1 00:00:00,090 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,690 continue to offer high-quality educational resources for free. 5 00:00:10,690 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,250 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,250 --> 00:00:18,210 at ocw.mit.edu. 8 00:00:21,130 --> 00:00:22,130 ROBERT FIELD: All right. 9 00:00:22,130 --> 00:00:30,990 So last time, I said that atomic sizes 10 00:00:30,990 --> 00:00:36,090 are interesting or useful to keep in mind, because you want 11 00:00:36,090 --> 00:00:42,960 numbers for them, which are somewhere between 0.1 and 100, 12 00:00:42,960 --> 00:00:44,790 or something like that. 13 00:00:44,790 --> 00:00:48,916 Because then you have a sense for how big everything is, 14 00:00:48,916 --> 00:00:50,290 and you're in the right ballpark. 15 00:00:50,290 --> 00:00:52,860 If you have to remember a number and an exponent, 16 00:00:52,860 --> 00:00:54,510 it's a little trickier. 17 00:00:54,510 --> 00:00:57,720 And so I'm going to say something about atomic sizes 18 00:00:57,720 --> 00:01:03,450 in just a few minutes, but the main things from the previous 19 00:01:03,450 --> 00:01:07,050 lecture were this relationship between the wavelength 20 00:01:07,050 --> 00:01:13,150 and the momentum, which is true for both waves and particles, 21 00:01:13,150 --> 00:01:15,820 or things that we think are wave-like, like light, 22 00:01:15,820 --> 00:01:19,300 and things that we think are particle-like, like electrons-- 23 00:01:19,300 --> 00:01:22,940 and so this unifying principle. 24 00:01:22,940 --> 00:01:24,440 Then I have a little question. 25 00:01:24,440 --> 00:01:29,540 And that is suppose the wavelength for a particle 26 00:01:29,540 --> 00:01:34,400 is known, and suppose we have n particles. 27 00:01:34,400 --> 00:01:40,540 And so if we say each particle requires a volume of lambda 28 00:01:40,540 --> 00:01:43,980 cubed, and there are n particles, 29 00:01:43,980 --> 00:01:48,820 and we stick them into a volume smaller than n times 30 00:01:48,820 --> 00:01:50,970 lambda cubed, what's going to happen? 31 00:01:54,631 --> 00:01:55,130 Yes? 32 00:01:55,130 --> 00:01:58,430 AUDIENCE: They interfere with the structure? 33 00:01:58,430 --> 00:02:01,130 ROBERT FIELD: Their identities get corrupted, 34 00:02:01,130 --> 00:02:04,580 and the person who does that for the first time 35 00:02:04,580 --> 00:02:08,240 gets a Nobel Prize, and that was Wolfgang Ketterle-- 36 00:02:08,240 --> 00:02:10,229 and others. 37 00:02:10,229 --> 00:02:13,190 So when you have quantum mechanical particles 38 00:02:13,190 --> 00:02:15,920 that are too close together, they 39 00:02:15,920 --> 00:02:18,870 lose their individual identity. 40 00:02:18,870 --> 00:02:20,810 And so this is a very simple thing 41 00:02:20,810 --> 00:02:24,410 that anybody who was beginning to understand quantum 42 00:02:24,410 --> 00:02:27,770 mechanics in the 1920s would say, 43 00:02:27,770 --> 00:02:30,740 this is a puzzling thing, maybe we should think about that. 44 00:02:30,740 --> 00:02:35,600 And it's really hard, so it took a long time. 45 00:02:35,600 --> 00:02:39,980 Then a whole bunch of experiments led to the idea 46 00:02:39,980 --> 00:02:47,210 that we need to have a way for atoms to fill space and satisfy 47 00:02:47,210 --> 00:02:52,640 all of the other stuff, and that was Rutherford's planetary atom 48 00:02:52,640 --> 00:02:55,010 picture. 49 00:02:55,010 --> 00:02:58,310 The problem with that is it does fill space, 50 00:02:58,310 --> 00:03:01,850 but there's no way for the electron 51 00:03:01,850 --> 00:03:05,900 to continue orbiting around a nucleus, because it 52 00:03:05,900 --> 00:03:09,890 will radiate its energy, and fall into the nucleus, and game 53 00:03:09,890 --> 00:03:11,060 over. 54 00:03:11,060 --> 00:03:17,980 And so Bohr and De Broglie both had ways of fixing this. 55 00:03:17,980 --> 00:03:24,260 And Bohr's way was simply to say the angular momentum is not 56 00:03:24,260 --> 00:03:27,760 just conserved, but it has certain values-- 57 00:03:27,760 --> 00:03:31,700 an integer times h bar, h bar is h over 2 pi. 58 00:03:34,380 --> 00:03:39,030 And De Broglie said in order to keep the electron 59 00:03:39,030 --> 00:03:42,210 from annihilating itself, since it's going around 60 00:03:42,210 --> 00:03:46,470 in a circular orbit of known circumference, 61 00:03:46,470 --> 00:03:48,870 there must be an integer number of wavelengths 62 00:03:48,870 --> 00:03:51,710 around that orbit. 63 00:03:51,710 --> 00:03:55,680 Now, that's a much more physical and reasonable ad 64 00:03:55,680 --> 00:03:58,200 hoc explanation, but it's still ad hoc. 65 00:04:00,880 --> 00:04:03,980 It assumes that the particles are 66 00:04:03,980 --> 00:04:07,560 moving around circular orbits. 67 00:04:07,560 --> 00:04:11,610 I hinted that the way out of this 68 00:04:11,610 --> 00:04:16,140 is going to be that the particles aren't moving. 69 00:04:16,140 --> 00:04:17,990 Then we could still have angular momentum. 70 00:04:17,990 --> 00:04:20,399 We can still have all sorts of useful stuff, 71 00:04:20,399 --> 00:04:23,340 but they're not moving, and we don't radiate 72 00:04:23,340 --> 00:04:28,290 the energy of the particles. 73 00:04:28,290 --> 00:04:30,990 But that requires a completely new way 74 00:04:30,990 --> 00:04:35,190 of looking at particles in quantum mechanics. 75 00:04:37,770 --> 00:04:42,360 But the thing about both the Bohr hypothesis and the De 76 00:04:42,360 --> 00:04:49,350 Broglie hypothesis is that any sophomore who 77 00:04:49,350 --> 00:04:52,530 made such a proposal would be laughed 78 00:04:52,530 --> 00:04:58,980 at in the 1910-1920 period, because it's just ridiculous. 79 00:04:58,980 --> 00:05:04,320 But the reason these hypotheses were taken seriously is that 80 00:05:04,320 --> 00:05:11,250 this planetary model with those corrections explain the spectra 81 00:05:11,250 --> 00:05:14,040 of all one-electron systems-- 82 00:05:14,040 --> 00:05:17,130 hydrogen, helium plus, lithium two plus, 83 00:05:17,130 --> 00:05:20,580 uranium 91 plus, all of them, to better 84 00:05:20,580 --> 00:05:22,860 than measurement accuracy at the time, 85 00:05:22,860 --> 00:05:25,710 and better than measurement accuracy for a long time 86 00:05:25,710 --> 00:05:28,410 after this was proposed. 87 00:05:28,410 --> 00:05:30,540 So getting a whole bunch-- 88 00:05:30,540 --> 00:05:32,980 now, a whole bunch is an infinite number, actually-- 89 00:05:32,980 --> 00:05:38,880 of 10 digit numbers makes you think there's something here, 90 00:05:38,880 --> 00:05:43,710 but nobody knows where here is except the numbers. 91 00:05:43,710 --> 00:05:49,080 And the idea of the planetary model with the fixes 92 00:05:49,080 --> 00:05:53,610 doesn't explain anything other than the spectral lines, which 93 00:05:53,610 --> 00:05:58,350 is a lot, but it tells you there's something really good 94 00:05:58,350 --> 00:06:00,090 there. 95 00:06:00,090 --> 00:06:02,700 I guess one thing I forgot to mention in the summary 96 00:06:02,700 --> 00:06:11,780 is that the energy levels that you get from the Bohr model 97 00:06:11,780 --> 00:06:15,560 are going to explain spectra if you 98 00:06:15,560 --> 00:06:20,675 say a spectrum are transitions between these energy levels. 99 00:06:23,210 --> 00:06:26,120 And that was also a brilliant suggestion, 100 00:06:26,120 --> 00:06:29,390 and it was suggested by numerology. 101 00:06:29,390 --> 00:06:34,610 But let us go back to work now. 102 00:06:34,610 --> 00:06:37,800 We're going to talk about the two-slit experiment. 103 00:06:37,800 --> 00:06:40,840 And I have a personal thing to report about that. 104 00:06:40,840 --> 00:06:43,720 The first time I gave a talk, it was 105 00:06:43,720 --> 00:06:47,990 going to be a 15-minute talk at a spectroscopy conference, 106 00:06:47,990 --> 00:06:52,540 and so I did a practice talk, and it 107 00:06:52,540 --> 00:06:56,140 had to do with the two-slit experiment in relationship 108 00:06:56,140 --> 00:06:57,910 to spectroscopy. 109 00:06:57,910 --> 00:07:02,990 But I did a practice talk, and it took two hours. 110 00:07:02,990 --> 00:07:08,830 So I have a thing about the two-slit experiment. 111 00:07:08,830 --> 00:07:13,450 And I think this lecture is going to be not two hours. 112 00:07:13,450 --> 00:07:15,400 It's going to be on time. 113 00:07:15,400 --> 00:07:17,980 So we're going to talk about the two-slit experiment, 114 00:07:17,980 --> 00:07:21,340 and the important thing about the two-slit experiment 115 00:07:21,340 --> 00:07:26,090 is that it's mostly ordinary wave interference. 116 00:07:26,090 --> 00:07:31,180 There is no quantum mechanics, and so most of the hard stuff 117 00:07:31,180 --> 00:07:35,690 to analyze is classical physics. 118 00:07:35,690 --> 00:07:38,950 And I'm going to do the best I can with that. 119 00:07:38,950 --> 00:07:44,440 But after we get to understanding this problem, 120 00:07:44,440 --> 00:07:46,630 there will be a surprise at the end, which 121 00:07:46,630 --> 00:07:49,420 is a quantum surprise. 122 00:07:49,420 --> 00:07:52,950 And it's something that absolutely requires 123 00:07:52,950 --> 00:07:55,930 a postulate, the first postulate of quantum mechanics, 124 00:07:55,930 --> 00:07:58,550 some idea of what we are talking about here. 125 00:07:58,550 --> 00:08:01,390 What are we allowed to know about a system? 126 00:08:01,390 --> 00:08:06,130 And it tells you there's something there. 127 00:08:06,130 --> 00:08:10,630 And then I'm going to give you a little description of something 128 00:08:10,630 --> 00:08:14,890 which I think I have to call a semi-classical optics 129 00:08:14,890 --> 00:08:16,710 uncertainty principle. 130 00:08:16,710 --> 00:08:21,430 Semi-classical or semi-anything is usually quantum mechanics, 131 00:08:21,430 --> 00:08:23,830 just a little bit, mixed into something 132 00:08:23,830 --> 00:08:26,930 that was well understood before, or that 133 00:08:26,930 --> 00:08:29,210 is very convenient to use, and you only 134 00:08:29,210 --> 00:08:31,670 bring in quantum mechanics when you have to. 135 00:08:31,670 --> 00:08:34,280 And it's the easiest thing to understand. 136 00:08:34,280 --> 00:08:38,840 And that introduces the uncertainty principle, 137 00:08:38,840 --> 00:08:45,350 and we get a taste of the first of several quantum mechanical 138 00:08:45,350 --> 00:08:48,470 postulates. 139 00:08:48,470 --> 00:08:50,270 So let's start with the sizes. 140 00:08:54,320 --> 00:08:59,540 I should have given you these sizes in the previous lecture, 141 00:08:59,540 --> 00:09:02,630 but I was being mostly number-free. 142 00:09:02,630 --> 00:09:07,730 So the radius of a Bohr orbit is given by n 143 00:09:07,730 --> 00:09:17,450 over z squared, where z is the integer charge on the nucleus, 144 00:09:17,450 --> 00:09:24,670 times 0.5292 angstroms. 145 00:09:24,670 --> 00:09:30,430 So half an angstrom is the radius of, basically, any atom. 146 00:09:30,430 --> 00:09:34,780 And this charge on the nucleus, it gets smaller. 147 00:09:34,780 --> 00:09:37,630 And this is a very useful thing. 148 00:09:37,630 --> 00:09:53,740 The wavelength is given by n over z times 3.32 angstroms. 149 00:09:53,740 --> 00:10:01,090 And that's the situation where n times 150 00:10:01,090 --> 00:10:06,810 the wavelength is equal to 2 pi rn. 151 00:10:06,810 --> 00:10:11,100 So we have n wavelengths around an orbit, 152 00:10:11,100 --> 00:10:14,880 and this is really the De Broglie hypothesis. 153 00:10:14,880 --> 00:10:18,630 But again, something which is on the order of something 154 00:10:18,630 --> 00:10:20,490 that you can remember. 155 00:10:20,490 --> 00:10:21,750 And the energy levels-- 156 00:10:33,480 --> 00:10:42,630 these energy levels are z squared times 13.6 electron 157 00:10:42,630 --> 00:10:44,820 volts over n squared. 158 00:10:48,030 --> 00:10:51,960 If we were talking about energies in joules, 159 00:10:51,960 --> 00:10:56,560 or in any units you want, it's likely to have a big exponent. 160 00:10:56,560 --> 00:10:59,749 But this one is in electron volts, 161 00:10:59,749 --> 00:11:01,290 and that's actually what's happening. 162 00:11:01,290 --> 00:11:05,430 An electron is being attracted to a positive thing, 163 00:11:05,430 --> 00:11:08,670 and there's, basically, a voltage difference. 164 00:11:08,670 --> 00:11:10,650 And so that's another useful thing. 165 00:11:10,650 --> 00:11:11,580 Now this z-- 166 00:11:14,850 --> 00:11:20,070 I left out the Rydberg. 167 00:11:20,070 --> 00:11:23,820 This Rydberg constant-- there's a bunch 168 00:11:23,820 --> 00:11:27,300 of fundamental constants, and since I'm a spectroscopist, 169 00:11:27,300 --> 00:11:29,520 I think in terms of wave numbers, 170 00:11:29,520 --> 00:11:31,500 reciprocal centimeters. 171 00:11:31,500 --> 00:11:35,190 And for me, that's energy, it's frequency, it's everything, 172 00:11:35,190 --> 00:11:48,420 but anyway, for hydrogen, it's 1097677.581. 173 00:11:48,420 --> 00:11:53,340 For something with an infinite mass nucleus, 174 00:11:53,340 --> 00:12:01,580 it's 109737.3153 reciprocal centimeters. 175 00:12:01,580 --> 00:12:04,440 And this is the number that I have in my head, 176 00:12:04,440 --> 00:12:07,140 and I use it in all sorts of places, 177 00:12:07,140 --> 00:12:09,450 and you can imagine where. 178 00:12:09,450 --> 00:12:16,300 And to get to any particular nucleus, 179 00:12:16,300 --> 00:12:22,860 this is just r infinity mu nucleus, or mu atom, 180 00:12:22,860 --> 00:12:24,510 over the mass of the electron. 181 00:12:28,040 --> 00:12:31,110 And it turns out that almost everything except hydrogen 182 00:12:31,110 --> 00:12:33,640 is very close to this number. 183 00:12:33,640 --> 00:12:35,700 And so this hardly matters, but it 184 00:12:35,700 --> 00:12:40,755 does give you a little bit of dependence on the nuclear mass. 185 00:12:44,130 --> 00:12:48,690 So I said before that this Rydberg equation, 186 00:12:48,690 --> 00:12:54,550 or this equation, tells you nothing. 187 00:12:54,550 --> 00:12:58,820 It tells you where all the energy levels are, 188 00:12:58,820 --> 00:13:02,240 and anyone could tell you where the rest are. 189 00:13:02,240 --> 00:13:06,200 So it's a pattern, which is nice, 190 00:13:06,200 --> 00:13:11,850 but a pattern which says if things are well-behaved, 191 00:13:11,850 --> 00:13:16,200 like hydrogen atom, this is what the energy levels will be. 192 00:13:16,200 --> 00:13:20,790 But life is difficult. Life is not with everything 193 00:13:20,790 --> 00:13:24,060 well-behaved, and so this is a pattern which 194 00:13:24,060 --> 00:13:27,120 says I'm interested in how the real life is 195 00:13:27,120 --> 00:13:29,370 different from that pattern. 196 00:13:29,370 --> 00:13:32,070 It's a way of thinking about structure 197 00:13:32,070 --> 00:13:34,260 and how we learn about structure. 198 00:13:34,260 --> 00:13:38,880 Information about the details of a molecule or an atom 199 00:13:38,880 --> 00:13:43,290 is encoded in the spectrum, and this is the magic decoder, 200 00:13:43,290 --> 00:13:45,660 or one of the magic decoders, we use 201 00:13:45,660 --> 00:13:50,100 to begin to assemble the new insights. 202 00:13:50,100 --> 00:13:52,410 This doesn't appear in textbooks. 203 00:13:52,410 --> 00:13:55,860 In textbooks, you get the equations, you get the truth, 204 00:13:55,860 --> 00:14:00,040 and you don't get any strange interpretations. 205 00:14:00,040 --> 00:14:01,816 That's what you're getting from me. 206 00:14:01,816 --> 00:14:03,440 You're getting strange interpretations, 207 00:14:03,440 --> 00:14:05,370 and you'll have them throughout the course. 208 00:14:08,640 --> 00:14:10,740 So now what we want to do is talk 209 00:14:10,740 --> 00:14:12,249 about the two-split experiment. 210 00:14:23,330 --> 00:14:25,860 Let's just begin. 211 00:14:25,860 --> 00:14:28,270 So here's a diagram. 212 00:14:28,270 --> 00:14:30,870 And here we have a source of light. 213 00:14:34,790 --> 00:14:36,710 It's a light bulb in your notes. 214 00:14:36,710 --> 00:14:38,450 It's a candle here. 215 00:14:38,450 --> 00:14:48,250 And then we have two slits, and the slits 216 00:14:48,250 --> 00:14:50,590 are separated by distance d. 217 00:14:56,170 --> 00:15:02,410 And so this is the first slit, s1, and the second slit, s2. 218 00:15:02,410 --> 00:15:07,600 And the distance between them is much larger 219 00:15:07,600 --> 00:15:08,860 than the width of each slit. 220 00:15:11,620 --> 00:15:15,880 And now we go down to the screen. 221 00:15:21,080 --> 00:15:28,700 And the distance from the slits to the screen is l, 222 00:15:28,700 --> 00:15:33,314 and l is much, much larger than d. 223 00:15:33,314 --> 00:15:34,730 This means, of course, we're going 224 00:15:34,730 --> 00:15:37,460 to be using small-angle approximation 225 00:15:37,460 --> 00:15:41,390 and simple solutions, because everything is much larger 226 00:15:41,390 --> 00:15:42,900 than something else. 227 00:15:42,900 --> 00:15:45,080 And that's very convenient. 228 00:15:45,080 --> 00:15:48,850 Now I want to just put on axes. 229 00:15:48,850 --> 00:15:55,880 So this is the x-axis, and this is 0, and this is l. 230 00:15:55,880 --> 00:15:57,900 And now the screen-- 231 00:15:57,900 --> 00:16:00,140 we're going to see something on the screen that 232 00:16:00,140 --> 00:16:03,800 looks like this. 233 00:16:03,800 --> 00:16:07,170 I'm giving it away. 234 00:16:07,170 --> 00:16:14,910 And the distance here is 0, and the distance here 235 00:16:14,910 --> 00:16:16,180 is the z-axis. 236 00:16:16,180 --> 00:16:20,760 So the distance to this slit is on the x-axis, 237 00:16:20,760 --> 00:16:27,230 and the pattern, the diffraction pattern, is on the z-axis. 238 00:16:27,230 --> 00:16:30,520 And this 0 is right in the middle of the pattern. 239 00:16:30,520 --> 00:16:33,530 It would correspond to this point, the midpoint here. 240 00:16:36,750 --> 00:16:38,630 So what we want to do is calculate 241 00:16:38,630 --> 00:16:41,880 what's going to appear on the screen, 242 00:16:41,880 --> 00:16:44,460 and I've already given it away. 243 00:16:44,460 --> 00:16:52,924 What you see is a bunch of equally-spaced intensity maxima 244 00:16:52,924 --> 00:16:54,590 where we have constructive interference. 245 00:16:54,590 --> 00:16:58,340 And in between, we have less constructive interference-- 246 00:16:58,340 --> 00:17:03,160 or destructive interference, and we want to understand that. 247 00:17:03,160 --> 00:17:04,290 Now this is optics. 248 00:17:04,290 --> 00:17:06,930 This is no quantum mechanics at all. 249 00:17:16,220 --> 00:17:20,099 Let's look at this in more detail. 250 00:17:20,099 --> 00:17:31,600 So we have the z-axis, horizontal, 251 00:17:31,600 --> 00:17:40,140 and we have a path to the screen, 252 00:17:40,140 --> 00:17:43,920 and another path to the screen. 253 00:17:43,920 --> 00:17:48,310 So what we're interested in is here's one slit, 254 00:17:48,310 --> 00:17:54,650 here's the other slit, and we have two parallel lines 255 00:17:54,650 --> 00:17:58,010 that meet at infinity, which is where the screen is-- 256 00:17:58,010 --> 00:18:00,230 and what we're going to be interested in 257 00:18:00,230 --> 00:18:07,200 is what is the path difference between this one and this one. 258 00:18:07,200 --> 00:18:13,500 So we have an angle which is given by the perpendicular 259 00:18:13,500 --> 00:18:14,890 to this right. 260 00:18:17,560 --> 00:18:20,860 So this distance is d. 261 00:18:20,860 --> 00:18:25,510 This distance is L. And this angle 262 00:18:25,510 --> 00:18:27,970 is theta, as is this angle. 263 00:18:31,140 --> 00:18:34,680 And what we're interested in is this-- 264 00:18:34,680 --> 00:18:41,350 the extra path traveled by the lower slit. 265 00:18:41,350 --> 00:18:44,480 So we use trigonometry, and we can figure that out. 266 00:18:44,480 --> 00:18:47,770 And so delta is the path difference. 267 00:18:52,580 --> 00:18:56,330 Delta is equal to d sine theta. 268 00:18:59,400 --> 00:19:04,340 And in order for it to be constructive interference, 269 00:19:04,340 --> 00:19:08,690 we have to have this path difference to be an integer 270 00:19:08,690 --> 00:19:12,410 number of wavelengths. 271 00:19:12,410 --> 00:19:17,810 Now this is optics, and so that's something 272 00:19:17,810 --> 00:19:20,080 that we don't need quantum mechanics for. 273 00:19:20,080 --> 00:19:23,370 We know we're going to get interference. 274 00:19:23,370 --> 00:19:28,190 So we can now solve for where the constructive interference 275 00:19:28,190 --> 00:19:30,000 occurs. 276 00:19:30,000 --> 00:19:36,920 And so we have theta n is equal to the inverse sine of n 277 00:19:36,920 --> 00:19:39,230 lambda over d. 278 00:19:39,230 --> 00:19:42,170 But this is a small angle, even though I drew it 279 00:19:42,170 --> 00:19:44,240 not as a small angle. 280 00:19:44,240 --> 00:19:55,280 And so we can replace sine x by x or inverse of sine y by y. 281 00:19:55,280 --> 00:20:00,380 And anyway, we can say that the angles 282 00:20:00,380 --> 00:20:06,300 for constructive interference are given by n lambda over d. 283 00:20:11,700 --> 00:20:16,620 So we've derived the diffraction equation. 284 00:20:16,620 --> 00:20:17,740 We've solved. 285 00:20:17,740 --> 00:20:21,240 And now what we want to know is where do the spots occur. 286 00:20:21,240 --> 00:20:25,830 They occur at z equals 0, z equals plus or minus l 287 00:20:25,830 --> 00:20:30,310 sine theta, which is approximately equal 288 00:20:30,310 --> 00:20:34,650 to l over d n lambda. 289 00:20:38,340 --> 00:20:43,560 So we have l times lambda over d times an integer. 290 00:20:43,560 --> 00:20:47,730 So what we're going to see on the screen 291 00:20:47,730 --> 00:20:55,740 is a series of bright lines for constructive interference-- 292 00:20:55,740 --> 00:20:59,730 and they're not lines. 293 00:20:59,730 --> 00:21:01,510 It's a curve. 294 00:21:01,510 --> 00:21:06,035 But we can measure the maximum of the intensity, 295 00:21:06,035 --> 00:21:07,410 and we can say they're like this, 296 00:21:07,410 --> 00:21:09,880 and they're equally spaced. 297 00:21:09,880 --> 00:21:12,990 And they tell us things we knew. 298 00:21:12,990 --> 00:21:13,710 We knew l. 299 00:21:13,710 --> 00:21:14,271 We knew d. 300 00:21:14,271 --> 00:21:14,895 We knew lambda. 301 00:21:19,700 --> 00:21:21,080 So there's nothing surprising. 302 00:21:21,080 --> 00:21:22,085 This is just optics. 303 00:21:28,240 --> 00:21:34,004 So now suppose we go in and we cover one slit. 304 00:21:34,004 --> 00:21:34,545 What happens? 305 00:21:38,580 --> 00:21:39,425 Yes? 306 00:21:39,425 --> 00:21:40,910 AUDIENCE: The interference stops? 307 00:21:40,910 --> 00:21:43,410 ROBERT FIELD: The interference goes away. 308 00:21:43,410 --> 00:21:49,650 And you could imagine that there would be a little sign. 309 00:21:49,650 --> 00:21:52,200 If you covered the top slit, the pattern 310 00:21:52,200 --> 00:21:55,170 would be skewed a little bit in the direction 311 00:21:55,170 --> 00:21:56,640 of the bottom slit. 312 00:21:56,640 --> 00:21:58,920 And so there'll be a little bit of asymmetry, 313 00:21:58,920 --> 00:22:03,150 but you could actually know which 314 00:22:03,150 --> 00:22:07,590 slit your colleague covered. 315 00:22:07,590 --> 00:22:11,270 So if both slits are open, you have interference. 316 00:22:11,270 --> 00:22:15,372 If one slit is covered, you have no interference. 317 00:22:25,040 --> 00:22:28,030 We're getting into the realm of quantum mechanics. 318 00:22:28,030 --> 00:22:32,670 In quantum mechanics, one of the things we do is we say, 319 00:22:32,670 --> 00:22:35,240 suppose we did a perfect experiment. 320 00:22:35,240 --> 00:22:37,700 Maybe it's an experiment that's beyond what you're 321 00:22:37,700 --> 00:22:41,490 capable of doing with the present technology, 322 00:22:41,490 --> 00:22:47,970 but you can say, I could measure positions in time 323 00:22:47,970 --> 00:22:53,550 as accurately as I want, and one could also 324 00:22:53,550 --> 00:22:58,980 say that I could do this an infinite number of times. 325 00:22:58,980 --> 00:23:04,130 I could do the same experiment an infinite number of times. 326 00:23:04,130 --> 00:23:07,364 Without quantum mechanics, if you did the same experiment 327 00:23:07,364 --> 00:23:08,780 an infinite number of times, you'd 328 00:23:08,780 --> 00:23:12,437 get the same answer an infinite number of times. 329 00:23:12,437 --> 00:23:14,770 But with quantum mechanics, you're going to discover you 330 00:23:14,770 --> 00:23:17,200 don't. 331 00:23:17,200 --> 00:23:20,730 It's probabilistic, not deterministic. 332 00:23:20,730 --> 00:23:26,130 And under certain conditions, the range over which you have 333 00:23:26,130 --> 00:23:27,870 a finite probability is very small, 334 00:23:27,870 --> 00:23:33,120 and it looks deterministic, but it isn't. 335 00:23:33,120 --> 00:23:39,840 The perfect experiment business is an interesting hypothesis, 336 00:23:39,840 --> 00:23:44,250 but you can imagine defining what 337 00:23:44,250 --> 00:23:47,130 is perfect in terms of what is intrinsically 338 00:23:47,130 --> 00:23:49,865 possible to achieve, even if it's not currently possible. 339 00:24:11,310 --> 00:24:15,300 So what we want to do is decrease the intensity 340 00:24:15,300 --> 00:24:18,840 of the light that's going into the apparatus, 341 00:24:18,840 --> 00:24:26,730 so that there is never more than one photon in the apparatus. 342 00:24:26,730 --> 00:24:30,010 Never is a strong word, and we could never do that, 343 00:24:30,010 --> 00:24:31,170 and so that's not legal. 344 00:24:31,170 --> 00:24:37,690 But we can say, suppose we decrease the intensity 345 00:24:37,690 --> 00:24:41,790 so that for the time it takes for the photon 346 00:24:41,790 --> 00:24:45,090 to go from the slit to the screen, which we know 347 00:24:45,090 --> 00:24:48,930 because we know the speed of light, 348 00:24:48,930 --> 00:24:51,960 and for the intensity of the light, which we can measure 349 00:24:51,960 --> 00:24:57,780 with an energy meter, we can say the probability of there being 350 00:24:57,780 --> 00:25:03,800 more than one photon at a time in the apparatus is small, as 351 00:25:03,800 --> 00:25:05,630 small as we want, but not zero. 352 00:25:08,590 --> 00:25:11,890 And so then we do the experiment. 353 00:25:11,890 --> 00:25:26,780 And what we discover when we do the experiment is instead 354 00:25:26,780 --> 00:25:34,280 of having a uniform intensity or some kind of continuously 355 00:25:34,280 --> 00:25:37,940 varying intensity on the detector screen, 356 00:25:37,940 --> 00:25:40,510 we get a series of dots-- 357 00:25:40,510 --> 00:25:43,100 events. 358 00:25:43,100 --> 00:25:46,170 The photon went in, and the photon 359 00:25:46,170 --> 00:25:47,660 was a wave when it went in. 360 00:25:51,240 --> 00:25:56,370 There was interference, maybe, and the photon 361 00:25:56,370 --> 00:25:59,600 died on this detector screen. 362 00:25:59,600 --> 00:26:02,350 This is an example of destructive detection, which 363 00:26:02,350 --> 00:26:05,470 is something that is very important in quantum mechanics, 364 00:26:05,470 --> 00:26:13,160 because in quantum mechanics, most measurements destroy 365 00:26:13,160 --> 00:26:17,270 the system, or destroy the state that the system was 366 00:26:17,270 --> 00:26:23,010 in during the experiment. 367 00:26:23,010 --> 00:26:26,210 So this business of what is the state of the system 368 00:26:26,210 --> 00:26:29,720 is a really important quantum mechanical concept, 369 00:26:29,720 --> 00:26:33,430 which you don't normally encounter in classic mechanics. 370 00:26:37,040 --> 00:26:40,920 We send photons one at a time through the apparatus, 371 00:26:40,920 --> 00:26:43,160 and we get something like this. 372 00:26:43,160 --> 00:26:47,090 And we get something like this whether both slits are opened 373 00:26:47,090 --> 00:26:48,230 or one slit is covered. 374 00:26:51,650 --> 00:26:55,600 So we do this, and we do this for a long time, 375 00:26:55,600 --> 00:27:02,300 and what we see is we see a lot of events, 376 00:27:02,300 --> 00:27:05,000 and they're starting to arrange themselves 377 00:27:05,000 --> 00:27:09,100 where the interference fringes were supposed to be. 378 00:27:09,100 --> 00:27:13,900 So this pattern only emerges after you'd 379 00:27:13,900 --> 00:27:21,020 allow a large number of photons to go into the apparatus. 380 00:27:21,020 --> 00:27:24,450 There is no way classical optics gives you that. 381 00:27:28,320 --> 00:27:31,050 And so if one slit is covered, you 382 00:27:31,050 --> 00:27:35,580 get a uniform distribution of dots. 383 00:27:35,580 --> 00:27:39,330 If both slits are open, you get this kind of a distribution. 384 00:27:51,010 --> 00:27:53,150 I'm going to ask you to vote on this. 385 00:27:53,150 --> 00:27:55,570 So we do the experiment-- 386 00:27:55,570 --> 00:27:57,340 and I've sort of given away the answer, 387 00:27:57,340 --> 00:28:00,040 but I still want you to vote on this. 388 00:28:00,040 --> 00:28:01,350 What are the possible things? 389 00:28:08,000 --> 00:28:12,460 So we know there's only one photon 390 00:28:12,460 --> 00:28:15,300 in the apparatus at a time. 391 00:28:15,300 --> 00:28:21,150 Our concept of interference is light interference with itself. 392 00:28:21,150 --> 00:28:25,310 And this is a possibility that says, I think I know. 393 00:28:25,310 --> 00:28:32,990 And here is another, weak interference 394 00:28:32,990 --> 00:28:40,302 on top of constant background. 395 00:28:43,470 --> 00:28:45,060 This would reflect. 396 00:28:45,060 --> 00:28:47,280 Even though we decided that we would 397 00:28:47,280 --> 00:28:50,220 have one photon in the apparatus, 398 00:28:50,220 --> 00:28:52,520 occasionally there are two. 399 00:28:52,520 --> 00:28:55,310 And when there's two, there could be interference, 400 00:28:55,310 --> 00:28:58,940 and so we'd have some weak interference superimposed 401 00:28:58,940 --> 00:29:03,472 on the constant background. 402 00:29:14,300 --> 00:29:18,620 Now we get this 100% modulated interference structure. 403 00:29:18,620 --> 00:29:20,510 And the last thing is something else. 404 00:29:28,580 --> 00:29:34,210 You have to transport yourself back in time to around 1910. 405 00:29:34,210 --> 00:29:37,410 You haven't heard this lecture, but you 406 00:29:37,410 --> 00:29:39,450 do know what the experiment is. 407 00:29:39,450 --> 00:29:43,380 And so what would you expect? 408 00:29:43,380 --> 00:29:50,040 First, no interference, raise your hand. 409 00:29:50,040 --> 00:29:51,180 I've got one-- two-- 410 00:29:51,180 --> 00:29:53,490 I've got a few votes for no interference. 411 00:29:53,490 --> 00:29:57,640 In 1910, that's what you would have said. 412 00:29:57,640 --> 00:30:01,751 Weak interference on top of constant background-- 413 00:30:06,720 --> 00:30:10,110 that would be when you're hedging your bets and saying, 414 00:30:10,110 --> 00:30:12,010 the experiment, it isn't perfect, 415 00:30:12,010 --> 00:30:14,230 and this is really the right answer, 416 00:30:14,230 --> 00:30:17,160 but if someone was a little sloppy, I'd get this. 417 00:30:20,560 --> 00:30:23,780 Raise your hands for this one. 418 00:30:23,780 --> 00:30:30,270 What would you have said in 1910? 419 00:30:30,270 --> 00:30:33,330 I got nobody with the courage to say this. 420 00:30:33,330 --> 00:30:35,340 You would have gotten a Nobel Prize 421 00:30:35,340 --> 00:30:38,680 if you could have defended it. 422 00:30:38,680 --> 00:30:41,260 Something else-- maybe something else-- maybe the photons 423 00:30:41,260 --> 00:30:47,500 come in pairs or something ridiculous. 424 00:30:47,500 --> 00:30:52,950 So the correct answer is 100% modulated. 425 00:30:52,950 --> 00:30:57,010 What people would have said in 1910 is this. 426 00:30:57,010 --> 00:31:00,910 Some curmudgeons, like the climate change deniers, 427 00:31:00,910 --> 00:31:03,570 would have said there is a little bit of this or maybe 428 00:31:03,570 --> 00:31:06,010 that, but I can't possibly accept 429 00:31:06,010 --> 00:31:07,330 that, which is the truth. 430 00:31:10,600 --> 00:31:12,915 We won't belabor that anymore. 431 00:31:15,950 --> 00:31:20,850 This means that one photon can interfere with itself. 432 00:31:20,850 --> 00:31:24,500 It's a very disturbing idea, but it 433 00:31:24,500 --> 00:31:30,590 leads to a critical idea in quantum mechanics. 434 00:31:30,590 --> 00:31:32,960 In quantum mechanics, the state of the system 435 00:31:32,960 --> 00:31:39,230 is described by some state function, which is 436 00:31:39,230 --> 00:31:41,970 a function of position in time. 437 00:31:41,970 --> 00:31:46,400 And so what happens is you prepare 438 00:31:46,400 --> 00:31:49,070 the system in some state. 439 00:31:49,070 --> 00:31:50,930 You do something to it, like force 440 00:31:50,930 --> 00:31:55,920 it to go through two slits. 441 00:31:55,920 --> 00:31:57,545 Then we get some new state function. 442 00:32:00,130 --> 00:32:04,570 And then we detect it, and we get something else. 443 00:32:04,570 --> 00:32:09,700 So the actual experiment is a click, the preparation, 444 00:32:09,700 --> 00:32:11,950 and the click, detection. 445 00:32:11,950 --> 00:32:16,060 And somehow, what is the nature of the experiment 446 00:32:16,060 --> 00:32:19,900 is expressed on this initial state of the system. 447 00:32:19,900 --> 00:32:26,540 This is all very abstract, but it's about interference. 448 00:32:26,540 --> 00:32:31,810 So this guy had better have phase. 449 00:32:31,810 --> 00:32:37,780 So we have a wave that can constructively or destructively 450 00:32:37,780 --> 00:32:39,295 interfere with itself. 451 00:32:42,190 --> 00:32:54,020 And so we start talking about things like amplitude, 452 00:32:54,020 --> 00:32:56,540 but the crucial word is amplitude. 453 00:32:56,540 --> 00:32:59,210 And mostly, when we detect things, 454 00:32:59,210 --> 00:33:00,650 we're detecting probability. 455 00:33:04,970 --> 00:33:07,830 This is always positive. 456 00:33:07,830 --> 00:33:11,160 These guys can be positive and negative. 457 00:33:11,160 --> 00:33:14,330 This is essential for quantum mechanics. 458 00:33:14,330 --> 00:33:18,050 It's essential for understanding the two-slit experiment, 459 00:33:18,050 --> 00:33:22,055 but we have to do an awful lot more to make all this concrete. 460 00:33:35,539 --> 00:33:37,580 Why don't we look at the classical wave equation? 461 00:33:45,780 --> 00:33:47,920 Actually, we'll look at the classical wave equation 462 00:33:47,920 --> 00:33:51,390 next time, but I will say that the solution to the wave 463 00:33:51,390 --> 00:33:55,380 equation is some function of x and t, 464 00:33:55,380 --> 00:34:02,670 which has the form a sine kx minus omega t. 465 00:34:02,670 --> 00:34:04,560 So I'm doing this to introduce you 466 00:34:04,560 --> 00:34:09,510 to the crucial actors in this game, which is amplitude, 467 00:34:09,510 --> 00:34:11,969 wave number, and frequency. 468 00:34:16,550 --> 00:34:19,900 And this is a probability amplitude. 469 00:34:19,900 --> 00:34:22,760 It can have either sine, because you can see the sine function. 470 00:34:25,290 --> 00:34:30,909 So this is a wave of frequency omega propagating 471 00:34:30,909 --> 00:34:34,210 in the plus x direction. 472 00:34:34,210 --> 00:34:40,670 Now let's just identify the crucial quantity. 473 00:34:43,730 --> 00:34:46,070 Wavelength is the repeat distance. 474 00:34:46,070 --> 00:34:51,110 So that if we said we have u of x and t, 475 00:34:51,110 --> 00:34:57,590 it has to be equal to u of x plus lambda and t. 476 00:34:57,590 --> 00:35:00,000 So that's how we define the wavelength. 477 00:35:00,000 --> 00:35:02,310 And we discover that the wavelength 478 00:35:02,310 --> 00:35:09,350 has to be related to k times lambda is equal to 2 pi. 479 00:35:09,350 --> 00:35:13,220 Because this part has to change by 2 pi in order 480 00:35:13,220 --> 00:35:19,950 for there to be an exact replica of what we had before. 481 00:35:19,950 --> 00:35:22,910 So we know that the wavelength is the repeat distance, 482 00:35:22,910 --> 00:35:28,740 and k is 2 pi over lambda. 483 00:35:28,740 --> 00:35:31,310 It's called wave number, and it's 484 00:35:31,310 --> 00:35:36,170 the number of waves, complete waves, that occur 485 00:35:36,170 --> 00:35:39,050 in 2 pi times the unit length. 486 00:35:43,920 --> 00:35:49,030 Now, in 3D, we have a vector as opposed to a number. 487 00:35:49,030 --> 00:35:52,240 And that points in the direction of propagation of the wave. 488 00:35:59,500 --> 00:36:02,770 We know from quantum mechanics-- 489 00:36:02,770 --> 00:36:04,330 or from the experiments-- 490 00:36:07,930 --> 00:36:11,800 that the wavelength is related to the momentum. 491 00:36:11,800 --> 00:36:17,290 There's several reasons for this for waves. 492 00:36:17,290 --> 00:36:19,120 This could still be optics, but it 493 00:36:19,120 --> 00:36:21,310 could have been relativistic optics, 494 00:36:21,310 --> 00:36:28,820 because Einstein proposed that the momentum is e over c. 495 00:36:31,990 --> 00:36:35,200 So we put that together, and we get the relationship 496 00:36:35,200 --> 00:36:38,830 between k and momentum. 497 00:36:45,470 --> 00:36:48,130 Now h bar is h over 2 pi. 498 00:36:48,130 --> 00:36:54,460 And so the wave number is large if the momentum is large, 499 00:36:54,460 --> 00:36:57,430 and the wavelength is small if the momentum is large. 500 00:37:01,460 --> 00:37:03,780 Now what about the velocity? 501 00:37:03,780 --> 00:37:06,930 So we have a wave. 502 00:37:06,930 --> 00:37:10,700 Let's sit on this wave here and say, 503 00:37:10,700 --> 00:37:17,690 we're now sitting on some point where the phase is constant. 504 00:37:17,690 --> 00:37:23,270 I like to call this the stationary phase point, 505 00:37:23,270 --> 00:37:25,940 but that has other meanings, so you just 506 00:37:25,940 --> 00:37:27,500 have to be careful with this. 507 00:37:27,500 --> 00:37:31,250 So we want to know how fast this moves in space. 508 00:37:31,250 --> 00:37:39,110 And so we say, the phase is the phase of this function here, 509 00:37:39,110 --> 00:37:40,670 and so it's k x-- 510 00:37:40,670 --> 00:37:43,620 I'll put a little phi on this-- 511 00:37:43,620 --> 00:37:46,160 minus omega t. 512 00:37:46,160 --> 00:37:49,100 And we want to find how this moves 513 00:37:49,100 --> 00:37:51,410 in time, this stationary phase. 514 00:37:51,410 --> 00:37:54,710 We could choose this to be zero. 515 00:37:54,710 --> 00:37:56,450 We could choose a phase. 516 00:37:56,450 --> 00:37:59,202 This is zero, this is some maximum, 517 00:37:59,202 --> 00:38:00,660 but we can choose anything we want. 518 00:38:00,660 --> 00:38:01,970 So let's make it zero. 519 00:38:01,970 --> 00:38:06,680 And so then we can solve for x phi is a function of t, 520 00:38:06,680 --> 00:38:14,060 and that's omega t over k. 521 00:38:20,160 --> 00:38:24,160 We want the phase velocity, the velocity of this wave. 522 00:38:24,160 --> 00:38:26,700 So we take the derivative with respect to t. 523 00:38:26,700 --> 00:38:34,160 And so the velocity, which we'll call c, 524 00:38:34,160 --> 00:38:36,800 is equal to omega over k. 525 00:38:39,430 --> 00:38:43,970 That's true for light traveling in a vacuum. 526 00:38:43,970 --> 00:38:46,750 This is called the dispersion relation. 527 00:38:46,750 --> 00:38:51,760 Whenever you do a calculation of waves in material, 528 00:38:51,760 --> 00:38:54,910 the goal is to get the dispersion relation. 529 00:38:54,910 --> 00:39:00,190 This is the simplest possible one, because it says everybody 530 00:39:00,190 --> 00:39:03,220 at the same frequency-- 531 00:39:03,220 --> 00:39:09,180 I'm sorry-- that there is a relationship between omega 532 00:39:09,180 --> 00:39:13,360 and k so that everybody travels with the same speed. 533 00:39:13,360 --> 00:39:15,649 If that didn't happen, the waves would get out 534 00:39:15,649 --> 00:39:16,690 of phase with each other. 535 00:39:16,690 --> 00:39:19,510 That's what's dispersion is. 536 00:39:19,510 --> 00:39:24,980 So we have from simple optics, basically, everything we need. 537 00:39:24,980 --> 00:39:31,210 And now, the last thing is that the intensity of this wave 538 00:39:31,210 --> 00:39:54,250 is given by kkix minus omega i t. 539 00:39:54,250 --> 00:39:59,000 So this is the wave function. 540 00:39:59,000 --> 00:40:03,340 It's something that has sines. 541 00:40:03,340 --> 00:40:06,850 And this is the intensity, which is 542 00:40:06,850 --> 00:40:08,950 proportional to this quantity. 543 00:40:08,950 --> 00:40:13,570 We sum the amplitudes and then square. 544 00:40:13,570 --> 00:40:17,080 We do this in quantum mechanics all the time. 545 00:40:17,080 --> 00:40:19,780 And this is like quantum mechanics in the sense 546 00:40:19,780 --> 00:40:22,720 that we have our fundamental building block, which 547 00:40:22,720 --> 00:40:24,640 is something with phase. 548 00:40:24,640 --> 00:40:26,500 And the relationship of the thing 549 00:40:26,500 --> 00:40:30,970 with phase to the thing which is probability 550 00:40:30,970 --> 00:40:34,290 is sum of the square. 551 00:40:34,290 --> 00:40:34,980 Yes? 552 00:40:34,980 --> 00:40:37,721 AUDIENCE: What's the first character after the capital 553 00:40:37,721 --> 00:40:38,220 sigma? 554 00:40:38,220 --> 00:40:39,220 ROBERT FIELD: I'm sorry? 555 00:40:39,220 --> 00:40:42,440 AUDIENCE: The first character after the sum there? 556 00:40:42,440 --> 00:40:43,420 ROBERT FIELD: This? 557 00:40:43,420 --> 00:40:44,150 No, this. 558 00:40:44,150 --> 00:40:45,750 That's a constant. 559 00:40:45,750 --> 00:40:51,750 That's the amplitude of that particular frequency and wave 560 00:40:51,750 --> 00:40:53,790 vector. 561 00:40:53,790 --> 00:40:55,590 I'm sorry about the mess on the board. 562 00:40:59,700 --> 00:41:00,600 We have enough time. 563 00:41:00,600 --> 00:41:01,860 Oh, good. 564 00:41:01,860 --> 00:41:04,650 So this is sort of a taste of what you're 565 00:41:04,650 --> 00:41:09,060 going to be doing, but now let's produce 566 00:41:09,060 --> 00:41:18,961 a form of the uncertainty principle. 567 00:41:32,380 --> 00:41:35,170 You've heard about the uncertainty principle, 568 00:41:35,170 --> 00:41:39,420 and it's a very important part of quantum mechanics, 569 00:41:39,420 --> 00:41:42,970 but it's also something that you can have in optics. 570 00:41:42,970 --> 00:41:49,160 And so again, we resort to this simple idea 571 00:41:49,160 --> 00:41:57,010 of sending a particle through a slit 572 00:41:57,010 --> 00:42:01,140 and looking at what happens over here. 573 00:42:01,140 --> 00:42:04,322 Instead of having two slits, we just have one. 574 00:42:04,322 --> 00:42:05,780 But there are two important things. 575 00:42:05,780 --> 00:42:11,430 The two edges of this slit are special, because they're edges. 576 00:42:11,430 --> 00:42:14,400 And so we can ask, what about the interference 577 00:42:14,400 --> 00:42:17,910 between particles that was diffracted 578 00:42:17,910 --> 00:42:21,290 by this edge versus that edge. 579 00:42:21,290 --> 00:42:22,990 And the same thing goes. 580 00:42:22,990 --> 00:42:26,320 You get constructive interference 581 00:42:26,320 --> 00:42:28,300 when the difference in path length 582 00:42:28,300 --> 00:42:32,350 is an integer number of waves, and destructive interference 583 00:42:32,350 --> 00:42:38,990 when it's an odd integer number of half waves. 584 00:42:38,990 --> 00:42:43,730 We're interested in the destructive interference. 585 00:42:43,730 --> 00:42:49,080 So we analyze this in exactly the same way 586 00:42:49,080 --> 00:42:51,520 we did the two-slit experiment. 587 00:42:51,520 --> 00:42:55,250 And now this is the z direction. 588 00:42:55,250 --> 00:42:56,938 And this is the x direction. 589 00:42:59,630 --> 00:43:02,920 And what you end up finding is that the uncertainty 590 00:43:02,920 --> 00:43:05,650 in the z direction is going to be 591 00:43:05,650 --> 00:43:14,620 related 2 times lambda over delta s over l. 592 00:43:14,620 --> 00:43:27,480 Delta s is the width of the slit, so width 593 00:43:27,480 --> 00:43:32,200 of the image along the z-axis. 594 00:43:32,200 --> 00:43:37,500 So we have a maximum here, and we have minima here and here, 595 00:43:37,500 --> 00:43:48,030 and so this delta z is the distance between minima. 596 00:43:48,030 --> 00:43:53,440 So we can say, between minima, we have the particle localized. 597 00:43:53,440 --> 00:43:58,890 Its position in space-- 598 00:43:58,890 --> 00:44:01,530 or the photon localized-- its position in space 599 00:44:01,530 --> 00:44:03,435 is uncertain by this quantity. 600 00:44:06,251 --> 00:44:07,250 What about its momentum? 601 00:44:12,940 --> 00:44:16,396 And so now we just have to draw a little bit of conservation 602 00:44:16,396 --> 00:44:16,895 of momentum. 603 00:44:28,240 --> 00:44:33,010 So here we have the magnitude of the momentum starting 604 00:44:33,010 --> 00:44:38,410 from the middle of the slit, and this magnitude 605 00:44:38,410 --> 00:44:40,120 is constant along a circle-- 606 00:44:43,690 --> 00:44:44,410 the magnitude. 607 00:44:47,600 --> 00:44:52,880 So if we do draw a line here, this has got p. 608 00:44:52,880 --> 00:44:57,370 And if we ask the length here, that's also p. 609 00:44:57,370 --> 00:44:59,060 But now what we want to know is what 610 00:44:59,060 --> 00:45:03,271 is the uncertainty of the momentum in the z direction-- 611 00:45:06,220 --> 00:45:07,560 delta pz. 612 00:45:12,820 --> 00:45:16,080 This is the same sort of calculation we did before, 613 00:45:16,080 --> 00:45:21,630 and what you find is delta pz is approximately 614 00:45:21,630 --> 00:45:29,790 equal to magnitude of p lambda or delta s. 615 00:45:29,790 --> 00:45:35,370 And now we put in the Bohr relationship here, 616 00:45:35,370 --> 00:45:39,650 and we get that this is equal to h 617 00:45:39,650 --> 00:45:46,980 over lambda, lambda over the ds, which is equal to h over ps. 618 00:45:49,960 --> 00:45:56,490 Or ds in the z direction-- 619 00:45:56,490 --> 00:45:58,200 no, ds is the slit width. 620 00:46:04,910 --> 00:46:07,630 I got something wrong here in my notes. 621 00:46:10,520 --> 00:46:12,980 The final result of this calculation 622 00:46:12,980 --> 00:46:22,280 is dz is dpz approximately equal to h. 623 00:46:22,280 --> 00:46:23,870 Sorry about the glitch here. 624 00:46:23,870 --> 00:46:25,850 I don't know how to fix it right now, 625 00:46:25,850 --> 00:46:29,450 but if this were done correctly, we 626 00:46:29,450 --> 00:46:31,580 would have gotten this result. 627 00:46:31,580 --> 00:46:33,410 This is an uncertainty principle. 628 00:46:33,410 --> 00:46:38,440 If you tried to measure z and pz simultaneously 629 00:46:38,440 --> 00:46:42,940 with classical optics, you still would get something like this. 630 00:46:42,940 --> 00:46:48,790 Using the relationship for momentum 631 00:46:48,790 --> 00:46:53,670 that lambda is equal h over p, we've got to use that. 632 00:46:53,670 --> 00:46:55,450 But because of that relationship, 633 00:46:55,450 --> 00:46:57,250 we get this result. 634 00:46:57,250 --> 00:47:00,520 If you do a perfect experiment, and you make the slit smaller 635 00:47:00,520 --> 00:47:03,340 and smaller, you make the uncertainty in the momentum 636 00:47:03,340 --> 00:47:05,140 larger and larger. 637 00:47:05,140 --> 00:47:06,850 The best you can do is this. 638 00:47:10,850 --> 00:47:14,540 Now, this is actually a reasonable and rigorous 639 00:47:14,540 --> 00:47:20,140 derivation if I had done it a little better. 640 00:47:20,140 --> 00:47:26,070 But I've never liked this introduction to the uncertainty 641 00:47:26,070 --> 00:47:30,600 principle, because it says, for the kind of experiment 642 00:47:30,600 --> 00:47:33,270 we thought about, you can't do better than this. 643 00:47:33,270 --> 00:47:36,260 Maybe there's a different kind of experiment. 644 00:47:36,260 --> 00:47:39,140 It's sort of an artifactual as opposed 645 00:47:39,140 --> 00:47:41,810 to a physical derivation. 646 00:47:41,810 --> 00:47:42,870 We care. 647 00:47:42,870 --> 00:47:47,150 We will do a physical derivation of the uncertainty principle. 648 00:47:47,150 --> 00:47:49,610 And it will have to do with the ability of two 649 00:47:49,610 --> 00:47:52,360 operates to commute with each other. 650 00:47:52,360 --> 00:47:56,250 That's a purely mathematical definition, 651 00:47:56,250 --> 00:48:03,870 but this is the first sign of the uncertainty principle. 652 00:48:12,830 --> 00:48:15,200 At the end of your notes, there is 653 00:48:15,200 --> 00:48:22,130 a set of postulates from which, essentially, all quantum 654 00:48:22,130 --> 00:48:24,210 mechanics can be derived. 655 00:48:24,210 --> 00:48:26,900 Now, there are different sets of postulates proposed 656 00:48:26,900 --> 00:48:28,670 by different people, but these are 657 00:48:28,670 --> 00:48:30,425 things that can't be proven. 658 00:48:30,425 --> 00:48:32,550 They are things that you think is going to be true, 659 00:48:32,550 --> 00:48:34,610 and then you look at the consequences. 660 00:48:34,610 --> 00:48:37,550 The first postulate says, the state 661 00:48:37,550 --> 00:48:39,650 of a quantum mechanical system is completely 662 00:48:39,650 --> 00:48:43,520 specified by a function, psi of r and t, 663 00:48:43,520 --> 00:48:47,030 that depends on the coordinates of the particle and on time. 664 00:48:47,030 --> 00:48:51,380 This function, called the wave function or the state function, 665 00:48:51,380 --> 00:48:55,730 has the important property that this quantity 666 00:48:55,730 --> 00:49:01,220 times its complex conjugate integrated over the volume 667 00:49:01,220 --> 00:49:04,880 element is the probability that the particle lies 668 00:49:04,880 --> 00:49:10,500 in the volume element centered at r at time t. 669 00:49:10,500 --> 00:49:14,460 So we are saying there is a way, a complete way, 670 00:49:14,460 --> 00:49:17,190 of describing the state of the system. 671 00:49:17,190 --> 00:49:18,850 It's a function. 672 00:49:18,850 --> 00:49:23,670 It's not a bunch of discrete quantities, 673 00:49:23,670 --> 00:49:25,950 like velocity and position. 674 00:49:25,950 --> 00:49:28,560 And things are smeared out. 675 00:49:28,560 --> 00:49:32,220 And what we want to do when we want to calculate anything, 676 00:49:32,220 --> 00:49:35,460 we're going to be using this function. 677 00:49:35,460 --> 00:49:39,350 And this function comes from the Schrodinger equation. 678 00:49:39,350 --> 00:49:42,530 And we're going to get to the Schrodinger equation-- 679 00:49:42,530 --> 00:49:43,860 not in the next lecture. 680 00:49:43,860 --> 00:49:46,760 I'm going to spend the next lecture really laboring 681 00:49:46,760 --> 00:49:51,880 the wave equation, because the Schrodinger equation is just 682 00:49:51,880 --> 00:49:54,430 a tiny step beyond the wave equation. 683 00:49:59,270 --> 00:50:02,060 I've given you the five postulates. 684 00:50:02,060 --> 00:50:03,740 You have not a clue what any of them 685 00:50:03,740 --> 00:50:07,450 mean, except maybe a little bit about the first one. 686 00:50:07,450 --> 00:50:09,670 But what I don't want to do is give a lecture 687 00:50:09,670 --> 00:50:11,770 on the postulates. 688 00:50:11,770 --> 00:50:15,610 I want to bring them into action when you need them, 689 00:50:15,610 --> 00:50:18,780 because they'll mean much more. 690 00:50:18,780 --> 00:50:22,150 And so you won't be asked to memorize the postulates. 691 00:50:22,150 --> 00:50:25,930 You'll know when they are applicable 692 00:50:25,930 --> 00:50:28,880 and how to apply them. 693 00:50:28,880 --> 00:50:30,910 So it's a little premature to say 694 00:50:30,910 --> 00:50:33,860 you understand the first postulate, 695 00:50:33,860 --> 00:50:37,260 but that's what's at play here in the two-slit experiment 696 00:50:37,260 --> 00:50:42,260 and in this hand waving derivation of the uncertainty 697 00:50:42,260 --> 00:50:43,990 principle. 698 00:50:43,990 --> 00:50:51,460 So next time, we will look at the wave equation. 699 00:50:51,460 --> 00:50:52,480 Thank you. 700 00:50:52,480 --> 00:50:54,430 Hey, I finished on time this time. 701 00:50:54,430 --> 00:50:56,460 That's a bad sign.