28:39,070 --> 00:28:41,180 was that at energy is equal to v0, 630 00:28:41,180 --> 00:28:43,102 we should have a step function. 631 00:28:43,102 --> 00:28:45,310 But now you can see that this is not a step function, 632 00:28:45,310 --> 00:28:50,610 and that's related to the fact that b was not actually 0, 633 00:28:50,610 --> 00:28:52,100 as you pointed out. 634 00:28:52,100 --> 00:28:53,460 Hold on one sec. 635 00:28:53,460 --> 00:28:54,675 Sorry, was it quick? 636 00:28:54,675 --> 00:28:56,261 AUDIENCE: The notes are wrong. 637 00:28:56,261 --> 00:28:58,990 PROFESSOR: OK, the notes are wrong, good, thank you. 638 00:28:58,990 --> 00:29:00,226 That's great. 639 00:29:00,226 --> 00:29:01,350 OK, so the notes are wrong. 640 00:29:01,350 --> 00:29:05,060 It should be v0 over e, not e over v0 on the notes 641 00:29:05,060 --> 00:29:08,090 that are posted. 642 00:29:08,090 --> 00:29:10,460 We're in the process of teching them up, 643 00:29:10,460 --> 00:29:14,240 so eventually a beautiful set of nice notes will be available. 644 00:29:19,200 --> 00:29:20,120 Extra elbow grease. 645 00:29:20,120 --> 00:29:21,710 OK. 646 00:29:21,710 --> 00:29:22,840 So what do we actually see? 647 00:29:22,840 --> 00:29:29,580 Here we got for e less than v0, we 648 00:29:29,580 --> 00:29:33,090 got in d that the transmission probability was exactly 0. 649 00:29:33,090 --> 00:29:36,350 So that's a result for the quantum result. 650 00:29:36,350 --> 00:29:38,720 But when e is equal to v0, what do we get? 651 00:29:38,720 --> 00:29:40,660 Well, when e is equal to v0, this is 1. 652 00:29:40,660 --> 00:29:42,270 And we get square root of 1 minus 1, 653 00:29:42,270 --> 00:29:44,520 which is square root of 0, over square root of 1 654 00:29:44,520 --> 00:29:48,970 plus square root of 1 minus 1, which is 0. 655 00:29:48,970 --> 00:29:50,200 That's 0 over-- OK, good. 656 00:29:50,200 --> 00:29:52,040 So that's not so bad. 657 00:29:52,040 --> 00:29:56,470 Except for this 1 over this e, this is a little bit worrying. 658 00:29:56,470 --> 00:30:00,340 If you actually plot this guy out, 659 00:30:00,340 --> 00:30:06,700 it does this, where it asymptotes to 1. 660 00:30:06,700 --> 00:30:10,710 And to see that it asymptotes to 1, 661 00:30:10,710 --> 00:30:14,250 to see that it asymptotes to 1, just take e0 gigantic. 662 00:30:14,250 --> 00:30:17,040 If you know it's gigantic, this becomes v0 over e, 663 00:30:17,040 --> 00:30:20,660 which goes to 0, and if e is much larger than the potential. 664 00:30:20,660 --> 00:30:22,427 So this becomes square root of 1. 665 00:30:22,427 --> 00:30:24,760 And in the denominator, this becomes square root of 1, 1 666 00:30:24,760 --> 00:30:28,370 plus 1, that's 2 squared, 4, 4 divided by 4, that goes to 1. 667 00:30:28,370 --> 00:30:33,920 So for large e much larger than v0, this goes to 1. 668 00:30:33,920 --> 00:30:37,440 So we do recover the classical prediction 669 00:30:37,440 --> 00:30:39,900 if we look at energy scales very large 670 00:30:39,900 --> 00:30:42,590 compared to the potential height. 671 00:30:42,590 --> 00:30:43,090 Yeah? 672 00:30:46,140 --> 00:30:47,070 OK. 673 00:30:47,070 --> 00:30:48,380 So that's nice. 674 00:30:48,380 --> 00:30:49,770 Another thing that's nice to note 675 00:30:49,770 --> 00:30:52,310 is that if you take this reflection probability 676 00:30:52,310 --> 00:30:55,890 and this transmission probability, 677 00:30:55,890 --> 00:30:57,930 then they sum up to 1. 678 00:31:01,052 --> 00:31:02,510 This turns out to be a general fact 679 00:31:02,510 --> 00:31:04,410 and it's a necessary condition in order for those 680 00:31:04,410 --> 00:31:05,784 to be interpretable as reflection 681 00:31:05,784 --> 00:31:07,290 in transition probabilities. 682 00:31:07,290 --> 00:31:08,095 The probability that it transmits 683 00:31:08,095 --> 00:31:10,650 and the probability that it reflects had better add to 1, 684 00:31:10,650 --> 00:31:12,514 or something is eating your particles, which 685 00:31:12,514 --> 00:31:14,180 is probably not what you're looking for. 686 00:31:14,180 --> 00:31:17,460 So this turns out to be true in this case, just explicitly. 687 00:31:17,460 --> 00:31:19,810 But you can also, as you will in your problem set, 688 00:31:19,810 --> 00:31:23,350 prove that from that definition r and t, it's always true. 689 00:31:23,350 --> 00:31:27,330 A check on the sensibility of our definition. 690 00:31:27,330 --> 00:31:29,182 If it weren't true, it wouldn't tell you 691 00:31:29,182 --> 00:31:30,640 that quantum mechanics is wrong, it 692 00:31:30,640 --> 00:31:33,040 would tell you that we chose a stupid definition 693 00:31:33,040 --> 00:31:35,320 of the transmission and reflection probabilities. 694 00:31:35,320 --> 00:31:38,924 So in this case, we actually chose quite an enlightened one. 695 00:31:38,924 --> 00:31:40,090 OK, questions at this point? 696 00:31:42,838 --> 00:31:43,754 Yeah. 697 00:31:43,754 --> 00:31:45,355 AUDIENCE: Does this analysis even 698 00:31:45,355 --> 00:31:47,082 hold for classical particles? 699 00:31:47,082 --> 00:31:49,248 If we're talking about the difference between-- like 700 00:31:49,248 --> 00:31:51,485 if I threw a baseball and I happened to throw it 701 00:31:51,485 --> 00:31:54,520 at a potential that had height one millionth of a joule 702 00:31:54,520 --> 00:31:56,186 less than the energy of the baseball, 703 00:31:56,186 --> 00:31:57,310 would we observe this also? 704 00:31:57,310 --> 00:31:58,026 PROFESSOR: No. 705 00:31:58,026 --> 00:31:58,800 AUDIENCE: No? 706 00:31:58,800 --> 00:32:01,550 PROFESSOR: No, for the following reason. 707 00:32:01,550 --> 00:32:03,870 So here's my classical system. 708 00:32:03,870 --> 00:32:08,020 Classical system is literally some hill. 709 00:32:08,020 --> 00:32:09,650 So where I was growing up as a kid, 710 00:32:09,650 --> 00:32:11,800 there was a hill not far from our house. 711 00:32:11,800 --> 00:32:14,050 I'm not even going to go into it, but it was horrible. 712 00:32:17,120 --> 00:32:19,300 If you have energy just ever so slightly greater 713 00:32:19,300 --> 00:32:22,230 than the hill-- he said from some experience-- 714 00:32:22,230 --> 00:32:26,118 if you have energy that's ever so slightly greater-- OK. 715 00:32:26,118 --> 00:32:27,930 I'm going resist the temptation. 716 00:32:27,930 --> 00:32:34,030 So one of these days. 717 00:32:34,030 --> 00:32:37,240 If you have ever so slightly greater than the hill, 718 00:32:37,240 --> 00:32:40,904 and you start up here, what does that tell you? 719 00:32:40,904 --> 00:32:42,570 What does it mean to say you have energy 720 00:32:42,570 --> 00:32:45,430 just ever so slightly greater than the potential energy 721 00:32:45,430 --> 00:32:46,060 at that point? 722 00:32:46,060 --> 00:32:47,287 AUDIENCE: Up more slowly. 723 00:32:47,287 --> 00:32:48,620 PROFESSOR: Little tiny velocity. 724 00:32:48,620 --> 00:32:50,870 OK, and I'm going to say I have a little tiny velocity 725 00:32:50,870 --> 00:32:52,580 this way. 726 00:32:52,580 --> 00:32:53,500 OK. 727 00:32:53,500 --> 00:32:55,890 So what happens? 728 00:32:55,890 --> 00:32:58,670 We follow Newton's equations. 729 00:32:58,670 --> 00:33:02,040 They are totally unambiguous, and what they tell you is, 730 00:33:02,040 --> 00:33:07,060 with 100% certainty, this thing will roll and roll and roll, 731 00:33:07,060 --> 00:33:08,770 and then all hell will break loose. 732 00:33:08,770 --> 00:33:10,970 And then if you're very lucky, your car 733 00:33:10,970 --> 00:33:14,880 gets caught in the trees on the side of the cliff, which 734 00:33:14,880 --> 00:33:17,330 is later referred to by the policeman who helps you tow it 735 00:33:17,330 --> 00:33:19,150 out as nature's guard rail. 736 00:33:22,860 --> 00:33:26,190 I was young, it won't happen again. 737 00:33:26,190 --> 00:33:30,760 And So with 100% certainty, Newton's equations 738 00:33:30,760 --> 00:33:32,800 send you right off the cliff. 739 00:33:32,800 --> 00:33:34,340 And for style points, backward. 740 00:33:39,930 --> 00:33:42,170 So now I'm going to do the second thing. 741 00:33:42,170 --> 00:33:46,430 Newton's laws satisfy time reversal invariance. 742 00:33:46,430 --> 00:33:49,350 So the time reversal of this is, this thing has 743 00:33:49,350 --> 00:33:52,160 this much energy, and it shoots up the cliff. 744 00:33:52,160 --> 00:33:55,250 And ever so slowly, it just eventually 745 00:33:55,250 --> 00:33:58,470 goes up to the top, where everything is fine. 746 00:33:58,470 --> 00:34:01,561 OK, but does it ever reflect back downhill? 747 00:34:01,561 --> 00:34:02,060 No. 748 00:34:02,060 --> 00:34:03,904 And does it ever-- classically, when 749 00:34:03,904 --> 00:34:05,820 you roll the thing-- does it ever actually not 750 00:34:05,820 --> 00:34:07,390 go off the cliff and hit the trees, 751 00:34:07,390 --> 00:34:09,719 but instead reflect backwards. 752 00:34:09,719 --> 00:34:13,610 I wish the answer were yes. 753 00:34:13,610 --> 00:34:15,550 But sadly, the answer is no. 754 00:34:15,550 --> 00:34:19,730 Now, if that car had been quantum mechanically small, 755 00:34:19,730 --> 00:34:23,670 my insurance would have been much more manageable. 756 00:34:23,670 --> 00:34:25,139 But sadly, it wasn't. 757 00:34:25,139 --> 00:34:27,969 OK, so this is a pretty stark and vivid-- this 758 00:34:27,969 --> 00:34:30,010 is a pretty stark difference between the quantum 759 00:34:30,010 --> 00:34:32,219 mechanical prediction and the classical prediction. 760 00:34:32,219 --> 00:34:33,710 Everyone cool with that? 761 00:34:33,710 --> 00:34:36,111 Other questions? 762 00:34:36,111 --> 00:34:36,610 Yeah. 763 00:34:36,610 --> 00:34:39,294 AUDIENCE: I understand why it works out that r plus t 764 00:34:39,294 --> 00:34:42,448 equals 1, but I'm not sure I understand 765 00:34:42,448 --> 00:34:44,489 the motivation to use the square with the ratios. 766 00:34:44,489 --> 00:34:45,969 PROFESSOR: Excellent, OK. 767 00:34:45,969 --> 00:34:51,050 So the reason to do it, so great, excellent. 768 00:34:51,050 --> 00:34:51,699 How to say. 769 00:35:02,416 --> 00:35:03,790 Here's one way to think about it. 770 00:35:08,914 --> 00:35:10,330 The first thing we want is we want 771 00:35:10,330 --> 00:35:11,560 a-- so this is a very good question, 772 00:35:11,560 --> 00:35:12,230 let me repeat the question. 773 00:35:12,230 --> 00:35:13,938 The question is, look, why is it squared? 774 00:35:16,470 --> 00:35:19,350 Why isn't it linear? 775 00:35:19,350 --> 00:35:22,390 So let's think about that. 776 00:35:22,390 --> 00:35:23,810 Ah, ah, ah. 777 00:35:28,450 --> 00:35:32,310 The reason it's squared is because of a typo. 778 00:35:32,310 --> 00:35:33,250 So let's think about-- 779 00:35:33,250 --> 00:35:35,090 [LAUGHTER] 780 00:35:35,090 --> 00:35:37,010 PROFESSOR: Let's step back for second 781 00:35:37,010 --> 00:35:39,750 and let's think, OK, it's a very good question. 782 00:35:39,750 --> 00:35:42,110 So let's think about what it should be. 783 00:35:42,110 --> 00:35:42,710 OK. 784 00:35:42,710 --> 00:35:46,210 For the moment, put an arbitrary power there. 785 00:35:46,210 --> 00:35:46,800 OK. 786 00:35:46,800 --> 00:35:47,780 And let's think about what it should be. 787 00:35:47,780 --> 00:35:49,430 Thank you so much for this question. 788 00:35:49,430 --> 00:35:54,300 I owe you, like, a plate of cheese or something. 789 00:35:54,300 --> 00:35:55,990 That's high praise, guys. 790 00:35:55,990 --> 00:35:57,500 France, cheese. 791 00:35:57,500 --> 00:36:01,000 So should it be linear or should it be quadratic? 792 00:36:01,000 --> 00:36:03,080 What is that supposed to represent? 793 00:36:03,080 --> 00:36:05,125 What is T, the capital T, supposed to represent? 794 00:36:05,125 --> 00:36:06,874 AUDIENCE: The probability of transmission. 795 00:36:06,874 --> 00:36:09,340 PROFESSOR: The probability of transmission, exactly. 796 00:36:09,340 --> 00:36:10,230 The wave function. 797 00:36:10,230 --> 00:36:12,390 Is the wave function, the value of the wave 798 00:36:12,390 --> 00:36:14,660 function at some point, is that a probability? 799 00:36:14,660 --> 00:36:15,330 It's a-- 800 00:36:15,330 --> 00:36:16,824 AUDIENCE: Probability of density. 801 00:36:16,824 --> 00:36:18,320 No, it's the square root. 802 00:36:18,320 --> 00:36:20,760 PROFESSOR: Is it a probability of density? 803 00:36:20,760 --> 00:36:22,860 It is a probability amplitude. 804 00:36:22,860 --> 00:36:25,560 It is a thing whose norm squared is a probability. 805 00:36:25,560 --> 00:36:29,000 So we want something that's quadratic in the wave function, 806 00:36:29,000 --> 00:36:30,190 right. 807 00:36:30,190 --> 00:36:33,860 Meanwhile, some time before, we wanted a definition 808 00:36:33,860 --> 00:36:36,320 of how much stuff is moving past a point in the given 809 00:36:36,320 --> 00:36:38,010 moment in time. 810 00:36:38,010 --> 00:36:41,250 What's the probability density moving past a point 811 00:36:41,250 --> 00:36:43,000 at a given moment in time. 812 00:36:43,000 --> 00:36:45,680 And that's where we got the current j from the first place. 813 00:36:45,680 --> 00:36:49,637 j was the probability density moving past a point 814 00:36:49,637 --> 00:36:50,720 at a given moment in time. 815 00:36:53,240 --> 00:36:57,160 So notice that j is quadratic in the wave function, 816 00:36:57,160 --> 00:37:01,110 so it's how much stuff is moving past any particular direction-- 817 00:37:01,110 --> 00:37:03,837 we chose it to be the incident to the reflected bit-- 818 00:37:03,837 --> 00:37:04,670 at a moment in time. 819 00:37:04,670 --> 00:37:08,444 So it's a probability density-- it's actually a current-- 820 00:37:08,444 --> 00:37:10,110 and it's quadratic in the wave function. 821 00:37:10,110 --> 00:37:12,290 So it has the right units and the right structure 822 00:37:12,290 --> 00:37:13,600 to be a probability. 823 00:37:13,600 --> 00:37:16,324 If we squared that, we would be in trouble, because it wouldn't 824 00:37:16,324 --> 00:37:18,490 be a probability, it would be a probability squared. 825 00:37:18,490 --> 00:37:21,370 And in particular, it wouldn't normalize correctly. 826 00:37:21,370 --> 00:37:23,320 So that typo was actually a bad typo. 827 00:37:23,320 --> 00:37:25,190 So what happened, now thinking back to-- you 828 00:37:25,190 --> 00:37:29,087 have a video someday, so you can scroll back. 829 00:37:29,087 --> 00:37:31,420 What happened was I didn't put the squared on the first, 830 00:37:31,420 --> 00:37:33,030 I put the squared on the second, and then someone said 831 00:37:33,030 --> 00:37:33,905 squared on the first? 832 00:37:33,905 --> 00:37:34,420 Like, yes. 833 00:37:34,420 --> 00:37:36,969 But the correct answer is no squared on the second. 834 00:37:36,969 --> 00:37:38,010 So there are no squareds. 835 00:37:38,010 --> 00:37:39,390 Thank you for the question. 836 00:37:39,390 --> 00:37:41,240 Very good question. 837 00:37:41,240 --> 00:37:42,615 In particular, the thing that was 838 00:37:42,615 --> 00:37:44,156 so awesome about that question was it 839 00:37:44,156 --> 00:37:45,329 was motivated by physics. 840 00:37:45,329 --> 00:37:47,870 It was like, look, why is this thing squared the right thing? 841 00:37:47,870 --> 00:37:49,460 That's two factors of the wave function, 842 00:37:49,460 --> 00:37:50,460 it's already quadratic. 843 00:37:50,460 --> 00:37:51,470 It should be like a probability. 844 00:37:51,470 --> 00:37:52,710 Why are you squaring it? 845 00:37:52,710 --> 00:37:53,971 Very good question. 846 00:37:53,971 --> 00:37:54,470 Yeah. 847 00:37:54,470 --> 00:37:57,540 AUDIENCE: So in the transmission probability, 848 00:37:57,540 --> 00:38:00,460 why do we know it's 0 if the energy is less than v0? 849 00:38:00,460 --> 00:38:02,890 I mean, is there any rule? 850 00:38:02,890 --> 00:38:04,909 PROFESSOR: Yeah, OK, good. 851 00:38:04,909 --> 00:38:06,950 We'll come back to that question in a little bit, 852 00:38:06,950 --> 00:38:09,380 but let me just quickly say that we calculated it, 853 00:38:09,380 --> 00:38:11,300 and we see explicitly that it's 0. 854 00:38:11,300 --> 00:38:13,466 Now you might say well, look, the true wave function 855 00:38:13,466 --> 00:38:15,626 has a little bit of a tail. 856 00:38:15,626 --> 00:38:17,250 So there's some [INAUDIBLE] probability 857 00:38:17,250 --> 00:38:19,140 you'll find it here if you do a measurement. 858 00:38:19,140 --> 00:38:21,098 But the question we want to ask in transmission 859 00:38:21,098 --> 00:38:23,590 is, if you send it in from far away over here, 860 00:38:23,590 --> 00:38:26,350 how likely are you to catch it far away over here? 861 00:38:26,350 --> 00:38:27,965 And the answer is, you are not. 862 00:38:27,965 --> 00:38:29,570 That make sense? 863 00:38:29,570 --> 00:38:32,910 OK, good. 864 00:38:32,910 --> 00:38:34,270 OK, so so much for that one. 865 00:38:34,270 --> 00:38:35,500 Now, here's a fun fact, I'm not going 866 00:38:35,500 --> 00:38:36,874 to go through this in any detail. 867 00:38:36,874 --> 00:38:40,996 We could have done exactly the same calculation-- OK, 868 00:38:40,996 --> 00:38:42,620 we could have done the same calculation 869 00:38:42,620 --> 00:38:45,310 sending something in from the right. 870 00:38:45,310 --> 00:38:50,370 And sending something in from the right, in on the right, 871 00:38:50,370 --> 00:38:52,549 corresponds to d not equal to 0 and nothing 872 00:38:52,549 --> 00:38:54,340 coming in from the left, that's a equals 0. 873 00:38:54,340 --> 00:38:56,089 So we could have done the case a equals 0, 874 00:38:56,089 --> 00:38:58,140 which is in from the right. 875 00:39:02,100 --> 00:39:03,850 And if you do that calculus, what you find 876 00:39:03,850 --> 00:39:12,270 is c is equal to k1 minus k2 upon k1 plus k2 d. 877 00:39:12,270 --> 00:39:23,160 And b is equal to 2k1 upon k1 plus k2 d. 878 00:39:23,160 --> 00:39:25,320 And this says that plugging these guys in, 879 00:39:25,320 --> 00:39:30,800 the reflection and the transmission are the same. 880 00:39:30,800 --> 00:39:33,531 In particular, physically, what does that mean? 881 00:39:33,531 --> 00:39:35,280 That means the reflection and transmission 882 00:39:35,280 --> 00:39:40,200 are the same uphill as downhill. 883 00:39:40,200 --> 00:39:44,970 Downhill is uphill, they're the same both ways. 884 00:39:44,970 --> 00:39:47,390 But that's truly weird, right? 885 00:39:47,390 --> 00:39:52,060 What's the probability to transmit quantum mechanically 886 00:39:52,060 --> 00:39:54,562 if you have just a little bit of energy 887 00:39:54,562 --> 00:39:55,770 and you send the particle in? 888 00:39:55,770 --> 00:39:57,395 How likely are you to go off the cliff. 889 00:40:01,130 --> 00:40:04,890 If we have energy ever so slightly greater the potential, 890 00:40:04,890 --> 00:40:07,610 and the transmission amplitude is the same downhill 891 00:40:07,610 --> 00:40:10,820 as it was uphill, how likely are you to fall off the potential? 892 00:40:10,820 --> 00:40:12,236 AUDIENCE: Always. 893 00:40:12,236 --> 00:40:14,420 PROFESSOR: Never. 894 00:40:14,420 --> 00:40:18,440 Because if energy is just slightly greater than v0, 895 00:40:18,440 --> 00:40:22,240 then the transmission probability is very, very low. 896 00:40:25,020 --> 00:40:28,340 The transmission to go from here to here 897 00:40:28,340 --> 00:40:30,450 is extremely low if the energy is 898 00:40:30,450 --> 00:40:32,900 close to the height of the barrier. 899 00:40:32,900 --> 00:40:36,994 So had I only been on a quantum mechanical hill, 900 00:40:36,994 --> 00:40:38,160 I would have been just fine. 901 00:40:40,780 --> 00:40:42,290 This is a really striking result, 902 00:40:42,290 --> 00:40:44,030 but this thing, the fact that you're 903 00:40:44,030 --> 00:40:46,730 unlikely to scatter uphill, that's maybe not so shocking. 904 00:40:46,730 --> 00:40:50,350 But you're really unlikely to scatter downhill, 905 00:40:50,350 --> 00:40:51,265 that is surprising. 906 00:40:54,290 --> 00:40:59,300 So I'll leave it to you to check that, in fact, 907 00:40:59,300 --> 00:41:01,860 the transmission, when we do it in from the right, 908 00:41:01,860 --> 00:41:06,050 the transmission and reflection probabilities are the same. 909 00:41:06,050 --> 00:41:08,160 Recalling that transmission means going this way, 910 00:41:08,160 --> 00:41:11,140 reflection means bouncing back to the right. 911 00:41:11,140 --> 00:41:11,650 Cool? 912 00:41:11,650 --> 00:41:12,013 Yeah. 913 00:41:12,013 --> 00:41:12,888 AUDIENCE: [INAUDIBLE] 914 00:41:16,668 --> 00:41:20,320 PROFESSOR: Uh, because, again. 915 00:41:20,320 --> 00:41:21,720 Jeez, today is just a disaster. 916 00:41:21,720 --> 00:41:23,390 Because there's a typo. 917 00:41:23,390 --> 00:41:25,008 This should be times a. 918 00:41:25,008 --> 00:41:25,917 God. 919 00:41:25,917 --> 00:41:28,250 And the reason you know it should be times a, first off, 920 00:41:28,250 --> 00:41:30,990 is that these should have the appropriate dimensions. 921 00:41:30,990 --> 00:41:32,975 So there should be an appropriate power of a. 922 00:41:32,975 --> 00:41:35,350 And the second thing is that if you double this stuff in, 923 00:41:35,350 --> 00:41:36,870 you'd better double this stuff out. 924 00:41:36,870 --> 00:41:38,800 So if you double a, you'd better double c. 925 00:41:38,800 --> 00:41:40,380 So you know there had better be a factor of a here. 926 00:41:40,380 --> 00:41:41,730 And that's just a typo again. 927 00:41:41,730 --> 00:41:46,124 I'm sorry, today is a bad day at the chalkboard. 928 00:41:46,124 --> 00:41:47,290 Thank you for that question. 929 00:41:47,290 --> 00:41:47,925 Yeah. 930 00:41:47,925 --> 00:41:49,790 AUDIENCE: Is there a physical reason 931 00:41:49,790 --> 00:41:55,657 for why r and t are the same and so different from what 932 00:41:55,657 --> 00:41:58,240 you would classically expect, or is that just the way the math 933 00:41:58,240 --> 00:41:58,700 works? 934 00:41:58,700 --> 00:41:59,533 PROFESSOR: There is. 935 00:41:59,533 --> 00:42:02,060 I want you to ponder that, and either at the end of today 936 00:42:02,060 --> 00:42:03,792 or at the beginning of tomorrow, we'll 937 00:42:03,792 --> 00:42:05,250 talk about that more in detail when 938 00:42:05,250 --> 00:42:06,962 we've done a little more technology. 939 00:42:06,962 --> 00:42:08,420 It's a good question and you should 940 00:42:08,420 --> 00:42:10,760 have that tingling sensation in your belly 941 00:42:10,760 --> 00:42:12,740 that something is confusing and surprising 942 00:42:12,740 --> 00:42:14,920 and requires more explanation, and we'll get there. 943 00:42:14,920 --> 00:42:16,800 But I want you to just think about it in the background 944 00:42:16,800 --> 00:42:17,299 first. 945 00:42:17,299 --> 00:42:18,114 Yeah. 946 00:42:18,114 --> 00:42:21,502 AUDIENCE: Is there any physical experiment 947 00:42:21,502 --> 00:42:23,922 where r and t don't sum up to 1? 948 00:42:23,922 --> 00:42:25,858 PROFESSOR: Mm mm. 949 00:42:25,858 --> 00:42:26,830 Nope. 950 00:42:26,830 --> 00:42:29,220 So the question is, is there any experiment 951 00:42:29,220 --> 00:42:32,120 in which r and t don't sum up to 1? 952 00:42:32,120 --> 00:42:35,715 And there are two ways to answer that. 953 00:42:35,715 --> 00:42:37,590 The first way to answer that is, it turns out 954 00:42:37,590 --> 00:42:41,369 that r and t adding up to 1 follows from the definitions. 955 00:42:41,369 --> 00:42:43,410 So one of the things you'll do on the problem set 956 00:42:43,410 --> 00:42:48,105 is you'll show that using these definitions of the transmission 957 00:42:48,105 --> 00:42:50,810 and reflection of probabilities, they necessarily 958 00:42:50,810 --> 00:42:52,300 add up to 1, strictly. 959 00:42:52,300 --> 00:42:52,970 They have to. 960 00:42:52,970 --> 00:42:55,120 It follows from the Schrodinger equation. 961 00:42:55,120 --> 00:42:56,639 The second answer is, let's think 962 00:42:56,639 --> 00:42:58,930 about what it would mean if r and t didn't add up to 1. 963 00:42:58,930 --> 00:43:00,230 If r and t didn't add up to 1, then 964 00:43:00,230 --> 00:43:01,646 that would say, look, if you throw 965 00:43:01,646 --> 00:43:05,270 a particle at a barrier, at some feature in a potential, 966 00:43:05,270 --> 00:43:07,282 some wall, then the probability that it 967 00:43:07,282 --> 00:43:09,240 goes this way at the end versus the probability 968 00:43:09,240 --> 00:43:12,540 that it goes that way at the end if you wait long enough 969 00:43:12,540 --> 00:43:13,370 is not 1. 970 00:43:13,370 --> 00:43:14,630 So what could have happened? 971 00:43:14,630 --> 00:43:16,213 What could have gotten stuck in there? 972 00:43:16,213 --> 00:43:19,970 But that doesn't even conserve momentum, 973 00:43:19,970 --> 00:43:21,100 not even approximately. 974 00:43:21,100 --> 00:43:22,933 And we know that the expectation of momentum 975 00:43:22,933 --> 00:43:28,780 is time independent for a free particle. 976 00:43:28,780 --> 00:43:30,780 In between while it's actually in the potential, 977 00:43:30,780 --> 00:43:31,860 it's not actually time independent, 978 00:43:31,860 --> 00:43:33,401 there's a potential, there's a force. 979 00:43:33,401 --> 00:43:36,540 AUDIENCE: I mean, if it's whole particles, can't some of them 980 00:43:36,540 --> 00:43:38,396 get annihilated or something? 981 00:43:38,396 --> 00:43:40,035 PROFESSOR: Ah, well OK. 982 00:43:40,035 --> 00:43:41,910 So if you're talking about multiple particles 983 00:43:41,910 --> 00:43:43,743 and interactions amongst multiple particles, 984 00:43:43,743 --> 00:43:45,920 then it's a slightly more complicated question. 985 00:43:45,920 --> 00:43:48,340 The answer there is still yes, it 986 00:43:48,340 --> 00:43:50,530 has to be that the total probability in 987 00:43:50,530 --> 00:43:51,990 is the total probability out. 988 00:43:51,990 --> 00:43:55,210 But we're only going to talk about single particles here. 989 00:43:55,210 --> 00:43:58,070 But it's always true that for-- we're 990 00:43:58,070 --> 00:43:59,570 going to take this as an assumption, 991 00:43:59,570 --> 00:44:02,617 that things don't just disappear, 992 00:44:02,617 --> 00:44:04,825 that the number of particles or the total probability 993 00:44:04,825 --> 00:44:06,110 is conserved. 994 00:44:06,110 --> 00:44:08,165 There's a third way to say this, this really 995 00:44:08,165 --> 00:44:09,540 isn't independent from the first. 996 00:44:09,540 --> 00:44:12,475 Remember that these were defined in terms of the current? 997 00:44:12,475 --> 00:44:15,490 The current satisfies-- the current concept 998 00:44:15,490 --> 00:44:18,790 of probability conservation equation dt of rho 999 00:44:18,790 --> 00:44:25,740 is equal to the gradient minus the gradient of the current. 1000 00:44:25,740 --> 00:44:27,820 So the probability is always conserved, 1001 00:44:27,820 --> 00:44:31,290 the integral of probability density is always conserved, 1002 00:44:31,290 --> 00:44:32,240 it's time independent. 1003 00:44:32,240 --> 00:44:33,680 AUDIENCE: So it's by [INAUDIBLE]. 1004 00:44:33,680 --> 00:44:34,650 PROFESSOR: Yeah, it's by construction. 1005 00:44:34,650 --> 00:44:35,150 Exactly. 1006 00:44:38,280 --> 00:44:39,059 Yeah? 1007 00:44:39,059 --> 00:44:41,350 AUDIENCE: Looking at your expression for the transition 1008 00:44:41,350 --> 00:44:45,770 probability, I'm having trouble seeing how that works out to 0 1009 00:44:45,770 --> 00:44:47,790 when e is less than v0? 1010 00:44:47,790 --> 00:44:50,460 PROFESSOR: Oh, this expression is only 1011 00:44:50,460 --> 00:44:52,394 defined when e is equal to v0. 1012 00:44:52,394 --> 00:44:54,310 Because we derived this-- excellent question-- 1013 00:44:54,310 --> 00:44:57,594 we derived this assuming that we had an energy greater than v0 1014 00:44:57,594 --> 00:44:59,510 and then that the wave function had this form. 1015 00:44:59,510 --> 00:45:02,093 You cannot use this form of the wave function if the energy is 1016 00:45:02,093 --> 00:45:02,730 less than v0. 1017 00:45:02,730 --> 00:45:04,050 If the energy is less than v0, you've 1018 00:45:04,050 --> 00:45:05,700 got use that form of the wave function. 1019 00:45:05,700 --> 00:45:06,660 And in this form of the wave function, 1020 00:45:06,660 --> 00:45:08,924 we derived that the transition amplitude is 0, 1021 00:45:08,924 --> 00:45:11,340 because the current on the right, the transmitted current, 1022 00:45:11,340 --> 00:45:12,510 is 0. 1023 00:45:12,510 --> 00:45:14,354 So this calculation is appropriate 1024 00:45:14,354 --> 00:45:16,770 when e is less than v0 and this calculation is appropriate 1025 00:45:16,770 --> 00:45:18,082 when e is greater than v0. 1026 00:45:18,082 --> 00:45:19,290 So you're absolutely correct. 1027 00:45:19,290 --> 00:45:21,160 You can't use this one e is less than v0, 1028 00:45:21,160 --> 00:45:23,030 it gives you not the same answer. 1029 00:45:23,030 --> 00:45:25,240 In fact, it gives you a complex-- 1030 00:45:25,240 --> 00:45:26,260 it's kind of confusing. 1031 00:45:26,260 --> 00:45:27,900 The factors cancel. 1032 00:45:27,900 --> 00:45:29,799 So it's not really a probability at all, 1033 00:45:29,799 --> 00:45:32,090 and indeed, this is just not the right quantity to use. 1034 00:45:32,090 --> 00:45:33,173 AUDIENCE: All right, cool. 1035 00:45:33,173 --> 00:45:33,930 PROFESSOR: Cool? 1036 00:45:33,930 --> 00:45:34,429 All right. 1037 00:45:38,000 --> 00:45:39,700 So there are a bunch of nice things 1038 00:45:39,700 --> 00:45:42,950 I want to deduce from what we've done so far. 1039 00:45:42,950 --> 00:45:46,660 So the first is, look, I pointed out 1040 00:45:46,660 --> 00:45:50,050 that this can be derived just explicitly 1041 00:45:50,050 --> 00:45:54,580 and it gives the same results as before. 1042 00:45:54,580 --> 00:45:55,860 That's not an accident. 1043 00:45:55,860 --> 00:45:58,490 If you take this system and you just 1044 00:45:58,490 --> 00:46:01,800 reverse the roles of k1 and k2, what happens? 1045 00:46:01,800 --> 00:46:04,040 Well that's just replaces e by e minus v0, 1046 00:46:04,040 --> 00:46:07,280 we can do that by doing this, replacing e by e minus v0 1047 00:46:07,280 --> 00:46:10,660 and e minus v0 by e, it swaps the role of a and d, 1048 00:46:10,660 --> 00:46:12,710 and it gives you exactly the same things back. 1049 00:46:12,710 --> 00:46:14,459 So if you're careful about that, you never 1050 00:46:14,459 --> 00:46:15,810 have to do this calculation. 1051 00:46:15,810 --> 00:46:17,420 You can just do the appropriate transformation 1052 00:46:17,420 --> 00:46:20,086 on that calculation and it gives you the exactly the same thing. 1053 00:46:23,210 --> 00:46:25,324 It's the same algebraic steps. 1054 00:46:25,324 --> 00:46:26,740 But the other thing that's nice is 1055 00:46:26,740 --> 00:46:29,960 that you can actually do the same thing from here. 1056 00:46:29,960 --> 00:46:31,632 So as long as you set the d equals 0-- 1057 00:46:31,632 --> 00:46:33,590 so there's another term here that we neglected, 1058 00:46:33,590 --> 00:46:36,600 the plus delta x, we got rid of it. 1059 00:46:36,600 --> 00:46:39,900 But you can analytically continue this calculation 1060 00:46:39,900 --> 00:46:42,500 by noting that look, if we just set alpha, 1061 00:46:42,500 --> 00:46:46,890 we want minus alpha equals ik2, then 1062 00:46:46,890 --> 00:46:49,170 the algebra is all going to be the same. 1063 00:46:49,170 --> 00:46:52,530 We just have ik2 instead of minus alpha. 1064 00:46:52,530 --> 00:46:56,450 So we can just replace alpha with minus ik2 1065 00:46:56,450 --> 00:46:58,510 everywhere in our expressions, being 1066 00:46:58,510 --> 00:47:00,620 careful about exactly how we do so, 1067 00:47:00,620 --> 00:47:02,980 being careful to take care of factors 1068 00:47:02,980 --> 00:47:04,600 of i and such correctly. 1069 00:47:04,600 --> 00:47:07,150 And you derive the same results for both cases, which 1070 00:47:07,150 --> 00:47:09,419 is a nice check on the calculation. 1071 00:47:09,419 --> 00:47:11,960 So often, when you get a little bit of experience with these, 1072 00:47:11,960 --> 00:47:13,410 you don't actually have to do the calculation. 1073 00:47:13,410 --> 00:47:14,910 Again, you can just take what you 1074 00:47:14,910 --> 00:47:17,010 know from a previous calculation and write down 1075 00:47:17,010 --> 00:47:18,360 the correct answer. 1076 00:47:18,360 --> 00:47:19,756 So it's a fun thing to play with, 1077 00:47:19,756 --> 00:47:20,880 exactly how do you do that. 1078 00:47:20,880 --> 00:47:23,950 So I invite you to think through that process 1079 00:47:23,950 --> 00:47:26,064 while you're doing your problem set. 1080 00:47:26,064 --> 00:47:27,730 Another thing is the reflection downhill 1081 00:47:27,730 --> 00:47:29,560 thing which is pretty surprising. 1082 00:47:29,560 --> 00:47:33,040 But here's the thing that I really want to emphasize. 1083 00:47:33,040 --> 00:47:35,870 What this calculation shows you is not so much 1084 00:47:35,870 --> 00:47:42,730 that-- it's not just that transmission downhill 1085 00:47:42,730 --> 00:47:45,390 is highly unlikely when the energy is 1086 00:47:45,390 --> 00:47:47,720 very close to the height of the potential barrier. 1087 00:47:47,720 --> 00:47:49,890 That's true, but it's not the most interesting thing 1088 00:47:49,890 --> 00:47:51,010 about this calculation. 1089 00:47:51,010 --> 00:47:53,950 The most interesting thing about this calculation, to my mind, 1090 00:47:53,950 --> 00:47:57,500 is the fact that from the detailed shape 1091 00:47:57,500 --> 00:48:03,710 of the transmission as a function of energy, 1092 00:48:03,710 --> 00:48:07,600 we can deduce what the potential is. 1093 00:48:07,600 --> 00:48:09,100 Think about what that tells you. 1094 00:48:09,100 --> 00:48:11,634 If you do an experiment, you have a barrier, 1095 00:48:11,634 --> 00:48:13,550 and you want to know the shape of the barrier. 1096 00:48:13,550 --> 00:48:15,630 Is it straight, is it wiggly, does 1097 00:48:15,630 --> 00:48:17,800 it have some complicated shape. 1098 00:48:17,800 --> 00:48:18,890 How do you measure that? 1099 00:48:18,890 --> 00:48:20,420 Well, you might measure it by just looking. 1100 00:48:20,420 --> 00:48:22,044 But imagine you can't, for some reason. 1101 00:48:22,044 --> 00:48:24,270 For whatever, it's in a box or you can't look at it. 1102 00:48:24,270 --> 00:48:26,010 Maybe it's just preposterously small. 1103 00:48:26,010 --> 00:48:29,356 How can you deduce what the shape of that hill is? 1104 00:48:29,356 --> 00:48:30,730 Well, one way to do it is to send 1105 00:48:30,730 --> 00:48:33,460 in particles as a function of energy, more and more energy, 1106 00:48:33,460 --> 00:48:36,360 and measure the probability that they transmit. 1107 00:48:36,360 --> 00:48:37,010 OK. 1108 00:48:37,010 --> 00:48:40,370 Now if you do so and you get this graph as a function of e, 1109 00:48:40,370 --> 00:48:42,700 what do you deduce? 1110 00:48:42,700 --> 00:48:45,480 You deduce that the barrier that you're scattering off of 1111 00:48:45,480 --> 00:48:48,960 is a square step with this height v0. 1112 00:48:52,070 --> 00:48:53,150 Are we cool with that? 1113 00:48:53,150 --> 00:48:55,290 So apparently, just look at the transmission 1114 00:48:55,290 --> 00:48:58,070 amplitudes, the transition probabilities, 1115 00:48:58,070 --> 00:49:00,270 you can deduce at least something 1116 00:49:00,270 --> 00:49:02,254 of the form of the potential. 1117 00:49:02,254 --> 00:49:03,170 Which is kind of cool. 1118 00:49:03,170 --> 00:49:04,545 If you didn't know the potential, 1119 00:49:04,545 --> 00:49:07,000 you could figure out what it was. 1120 00:49:07,000 --> 00:49:09,696 And this turns out to be a very general statement that you 1121 00:49:09,696 --> 00:49:11,696 can deduce an enormous amount, and as we'll see, 1122 00:49:11,696 --> 00:49:13,310 you can, in fact, deduce basically 1123 00:49:13,310 --> 00:49:15,230 everything you want of the potential 1124 00:49:15,230 --> 00:49:18,840 from knowing about the transmission probabilities 1125 00:49:18,840 --> 00:49:22,240 as well as the phase shift, the transmission amplitudes. 1126 00:49:22,240 --> 00:49:24,334 So this is the basic goal of scattering. 1127 00:49:24,334 --> 00:49:25,750 And so the way I want you to think 1128 00:49:25,750 --> 00:49:28,390 about it is imagine, for example, that someone 1129 00:49:28,390 --> 00:49:29,660 hands you an object. 1130 00:49:29,660 --> 00:49:30,890 A box. 1131 00:49:30,890 --> 00:49:34,610 And the box has an in port and it has an out port. 1132 00:49:34,610 --> 00:49:36,380 And they allow you to send in particles 1133 00:49:36,380 --> 00:49:38,535 as a function of energy and measure transmission 1134 00:49:38,535 --> 00:49:41,035 and reflection, you can measure transmission and reflection. 1135 00:49:41,035 --> 00:49:44,350 Just like I'm measuring transmission off of your faces 1136 00:49:44,350 --> 00:49:47,760 right now, from the light from above. 1137 00:49:47,760 --> 00:49:51,725 So suppose that you do so, put it on your test stand and you 1138 00:49:51,725 --> 00:49:52,600 measure transmission. 1139 00:49:52,600 --> 00:49:54,750 You measure transmission as a function of energy, 1140 00:49:54,750 --> 00:49:56,800 and you observe the following. 1141 00:49:56,800 --> 00:49:58,500 The transmission as a function of energy 1142 00:49:58,500 --> 00:50:03,770 is small up until some point, and then at some point, 1143 00:50:03,770 --> 00:50:07,260 which may be the minimum energy you can meaningfully probe, 1144 00:50:07,260 --> 00:50:08,620 you get something like this. 1145 00:50:19,380 --> 00:50:21,987 So here's the transmission as a function of energy. 1146 00:50:21,987 --> 00:50:22,820 So what can you say? 1147 00:50:25,534 --> 00:50:26,950 If this is all the information you 1148 00:50:26,950 --> 00:50:28,470 have about what's going on inside the box, 1149 00:50:28,470 --> 00:50:30,620 what can you deduce about the thing inside the box? 1150 00:50:33,340 --> 00:50:34,900 One thing you can deduce is that it 1151 00:50:34,900 --> 00:50:42,130 looks kind of like a potential with height around v0. 1152 00:50:42,130 --> 00:50:44,500 It looks kind of like a potential step with some height 1153 00:50:44,500 --> 00:50:45,380 v0. 1154 00:50:45,380 --> 00:50:48,460 This is asymptoting to 1. 1155 00:50:48,460 --> 00:50:53,070 However, it's not, because it has these oscillations in it. 1156 00:50:53,070 --> 00:50:55,720 So there's more to the potential than just a barrier of height 1157 00:50:55,720 --> 00:50:56,885 v0. 1158 00:50:56,885 --> 00:50:59,260 What I want to show you is that you can deduce everything 1159 00:50:59,260 --> 00:51:02,790 about that potential, and that's the point of scattering. 1160 00:51:02,790 --> 00:51:05,010 So let's do it. 1161 00:51:05,010 --> 00:51:11,600 So the goal here, again, to say it differently, is what's v0? 1162 00:51:11,600 --> 00:51:13,866 What is v of x? 1163 00:51:13,866 --> 00:51:15,740 Not just the height, but the total potential. 1164 00:51:18,152 --> 00:51:19,860 So another way to say this, let me set up 1165 00:51:19,860 --> 00:51:22,457 a precise version of this question. 1166 00:51:22,457 --> 00:51:24,040 I want to be able to do the following. 1167 00:51:24,040 --> 00:51:26,623 I want to take a system that has a potential which is constant 1168 00:51:26,623 --> 00:51:31,670 up to some point which I'll call 0, 1169 00:51:31,670 --> 00:51:34,660 and then again from some point, which I'll call L, 1170 00:51:34,660 --> 00:51:35,860 is constant again. 1171 00:51:35,860 --> 00:51:39,940 And inside, I don't know what the potential is. 1172 00:51:43,569 --> 00:51:45,360 So in here, there's some unknown potential, 1173 00:51:45,360 --> 00:51:47,480 v of x, which is some crazy thing. 1174 00:51:47,480 --> 00:51:48,660 It could be doing anything. 1175 00:51:48,660 --> 00:51:54,230 It could be some crazy-- it could have horns and whatever. 1176 00:51:54,230 --> 00:51:57,287 It could be awful. 1177 00:51:57,287 --> 00:51:59,620 But the potential is constant if you go far enough away, 1178 00:51:59,620 --> 00:52:01,953 and the potential is constant if you go far enough away. 1179 00:52:01,953 --> 00:52:05,980 A good example of this is a hydrogen atom. 1180 00:52:05,980 --> 00:52:08,380 It's neutral but there's a clearly and complicated 1181 00:52:08,380 --> 00:52:10,550 potential inside because the proton and the electron 1182 00:52:10,550 --> 00:52:13,350 are moving around in there in some quantum state, anyway, 1183 00:52:13,350 --> 00:52:15,366 and if you send something at it, far away, it's 1184 00:52:15,366 --> 00:52:16,240 as if it's not there. 1185 00:52:16,240 --> 00:52:18,950 But close by, you know there are strong electrostatic forces. 1186 00:52:18,950 --> 00:52:21,325 And so the question is what you learn about those forces, 1187 00:52:21,325 --> 00:52:23,260 what can you learn about the potential 1188 00:52:23,260 --> 00:52:26,120 by throwing things in from far away, from either side. 1189 00:52:26,120 --> 00:52:28,660 Now one thing we know already is that out here, the wave 1190 00:52:28,660 --> 00:52:31,170 function always-- because it's a constant potential-- always 1191 00:52:31,170 --> 00:52:36,750 takes the form e to the ikx plus b e to the minus ikx. 1192 00:52:36,750 --> 00:52:41,740 And out here it takes the form c e to the ikx plus d 1193 00:52:41,740 --> 00:52:44,550 e to the minus ikx. 1194 00:52:44,550 --> 00:52:47,350 And again, this corresponds to moving in. 1195 00:52:47,350 --> 00:52:51,010 d is in from the right. 1196 00:52:51,010 --> 00:52:54,330 c is out to the right. 1197 00:52:54,330 --> 00:52:56,580 b is out to the left. 1198 00:52:56,580 --> 00:53:00,735 And a is in from the left. 1199 00:53:00,735 --> 00:53:02,360 So again, there are basically four kind 1200 00:53:02,360 --> 00:53:03,430 of scattering experiments we can do. 1201 00:53:03,430 --> 00:53:05,250 We can send things in from the right, which 1202 00:53:05,250 --> 00:53:06,680 corresponds to setting a to 0. 1203 00:53:06,680 --> 00:53:08,430 We can send things in from the left, which 1204 00:53:08,430 --> 00:53:10,460 corresponds to setting t equal to 0. 1205 00:53:10,460 --> 00:53:13,405 And all the information about what happens in v 1206 00:53:13,405 --> 00:53:15,740 is going to be encoded in what's coming out, 1207 00:53:15,740 --> 00:53:17,787 the b and c coefficients. 1208 00:53:17,787 --> 00:53:19,370 And the way to make that sharp is just 1209 00:53:19,370 --> 00:53:22,817 to notice that the transmission probability, if we compute 1210 00:53:22,817 --> 00:53:25,150 for this system, assuming it's forming the wave function 1211 00:53:25,150 --> 00:53:28,450 asymptotically away from the potential, the transmission 1212 00:53:28,450 --> 00:53:31,800 amplitude is just c over a squared 1213 00:53:31,800 --> 00:53:33,500 when you're sending in from the left. 1214 00:53:33,500 --> 00:53:41,260 And the reflection is equal to b over a norm squared. 1215 00:53:41,260 --> 00:53:42,987 This squared is a squared. 1216 00:53:42,987 --> 00:53:45,070 [INAUDIBLE] function over amplitude squared, good. 1217 00:53:49,540 --> 00:53:52,280 To learn about the transmission and reflection coefficients, 1218 00:53:52,280 --> 00:54:01,840 it's enough-- suffices-- to compute, to know b and c 1219 00:54:01,840 --> 00:54:05,905 as a function of a and d. 1220 00:54:05,905 --> 00:54:07,280 All of the scattering information 1221 00:54:07,280 --> 00:54:09,230 is in those coefficients b and c for a and d. 1222 00:54:09,230 --> 00:54:12,380 And here I'm assuming that I'm sending in a monochromatic wave 1223 00:54:12,380 --> 00:54:15,040 with a single, well defined energy. 1224 00:54:15,040 --> 00:54:18,446 I'm sending in a beam of particles with energy e. 1225 00:54:18,446 --> 00:54:20,070 I don't know where they are, but I sure 1226 00:54:20,070 --> 00:54:22,360 know what their momentum is. 1227 00:54:22,360 --> 00:54:28,350 So some well defined beam of particles with energy e. 1228 00:54:28,350 --> 00:54:30,490 And these probabilities are going 1229 00:54:30,490 --> 00:54:33,910 to contain all the data I want. 1230 00:54:33,910 --> 00:54:36,560 So this is the basic project of scattering. 1231 00:54:36,560 --> 00:54:37,250 Questions? 1232 00:54:37,250 --> 00:54:40,040 AUDIENCE: So basically, it only depends 1233 00:54:40,040 --> 00:54:43,414 on the transmission-- it only depends on the edges? 1234 00:54:43,414 --> 00:54:44,830 PROFESSOR: That's a good question. 1235 00:54:44,830 --> 00:54:46,913 The question is, does the transmission depend only 1236 00:54:46,913 --> 00:54:48,430 on what you do with the edges. 1237 00:54:48,430 --> 00:54:49,930 And here's the important thing. 1238 00:54:49,930 --> 00:54:51,640 The transmission depends crucially 1239 00:54:51,640 --> 00:54:52,785 on what happens in here. 1240 00:54:52,785 --> 00:54:54,910 For example, if this is an infinitely high barrier, 1241 00:54:54,910 --> 00:54:56,940 nothing's going across. 1242 00:54:56,940 --> 00:54:59,360 So this transmission depends on what's in here. 1243 00:54:59,360 --> 00:55:04,870 But the point is we can deduce just by looking far away, 1244 00:55:04,870 --> 00:55:06,890 we can deduce the transmission probability 1245 00:55:06,890 --> 00:55:09,740 and amplitude just by measuring b and c far away, 1246 00:55:09,740 --> 00:55:11,390 b and c far away. 1247 00:55:11,390 --> 00:55:12,180 OK. 1248 00:55:12,180 --> 00:55:13,530 So the transmission amplitude is something 1249 00:55:13,530 --> 00:55:14,920 you measure when very far away. 1250 00:55:14,920 --> 00:55:17,045 You measure-- if I throw something in from very far 1251 00:55:17,045 --> 00:55:19,740 out here to the left, how likely is it to get out here very 1252 00:55:19,740 --> 00:55:21,120 far to the right? 1253 00:55:21,120 --> 00:55:23,194 And in order to answer that, if someone 1254 00:55:23,194 --> 00:55:24,610 hands you the answer to that, they 1255 00:55:24,610 --> 00:55:26,930 must have solved for what's inside. 1256 00:55:26,930 --> 00:55:27,810 That the point. 1257 00:55:27,810 --> 00:55:29,810 So knowing the answer to that question 1258 00:55:29,810 --> 00:55:34,282 encodes information about what happened in between. 1259 00:55:34,282 --> 00:55:37,560 AUDIENCE: So I guess initially, your potential's 1260 00:55:37,560 --> 00:55:40,511 going to-- so say it's stuff you did earlier. 1261 00:55:40,511 --> 00:55:42,946 The potential drops down to the [INAUDIBLE] in the box. 1262 00:55:42,946 --> 00:55:43,920 Is that going to be problematic? 1263 00:55:43,920 --> 00:55:45,520 PROFESSOR: Yeah, for simplicity, I'm 1264 00:55:45,520 --> 00:55:47,370 going to assume that the potential always goes to 0 when 1265 00:55:47,370 --> 00:55:49,953 we're far away, because that's going to be useful for modeling 1266 00:55:49,953 --> 00:55:55,070 things like hydrogen and, exactly. 1267 00:55:55,070 --> 00:55:57,050 Carbon, we're going to do diamond later 1268 00:55:57,050 --> 00:55:59,540 in the semester, that'll be useful. 1269 00:55:59,540 --> 00:56:01,020 But we could repeat this analysis 1270 00:56:01,020 --> 00:56:04,020 by adding an extra change in the asymptotic potential, 1271 00:56:04,020 --> 00:56:05,870 it doesn't really change anything important. 1272 00:56:05,870 --> 00:56:06,828 Yeah. 1273 00:56:06,828 --> 00:56:09,702 AUDIENCE: It looks like even for just the simple step up 1274 00:56:09,702 --> 00:56:12,256 [INAUDIBLE] you can't tell from just the probabilities why 1275 00:56:12,256 --> 00:56:14,360 the step is going up or going down. 1276 00:56:14,360 --> 00:56:15,760 PROFESSOR: Ah, excellent. 1277 00:56:15,760 --> 00:56:16,680 Excellent, excellent. 1278 00:56:16,680 --> 00:56:19,430 So good, thank you for that question, that's really great. 1279 00:56:19,430 --> 00:56:22,700 So already, it seems like we can't uniquely 1280 00:56:22,700 --> 00:56:27,749 identify the potential from the transmission probability 1281 00:56:27,749 --> 00:56:30,040 if the transmission probability is the same for step up 1282 00:56:30,040 --> 00:56:32,120 or step down. 1283 00:56:32,120 --> 00:56:34,376 So what's missing? 1284 00:56:34,376 --> 00:56:37,777 AUDIENCE: Maybe the energy [INAUDIBLE] 1285 00:56:37,777 --> 00:56:40,110 PROFESSOR: But we're working with an energy [INAUDIBLE]. 1286 00:56:40,110 --> 00:56:42,670 So the energy is just a global constant. 1287 00:56:42,670 --> 00:56:44,330 We'll see what's missing, and what's 1288 00:56:44,330 --> 00:56:47,130 missing is something called the phase shift. 1289 00:56:47,130 --> 00:56:50,390 So very good question, yes. 1290 00:56:50,390 --> 00:56:56,078 AUDIENCE: It looks like when we did the example for the step, 1291 00:56:56,078 --> 00:56:58,863 that t equals [INAUDIBLE] over a squared. 1292 00:56:58,863 --> 00:57:01,950 PROFESSOR: Yeah, it is, because it's the ratio. 1293 00:57:01,950 --> 00:57:03,680 That's why I wrote t is c over here. 1294 00:57:03,680 --> 00:57:06,150 So t is the ratio of j over ji. 1295 00:57:06,150 --> 00:57:14,220 And in fact, jt here is equal to c squared times-- 1296 00:57:14,220 --> 00:57:15,834 so this is the probability density, 1297 00:57:15,834 --> 00:57:18,250 so it's the probability density times the velocity, what's 1298 00:57:18,250 --> 00:57:19,480 the effect of velocity here? 1299 00:57:19,480 --> 00:57:26,560 H bar k 2 over m, whereas j incident is equal 1300 00:57:26,560 --> 00:57:29,450 to a squared-- probability density 1301 00:57:29,450 --> 00:57:31,480 times the momentum there, the velocity there-- 1302 00:57:31,480 --> 00:57:34,720 which is h bar k1 upon m. 1303 00:57:34,720 --> 00:57:38,600 So here the ratio of j transmitted to j incident 1304 00:57:38,600 --> 00:57:43,265 is norm c squared k2 over norm a squared k1. 1305 00:57:43,265 --> 00:57:44,806 AUDIENCE: So because it's not level-- 1306 00:57:44,806 --> 00:57:46,340 PROFESSOR: Exactly, it's because it's not level. 1307 00:57:46,340 --> 00:57:47,800 So here, they happen to be level, 1308 00:57:47,800 --> 00:57:50,390 so they're only [INAUDIBLE] factor, cancel, 1309 00:57:50,390 --> 00:57:52,900 and the only thing that survives is the amplitude. 1310 00:57:52,900 --> 00:57:53,947 Good question. 1311 00:57:53,947 --> 00:57:55,438 Yeah. 1312 00:57:55,438 --> 00:57:59,414 AUDIENCE: For the lead box example, 1313 00:57:59,414 --> 00:58:02,010 is it sufficient just to know what's 1314 00:58:02,010 --> 00:58:05,047 reflected back to solve the situation? 1315 00:58:05,047 --> 00:58:06,880 PROFESSOR: So for precisely for this reason, 1316 00:58:06,880 --> 00:58:08,961 it's not sufficient to know t. 1317 00:58:08,961 --> 00:58:10,460 It's not sufficient to know t and r. 1318 00:58:10,460 --> 00:58:12,431 But, of course, once you know r, you know t, 1319 00:58:12,431 --> 00:58:13,430 so you're exactly right. 1320 00:58:13,430 --> 00:58:14,665 Once you know the reflection probability, 1321 00:58:14,665 --> 00:58:16,336 you know the transmission probability, 1322 00:58:16,336 --> 00:58:17,960 but there's one more bit of information 1323 00:58:17,960 --> 00:58:19,584 which we're going to also need in order 1324 00:58:19,584 --> 00:58:23,565 to specify the potential, which is going to be the phase shift. 1325 00:58:23,565 --> 00:58:25,940 But you're right, you don't need to independently compute 1326 00:58:25,940 --> 00:58:27,290 r and t, you can just compute one. 1327 00:58:27,290 --> 00:58:29,456 AUDIENCE: You need sensors on both sides of the box, 1328 00:58:29,456 --> 00:58:30,421 to answer my question. 1329 00:58:30,421 --> 00:58:32,722 PROFESSOR: You don't need sensors 1330 00:58:32,722 --> 00:58:35,180 on both sides of the box, but you need to do more than just 1331 00:58:35,180 --> 00:58:37,020 do the counting problem. 1332 00:58:37,020 --> 00:58:39,710 We'll see that. 1333 00:58:39,710 --> 00:58:41,040 OK. 1334 00:58:41,040 --> 00:58:44,390 So let's work out a simple example, the simplest 1335 00:58:44,390 --> 00:58:46,520 example of a barrier of this kind. 1336 00:58:46,520 --> 00:58:51,810 We want constant potential, and then ending at 0, 1337 00:58:51,810 --> 00:58:57,160 and we want a constant again from L going off to infinity. 1338 00:58:57,160 --> 00:58:59,930 So what's the easiest possible thing we could do? 1339 00:58:59,930 --> 00:59:01,200 Step, step. 1340 00:59:01,200 --> 00:59:04,050 We're just doing what we've done before twice. 1341 00:59:04,050 --> 00:59:06,520 So this is an example of this kind of potential. 1342 00:59:06,520 --> 00:59:10,390 It's sort of ridiculously simple, but let's work it out. 1343 00:59:10,390 --> 00:59:12,035 So we want scattering, let's start out, 1344 00:59:12,035 --> 00:59:13,090 we could do either scattering from the left 1345 00:59:13,090 --> 00:59:14,220 or scattering from the right. 1346 00:59:14,220 --> 00:59:16,600 Let's start out scattering from the left, so d equals 0, 1347 00:59:16,600 --> 00:59:19,550 and let's study this problem. 1348 00:59:19,550 --> 00:59:25,291 So what we know-- and let's also note 1349 00:59:25,291 --> 00:59:26,540 that we have a choice to make. 1350 00:59:26,540 --> 00:59:28,555 We could either study energy below the height 1351 00:59:28,555 --> 00:59:30,180 of the potential or we can study energy 1352 00:59:30,180 --> 00:59:32,749 above the height of the potential. 1353 00:59:32,749 --> 00:59:34,290 And so for simplicity, I'm also going 1354 00:59:34,290 --> 00:59:36,123 to start with energy greater than the height 1355 00:59:36,123 --> 00:59:40,511 of the potential, v0, and then we'll do e less than v0 1356 00:59:40,511 --> 00:59:41,010 afterwards. 1357 00:59:41,010 --> 00:59:42,926 It'll be an easy extension of what we've done. 1358 00:59:46,760 --> 00:59:50,217 OK, so this will be our first case to study. 1359 00:59:50,217 --> 00:59:52,300 So we know the form of the wave function out here, 1360 00:59:52,300 --> 00:59:54,815 it's a e to the ikx, b to the minus ikx. 1361 00:59:54,815 --> 00:59:56,440 We know that for the potential out here 1362 00:59:56,440 --> 00:59:58,499 it's c e to the ikx and d e to the minus ikx. 1363 00:59:58,499 --> 01:00:00,165 The only thing we don't know is the form 1364 01:00:00,165 --> 01:00:02,117 of the potential in here. 1365 01:00:02,117 --> 01:00:03,700 And in here it's actually very simple. 1366 01:00:03,700 --> 01:00:05,430 It's got to be something of the form-- I 1367 01:00:05,430 --> 01:00:07,920 think I called it f and g, I did. 1368 01:00:07,920 --> 01:00:13,460 f e to the ik prime x-- I'm calling this k, 1369 01:00:13,460 --> 01:00:16,396 so I'll just call this k prime x-- plus g 1370 01:00:16,396 --> 01:00:19,090 e to the minus ik prime x. 1371 01:00:19,090 --> 01:00:20,690 And the reason I chose k prime is 1372 01:00:20,690 --> 01:00:23,350 because we're working with energy greater 1373 01:00:23,350 --> 01:00:25,850 than the potential, so this is a classically allowed region. 1374 01:00:25,850 --> 01:00:28,980 It's an oscillatory domain but with a different k prime. 1375 01:00:28,980 --> 01:00:34,850 So here, k squared, h bar squared over 2m is equal to e, 1376 01:00:34,850 --> 01:00:39,800 and h bar squared k prime squared over 2m 1377 01:00:39,800 --> 01:00:42,510 is equal to e minus v0, which is positive 1378 01:00:42,510 --> 01:00:44,270 when e is greater than v0. 1379 01:00:44,270 --> 01:00:46,500 This analysis will not obtain when e is less than v0, 1380 01:00:46,500 --> 01:00:49,200 we'll have to treat it separately. 1381 01:00:49,200 --> 01:00:50,407 So now what do we do? 1382 01:00:50,407 --> 01:00:52,740 We do the same thing we did before, we just do it twice. 1383 01:00:52,740 --> 01:00:54,080 We'll do the matching conditions here, 1384 01:00:54,080 --> 01:00:55,410 the matching conditions here. 1385 01:00:55,410 --> 01:00:58,520 That's going to give us 1, 2, 3, 4 matching conditions. 1386 01:00:58,520 --> 01:01:01,922 We have 1, 2, 3, 4, 5, 6 unknown coefficients, 1387 01:01:01,922 --> 01:01:03,380 so we'll have two independent ones. 1388 01:01:03,380 --> 01:01:04,170 That's great. 1389 01:01:04,170 --> 01:01:07,240 We set d equal to 0 to specify that it's 1390 01:01:07,240 --> 01:01:10,949 coming in from the left and not from the right, that's 5. 1391 01:01:10,949 --> 01:01:12,740 And then we have normalization, which is 6, 1392 01:01:12,740 --> 01:01:15,850 so this should uniquely specify our wave function. 1393 01:01:15,850 --> 01:01:17,110 Yeah. 1394 01:01:17,110 --> 01:01:19,295 Once we've fixed e, we have enough conditions. 1395 01:01:19,295 --> 01:01:21,170 So I'm not going to go through the derivation 1396 01:01:21,170 --> 01:01:23,430 because it's just an extension of what we did for the first. 1397 01:01:23,430 --> 01:01:24,888 It's just a whole bunch of algebra. 1398 01:01:24,888 --> 01:01:26,200 And let me just emphasize this. 1399 01:01:26,200 --> 01:01:27,650 The algebra is not interesting. 1400 01:01:27,650 --> 01:01:28,465 It's just algebra. 1401 01:01:28,465 --> 01:01:29,840 You have to be able to do it, you 1402 01:01:29,840 --> 01:01:31,760 have to develop some familiarity with it, 1403 01:01:31,760 --> 01:01:33,380 and it's easy to get good at this. 1404 01:01:33,380 --> 01:01:34,370 You just practice. 1405 01:01:34,370 --> 01:01:36,910 It's just algebra. 1406 01:01:36,910 --> 01:01:40,260 But once you get the idea, don't ever do it again. 1407 01:01:40,260 --> 01:01:42,270 Once you get reasonably quick at it, 1408 01:01:42,270 --> 01:01:44,750 learn to use Mathematica, Maple, whatever package you want, 1409 01:01:44,750 --> 01:01:47,520 and use computer algebra to check your analysis. 1410 01:01:47,520 --> 01:01:49,540 And use your physics to check the answer 1411 01:01:49,540 --> 01:01:52,570 you get from Mathematica or Maple or whatever you use. 1412 01:01:52,570 --> 01:01:55,110 Always check against physical reasonability, 1413 01:01:55,110 --> 01:01:56,230 but use Mathematica. 1414 01:01:56,230 --> 01:01:59,097 So posted on the website are Mathematica files 1415 01:01:59,097 --> 01:02:01,305 that walk through the computation of the transmission 1416 01:02:01,305 --> 01:02:03,900 and reflection amplitudes and probabilities 1417 01:02:03,900 --> 01:02:06,690 for this potential, and I think maybe another one, 1418 01:02:06,690 --> 01:02:08,370 I don't remember exactly. 1419 01:02:08,370 --> 01:02:11,546 But I encourage you strongly to use computer algebra tools, 1420 01:02:11,546 --> 01:02:12,920 because it's just a waste of time 1421 01:02:12,920 --> 01:02:15,003 to spend three hours doing an algebra calculation. 1422 01:02:15,003 --> 01:02:18,380 In particular, on your problem set this week, 1423 01:02:18,380 --> 01:02:22,620 you will do a scattering problem similar to the bound state 1424 01:02:22,620 --> 01:02:25,120 probably you did last week, the quantum glue problem-- which 1425 01:02:25,120 --> 01:02:27,935 you may be doing tonight, the one due tomorrow. 1426 01:02:27,935 --> 01:02:29,310 Which is two delta function wells 1427 01:02:29,310 --> 01:02:31,210 and find the bound states, so that 1428 01:02:31,210 --> 01:02:32,650 involves a fair amount of algebra. 1429 01:02:32,650 --> 01:02:34,185 The scattering problem will involve 1430 01:02:34,185 --> 01:02:35,351 a similar amount of algebra. 1431 01:02:35,351 --> 01:02:37,490 Do not do it. 1432 01:02:37,490 --> 01:02:38,966 Use Mathematica or computer algebra 1433 01:02:38,966 --> 01:02:40,090 just to simplify your life. 1434 01:02:46,430 --> 01:02:49,405 So if we go through and compute the-- so what 1435 01:02:49,405 --> 01:02:51,530 are we going to do, we're going to use the matching 1436 01:02:51,530 --> 01:02:54,330 conditions here to determine f and g in terms of a and b, 1437 01:02:54,330 --> 01:02:56,150 then we'll use the matching conditions here 1438 01:02:56,150 --> 01:03:02,480 to determine c-- d is 0-- c in terms of f and g. 1439 01:03:02,480 --> 01:03:04,970 So that's going to give us an effective constraint relating 1440 01:03:04,970 --> 01:03:07,600 a and b, leaving us with an overall unknown coefficient 1441 01:03:07,600 --> 01:03:09,470 a, which we'll use for normalization. 1442 01:03:09,470 --> 01:03:13,010 The upshot of all of which is the answers 1443 01:03:13,010 --> 01:03:15,300 are that, I'm not even going to write down-- they're 1444 01:03:15,300 --> 01:03:16,790 in the notes. 1445 01:03:16,790 --> 01:03:18,330 Should I write this down? 1446 01:03:18,330 --> 01:03:18,940 I will skip. 1447 01:03:18,940 --> 01:03:21,160 So the upshot is that the transmission amplitude, 1448 01:03:21,160 --> 01:03:24,840 as a function of k and k prime-- the transmission probability, I 1449 01:03:24,840 --> 01:03:26,437 should say, is 1 over-- actually, 1450 01:03:26,437 --> 01:03:28,020 I'm going to need the whole amplitude. 1451 01:03:28,020 --> 01:03:28,520 Shoot. 1452 01:03:36,740 --> 01:03:40,210 The transmission probability is equal to-- 1453 01:03:40,210 --> 01:03:51,190 and this is a horribly long expression-- 1454 01:03:51,190 --> 01:03:57,200 the transmission probability, which is c over a norm squared, 1455 01:03:57,200 --> 01:04:08,110 is equal to 4 k squared k prime squared cosine squared of k 1456 01:04:08,110 --> 01:04:19,890 prime l plus k squared plus k prime squared sine squared 1457 01:04:19,890 --> 01:04:30,945 of k prime l under 4k squared k prime squared. 1458 01:04:30,945 --> 01:04:31,445 Seriously. 1459 01:04:34,450 --> 01:04:36,970 So we can simplify this out, so you can do some algebra. 1460 01:04:36,970 --> 01:04:38,595 This is just what you get when you just 1461 01:04:38,595 --> 01:04:40,950 naively do the algebra. 1462 01:04:40,950 --> 01:04:41,960 I want to do two things. 1463 01:04:41,960 --> 01:04:43,189 First off, this is horrible. 1464 01:04:43,189 --> 01:04:45,230 There's a cosine squared, there's a sine squared, 1465 01:04:45,230 --> 01:04:47,730 surely we can all be friends and put it together. 1466 01:04:47,730 --> 01:04:48,915 So let's use some trig. 1467 01:04:48,915 --> 01:04:51,040 But the second thing, and the more important thing, 1468 01:04:51,040 --> 01:04:53,150 is I want to put this in dimensionless form. 1469 01:04:53,150 --> 01:04:53,870 This is horrible. 1470 01:04:53,870 --> 01:04:58,040 Here we have ks and we have ls, and these all have dimensions, 1471 01:04:58,040 --> 01:05:00,939 and they're inside the sines and the cosines it's kl. 1472 01:05:00,939 --> 01:05:03,230 That's good, because this has units of one over length, 1473 01:05:03,230 --> 01:05:04,990 this has units of length, so that's dimensions. 1474 01:05:04,990 --> 01:05:06,890 Let's put everything in dimensionless form. 1475 01:05:06,890 --> 01:05:09,410 And in particular, what are the parameters of my system? 1476 01:05:09,410 --> 01:05:11,660 The parameters of my system are, well, there's a mass, 1477 01:05:11,660 --> 01:05:15,400 there's an h bar, there's a v0, and then there's an energy, 1478 01:05:15,400 --> 01:05:17,590 and there's a length l. 1479 01:05:17,590 --> 01:05:20,500 So it's easy to make a dimensionless parameter out 1480 01:05:20,500 --> 01:05:24,341 of these guys, and a ratio of energies-- 1481 01:05:24,341 --> 01:05:26,590 a dimensionless ratio of energies-- out of these guys. 1482 01:05:26,590 --> 01:05:29,750 So I'm going to do that, and the parameters I'm going to use 1483 01:05:29,750 --> 01:05:32,540 are coming from here. 1484 01:05:40,550 --> 01:05:46,630 I'm going to define the parameters g0 squared, which 1485 01:05:46,630 --> 01:05:49,450 is a dimensionless measure of the depth of the potential. 1486 01:05:49,450 --> 01:05:51,890 We've actually run into this guy before. 1487 01:05:51,890 --> 01:05:57,600 2m l squared v0 over h bar squared. 1488 01:05:57,600 --> 01:06:01,237 So this is h bar squared, 1 over l squared is k squared over 2m, 1489 01:06:01,237 --> 01:06:02,070 so that's an energy. 1490 01:06:02,070 --> 01:06:04,250 So this is a ratio of the height of the potential 1491 01:06:04,250 --> 01:06:07,030 to the characteristic energy corresponding to length scale 1492 01:06:07,030 --> 01:06:08,295 l. 1493 01:06:08,295 --> 01:06:10,420 So the width, there's an energy corresponding to it 1494 01:06:10,420 --> 01:06:13,061 because you take a momentum which has 1 over that width. 1495 01:06:13,061 --> 01:06:15,560 You can build an energy out of that h bar k squared over 2m. 1496 01:06:15,560 --> 01:06:17,935 And we have an energy which is the height of the barrier, 1497 01:06:17,935 --> 01:06:19,600 we take the ratio of those. 1498 01:06:19,600 --> 01:06:21,450 So that's a dimensionless quantity, g0. 1499 01:06:21,450 --> 01:06:23,860 And the other dimensionless quantity I want to consider 1500 01:06:23,860 --> 01:06:26,050 is a ratio of the energy, e, to v0, 1501 01:06:26,050 --> 01:06:29,315 which is what showed up before in our energy plot. 1502 01:06:32,850 --> 01:06:35,820 Or in our transmission plot over there, e over v0. 1503 01:06:35,820 --> 01:06:38,230 So when we take this and we do a little bit of algebra 1504 01:06:38,230 --> 01:06:41,740 to simplify our life, again, use Mathematica, it's your friend. 1505 01:06:41,740 --> 01:06:45,660 The result is much more palatable. 1506 01:06:45,660 --> 01:06:50,990 It's t-- again, for the energy greater than v0-- 1507 01:06:50,990 --> 01:06:54,490 is equal to, still long. 1508 01:06:54,490 --> 01:07:03,230 But 1 plus 1 over 4 epsilon, epsilon minus 1 sine 1509 01:07:03,230 --> 01:07:10,960 squared of g0 square root of epsilon minus 1. 1510 01:07:15,400 --> 01:07:16,645 And upstairs is a one. 1511 01:07:21,900 --> 01:07:24,270 So remember this is only valid for e greater than v0, 1512 01:07:24,270 --> 01:07:26,630 or equivalently, epsilon greater than 1. 1513 01:07:26,630 --> 01:07:28,640 And I guess we can put an equal in. 1514 01:07:32,590 --> 01:07:34,267 So when you get an expression like this, 1515 01:07:34,267 --> 01:07:36,600 this is as easy as you're going to make this expression. 1516 01:07:36,600 --> 01:07:37,974 It's not going to get any easier. 1517 01:07:37,974 --> 01:07:41,430 It's 1 over a sine times a function plus 1. 1518 01:07:41,430 --> 01:07:43,542 There's really no great way to simplify this. 1519 01:07:43,542 --> 01:07:45,750 So what you need when you get an expression like this 1520 01:07:45,750 --> 01:07:47,549 is try to figure out what it's telling you. 1521 01:07:47,549 --> 01:07:49,090 The useful thing to do is to plot it. 1522 01:07:49,090 --> 01:07:50,682 So let's just look at this function 1523 01:07:50,682 --> 01:07:51,890 and see what it's telling us. 1524 01:07:51,890 --> 01:07:55,560 Let's plot this t as a function of epsilon 1525 01:07:55,560 --> 01:07:56,750 and for some fixed g0. 1526 01:07:59,460 --> 01:08:01,960 Keep in mind that this only makes sense for e greater 1527 01:08:01,960 --> 01:08:04,564 than v0 or for epsilon greater than 1, so here's 1. 1528 01:08:04,564 --> 01:08:05,980 And we're going to remain agnostic 1529 01:08:05,980 --> 01:08:08,780 as to what happens below 1. 1530 01:08:08,780 --> 01:08:11,310 And just for normalization, we know that the transmission 1531 01:08:11,310 --> 01:08:12,976 probability can never be greater than 1, 1532 01:08:12,976 --> 01:08:15,180 so it's got to be between 1 and 0. 1533 01:08:15,180 --> 01:08:19,260 So here's 0 and 0. 1534 01:08:19,260 --> 01:08:22,207 We're remaining agnostic about this for the moment. 1535 01:08:22,207 --> 01:08:24,540 So let's start thinking about what this plot looks like. 1536 01:08:24,540 --> 01:08:26,620 First off, what does it look like at 1? 1537 01:08:26,620 --> 01:08:29,020 So when epsilon goes to 1, this is 1538 01:08:29,020 --> 01:08:31,380 going to 0, that's bad because it's in a denominator. 1539 01:08:31,380 --> 01:08:34,130 But upstairs, this is going to 0, 1540 01:08:34,130 --> 01:08:37,149 and sine squared of something when it's becoming small 1541 01:08:37,149 --> 01:08:39,210 goes like-- well, sine goes like that thing. 1542 01:08:39,210 --> 01:08:41,840 So sine squared goes like this quantity squared. 1543 01:08:41,840 --> 01:08:44,100 So sine squared goes like g0 squared, 1544 01:08:44,100 --> 01:08:47,260 this goes like 1 over-- this is going 1545 01:08:47,260 --> 01:08:50,270 like g0 squared epsilon minus 1 square root quantity 1546 01:08:50,270 --> 01:08:52,040 squared, which is epsilon minus 1. 1547 01:08:52,040 --> 01:08:54,290 So this is going to 0 and this, the denominator, 1548 01:08:54,290 --> 01:08:56,290 is going to 0 exactly in the same way. 1549 01:08:56,290 --> 01:08:58,359 Epsilon minus 1 from here, epsilon minus 1 1550 01:08:58,359 --> 01:08:59,240 downstairs from here. 1551 01:08:59,240 --> 01:09:01,770 So the epsilon minus ones precisely cancel. 1552 01:09:01,770 --> 01:09:04,810 From the sine squared we get a g0 squared, 1553 01:09:04,810 --> 01:09:06,300 and from here we get a 4 epsilon. 1554 01:09:06,300 --> 01:09:10,840 So we get 1 plus g0 squared over 4 epsilon. 1555 01:09:10,840 --> 01:09:14,010 But 4 epsilon, what was epsilon here? 1556 01:09:14,010 --> 01:09:14,672 1. 1557 01:09:14,672 --> 01:09:16,130 We're looking at epsilon goes to 1, 1558 01:09:16,130 --> 01:09:18,109 so this is just g0 squared over 4. 1559 01:09:18,109 --> 01:09:22,779 So the height, the value of t-- so we're not looking in here. 1560 01:09:22,779 --> 01:09:26,399 But at energy is equal to v0 or epsilon is equal to 1, 1561 01:09:26,399 --> 01:09:29,220 we know that the transmission amplitude is not 0, 1562 01:09:29,220 --> 01:09:32,091 but it's also strictly smaller than 1. 1563 01:09:32,091 --> 01:09:33,590 Which is good, because if it were 6, 1564 01:09:33,590 --> 01:09:35,630 you'd be really worried. 1565 01:09:35,630 --> 01:09:39,600 So g0 squared, if g0 squared is 0, what do we get? 1566 01:09:39,600 --> 01:09:40,100 1. 1567 01:09:40,100 --> 01:09:41,189 Fantastic, there's no barrier. 1568 01:09:41,189 --> 01:09:43,700 We just keep right on going through, perfect transmission. 1569 01:09:43,700 --> 01:09:47,510 If g0 is not zero, however, the transmission is suppressed. 1570 01:09:47,510 --> 01:09:48,660 Like 1 over g0 squared. 1571 01:09:51,240 --> 01:09:55,320 What does this actually look like with some value. 1572 01:09:55,320 --> 01:09:59,400 And what is this value, it's just 1 over 1 1573 01:09:59,400 --> 01:10:03,620 plus 1/4 g0 squared. 1574 01:10:03,620 --> 01:10:05,060 Cool? 1575 01:10:05,060 --> 01:10:05,824 OK. 1576 01:10:05,824 --> 01:10:07,240 And now what happens, for example, 1577 01:10:07,240 --> 01:10:09,780 for a very large epsilon? 1578 01:10:09,780 --> 01:10:14,430 Well, when epsilon is gigantic, sine squared of this-- well, 1579 01:10:14,430 --> 01:10:15,930 sine squared is oscillating rapidly, 1580 01:10:15,930 --> 01:10:16,850 so that's kind of worrying. 1581 01:10:16,850 --> 01:10:18,390 If this is very, very large, this 1582 01:10:18,390 --> 01:10:19,960 is a rapidly oscillating function. 1583 01:10:19,960 --> 01:10:22,796 However, it's being divided by roughly epsilon squared. 1584 01:10:22,796 --> 01:10:25,170 So something that goes between 0 and 1 divided by epsilon 1585 01:10:25,170 --> 01:10:28,090 squared, as epsilon gets large, becomes 0. 1586 01:10:28,090 --> 01:10:30,560 And so we get 1 over 1 plus 0, we get 1. 1587 01:10:30,560 --> 01:10:33,780 So for very large values it's asymptoting to 1. 1588 01:10:33,780 --> 01:10:36,760 So naively, it's going to do something like this. 1589 01:10:36,760 --> 01:10:38,425 However, there's this sine squared. 1590 01:10:38,425 --> 01:10:39,800 And in fact, this is exactly what 1591 01:10:39,800 --> 01:10:43,900 it would do if we just had the 1 over epsilon times 1592 01:10:43,900 --> 01:10:44,577 some constant. 1593 01:10:44,577 --> 01:10:46,160 But in fact, we have the sine squared, 1594 01:10:46,160 --> 01:10:47,868 and the sine squared is making it wiggle. 1595 01:10:47,868 --> 01:10:51,875 And the frequency of the wiggle is g0, 1596 01:10:51,875 --> 01:10:53,500 except that it's not linear in epsilon, 1597 01:10:53,500 --> 01:10:55,342 it's linear in square root of epsilon. 1598 01:10:55,342 --> 01:10:57,550 So as epsilon gets larger, the square root of epsilon 1599 01:10:57,550 --> 01:11:00,740 is getting larger less than linearly. 1600 01:11:00,740 --> 01:11:03,650 So what that's telling you is if you looked at root epsilon, 1601 01:11:03,650 --> 01:11:05,290 you would see it with even period. 1602 01:11:05,290 --> 01:11:06,840 But we don't have root epsilon, we're 1603 01:11:06,840 --> 01:11:08,464 plugging this as a function of epsilon. 1604 01:11:08,464 --> 01:11:10,910 So it's not even period, it's getting wider and wider. 1605 01:11:10,910 --> 01:11:12,780 Meanwhile, there's a nice fact about this. 1606 01:11:12,780 --> 01:11:17,984 For special values of epsilon, what happens to sine squared? 1607 01:11:17,984 --> 01:11:18,650 It goes to zero. 1608 01:11:18,650 --> 01:11:20,108 And what happens when this is zero? 1609 01:11:22,280 --> 01:11:23,700 Yeah, t is 1. 1610 01:11:23,700 --> 01:11:27,190 So every time sine is 0, i.e., for sufficient values 1611 01:11:27,190 --> 01:11:31,282 when root epsilon is a multiple of pi determined by g0, 1612 01:11:31,282 --> 01:11:32,240 transmission goes to 1. 1613 01:11:32,240 --> 01:11:33,790 It becomes perfect. 1614 01:11:33,790 --> 01:11:35,880 So in fact, instead of doing-- wow. 1615 01:11:39,210 --> 01:11:45,780 Instead of doing this, what it does is it does this. 1616 01:11:50,000 --> 01:11:52,760 So let's check that I'm not lying to you. 1617 01:11:52,760 --> 01:11:55,510 So let me draw it slightly different. 1618 01:11:55,510 --> 01:11:58,659 To check, so it's going to 0, and the period of the 0 1619 01:11:58,659 --> 01:12:00,200 is getting further and further along. 1620 01:12:00,200 --> 01:12:03,570 That's because this is square root, not squared. 1621 01:12:03,570 --> 01:12:06,600 So epsilon has to get much larger to hit the next period. 1622 01:12:06,600 --> 01:12:08,760 That's why it's getting larger and larger spacing. 1623 01:12:08,760 --> 01:12:11,840 However, the amplitude goes down from 1, 1624 01:12:11,840 --> 01:12:13,200 that's when this gets largest. 1625 01:12:13,200 --> 01:12:15,610 Well, at large values of epsilon, this is suppressed. 1626 01:12:15,610 --> 01:12:18,040 As epsilon goes larger and larger, 1627 01:12:18,040 --> 01:12:21,440 this deviation become smaller and smaller. 1628 01:12:21,440 --> 01:12:23,250 So that's what this plot is telling us. 1629 01:12:23,250 --> 01:12:25,400 This plot is telling us a bunch of things. 1630 01:12:25,400 --> 01:12:31,440 First off, we see that at large energies, 1631 01:12:31,440 --> 01:12:32,462 we transmit perfectly. 1632 01:12:32,462 --> 01:12:33,170 That makes sense. 1633 01:12:33,170 --> 01:12:34,590 This was a finitely high barrier. 1634 01:12:34,590 --> 01:12:37,191 Large energies, we don't even notice it. 1635 01:12:37,191 --> 01:12:39,210 It tells you at low energies-- well, 1636 01:12:39,210 --> 01:12:41,570 we don't know yet what happens at very low energies. 1637 01:12:41,570 --> 01:12:44,805 But at reasonably low energies, the transmission is suppressed. 1638 01:12:44,805 --> 01:12:46,430 And if you sort of squint, this roughly 1639 01:12:46,430 --> 01:12:48,530 does what we'd expect from the step barrier. 1640 01:12:48,530 --> 01:12:50,460 However, something really special 1641 01:12:50,460 --> 01:12:52,472 is happening at special values of epsilon. 1642 01:12:52,472 --> 01:12:53,930 We're getting perfect transmission. 1643 01:12:56,710 --> 01:13:01,390 We get perfect transmission of these points, t is equal to 1. 1644 01:13:01,390 --> 01:13:02,550 Perfect. 1645 01:13:02,550 --> 01:13:03,050 Star. 1646 01:13:05,890 --> 01:13:06,860 Happy with a big nose. 1647 01:13:06,860 --> 01:13:07,840 I don't know. 1648 01:13:07,840 --> 01:13:10,560 Perfect transmission at all these points. 1649 01:13:10,560 --> 01:13:16,042 So this leaves us with a question of why. 1650 01:13:16,042 --> 01:13:17,620 Why the perfect transmission, what's 1651 01:13:17,620 --> 01:13:20,320 the mechanism making transmission perfect. 1652 01:13:20,320 --> 01:13:22,630 But it also does something really lovely for us. 1653 01:13:22,630 --> 01:13:26,280 Suppose you see a spectrum, you see a transmission amplitude 1654 01:13:26,280 --> 01:13:28,630 or transmission probability that looks like this. 1655 01:13:28,630 --> 01:13:29,900 You may get crappy data. 1656 01:13:29,900 --> 01:13:31,289 You may see that it's smudged out 1657 01:13:31,289 --> 01:13:32,830 and you see all sorts of messy stuff. 1658 01:13:32,830 --> 01:13:35,210 But if you know that there's perfect transmission 1659 01:13:35,210 --> 01:13:37,430 at some particular epsilon 1, there's 1660 01:13:37,430 --> 01:13:40,180 more perfect transmission at another epsilon 2, more perfect 1661 01:13:40,180 --> 01:13:41,900 transmission at epsilon 3, and they 1662 01:13:41,900 --> 01:13:43,750 scale, they fit to this prediction. 1663 01:13:43,750 --> 01:13:45,487 What do you know? 1664 01:13:45,487 --> 01:13:47,070 That you've got a finite high barrier, 1665 01:13:47,070 --> 01:13:52,047 and it's probably pretty well approximated by a square step. 1666 01:13:52,047 --> 01:13:54,630 So in your problem set, you're going to get experimental data, 1667 01:13:54,630 --> 01:13:57,882 and you're going to have to match to the experimental data. 1668 01:13:57,882 --> 01:13:59,590 You're going to have to predict something 1669 01:13:59,590 --> 01:14:02,320 about the potential that created a particular transmission 1670 01:14:02,320 --> 01:14:03,730 probability distribution. 1671 01:14:03,730 --> 01:14:05,449 And knowing where these resonances are, 1672 01:14:05,449 --> 01:14:07,240 where these points of perfect transmission, 1673 01:14:07,240 --> 01:14:08,490 is going to be very useful for you. 1674 01:14:08,490 --> 01:14:08,989 Yeah. 1675 01:14:08,989 --> 01:14:10,866 AUDIENCE: Wasn't there another potential 1676 01:14:10,866 --> 01:14:13,920 that will also create periodic 1s? 1677 01:14:13,920 --> 01:14:15,370 PROFESSOR: A very good question. 1678 01:14:15,370 --> 01:14:17,520 So I'm going to turn that around to you. 1679 01:14:17,520 --> 01:14:20,790 Can you orchestrate a potential that gives you the same thing? 1680 01:14:24,092 --> 01:14:25,800 That's an interesting empirical question. 1681 01:14:25,800 --> 01:14:30,350 So, on the problem set, you'll study a double well potential. 1682 01:14:30,350 --> 01:14:32,121 So the question was, how do I know 1683 01:14:32,121 --> 01:14:34,370 isn't another potential that gives me the same answer? 1684 01:14:34,370 --> 01:14:35,200 At this point, we don't know. 1685 01:14:35,200 --> 01:14:36,300 Maybe there is another potential. 1686 01:14:36,300 --> 01:14:37,850 In fact, you can do all sorts of things 1687 01:14:37,850 --> 01:14:40,100 to make a potential that's very arbitrarily close that 1688 01:14:40,100 --> 01:14:43,000 gives you an arbitrarily similar profile. 1689 01:14:43,000 --> 01:14:45,296 So if they're significantly different, 1690 01:14:45,296 --> 01:14:46,615 how different do they look? 1691 01:14:46,615 --> 01:14:48,490 So then the students say, well, I don't know. 1692 01:14:48,490 --> 01:14:49,650 What about a double well potential? 1693 01:14:49,650 --> 01:14:51,774 So, in fact, we'll be doing a double well potential 1694 01:14:51,774 --> 01:14:54,600 on the problem set. 1695 01:14:54,600 --> 01:14:55,370 A good question. 1696 01:14:55,370 --> 01:14:56,640 And we certainly haven't proven anything 1697 01:14:56,640 --> 01:14:57,900 like this is the only potentially 1698 01:14:57,900 --> 01:14:59,420 that gives you this transmission amplitude. 1699 01:14:59,420 --> 01:15:01,794 What we can say is if you get transmission amplitude that 1700 01:15:01,794 --> 01:15:04,550 looks like this, it's probably pretty reasonable 1701 01:15:04,550 --> 01:15:07,796 to say it's probably well modeled. 1702 01:15:07,796 --> 01:15:09,920 We seem to be reproducing the data reasonably well. 1703 01:15:09,920 --> 01:15:12,600 So it's not a proof of anything, but it's a nice model. 1704 01:15:12,600 --> 01:15:15,200 And we're physicists, we build models. 1705 01:15:15,200 --> 01:15:17,990 We don't tell you what's true, we 1706 01:15:17,990 --> 01:15:20,594 tell you what are good models. 1707 01:15:20,594 --> 01:15:22,010 If a theorist ever walks up on you 1708 01:15:22,010 --> 01:15:27,410 and he says, here's the truth, punch him in the gut. 1709 01:15:27,410 --> 01:15:28,960 That's not how it works. 1710 01:15:28,960 --> 01:15:30,852 Experimentalists, on the other hand. 1711 01:15:30,852 --> 01:15:32,310 We'll just punch them too, I guess. 1712 01:15:32,310 --> 01:15:36,264 [LAUGHTER] 1713 01:15:36,264 --> 01:15:38,330 PROFESSOR: We build models. 1714 01:15:38,330 --> 01:15:42,580 OK, so this leaves us with an obvious question, 1715 01:15:42,580 --> 01:15:47,210 which we've answer in the case of energy 1716 01:15:47,210 --> 01:15:50,030 grade in the potential. 1717 01:15:50,030 --> 01:15:53,571 What about the case of the energy less than the potential. 1718 01:15:53,571 --> 01:15:55,570 So we haven't filled out that part of the graph. 1719 01:15:55,570 --> 01:15:57,153 Let's fill out that part of the graph. 1720 01:15:57,153 --> 01:16:00,020 What happens if the energy is less than the potential? 1721 01:16:00,020 --> 01:16:03,190 And for this, I'm going to use the trick that I mentioned 1722 01:16:03,190 --> 01:16:04,696 earlier that, look, if the energy is 1723 01:16:04,696 --> 01:16:06,570 less than the potential, that just means that 1724 01:16:06,570 --> 01:16:09,574 in the intervening regime, in here, 1725 01:16:09,574 --> 01:16:10,990 instead of being oscillatory, it's 1726 01:16:10,990 --> 01:16:13,550 going to be exponentially growing [INAUDIBLE] 1727 01:16:13,550 --> 01:16:15,940 because we're in a classically disallowed region. 1728 01:16:15,940 --> 01:16:17,410 So if the energy is less than v0, 1729 01:16:17,410 --> 01:16:21,200 here we have instead of ikx, we have minus alpha x. 1730 01:16:21,200 --> 01:16:25,750 And instead of minus ikx we have plus alpha x. 1731 01:16:25,750 --> 01:16:28,010 So if you go through that whole analysis 1732 01:16:28,010 --> 01:16:32,400 and plug in those values for the k prime, the answer you get 1733 01:16:32,400 --> 01:16:33,570 is really quite nice. 1734 01:16:33,570 --> 01:16:38,174 We find that t for energy less than v0 is equal to-- 1735 01:16:38,174 --> 01:16:40,340 and it's just a direct analytic continuation of what 1736 01:16:40,340 --> 01:16:51,571 you get here-- 1 plus 1 over 4 epsilon 1 minus epsilon. 1737 01:16:51,571 --> 01:16:52,820 So epsilon is less than 1 now. 1738 01:16:56,516 --> 01:17:08,050 1 sinh squared of g0 1 minus epsilon under 1. 1739 01:17:08,050 --> 01:17:09,965 OK, so now this lets us complete the plot. 1740 01:17:13,812 --> 01:17:15,770 And you can check, it's pretty straightforward, 1741 01:17:15,770 --> 01:17:18,230 that because sinh, again, goes like its argument 1742 01:17:18,230 --> 01:17:20,020 at very small values of its argument, 1743 01:17:20,020 --> 01:17:23,000 we get g0 squared 1 minus epsilon. 1744 01:17:23,000 --> 01:17:27,130 Sorry, this should be square root 1 minus epsilon. 1745 01:17:27,130 --> 01:17:27,680 Really? 1746 01:17:27,680 --> 01:17:28,896 Is that a typo? 1747 01:17:28,896 --> 01:17:29,770 Oh no, it's in there. 1748 01:17:29,770 --> 01:17:32,510 OK, good. 1749 01:17:32,510 --> 01:17:34,580 So again, we get g0 squared 1 minus epsilon, 1750 01:17:34,580 --> 01:17:38,282 the 1 minus epsilon cancels, and we get 1 over 4 g0 squared. 1751 01:17:38,282 --> 01:17:39,990 Which is good, because if they disagreed, 1752 01:17:39,990 --> 01:17:42,050 we'd be in real trouble. 1753 01:17:42,050 --> 01:17:44,740 So they agree, but the sinh squared is just a strictly-- 1754 01:17:44,740 --> 01:17:46,920 or that function is just a strictly decreasing 1755 01:17:46,920 --> 01:17:48,670 function as we approach epsilon goes to 0. 1756 01:17:48,670 --> 01:17:50,320 It goes mostly to 0. 1757 01:17:50,320 --> 01:17:51,300 So this is what we see. 1758 01:17:51,300 --> 01:17:53,890 We see an exponential region and then 1759 01:17:53,890 --> 01:17:57,820 we see oscillations with residences 1760 01:17:57,820 --> 01:18:01,780 where the period is getting wider and wider. 1761 01:18:01,780 --> 01:18:04,379 But this should trouble you a little bit. 1762 01:18:04,379 --> 01:18:05,420 Here, what are we saying? 1763 01:18:05,420 --> 01:18:09,570 We're saying look, if we have a barrier, 1764 01:18:09,570 --> 01:18:12,010 and we send in a particle with energy way 1765 01:18:12,010 --> 01:18:18,012 below the barrier, that's kind of troubling. 1766 01:18:18,012 --> 01:18:20,470 If we send in a particle with energy way below the barrier, 1767 01:18:20,470 --> 01:18:22,860 there's some probability that it gets out. 1768 01:18:22,860 --> 01:18:23,610 It goes across. 1769 01:18:27,980 --> 01:18:35,340 So for e less than v0, classically no transmission, 1770 01:18:35,340 --> 01:18:36,800 we get transmission. 1771 01:18:36,800 --> 01:18:44,447 Now, do we get resonances when e is less than v0? 1772 01:18:44,447 --> 01:18:45,780 No, that's an interesting thing. 1773 01:18:45,780 --> 01:18:47,900 We'll talk about resonances and where they come from, 1774 01:18:47,900 --> 01:18:50,360 and we'll talk about the more generalized notion of a nest 1775 01:18:50,360 --> 01:18:52,150 matrix in the next lecture. 1776 01:18:52,150 --> 01:18:55,860 But for now I just want to say a couple of things about it. 1777 01:18:55,860 --> 01:19:00,700 So I want to ask-- so this is called tunneling, 1778 01:19:00,700 --> 01:19:06,240 this transmission across a disallowed barrier, 1779 01:19:06,240 --> 01:19:14,841 a classically disallowed barrier-- 1780 01:19:14,841 --> 01:19:16,840 so this transmission across a disallowed barrier 1781 01:19:16,840 --> 01:19:19,910 is called tunneling. 1782 01:19:19,910 --> 01:19:26,000 And just a quick thing to notice is that if we hold e fixed 1783 01:19:26,000 --> 01:19:29,496 and we vary l, we vary the width of the barrier, 1784 01:19:29,496 --> 01:19:31,120 how does the tunneling amplitude depend 1785 01:19:31,120 --> 01:19:33,120 on the width of the barrier? 1786 01:19:33,120 --> 01:19:37,140 At fixed energy, if we vary the width of the barrier-- which 1787 01:19:37,140 --> 01:19:40,527 is only contained in g0-- how does 1788 01:19:40,527 --> 01:19:41,860 the transmission amplitude vary? 1789 01:19:41,860 --> 01:19:45,450 And we find that for large l-- for l much greater than 1 1790 01:19:45,450 --> 01:19:46,950 for a typical scale in the problem-- 1791 01:19:46,950 --> 01:19:54,180 the tunneling amplitude goes like e to the minus 2 alpha l 1792 01:19:54,180 --> 01:19:58,049 where alpha just depends on the energy in the potential. 1793 01:19:58,049 --> 01:20:00,340 So what we see is that the probability of transmitting, 1794 01:20:00,340 --> 01:20:03,170 of tunneling through a wall, depends 1795 01:20:03,170 --> 01:20:04,610 on the width of that wall. 1796 01:20:04,610 --> 01:20:06,830 And it depends on it exponentially. 1797 01:20:06,830 --> 01:20:09,300 The wider the wall, the exponentially less likely you 1798 01:20:09,300 --> 01:20:10,420 are to tunnel across it. 1799 01:20:10,420 --> 01:20:12,280 And this fits the simulation we ran 1800 01:20:12,280 --> 01:20:15,530 where we saw that tunnelling through a very thin wall 1801 01:20:15,530 --> 01:20:17,590 was actually quite efficient. 1802 01:20:17,590 --> 01:20:20,260 And we also saw that tunnelling across a wall 1803 01:20:20,260 --> 01:20:21,260 can have resonant peaks. 1804 01:20:21,260 --> 01:20:23,584 At special values, it transmits perfectly. 1805 01:20:23,584 --> 01:20:24,875 We saw that in the simulations. 1806 01:20:29,396 --> 01:20:32,020 So at this point I can't resist telling you a very short story. 1807 01:20:32,020 --> 01:20:35,620 I have a very good friend who is not a physicist, 1808 01:20:35,620 --> 01:20:37,880 but who is a professor at MIT. 1809 01:20:37,880 --> 01:20:41,810 So a smart person, and very smart. 1810 01:20:41,810 --> 01:20:43,890 She's one of the smartest people I know. 1811 01:20:43,890 --> 01:20:47,170 She broke her ankle. 1812 01:20:47,170 --> 01:20:49,880 She's always losing stuff, she's constantly losing stuff. 1813 01:20:49,880 --> 01:20:54,840 And she broke her ankle, and she was going to the rehab place, 1814 01:20:54,840 --> 01:20:58,635 and she parked her car right in front of the rehab place. 1815 01:20:58,635 --> 01:21:00,010 She parked her car right in front 1816 01:21:00,010 --> 01:21:01,250 because she's got the broken ankle 1817 01:21:01,250 --> 01:21:03,130 and she doesn't want to have to walk a long way to get the car, 1818 01:21:03,130 --> 01:21:04,370 she parked right in front. 1819 01:21:04,370 --> 01:21:06,660 She went in, she did her two hours of rehab, 1820 01:21:06,660 --> 01:21:10,740 and she came outside, and her car was gone. 1821 01:21:10,740 --> 01:21:12,575 And she's always misplacing her car, 1822 01:21:12,575 --> 01:21:13,950 she's always misplacing her keys, 1823 01:21:13,950 --> 01:21:15,241 she's always losing everything. 1824 01:21:15,241 --> 01:21:19,750 But this time, this time she knew where it was. 1825 01:21:19,750 --> 01:21:21,380 And it wasn't there. 1826 01:21:21,380 --> 01:21:24,777 So first she thought, oh crap, my car's been stolen. 1827 01:21:24,777 --> 01:21:26,360 How annoying, I've got a broken ankle. 1828 01:21:26,360 --> 01:21:28,262 But she looks around, and there's 1829 01:21:28,262 --> 01:21:30,220 her car on the other side of the street pointed 1830 01:21:30,220 --> 01:21:32,700 the opposite direction. 1831 01:21:32,700 --> 01:21:36,340 And she's like, finally. 1832 01:21:36,340 --> 01:21:36,910 Finally. 1833 01:21:36,910 --> 01:21:39,235 She was explaining this to me and a friend of mine who 1834 01:21:39,235 --> 01:21:41,760 is also a physicist, she said finally, I 1835 01:21:41,760 --> 01:21:45,710 knew I had incontrovertible proof that quantum 1836 01:21:45,710 --> 01:21:47,990 effects happen to macroscopic objects. 1837 01:21:47,990 --> 01:21:50,300 My car tunnelled across the street. 1838 01:21:50,300 --> 01:21:51,890 [LAUGHTER] 1839 01:21:51,890 --> 01:21:53,390 PROFESSOR: At this point, of course, 1840 01:21:53,390 --> 01:21:55,490 my friend and I are just dying of laughter. 1841 01:21:55,490 --> 01:21:57,560 But she said my car tunnelled across the street. 1842 01:21:57,560 --> 01:22:00,290 But here's the problem, I couldn't tell anyone. 1843 01:22:00,290 --> 01:22:03,340 Because if I told anyone, they would know that I was crazy. 1844 01:22:03,340 --> 01:22:07,000 Clearly I'm crazy, because I lose my stuff all the time, 1845 01:22:07,000 --> 01:22:08,870 but surely it's just gotten misplaced. 1846 01:22:08,870 --> 01:22:10,820 So I went home, and I got home and I 1847 01:22:10,820 --> 01:22:13,400 was sitting down to dinner with my partner and our daughter 1848 01:22:13,400 --> 01:22:16,120 and I was burning up inside because I wanted 1849 01:22:16,120 --> 01:22:18,020 to tell them this crazy thing happened. 1850 01:22:18,020 --> 01:22:20,817 I know quantum mechanics, it happens. 1851 01:22:20,817 --> 01:22:22,650 I couldn't tell them anything, so I was just 1852 01:22:22,650 --> 01:22:23,530 fidgeting and dying. 1853 01:22:23,530 --> 01:22:26,380 And finally my daughter said, mom, 1854 01:22:26,380 --> 01:22:29,190 didn't you notice that we moved the car across the street? 1855 01:22:29,190 --> 01:22:31,520 [LAUGHTER] 1856 01:22:31,520 --> 01:22:33,660 PROFESSOR: See you next time.