1 00:00:00 --> 00:00:01 2 00:00:01 --> 00:00:02 The following content is provided under a Creative 3 00:00:02 --> 00:00:03 Commons license. 4 00:00:03 --> 00:00:06 Your support will help MIT OpenCourseWare continue to 5 00:00:06 --> 00:00:10 offer high-quality educational resources for free. 6 00:00:10 --> 00:00:13 To make a donation or view additional materials from 7 00:00:13 --> 00:00:15 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:15 --> 00:00:17 at ocw.mit.edu. 9 00:00:17 --> 00:00:25 PROFESSOR: And this is a question based on where we left 10 00:00:25 --> 00:00:28 off on Wednesday -- we were talking about Coulomb's force 11 00:00:28 --> 00:00:32 law to describe the interaction between two particles, and 12 00:00:32 --> 00:00:35 good job, most of you got this correct. 13 00:00:35 --> 00:00:39 So, what we're looking at here is the force when we have 14 00:00:39 --> 00:00:42 two charged particles, one positive, one negative -- here, 15 00:00:42 --> 00:00:46 the nucleus and an electron. 16 00:00:46 --> 00:00:49 So, I know this is a simple example and I can see everyone 17 00:00:49 --> 00:00:52 pretty much got it right, and probably those that didn't 18 00:00:52 --> 00:00:55 actually made some sort of clicker error is my guess. 19 00:00:55 --> 00:00:59 But I wanted to use this to point out that in this class in 20 00:00:59 --> 00:01:03 general, any time you see an equation to explain a certain 21 00:01:03 --> 00:01:06 phenomenon, such as here looking at force, it's a good 22 00:01:06 --> 00:01:09 idea to check yourself by first plugging it into the actual 23 00:01:09 --> 00:01:12 equation, so you can plug in infinity and this equation 24 00:01:12 --> 00:01:15 here, and what you would see is, of course, the force, if 25 00:01:15 --> 00:01:18 you just solve the math problem goes to zero. 26 00:01:18 --> 00:01:21 But you can also look at it qualitatively, so, if you think 27 00:01:21 --> 00:01:25 about the force between the electron and the proton, you 28 00:01:25 --> 00:01:27 could just qualitatively think about what's happening. 29 00:01:27 --> 00:01:29 If they're close together there's a certain force -- 30 00:01:29 --> 00:01:32 they're attracted because they have opposite charges, but as 31 00:01:32 --> 00:01:35 that gets further and further away, that force is going to 32 00:01:35 --> 00:01:38 get smaller and smaller, and eventually the force is 33 00:01:38 --> 00:01:39 going to approach zero. 34 00:01:39 --> 00:01:42 So, it's a good kind of mental check as we go through this 35 00:01:42 --> 00:01:45 course to remember every time there's an equation, usually 36 00:01:45 --> 00:01:47 there's a very good reason for that equation, and you can go 37 00:01:47 --> 00:01:50 ahead and just use your qualitative knowledge, you 38 00:01:50 --> 00:01:52 don't have to just always stick with the math to check 39 00:01:52 --> 00:01:54 and justify your answers. 40 00:01:54 --> 00:01:58 So, we can get started with today's lecture notes. 41 00:01:58 --> 00:02:03 And, as I mentioned, we left off and as we started back here 42 00:02:03 --> 00:02:07 to describe the atom and how the atom holds together the 43 00:02:07 --> 00:02:11 nucleus and the electron using classical mechanics. 44 00:02:11 --> 00:02:13 And today we'll finish that discussion, and, of course, 45 00:02:13 --> 00:02:17 point out actually the failure of classical mechanics to 46 00:02:17 --> 00:02:21 appropriately describe what's going on in an atom. 47 00:02:21 --> 00:02:24 So, then we'll get to turn to a new kind of mechanics or 48 00:02:24 --> 00:02:28 quantum mechanics, which will in fact be able to describe 49 00:02:28 --> 00:02:31 what's happening on this very, very small size scale -- so on 50 00:02:31 --> 00:02:33 the atomic size scale on the order of nanometers 51 00:02:33 --> 00:02:37 or angstroms, very small particles. 52 00:02:37 --> 00:02:39 And the reason that quantum mechanics is going to work 53 00:02:39 --> 00:02:42 where classical mechanics fails is that classical mechanics did 54 00:02:42 --> 00:02:47 not take into account the fact that matter has both wave-like 55 00:02:47 --> 00:02:50 and particle-like properties, and light has both wave-like 56 00:02:50 --> 00:02:52 and particle-like properties. 57 00:02:52 --> 00:02:55 So, we'll take a little bit of a step back after we introduce 58 00:02:55 --> 00:02:58 quantum mechanics, and talk about light as a wave, and the 59 00:02:58 --> 00:03:02 characteristic of waves, and then light as a particle. 60 00:03:02 --> 00:03:08 And one example of this is in the photoelectric effect. 61 00:03:08 --> 00:03:12 So, we just talked about the force law to describe the 62 00:03:12 --> 00:03:15 interaction between a proton and an electron. 63 00:03:15 --> 00:03:17 You told me that when the distance went to infinity, 64 00:03:17 --> 00:03:19 the force went to zero. 65 00:03:19 --> 00:03:23 What happens instead when the distance goes to zero? 66 00:03:23 --> 00:03:26 What happens to the force? 67 00:03:26 --> 00:03:27 Yeah. 68 00:03:27 --> 00:03:30 So, the force actually goes to infinity, and specifically it 69 00:03:30 --> 00:03:31 goes to negative infinity. 70 00:03:31 --> 00:03:34 Infinity is the force when we're thinking about it and our 71 00:03:34 --> 00:03:37 brains, negative infinity is when we actually plug it into 72 00:03:37 --> 00:03:40 the equation here, and the reason is the convention that 73 00:03:40 --> 00:03:42 the negative sign is just telling us the direction that 74 00:03:42 --> 00:03:46 the force is coming together instead of pushing apart. 75 00:03:46 --> 00:03:49 So, we can use Coulomb's force law to think about the force 76 00:03:49 --> 00:03:52 between these two particles -- and it does that, it tells us 77 00:03:52 --> 00:03:54 the force is a function of that distance. 78 00:03:54 --> 00:03:57 But what it does not tell us, which if we're trying to 79 00:03:57 --> 00:04:02 describe an atom we really want to know, is what happens to the 80 00:04:02 --> 00:04:04 distance as time passes? 81 00:04:04 --> 00:04:07 So, r is a function of time. 82 00:04:07 --> 00:04:10 But luckily for us, there's a classical equation of motion 83 00:04:10 --> 00:04:14 that will, in fact, describe how the electron and nucleus 84 00:04:14 --> 00:04:19 change position or change their radius as a function of time. 85 00:04:19 --> 00:04:21 So, that's -- does anyone know which classical law 86 00:04:21 --> 00:04:25 of motion that would be? 87 00:04:25 --> 00:04:28 Yup, so it's going to be Newton's second law, force 88 00:04:28 --> 00:04:31 equals mass times acceleration -- those of you that are quick 89 00:04:31 --> 00:04:35 page-turners, have a little one-up on answering that. 90 00:04:35 --> 00:04:39 And that tells us force as a function of acceleration, we 91 00:04:39 --> 00:04:42 want to know it though as a function of radius, so we can 92 00:04:42 --> 00:04:45 just take the first derivative and get ourselves to velocity. 93 00:04:45 --> 00:04:47 So, force is equal to mass times dv 94 00:04:47 --> 00:04:49 /dt. 95 00:04:49 --> 00:04:52 But, of course, we want to go all the way to distance, so we 96 00:04:52 --> 00:04:54 take the second derivative and we have this equation 97 00:04:54 --> 00:04:56 for force here. 98 00:04:56 --> 00:04:59 And what we can do in order to bring the two equations 99 00:04:59 --> 00:05:04 together, is to plug in the Coulomb force law right here. 100 00:05:04 --> 00:05:08 So, now we have our Coulomb force law all plugged in here, 101 00:05:08 --> 00:05:10 and we have this differential equation that we could solve, 102 00:05:10 --> 00:05:14 if we wanted to figure out what the force was at different 103 00:05:14 --> 00:05:17 times t, or at different positions of r. 104 00:05:17 --> 00:05:19 So, all you will have the opportunity to solve 105 00:05:19 --> 00:05:21 differential equations in your math courses here. 106 00:05:21 --> 00:05:24 We won't do it in this chemistry course. 107 00:05:24 --> 00:05:26 In later chemistry courses, you'll also get to solve 108 00:05:26 --> 00:05:27 differential equations. 109 00:05:27 --> 00:05:30 But instead in this chemistry course, I will just tell 110 00:05:30 --> 00:05:33 you the solutions to differential equations. 111 00:05:33 --> 00:05:36 And what we can do is we can start with some initial value 112 00:05:36 --> 00:05:39 of r, and here I write r being ten angstroms. 113 00:05:39 --> 00:05:41 That's a good approximation when we're talking about 114 00:05:41 --> 00:05:43 atoms, because that's about the size of and atom. 115 00:05:43 --> 00:05:46 So, let's say we start off at the distance 116 00:05:46 --> 00:05:47 being ten angstroms. 117 00:05:47 --> 00:05:50 We can plug that into this differential equation that 118 00:05:50 --> 00:05:54 we'll have and solve it, and what we find out is that r 119 00:05:54 --> 00:05:58 actually goes to zero at a time that's equal to 10 to 120 00:05:58 --> 00:06:00 the negative 10 seconds. 121 00:06:00 --> 00:06:04 So, let's think qualitatively for a second about what that 122 00:06:04 --> 00:06:06 means or what the real meaning of that is. 123 00:06:06 --> 00:06:09 What that is telling us is that according to Newtonian 124 00:06:09 --> 00:06:13 mechanics and Coulomb's force law, is that the electron 125 00:06:13 --> 00:06:16 should actually plummet into the nucleus in 0.1 nanoseconds. 126 00:06:16 --> 00:06:22 So, we have a little bit of a problem here. 127 00:06:22 --> 00:06:26 And the problem that we have is that what we're figuring out 128 00:06:26 --> 00:06:30 mathematically is not exactly matching up with what we're 129 00:06:30 --> 00:06:33 observing experimentally. 130 00:06:33 --> 00:06:36 And, in fact, it's often kind of difficult to experimentally 131 00:06:36 --> 00:06:39 test your mathematical predictions -- a lot of people 132 00:06:39 --> 00:06:42 spend many, many years testing one single mathematical 133 00:06:42 --> 00:06:43 prediction. 134 00:06:43 --> 00:06:46 But, I think all of us right now can probably test this 135 00:06:46 --> 00:06:49 prediction right here, and we're observing that, in fact, 136 00:06:49 --> 00:06:52 all of us and all the atoms we can see are not immediately 137 00:06:52 --> 00:06:56 collapsing in less than a nanosecond. 138 00:06:56 --> 00:06:59 So, just, if you can take what I'm saying for a moment right 139 00:06:59 --> 00:07:03 now that in fact this should collapse in this very small 140 00:07:03 --> 00:07:06 time frame, we have to see that there's a problem with one of 141 00:07:06 --> 00:07:10 these two things, either the Coulomb force law or 142 00:07:10 --> 00:07:12 Newtonian mechanics. 143 00:07:12 --> 00:07:18 So, what do you guys think is probably the issue here? 144 00:07:18 --> 00:07:23 So, it's Newtonian mechanics, and the reason for this is 145 00:07:23 --> 00:07:26 because Newtonian mechanics does not work on this very, 146 00:07:26 --> 00:07:27 very small size scale. 147 00:07:27 --> 00:07:32 As we said, Newtonian mechanics does work in most cases, it 148 00:07:32 --> 00:07:35 does work when we're discussing things that we can see, it 149 00:07:35 --> 00:07:38 does work even on things that are too small to measure. 150 00:07:38 --> 00:07:41 But once we got to the atomic size scale, what happens is we 151 00:07:41 --> 00:07:45 need to be taking into account the fact that matter has these 152 00:07:45 --> 00:07:49 wave-like properties, and we'll learn more about that later, 153 00:07:49 --> 00:07:51 but essentially classical mechanics does not take 154 00:07:51 --> 00:07:52 that into account at all. 155 00:07:52 --> 00:07:55 So, we need a new kind of mechanics, which is quantum 156 00:07:55 --> 00:07:57 mechanics, which will accurately explain the 157 00:07:57 --> 00:08:01 behavior of molecules on this small scale. 158 00:08:01 --> 00:08:05 So, as I mentioned, the real key to quantum mechanics is 159 00:08:05 --> 00:08:08 that it's treating matter not just like it's a particle, 160 00:08:08 --> 00:08:10 which is what we were just doing, but also like it's 161 00:08:10 --> 00:08:13 a wave, and it treats light that way, too. 162 00:08:13 --> 00:08:16 The second important point to quantum mechanics is that it 163 00:08:16 --> 00:08:20 actually considers the fact that light consists of 164 00:08:20 --> 00:08:24 these discrete packets or particle-like pieces of energy, 165 00:08:24 --> 00:08:25 which are called photons. 166 00:08:25 --> 00:08:28 And if you think about what's actually happening here, this 167 00:08:28 --> 00:08:31 second point that light consists of photons is actually 168 00:08:31 --> 00:08:34 the same thing as saying that light shows particle-like 169 00:08:34 --> 00:08:38 properties, but that's such an important point that I put it 170 00:08:38 --> 00:08:43 separately, and we'll cover that separately as we go along. 171 00:08:43 --> 00:08:49 So, we now have this new way of thinking about how a nucleus 172 00:08:49 --> 00:08:51 and an electron can hang together, and this is quantum 173 00:08:51 --> 00:08:54 mechanics, and we can use this to come up with a new way 174 00:08:54 --> 00:08:58 to describe our atom and the behavior of atoms. 175 00:08:58 --> 00:09:01 But the problem is before we do this, it makes sense to take a 176 00:09:01 --> 00:09:03 little bit of a step back and actually make sure we're all on 177 00:09:03 --> 00:09:06 the same page and understanding why quantum mechanics is so 178 00:09:06 --> 00:09:09 important and how it works, and specifically understanding what 179 00:09:09 --> 00:09:13 we mean when we say that light is both a particle and a wave, 180 00:09:13 --> 00:09:15 and that matter is both a particle and a wave. 181 00:09:15 --> 00:09:19 So, we'll move on to this discussion of light as a wave, 182 00:09:19 --> 00:09:22 and we really won't pick up into going back to applying 183 00:09:22 --> 00:09:26 quantum mechanics to the atom until Friday, but in the 184 00:09:26 --> 00:09:29 meantime, we'll really get to understand the wave particle 185 00:09:29 --> 00:09:33 duality of light and of matter. 186 00:09:33 --> 00:09:36 So, we'll start with thinking about some properties of waves 187 00:09:36 --> 00:09:38 that are going to be applicable to all waves that we're talking 188 00:09:38 --> 00:09:40 about, including light waves. 189 00:09:40 --> 00:09:44 The easiest kind of waves for us to picture are ocean waves 190 00:09:44 --> 00:09:47 or water waves, because we can, in fact, see them, but they 191 00:09:47 --> 00:09:50 have similar properties to all waves. 192 00:09:50 --> 00:09:53 And those properties include that you have this periodic 193 00:09:53 --> 00:09:55 variation of some property. 194 00:09:55 --> 00:09:58 So, when we're talking about water waves, the property 195 00:09:58 --> 00:10:01 we're discussing is just the water level. 196 00:10:01 --> 00:10:04 So, for example, we have this average level, and then it can 197 00:10:04 --> 00:10:08 go high where we have the peak, or it can go very low. 198 00:10:08 --> 00:10:10 We can also discuss sound waves, so again it's just the 199 00:10:10 --> 00:10:14 periodic variation of some property -- in this case we're 200 00:10:14 --> 00:10:17 talking about density, so we have high density areas 201 00:10:17 --> 00:10:20 and low density areas. 202 00:10:20 --> 00:10:23 So, regardless of the type of wave that we're talking about, 203 00:10:23 --> 00:10:27 there's some common definitions that we want to make sure that 204 00:10:27 --> 00:10:28 we're all able to use, and the first is amplitude. 205 00:10:28 --> 00:10:32 And when we're talking about the amplitude of the wave, 206 00:10:32 --> 00:10:34 we're talking about the deviation from that 207 00:10:34 --> 00:10:35 average level. 208 00:10:35 --> 00:10:38 So, if we define the average level as zero, you can have 209 00:10:38 --> 00:10:41 either a positive amplitude or a negative amplitude. 210 00:10:41 --> 00:10:43 So, sometimes people get confused when they're solving 211 00:10:43 --> 00:10:46 problems and call the amplitude this distance all the way from 212 00:10:46 --> 00:10:50 the max to the min, but it's only half of that because we're 213 00:10:50 --> 00:10:53 only going back to the average level. 214 00:10:53 --> 00:10:56 So, what we really want to talk about here is light waves, and 215 00:10:56 --> 00:10:59 light waves have the same properties as these other kind 216 00:10:59 --> 00:11:02 of waves in that they're the periodic variation 217 00:11:02 --> 00:11:04 of some property. 218 00:11:04 --> 00:11:07 So, when we're discussing light waves, what we're talking about 219 00:11:07 --> 00:11:11 is actually light or electromagnetic radiation, is 220 00:11:11 --> 00:11:13 what we'll be calling it throughout the course. 221 00:11:13 --> 00:11:17 And that's the periodic variation of an electric field. 222 00:11:17 --> 00:11:20 So, instead of having the periodic variation of water, or 223 00:11:20 --> 00:11:24 the periodic variation of air density, here we're talking 224 00:11:24 --> 00:11:25 about an electric field. 225 00:11:25 --> 00:11:28 We know what an electric field is, it's just a space through 226 00:11:28 --> 00:11:30 which a Coulomb force operates. 227 00:11:30 --> 00:11:33 And the important thing to think about when you're talking 228 00:11:33 --> 00:11:35 about the fact that it's a periodic variation, is if 229 00:11:35 --> 00:11:38 you put a charged particle somewhere into an electric 230 00:11:38 --> 00:11:42 field, it will, of course, go in a certain direction toward 231 00:11:42 --> 00:11:43 the charge it's attracted to. 232 00:11:43 --> 00:11:46 But you need to think about the difference, if you have a 233 00:11:46 --> 00:11:49 particle here on your wave, it will go in one direction. 234 00:11:49 --> 00:11:52 But remember, waves don't just have magnitude, they 235 00:11:52 --> 00:11:55 also do have direction. 236 00:11:55 --> 00:11:58 So, if instead you put your particle somewhere down here on 237 00:11:58 --> 00:12:01 the electric field, or on the wave, the electric field will 238 00:12:01 --> 00:12:04 now be in the other direction, so your particle will be 239 00:12:04 --> 00:12:06 pushed the other way. 240 00:12:06 --> 00:12:09 And from physics you know that, of course, if we have a 241 00:12:09 --> 00:12:12 propagating electric field, we also have a perpendicular 242 00:12:12 --> 00:12:14 magnetic field that's going back and forth. 243 00:12:14 --> 00:12:18 But in terms of worrying about using the concepts of a wave to 244 00:12:18 --> 00:12:21 solve chemistry problems in this course, we can actually 245 00:12:21 --> 00:12:26 put aside the fact, and only focus on the electric field 246 00:12:26 --> 00:12:28 part of things, because that's what's going to be interacting 247 00:12:28 --> 00:12:32 with our charged particles, such as our electrons. 248 00:12:32 --> 00:12:36 So, other properties of waves that you probably are all 249 00:12:36 --> 00:12:39 familiar with but I just want to review is the idea 250 00:12:39 --> 00:12:40 of a wavelength. 251 00:12:40 --> 00:12:43 If we're talking about the wavelength of a wave, we're 252 00:12:43 --> 00:12:45 just talking about the distance that there is between 253 00:12:45 --> 00:12:50 successive maxima, or of course, we can also be talking 254 00:12:50 --> 00:12:52 about the distance between successive minima. 255 00:12:52 --> 00:12:55 Basically, we can take any point on the wave, and it's the 256 00:12:55 --> 00:12:59 distance to that same point later on in the wave. 257 00:12:59 --> 00:13:01 So, that's what we call one wavelength. 258 00:13:01 --> 00:13:05 We also commonly discuss the frequency of a wave, and the 259 00:13:05 --> 00:13:07 frequency is just the number of cycles that that wave 260 00:13:07 --> 00:13:10 goes through per unit time. 261 00:13:10 --> 00:13:13 So, by a cycle we'd basically mean how many times we cycle 262 00:13:13 --> 00:13:15 through a complete wavelength. 263 00:13:15 --> 00:13:19 So, if something cycles through five wavelengths in a single 264 00:13:19 --> 00:13:21 second, we would just say that the frequency of that 265 00:13:21 --> 00:13:27 wave is five per second. 266 00:13:27 --> 00:13:30 We can also mathematically describe what's going on here 267 00:13:30 --> 00:13:32 other than just graphing it. 268 00:13:32 --> 00:13:35 So, if we want to look at the mathematical equation of a 269 00:13:35 --> 00:13:38 wave, we want to describe -- again as I mention, what we're 270 00:13:38 --> 00:13:41 describing is the electric field, we're not worrying about 271 00:13:41 --> 00:13:46 the magnetic field here, as a function of x and t that's 272 00:13:46 --> 00:13:57 equal to a cosine [ 2 pi x over wavelength, minus 2 pi nu t ]. 273 00:13:57 --> 00:13:59 And note this is the Greek letter nu. 274 00:13:59 --> 00:14:01 This is not a v. 275 00:14:01 --> 00:14:07 Where we have E, which is equal to the electric 276 00:14:07 --> 00:14:13 field, what is x? 277 00:14:13 --> 00:14:16 STUDENT: Position. 278 00:14:16 --> 00:14:19 PROFESSOR: Yup, the position of the wave. 279 00:14:19 --> 00:14:21 And what about t? 280 00:14:21 --> 00:14:26 Yeah, so we're talking about both position and time. 281 00:14:26 --> 00:14:29 So what we can do if we're talking about a wave is think 282 00:14:29 --> 00:14:32 of it both in terms of position time, but if we're trying to 283 00:14:32 --> 00:14:36 visualize this -- for example if we're actually to graph this 284 00:14:36 --> 00:14:40 out, the easiest thing to do is keep one of these two variables 285 00:14:40 --> 00:14:47 constant, either the x or the t, and then just consider 286 00:14:47 --> 00:14:49 the other variable. 287 00:14:49 --> 00:14:52 So, for example, if we're to hold the time constant, this 288 00:14:52 --> 00:14:55 makes it a lot simpler of an equation, because what we can 289 00:14:55 --> 00:14:58 end up doing is actually crossing out this 290 00:14:58 --> 00:14:59 whole term here. 291 00:14:59 --> 00:15:03 So what we're left with is just that the electric field as a 292 00:15:03 --> 00:15:09 function of distance is a times cosine of the argument there, 293 00:15:09 --> 00:15:13 which is now just 2 pi x over wavelength. 294 00:15:13 --> 00:15:15 So, what we want to be able to do, either when we're looking 295 00:15:15 --> 00:15:19 at the graph or looking at the equation up there, is to 296 00:15:19 --> 00:15:21 think about different properties of the wave. 297 00:15:21 --> 00:15:25 For example, to think about at what point do we have 298 00:15:25 --> 00:15:29 the wave where it's at its maximum amplitude? 299 00:15:29 --> 00:15:32 So, if we think about that, we need to have a point where 300 00:15:32 --> 00:15:36 we're making this argument of the cosine such that the cosine 301 00:15:36 --> 00:15:39 is going to all be equal to one, so all we're left 302 00:15:39 --> 00:15:40 with is that a term. 303 00:15:40 --> 00:15:45 So, we can do that basically any time that we have an 304 00:15:45 --> 00:15:50 integer variable that is either zero or an integer variable 305 00:15:50 --> 00:15:51 of the wavelength. 306 00:15:51 --> 00:15:53 So, for example, negative wavelength or positive 307 00:15:53 --> 00:15:56 wavelength are two times the wavelength, because that lets 308 00:15:56 --> 00:15:59 us cross out the term with the wavelength here, and we're 309 00:15:59 --> 00:16:04 left with some integer multiple of just pi. 310 00:16:04 --> 00:16:07 So, that's sort of the mathematically how we get to a, 311 00:16:07 --> 00:16:10 but we can also just look at the graph here, because every 312 00:16:10 --> 00:16:12 time we go one wavelength, we can see that we're 313 00:16:12 --> 00:16:14 back in a maximum. 314 00:16:14 --> 00:16:18 So, I mentioned we should be able to figure out where 315 00:16:18 --> 00:16:19 the maximum amplitude is. 316 00:16:19 --> 00:16:22 You should also just looking at an equation, immediately be 317 00:16:22 --> 00:16:26 able to figure out what that maximum amplitude is in terms 318 00:16:26 --> 00:16:30 of the height of it just by looking at that a-term, here we 319 00:16:30 --> 00:16:33 should also be able to know the intensity of any light wave, 320 00:16:33 --> 00:16:36 because intensity is just the amplitude squared. 321 00:16:36 --> 00:16:39 So, we should immediately be able to know how bright or how 322 00:16:39 --> 00:16:42 intense a light is just looking at the wave equation, or 323 00:16:42 --> 00:16:47 just by looking at a graph. 324 00:16:47 --> 00:16:50 We can also do a similar thing, and I'll keep my distance from 325 00:16:50 --> 00:16:55 the board, but we can instead be holding x constant, for 326 00:16:55 --> 00:16:58 example, putting x to be equal to zero, and then all we're 327 00:16:58 --> 00:17:02 doing is considering the electric field as 328 00:17:02 --> 00:17:03 a function of t. 329 00:17:03 --> 00:17:06 So, in this case we're crossing out the first term there, and 330 00:17:06 --> 00:17:13 we're left with amplitude times the cosine of 2 pi nu times t. 331 00:17:13 --> 00:17:15 And, of course, we can do the same thing again, we can think 332 00:17:15 --> 00:17:18 about when the amplitude is going to be at its maximum, and 333 00:17:18 --> 00:17:21 it's going to be any time cosine of this term 334 00:17:21 --> 00:17:23 now is equal to one. 335 00:17:23 --> 00:17:25 So that will be at, for example, negative 1 over 336 00:17:25 --> 00:17:28 nu, or 0, or 1 over nu. 337 00:17:28 --> 00:17:31 And again, we can just look at our graph to figure that out, 338 00:17:31 --> 00:17:35 that's exactly where we're at a maximum. 339 00:17:35 --> 00:17:39 So, 1 over nu is another term we use and we call it the 340 00:17:39 --> 00:17:42 period of a wave, and the period is just the inverse 341 00:17:42 --> 00:17:44 of the frequency. 342 00:17:44 --> 00:17:46 And if we think about frequency, that's number 343 00:17:46 --> 00:17:48 of cycles per unit time. 344 00:17:48 --> 00:17:51 So, for example, number of cycles per second, whereas the 345 00:17:51 --> 00:17:58 period is how much time it takes for one cycle to occur. 346 00:17:58 --> 00:18:01 And when we talk about units of frequency, in almost every 347 00:18:01 --> 00:18:05 case, you'll be talking about number of cycles per second. 348 00:18:05 --> 00:18:06 So, you can just write inverse second, the 349 00:18:06 --> 00:18:07 cycle part is assumed. 350 00:18:07 --> 00:18:13 But you'll also frequently see it called Hertz, so, Hz here. 351 00:18:13 --> 00:18:15 So, if you're talking about five cycles per second, you 352 00:18:15 --> 00:18:18 can write five per second, or you can write five Hertz. 353 00:18:18 --> 00:18:21 The one thing you want to keep in mind though is that Hertz 354 00:18:21 --> 00:18:23 does not actually mean inverse seconds, it 355 00:18:23 --> 00:18:24 means cycles per second. 356 00:18:24 --> 00:18:27 So, if you're talking about a car going so many meters per 357 00:18:27 --> 00:18:30 second, you can't say it's going meter Hertz, you have 358 00:18:30 --> 00:18:31 to say meters per second. 359 00:18:31 --> 00:18:33 So, this really just means for frequency, it's 360 00:18:33 --> 00:18:37 a frequency label. 361 00:18:37 --> 00:18:37 Alright. 362 00:18:38 --> 00:18:40 So, since we have these terms defined, we know the frequency 363 00:18:40 --> 00:18:43 and the wavelength, it turns out we can also think about the 364 00:18:43 --> 00:18:47 speed of the wave, and specifically of a light wave, 365 00:18:47 --> 00:18:49 and speed and is just equal to the distance that's 366 00:18:49 --> 00:18:53 traveled divided by the time the elapsed. 367 00:18:53 --> 00:18:56 And because we've defined these terms, we have ways 368 00:18:56 --> 00:18:57 to describe these things. 369 00:18:57 --> 00:19:00 So, we can describe the distance that's traveled, 370 00:19:00 --> 00:19:03 it's just a wavelength here. 371 00:19:03 --> 00:19:06 And we can think about how long it takes for a wave, because 372 00:19:06 --> 00:19:09 waves are, we know not just changing in position, but the 373 00:19:09 --> 00:19:12 whole wave is moving forward with time, we can think about 374 00:19:12 --> 00:19:15 how long it takes for wave to go one wavelength. 375 00:19:15 --> 00:19:18 So, one distance that's equal to lambda. 376 00:19:18 --> 00:19:25 So, how much time would that take, does anyone know? 377 00:19:25 --> 00:19:29 So, would it take, for example, the same amount 378 00:19:29 --> 00:19:33 of time as the frequency? 379 00:19:33 --> 00:19:34 The period, that's right. 380 00:19:34 --> 00:19:38 So, it's going to take one period to move that long. 381 00:19:38 --> 00:19:42 And another way we can say period is just 1 over nu or 1 382 00:19:42 --> 00:19:46 over the frequency So, now we know both the distance traveled 383 00:19:46 --> 00:19:48 and the time the elapsed. 384 00:19:48 --> 00:19:49 So, we can just plug it in. 385 00:19:49 --> 00:19:53 Speed is equal to the distance traveled, which is lambda over 386 00:19:53 --> 00:19:56 the time elapsed, which is 1 over nu. so, we can re-write 387 00:19:56 --> 00:20:02 that as speed is equal to lambda times nu, and it turns 388 00:20:02 --> 00:20:04 out typically this is reported in meters per second or 389 00:20:04 --> 00:20:06 nanometers per second. 390 00:20:06 --> 00:20:09 So, now we have an equation where we know the relationship 391 00:20:09 --> 00:20:14 between speed and wavelength and frequency, and it turns out 392 00:20:14 --> 00:20:17 that we could take any wave, and as long as we know the 393 00:20:17 --> 00:20:19 frequency and the wavelength, we'll be able to 394 00:20:19 --> 00:20:20 figure out the speed. 395 00:20:20 --> 00:20:22 But, of course, there's something very special about 396 00:20:22 --> 00:20:25 electromagnetic waves, electromagnetic radiation 397 00:20:25 --> 00:20:28 and the speed. 398 00:20:28 --> 00:20:30 And it's not really surprising for me to tell you that 399 00:20:30 --> 00:20:34 electromagnetic radiation has a constant speed, and that speed 400 00:20:34 --> 00:20:37 is what we call the speed of light, and typically we 401 00:20:37 --> 00:20:41 abbreviate that as c, and that's from the Latin term 402 00:20:41 --> 00:20:45 celeritas, which means speed in Latin. 403 00:20:45 --> 00:20:47 That's one of four or five Latin words I remember from 404 00:20:47 --> 00:20:50 four years of high school Latin, but it comes in handy 405 00:20:50 --> 00:20:53 to remember speed of light. 406 00:20:53 --> 00:20:56 And some of you may have memorized what the speed of 407 00:20:56 --> 00:20:59 light is in high school -- it's about 3 times 10 to 408 00:20:59 --> 00:21:01 the 8 meters per second. 409 00:21:01 --> 00:21:03 This is another example of a constant that you will 410 00:21:03 --> 00:21:06 accidentally memorize in this course as you use it 411 00:21:06 --> 00:21:07 over and over again. 412 00:21:07 --> 00:21:10 But again, that we will supply for you on the exam just in 413 00:21:10 --> 00:21:13 case you forget it at that moment. 414 00:21:13 --> 00:21:16 And this is a very fast speed, of course, it's about 700 415 00:21:16 --> 00:21:20 million miles per hour. 416 00:21:20 --> 00:21:22 So, one way to put that in perspective is to think about 417 00:21:22 --> 00:21:27 how long it takes for a light beam to get from 418 00:21:27 --> 00:21:28 earth to the moon. 419 00:21:28 --> 00:21:32 Does anyone have any guesses? 420 00:21:32 --> 00:21:33 Eight seconds, that sounds good. 421 00:21:33 --> 00:21:37 Anyone else? 422 00:21:37 --> 00:21:39 These are all really good guesses, so it actually takes 423 00:21:39 --> 00:21:43 1.2 seconds for light to travel from the earth to the moon. 424 00:21:43 --> 00:21:46 So, we're talking pretty fast, so that's nice to 425 00:21:46 --> 00:21:48 appreciate in itself. 426 00:21:48 --> 00:21:53 But other than that point, we can also think about the fact 427 00:21:53 --> 00:22:00 that frequency and wavelength are related in a way that now 428 00:22:00 --> 00:22:02 since we know the speed of light, if we know one 429 00:22:02 --> 00:22:03 we can tell the other. 430 00:22:03 --> 00:22:05 So, you can go ahead and switch us to our 431 00:22:05 --> 00:22:08 clicker question here. 432 00:22:08 --> 00:22:13 So, we should be able to look at different types of waves and 433 00:22:13 --> 00:22:17 be able to figure out something about both their frequency and 434 00:22:17 --> 00:22:20 their wavelength, and know the relationship between the two. 435 00:22:20 --> 00:22:23 So, it's up on this screen here now, so we'll 436 00:22:23 --> 00:22:25 work on the other one. 437 00:22:25 --> 00:22:28 If you can identify which of these statements is correct 438 00:22:28 --> 00:22:30 based on what you know about the relationship between 439 00:22:30 --> 00:22:42 frequency and wavelength and also just looking at the waves. 440 00:22:42 --> 00:22:42 Alright. 441 00:22:42 --> 00:22:49 So, let's give ten more seconds on that. 442 00:22:49 --> 00:23:06 So, ten seconds on that. 443 00:23:06 --> 00:23:06 Alright. 444 00:23:07 --> 00:23:08 So, good job. 445 00:23:08 --> 00:23:12 So, most people could recognize that light wave a has 446 00:23:12 --> 00:23:13 the shorter wavelength. 447 00:23:13 --> 00:23:18 We can see that just by looking at the graph itself -- we can 448 00:23:18 --> 00:23:20 see, certainly, this is shorter from maxima to maxima. 449 00:23:20 --> 00:23:24 This we can't even see the next maxima, so it's much longer. 450 00:23:24 --> 00:23:26 And then, we also know that means that it has the higher 451 00:23:26 --> 00:23:29 frequency, because our relationship between 452 00:23:29 --> 00:23:32 wavelength and frequency are inversely related. 453 00:23:32 --> 00:23:35 And also, we know the speed of light. 454 00:23:35 --> 00:23:38 So, if we think about if it's a shorter wavelength, we'll be 455 00:23:38 --> 00:23:40 able to get a lot more wavelengths in, in a given 456 00:23:40 --> 00:23:43 time, than we would for a longer wavelength. 457 00:23:43 --> 00:23:47 So, we can switch back to the notes and think about what this 458 00:23:47 --> 00:23:50 means, and what this means when we're talking about all the 459 00:23:50 --> 00:23:53 different kinds of light waves we have, and I've shown a bunch 460 00:23:53 --> 00:23:57 here, is that if we have the wavelength, we also know the 461 00:23:57 --> 00:23:59 frequency of these wavelengths. 462 00:23:59 --> 00:24:03 So, for example, radio waves, which have very long 463 00:24:03 --> 00:24:07 wavelengths have very low frequencies. 464 00:24:07 --> 00:24:12 Whereas where we go to waves that have very short 465 00:24:12 --> 00:24:16 wavelengths, such a x-rays or cosmic rays, they, in turn, 466 00:24:16 --> 00:24:19 have very high frequencies. 467 00:24:19 --> 00:24:22 So, it's important to get a little bit of a sense of 468 00:24:22 --> 00:24:24 what all these different kinds of lights do. 469 00:24:24 --> 00:24:27 You're absolutely not responsible to memorize what 470 00:24:27 --> 00:24:29 the wavelengths of the different types of lights are, 471 00:24:29 --> 00:24:34 but you do want to be able to know the general order of them. 472 00:24:34 --> 00:24:37 So, if someone tells you they're using UV light versus 473 00:24:37 --> 00:24:40 x-ray light, you know that the x-ray light is, in fact, 474 00:24:40 --> 00:24:41 at a higher frequency. 475 00:24:41 --> 00:24:43 So that's the important take-away message 476 00:24:43 --> 00:24:44 from this slide. 477 00:24:44 --> 00:24:47 If we think about these different types of lights, 478 00:24:47 --> 00:24:50 microwave light, if it's absorbed by a molecule, is a 479 00:24:50 --> 00:24:53 sufficient amount of frequency and energy to get those 480 00:24:53 --> 00:24:55 molecules to rotate. 481 00:24:55 --> 00:24:57 That, of course, generates heat, so that's how 482 00:24:57 --> 00:24:58 your microwaves work. 483 00:24:58 --> 00:25:02 If we talk about infrared light, which is at a higher 484 00:25:02 --> 00:25:06 frequency here and a shorter wavelength, infrared light when 485 00:25:06 --> 00:25:09 it's absorbed by molecules actually is enough to cause 486 00:25:09 --> 00:25:12 molecules now to vibrate. 487 00:25:12 --> 00:25:15 If we move up to the more high-frequency and divisible 488 00:25:15 --> 00:25:19 light and all the way into UV light, if you shine UV light at 489 00:25:19 --> 00:25:21 certain molecules, it's going to have enough energy to 490 00:25:21 --> 00:25:26 actually pop those electrons in that molecule up to a higher 491 00:25:26 --> 00:25:28 energy level, which will make more sense once we talk about 492 00:25:28 --> 00:25:32 energy levels in atoms, but that's what UV light can do. 493 00:25:32 --> 00:25:35 And actually, that's responsible for fluorescence 494 00:25:35 --> 00:25:37 and phosphorescence that you see where typically 495 00:25:37 --> 00:25:39 UV light comes in. 496 00:25:39 --> 00:25:42 So, if you use a black lamp or something and you excite 497 00:25:42 --> 00:25:45 something up to a higher energy level and then it relaxes back 498 00:25:45 --> 00:25:48 down to its lower energy state, it's going to emit a new 499 00:25:48 --> 00:25:51 wavelength of light, which is going to be visible to you. 500 00:25:51 --> 00:25:55 X-rays are at even a higher frequency, and those are 501 00:25:55 --> 00:25:59 sufficient to actually be absorbed by a molecule and 502 00:25:59 --> 00:26:01 pop an electron all the way out of that molecule. 503 00:26:01 --> 00:26:04 You can see how that would be damaging to the integrity of 504 00:26:04 --> 00:26:08 that molecule, that's why x-rays are so damaging -- you 505 00:26:08 --> 00:26:11 don't want to have electrons disappearing for no good reason 506 00:26:11 --> 00:26:13 from your molecules that can cause the kind of mutations 507 00:26:13 --> 00:26:16 we don't want to be seeing in ourselves. 508 00:26:16 --> 00:26:18 And then also as we go higher, we have gamma 509 00:26:18 --> 00:26:21 rays and cosmic rays. 510 00:26:21 --> 00:26:25 Within the visible range of what we can see, you also want 511 00:26:25 --> 00:26:29 to know this relative order that's pretty easy -- most of 512 00:26:29 --> 00:26:31 us have memorized that in kindergarten, so that 513 00:26:31 --> 00:26:32 should be fine. 514 00:26:32 --> 00:26:36 Just remembering that violet is the end that actually has the 515 00:26:36 --> 00:26:39 shortest wavelength, which means that it also has, of 516 00:26:39 --> 00:26:42 course, the highest frequency. 517 00:26:42 --> 00:26:45 So, just an interesting fact about this set of light, which 518 00:26:45 --> 00:26:48 we're most familiar with, if we think about our vision, it 519 00:26:48 --> 00:26:51 turns out that our vision's actually logarithmic and it's 520 00:26:51 --> 00:26:54 centered around this green frequency. 521 00:26:54 --> 00:26:57 So, if instead of a red laser pointer here, I had a green 522 00:26:57 --> 00:27:00 one, you'd actually, to our eyes, it would seem like the 523 00:27:00 --> 00:27:04 green one was brighter, even if the intensity was the same, and 524 00:27:04 --> 00:27:07 that's just because our eyes are centered and logarithmic 525 00:27:07 --> 00:27:11 around this green frequency set. 526 00:27:11 --> 00:27:15 So, using the relationship between frequency and 527 00:27:15 --> 00:27:18 wavelength, we can actually understand a lot about what's 528 00:27:18 --> 00:27:21 going on, and pretty soon we'll also draw the relationship very 529 00:27:21 --> 00:27:24 soon to energy, so it will be even more informative then. 530 00:27:24 --> 00:27:27 But I just want to point out one of the many, many groups at 531 00:27:27 --> 00:27:32 MIT that works with different fluorescing types of molecules, 532 00:27:32 --> 00:27:36 and this is Professor Bawendi's laboratory at MIT, and he 533 00:27:36 --> 00:27:38 works with quantum dots. 534 00:27:38 --> 00:27:42 And quantum dots are these just very tiny, tiny crystals 535 00:27:42 --> 00:27:44 of semiconductor material. 536 00:27:44 --> 00:27:48 They're on the order of one to ten nanometers, and these can 537 00:27:48 --> 00:27:51 be shined on with UV light -- they have a lot of different 538 00:27:51 --> 00:27:53 interesting properties, but one I'll mention is that if you 539 00:27:53 --> 00:27:57 excite them with UV light, they will have some of the electrons 540 00:27:57 --> 00:28:01 move to a higher energy state, and when they drop back down, 541 00:28:01 --> 00:28:06 they actually emit light with a wavelength that corresponds 542 00:28:06 --> 00:28:09 with the size of the actual quantum dot. 543 00:28:09 --> 00:28:13 So, from what we know so far, we should be able to look at 544 00:28:13 --> 00:28:15 any of these quantum dots, which are depicted as a cartoon 545 00:28:15 --> 00:28:19 here, but here we have an actual picture of the quantum 546 00:28:19 --> 00:28:24 dots suspended in some sort of solution and shone on with UV 547 00:28:24 --> 00:28:27 light, and you can see that you can achieve this whole 548 00:28:27 --> 00:28:32 beautiful range of colors just by modulating the size 549 00:28:32 --> 00:28:33 of the different dots. 550 00:28:33 --> 00:28:36 And we should be able to know if we're looking at a red dot 551 00:28:36 --> 00:28:41 -- is a red dot, it's going to have a longer wavelength, so is 552 00:28:41 --> 00:28:44 this a higher or lower frequency? 553 00:28:44 --> 00:28:47 Yeah, and similarly, if someone tells us that their dot is 554 00:28:47 --> 00:28:50 blue-shifted, that should automatically in our heads tell 555 00:28:50 --> 00:28:52 us, oh it shifted to a higher frequency. 556 00:28:52 --> 00:28:55 And these dots are really interesting in that you can, 557 00:28:55 --> 00:28:58 I'm sure by looking at this picture, already imagine just 558 00:28:58 --> 00:29:01 a whole slew of different biological or sensing 559 00:29:01 --> 00:29:02 applications that you could think of. 560 00:29:02 --> 00:29:05 For example, if you were trying to study different protein 561 00:29:05 --> 00:29:07 interactions, you could think about labeling them with 562 00:29:07 --> 00:29:10 different colored dots, or there's also a bunch of 563 00:29:10 --> 00:29:12 different fluorescent techniques that you could apply 564 00:29:12 --> 00:29:16 using these dots, or you could think of in-vivo sensing, how 565 00:29:16 --> 00:29:19 useful these could be if you could think of a way to get 566 00:29:19 --> 00:29:22 them into your body without being too toxic, for example. 567 00:29:22 --> 00:29:25 These are all things that the Bawendi group is working on. 568 00:29:25 --> 00:29:28 What they are real experts in is synthesizing many different 569 00:29:28 --> 00:29:31 kinds of these dots, and they have a synthetic scheme that's 570 00:29:31 --> 00:29:33 used by research groups around the world. 571 00:29:33 --> 00:29:36 The Bawendi group also collaborates with people, 572 00:29:36 --> 00:29:39 both at different schools and at MIT. 573 00:29:39 --> 00:29:43 One example, on some of their biochemistry applications is 574 00:29:43 --> 00:29:47 with another Professor at MIT, Alice Ting and her lab. 575 00:29:47 --> 00:29:52 So really what I want to point out here is as we get more into 576 00:29:52 --> 00:29:55 describing quantum mechanics, these quantum dots are one 577 00:29:55 --> 00:29:58 really good example where a lot of the properties of quantum 578 00:29:58 --> 00:29:59 mechanics apply directly. 579 00:29:59 --> 00:30:02 So, if you're interested, I put the Bawendi lab research 580 00:30:02 --> 00:30:04 website onto your notes. 581 00:30:04 --> 00:30:08 And also, Professor Bawendi recently did an interview with 582 00:30:08 --> 00:30:11 "The Tech." Did anyone see that interview in the paper? 583 00:30:11 --> 00:30:16 So, three or four -- a few of you read the paper last week. 584 00:30:16 --> 00:30:18 So, you can either pick up an old issue or I put the 585 00:30:18 --> 00:30:19 link on the website, too. 586 00:30:19 --> 00:30:23 And that's not just about his research, it's also about some 587 00:30:23 --> 00:30:26 of his memories as a student and advice to all of you. 588 00:30:26 --> 00:30:29 So, it's interesting to read and get to know some of these 589 00:30:29 --> 00:30:32 Professors at MIT a little bit better. 590 00:30:32 --> 00:30:35 So, one property that was important we talked about with 591 00:30:35 --> 00:30:38 waves is the relationship between frequency 592 00:30:38 --> 00:30:39 and wavelength. 593 00:30:39 --> 00:30:42 Another very important property of waves that's true of all 594 00:30:42 --> 00:30:47 waves, is that you can have superposition or interference 595 00:30:47 --> 00:30:48 between two waves. 596 00:30:48 --> 00:30:52 So, if we're looking at waves and they're in-phase, and when 597 00:30:52 --> 00:30:55 I talk about in-phase, what I mean is that they're lined up, 598 00:30:55 --> 00:30:58 so that the maxima are in the same position and the minima 599 00:30:58 --> 00:31:01 are in the same position, what we can have a something called 600 00:31:01 --> 00:31:03 constructive interference. 601 00:31:03 --> 00:31:06 And all we mean by constructive interference is that literally 602 00:31:06 --> 00:31:10 those two waves add together, such as the maxima are now 603 00:31:10 --> 00:31:13 twice as high, and the minima are now twice as low. 604 00:31:13 --> 00:31:17 So, you can also imagine a situation where instead of 605 00:31:17 --> 00:31:21 being perfectly lined up, now we have the minima being lined 606 00:31:21 --> 00:31:23 up with the maxima here. 607 00:31:23 --> 00:31:27 So, if we switch over to a clicker question maybe on this 608 00:31:27 --> 00:31:41 screen -- okay, can it be done up there to switch? 609 00:31:41 --> 00:31:44 So, we're still settling in with the renovations 610 00:31:44 --> 00:31:46 here in this room. 611 00:31:46 --> 00:31:48 So, why don't you all go ahead and tell me what happens if 612 00:31:48 --> 00:31:56 you combine these two waves, which are now out of phase? 613 00:31:56 --> 00:32:07 So, let's -- okay, so, why don't you all think about would 614 00:32:07 --> 00:32:10 happen -- we'll start with the thought exercise. 615 00:32:10 --> 00:32:12 You can switch back to my lecture notes then 616 00:32:12 --> 00:32:18 if this isn't going. 617 00:32:18 --> 00:32:18 Alright. 618 00:32:18 --> 00:32:21 So, hopefully what everyone came up with is the straight 619 00:32:21 --> 00:32:23 line, is that what you answered? 620 00:32:23 --> 00:32:24 STUDENT: Yeah. 621 00:32:24 --> 00:32:25 PROFESSOR: OK, very good. 622 00:32:25 --> 00:32:28 And I didn't make you try to draw the added, the 623 00:32:28 --> 00:32:33 superimposed positive construction in your notes, but 624 00:32:33 --> 00:32:36 I think everyone can handle drawing a straight line. 625 00:32:36 --> 00:32:39 So, you can go ahead and draw what happens when we have 626 00:32:39 --> 00:32:42 destructive interference. 627 00:32:42 --> 00:32:44 And destructive interference, of course, is the extreme, but 628 00:32:44 --> 00:32:47 you can picture also a case where you have waves that are 629 00:32:47 --> 00:32:50 not quite lined up, but they're also not completely 630 00:32:50 --> 00:32:50 out of phase. 631 00:32:50 --> 00:32:53 So in that case, you're either going to have the wave get a 632 00:32:53 --> 00:32:56 little bigger, but not twice as big or a little bit smaller. 633 00:32:56 --> 00:32:59 So, I think the easiest way to think about interference is not 634 00:32:59 --> 00:33:02 actually with light, but sometimes it's easiest to think 635 00:33:02 --> 00:33:04 about with sound, especially when you're dealing with times 636 00:33:04 --> 00:33:07 where you have destructive interference. 637 00:33:07 --> 00:33:10 Has anyone here ever been in a concert hall where they feel 638 00:33:10 --> 00:33:13 like they're kind of in a dead spot, or you don't quite hear 639 00:33:13 --> 00:33:16 as well, and if you move down just two seats all of a sudden 640 00:33:16 --> 00:33:19 it's just blasting at you -- hopefully not in this room. 641 00:33:19 --> 00:33:22 But have people experienced that before? 642 00:33:22 --> 00:33:24 Yeah, I definitely experienced it, too. 643 00:33:24 --> 00:33:26 And really, all you're experiencing there is 644 00:33:26 --> 00:33:30 destructive interference in a very bad way. 645 00:33:30 --> 00:33:34 Halls, they try to design halls such that that doesn't happen, 646 00:33:34 --> 00:33:38 and I show an example of a concert hall here -- this is 647 00:33:38 --> 00:33:41 Symphony Hall in Boston, and I can pretty much guarantee you 648 00:33:41 --> 00:33:45 if you do go to this Symphony Hall, you will not experience 649 00:33:45 --> 00:33:46 a bad seat or a dead seat. 650 00:33:46 --> 00:33:50 This is described as actually one of the top two or three 651 00:33:50 --> 00:33:53 acoustic concert halls in the whole world. 652 00:33:53 --> 00:33:57 So, it's very well designed such that they've minimized 653 00:33:57 --> 00:34:01 any of these destructive interference dead sounds. 654 00:34:01 --> 00:34:03 So, it's nice, on a student budget you can go and get the 655 00:34:03 --> 00:34:05 worst seat in the house and you can hear just as well as they 656 00:34:05 --> 00:34:08 can hear up front, even if you can't actually see 657 00:34:08 --> 00:34:11 what's going on. 658 00:34:11 --> 00:34:14 So, another example of destructive interference is 659 00:34:14 --> 00:34:16 just with the Bose headphones. 660 00:34:16 --> 00:34:18 I've never actually tried these on, but you see people with 661 00:34:18 --> 00:34:22 them, and what happens here is it's supposed to be those noise 662 00:34:22 --> 00:34:24 cancellation headphones. 663 00:34:24 --> 00:34:27 All they do is they take in the ambient noise that's around it, 664 00:34:27 --> 00:34:29 and there's actually battery in the headphones, that then 665 00:34:29 --> 00:34:33 produces waves that are going to destructively interfere 666 00:34:33 --> 00:34:34 with that ambient noise. 667 00:34:34 --> 00:34:38 And that's how it actually gets to be so quiet when you have 668 00:34:38 --> 00:34:43 on, supposedly, these quite expensive headphones. 669 00:34:43 --> 00:34:47 So, that's light as a wave, and the reason -- well, that was 670 00:34:47 --> 00:34:49 sound as a wave, but light as a wave is the same idea. 671 00:34:49 --> 00:34:53 And it was really established by the early 1900s that, in 672 00:34:53 --> 00:34:55 fact, light behaved as a wave. 673 00:34:55 --> 00:34:58 And the reason that it was so certain that light was a wave 674 00:34:58 --> 00:35:00 was because we could observe these things -- we could see, 675 00:35:00 --> 00:35:03 for example, that light defracted, and we could see 676 00:35:03 --> 00:35:07 that light constructively or destructively could interfere 677 00:35:07 --> 00:35:11 with other light waves, and this was all confirmed 678 00:35:11 --> 00:35:12 and visualized. 679 00:35:12 --> 00:35:17 But also, around the time that Thomson was discovering the 680 00:35:17 --> 00:35:20 electron, there were some other observations that were going 681 00:35:20 --> 00:35:24 on, and the most disturbing to kind of the understanding of 682 00:35:24 --> 00:35:27 the universe was the fact that there were some observations 683 00:35:27 --> 00:35:30 about light that didn't make sense with this idea that 684 00:35:30 --> 00:35:32 light is a particle. 685 00:35:32 --> 00:35:35 And the photoelectric effect is maybe the most 686 00:35:35 --> 00:35:37 clear example of this. 687 00:35:37 --> 00:35:41 So, the photoelectric effect is the effect that if you have 688 00:35:41 --> 00:35:44 some metal, and you can pick essentially any metal you want, 689 00:35:44 --> 00:35:47 and you shine light of a certain frequency onto that 690 00:35:47 --> 00:35:52 metal, you can actually pop off an electron, and you can go 691 00:35:52 --> 00:35:54 ahead and measure what the kinetic energy of that electron 692 00:35:54 --> 00:35:58 that comes off is, because we can measure the velocity and we 693 00:35:58 --> 00:36:01 know that kinetic energy equals 1/2 m b squared, and thanks to 694 00:36:01 --> 00:36:05 Thomson we also know the mass of an electron. 695 00:36:05 --> 00:36:09 So, this is an interesting observation, and in itself not 696 00:36:09 --> 00:36:14 too disturbing, yet but the important thing to point out is 697 00:36:14 --> 00:36:18 that there's this threshold frequency that is of the metal, 698 00:36:18 --> 00:36:21 and each metal has a different threshold frequency, such as if 699 00:36:21 --> 00:36:23 you shine light on the metal where the frequency of the 700 00:36:23 --> 00:36:27 light is less than the threshold frequency, nothing 701 00:36:27 --> 00:36:31 will happen -- no electron will pop off of that metal. 702 00:36:31 --> 00:36:34 However, if you shine a light with a frequency that's greater 703 00:36:34 --> 00:36:37 than the threshold frequency, you will be able to 704 00:36:37 --> 00:36:39 pop off an electron. 705 00:36:39 --> 00:36:42 So, people were making this observation, but this wasn't 706 00:36:42 --> 00:36:45 making any sense at all because there was nothing in classical 707 00:36:45 --> 00:36:48 physics that described any sort of relationship between the 708 00:36:48 --> 00:36:53 frequency of light and the energy, much less the energy of 709 00:36:53 --> 00:36:56 an electron that would get popped off of a metal that 710 00:36:56 --> 00:37:01 would basically come off only when we're hitting this 711 00:37:01 --> 00:37:03 threshold frequency. 712 00:37:03 --> 00:37:07 So, what they could do was actually graph what was 713 00:37:07 --> 00:37:10 happening here, so we can also graph what was happening, and 714 00:37:10 --> 00:37:13 what they found was that if we were at any point below the 715 00:37:13 --> 00:37:16 threshold frequency and we were counting the numbers of 716 00:37:16 --> 00:37:20 electrons that were popping off of our metal, we weren't 717 00:37:20 --> 00:37:21 seeing anything at all. 718 00:37:21 --> 00:37:25 But if you go up the threshold frequency, suddenly you see 719 00:37:25 --> 00:37:28 that there's some number of electrons that comes off, and 720 00:37:28 --> 00:37:31 amazingly, the number of electrons actually had no 721 00:37:31 --> 00:37:35 relationship at all to the frequency of the light. 722 00:37:35 --> 00:37:38 And this didn't make a lot of sense to people at the time 723 00:37:38 --> 00:37:42 because they thought that the frequency should be related to 724 00:37:42 --> 00:37:44 the number of electrons that are coming off, because you 725 00:37:44 --> 00:37:48 have more frequency coming in, you'd expect more electrons 726 00:37:48 --> 00:37:50 that are coming off -- this wasn't what people were seeing. 727 00:37:50 --> 00:37:53 So, what they decided to do is just study absolutely 728 00:37:53 --> 00:37:56 everything they could about the photoelectric effect and hope, 729 00:37:56 --> 00:37:58 at some point, someone would piece something together that 730 00:37:58 --> 00:38:00 could explain what's going on or shed some light 731 00:38:00 --> 00:38:02 on this effect. 732 00:38:02 --> 00:38:04 So, one thing they did, because it was so easy to measure 733 00:38:04 --> 00:38:09 kinetic energy of electrons, is plot the frequency of the light 734 00:38:09 --> 00:38:12 against the kinetic energy of the electron that's 735 00:38:12 --> 00:38:13 coming off here. 736 00:38:13 --> 00:38:16 And in your notes and on these slides here, just for your 737 00:38:16 --> 00:38:18 reference, I'm just pointing out what's going to 738 00:38:18 --> 00:38:19 be predicted from classical physics. 739 00:38:19 --> 00:38:22 You're not responsible for that and we won't really discuss it, 740 00:38:22 --> 00:38:25 but it just gives you the contrast of the surprise that 741 00:38:25 --> 00:38:27 comes up when people make these observations. 742 00:38:27 --> 00:38:31 And the first observation was that the frequency of the light 743 00:38:31 --> 00:38:35 had a linear relationship to the kinetic energy of the 744 00:38:35 --> 00:38:38 electrons that are ejected here. 745 00:38:38 --> 00:38:42 This made no sense at all to people, and again they saw this 746 00:38:42 --> 00:38:45 effect where if you were below that threshold frequency, 747 00:38:45 --> 00:38:48 you saw nothing at all. 748 00:38:48 --> 00:38:51 So, that was frequency with kinetic energy. 749 00:38:51 --> 00:38:54 The next thing that they wanted to look at was the actual 750 00:38:54 --> 00:38:58 intensity of the light and see what the relationship of 751 00:38:58 --> 00:39:00 intensity to kinetic energy is. 752 00:39:00 --> 00:39:03 So, what we would expect is that there is a relationship 753 00:39:03 --> 00:39:06 between intensity in kinetic energy, because it was 754 00:39:06 --> 00:39:09 understood that however intense the light was, if you had a 755 00:39:09 --> 00:39:12 more intense light, it was a higher energy light beam. 756 00:39:12 --> 00:39:14 So that should mean that the energy that's transferred to 757 00:39:14 --> 00:39:18 the electron should be greater, but that's not what you saw at 758 00:39:18 --> 00:39:21 all, and what you saw is that if you kept the frequency 759 00:39:21 --> 00:39:25 constant, there was absolutely no change in the kinetic energy 760 00:39:25 --> 00:39:29 of the electrons, no matter how high up you had the 761 00:39:29 --> 00:39:30 intensity of the light go. 762 00:39:30 --> 00:39:33 You could keep increasing the intensity and nothing 763 00:39:33 --> 00:39:36 was going to happen. 764 00:39:36 --> 00:39:40 So, we could also plot the number of electrons that are 765 00:39:40 --> 00:39:43 ejected as a relationship to the intensity, so that was 766 00:39:43 --> 00:39:46 yet another experiment they could do. 767 00:39:46 --> 00:39:48 And this is what they had expected that there would be no 768 00:39:48 --> 00:39:53 relationship, but instead here they saw that there was a 769 00:39:53 --> 00:39:56 linear relationship not to the intensity and the kinetic 770 00:39:56 --> 00:39:59 energy of the electrons, but to the intensity and the 771 00:39:59 --> 00:40:00 number of electrons. 772 00:40:00 --> 00:40:04 So, none of these observations made sense to any scientists at 773 00:40:04 --> 00:40:06 the time, and really all of these observations were made 774 00:40:06 --> 00:40:10 and somewhat put aside for several years before someone 775 00:40:10 --> 00:40:13 that could kind of process everything that was going on at 776 00:40:13 --> 00:40:17 once came along, and that person was Einstein, 777 00:40:17 --> 00:40:19 conveniently enough -- if anyone could put it together, 778 00:40:19 --> 00:40:22 we would hope that he could, and he did. 779 00:40:22 --> 00:40:26 And what he did in a way that made sense when all of us look 780 00:40:26 --> 00:40:30 at it, is he plotted all of these different metals on the 781 00:40:30 --> 00:40:32 same graph and made some observations. 782 00:40:32 --> 00:40:35 So, for example, here we're showing rubidium and potassium 783 00:40:35 --> 00:40:40 and sodium plotted where we're plotting the frequency -- 784 00:40:40 --> 00:40:43 that's the frequency of that light that's coming into the 785 00:40:43 --> 00:40:48 metal versus the kinetic energy of the electron that's ejected 786 00:40:48 --> 00:40:50 from the surface of the metal. 787 00:40:50 --> 00:40:53 And what he found here, which is what you can see and we can 788 00:40:53 --> 00:40:57 all see pretty clearly, is the slope of all of these lines is 789 00:40:57 --> 00:41:01 the same regardless of what the type of metal is. 790 00:41:01 --> 00:41:05 So, he fit all these to the equation of the line, and what 791 00:41:05 --> 00:41:09 he noticed was the slope was specifically this number, 6.626 792 00:41:09 --> 00:41:13 times 10 to the negative 34, joules times seconds. 793 00:41:13 --> 00:41:18 And he also found that the y intercept for each one of these 794 00:41:18 --> 00:41:24 metals was equal to basically this number here, which was the 795 00:41:24 --> 00:41:32 slope times the minimum frequency required of each 796 00:41:32 --> 00:41:37 specific metal, so that's of the threshold frequency. 797 00:41:37 --> 00:41:40 And he actually knew that this number had popped up before, 798 00:41:40 --> 00:41:44 and a lot of you are familiar with this number also, and 799 00:41:44 --> 00:41:45 this is Planck's constant. 800 00:41:45 --> 00:41:48 Planck had observed this number as a fitting constant years 801 00:41:48 --> 00:41:51 earlier when he looked at some phenomena, and you can read 802 00:41:51 --> 00:41:53 about in your book, such as black body radiation. 803 00:41:53 --> 00:41:57 And what he found was he needed this constant to fit his 804 00:41:57 --> 00:42:00 data to what was observed. 805 00:42:00 --> 00:42:03 And this is the same thing that Einstein was observing, that he 806 00:42:03 --> 00:42:07 needed this fitting constant, that this constant was just 807 00:42:07 --> 00:42:09 falling right out of, for example, this slope and 808 00:42:09 --> 00:42:11 also the y intercept. 809 00:42:11 --> 00:42:16 So he decided to go ahead and define exactly what it is, this 810 00:42:16 --> 00:42:19 line, in terms of these new constants, this constant he's 811 00:42:19 --> 00:42:22 calling h, which is Planck's constant. 812 00:42:22 --> 00:42:25 So, on the y axis we have kinetic energy, so 813 00:42:25 --> 00:42:27 we can plug that in. 814 00:42:27 --> 00:42:29 If we talk about what the x axis is, that's just the 815 00:42:29 --> 00:42:32 frequency of the light that's coming in. 816 00:42:32 --> 00:42:36 We know what m is, m is equal to h. 817 00:42:36 --> 00:42:41 And then we can plug in what b is, the y intercept, because 818 00:42:41 --> 00:42:47 that's just the negative of h times that threshold frequency. 819 00:42:47 --> 00:42:50 So we have this new equation here when we're considering 820 00:42:50 --> 00:42:53 this photoelectric effect, which is that the kinetic 821 00:42:53 --> 00:42:59 energy is equal to h nu minus h nu threshold of the metal. 822 00:42:59 --> 00:43:03 And what Einstein concluded and observed is that well, kinetic 823 00:43:03 --> 00:43:07 energy, of course, that's an energy term, and h times nu, 824 00:43:07 --> 00:43:10 well that has to be energy also, because energy has to be 825 00:43:10 --> 00:43:13 equal to energy -- there's no other way about it. 826 00:43:13 --> 00:43:15 And this worked out with units as well because we're talking 827 00:43:15 --> 00:43:18 about joules for kinetic energy, and when we're talking 828 00:43:18 --> 00:43:21 about h times nu, we're talking about joules times second 829 00:43:21 --> 00:43:23 times inverse seconds. 830 00:43:23 --> 00:43:26 So, the very important conclusion that Einstein made 831 00:43:26 --> 00:43:32 here is that energy is equal to h times nu, or that h times nu 832 00:43:32 --> 00:43:35 is an actual energy term. 833 00:43:35 --> 00:43:39 And this kind of went along with two observations. 834 00:43:39 --> 00:43:42 The first is that energy of a photon is proportional 835 00:43:42 --> 00:43:43 to its frequency. 836 00:43:43 --> 00:43:47 So this was never recognized before that if we know the 837 00:43:47 --> 00:43:52 frequency of a photon or a wave of light, we can know the 838 00:43:52 --> 00:43:54 energy of that light. 839 00:43:54 --> 00:43:57 So, since we know that there's relationship also between 840 00:43:57 --> 00:43:59 frequency and wavelength, we can do the same thing -- if we 841 00:43:59 --> 00:44:02 know the wavelength, we can know the energy of the light. 842 00:44:02 --> 00:44:06 And I use the term photon here, and that's because he also 843 00:44:06 --> 00:44:08 concluded that light must be made up of these energy 844 00:44:08 --> 00:44:11 packets, and each packet has that h, that Planck's 845 00:44:11 --> 00:44:15 constant's worth of energy in it, so that's why you have to 846 00:44:15 --> 00:44:17 multiply Planck's constant times the frequency. 847 00:44:17 --> 00:44:21 Any frequency can't have an energy, you have to -- you 848 00:44:21 --> 00:44:24 don't have a continuum of frequencies that are of a 849 00:44:24 --> 00:44:28 certain energy, it's actually punctuated into these packets 850 00:44:28 --> 00:44:30 that are called photons. 851 00:44:30 --> 00:44:33 And, as you know, Einstein made many, many, many very important 852 00:44:33 --> 00:44:38 contributions to science and relativity, but he called this 853 00:44:38 --> 00:44:42 his one single most important contribution to science, the 854 00:44:42 --> 00:44:46 relationship between energy and frequency and the 855 00:44:46 --> 00:44:48 idea of photons. 856 00:44:48 --> 00:44:52 So this means we now have a new way of thinking about the 857 00:44:52 --> 00:44:56 photoelectric effect, and that is the idea that h times 858 00:44:56 --> 00:44:59 nu is actually an energy. 859 00:44:59 --> 00:45:02 So, it's the energy of an incident photon if we're 860 00:45:02 --> 00:45:04 talking about nu where we're talking about the energy of the 861 00:45:04 --> 00:45:08 photon going in, so we can abbreviate that as e sub i, 862 00:45:08 --> 00:45:11 energy of the incident photon. 863 00:45:11 --> 00:45:16 We can talk about also h times nu nought, which is that 864 00:45:16 --> 00:45:17 threshold frequency. 865 00:45:17 --> 00:45:20 So this is a term we're going to see a lot, especially in 866 00:45:20 --> 00:45:23 your problem sets, it's called the work function, and the work 867 00:45:23 --> 00:45:28 function is the same thing as the threshold frequency of a 868 00:45:28 --> 00:45:30 metal, except, of course, that it's multiplied by 869 00:45:30 --> 00:45:31 Planck's constant. 870 00:45:31 --> 00:45:34 So, it's the minimum energy that a certain metal requires 871 00:45:34 --> 00:45:36 in order to pop a photon out of it -- in order to eject 872 00:45:36 --> 00:45:38 an electron from the surface of that metal. 873 00:45:38 --> 00:45:46 So this is our new kind of schematic way that we can 874 00:45:46 --> 00:45:50 think about looking at the photoelectric effect, so if 875 00:45:50 --> 00:45:53 this is the total amount of energy that we put into the 876 00:45:53 --> 00:45:57 system, where here we have the energy of a free electron. 877 00:45:57 --> 00:46:01 We have this much energy going in, the metal itself requires 878 00:46:01 --> 00:46:05 this much energy, the work function, in order to 879 00:46:05 --> 00:46:06 eject an electron. 880 00:46:06 --> 00:46:09 So that much energy is going to be used up just ejecting it. 881 00:46:09 --> 00:46:12 And what we have left over is this amount of energy here, 882 00:46:12 --> 00:46:15 which is going to be the kinetic energy of the 883 00:46:15 --> 00:46:17 ejected electron. 884 00:46:17 --> 00:46:20 So, therefore, we can rewrite our equation in two ways. 885 00:46:20 --> 00:46:24 One is just talking about it in terms only of energy where our 886 00:46:24 --> 00:46:27 kinetic energy here is going to be equal to the total energy 887 00:46:27 --> 00:46:32 going in -- the energy initial minus this energy of the 888 00:46:32 --> 00:46:34 work function here. 889 00:46:34 --> 00:46:38 We can also talk about it in terms of if we want to solve, 890 00:46:38 --> 00:46:41 if we, for example, we want to find out what that initial 891 00:46:41 --> 00:46:44 energy was, we can just rearrange our equation, or we 892 00:46:44 --> 00:46:48 can look at this here where the initial energy is equal to 893 00:46:48 --> 00:46:50 kinetic energy plus the work function. 894 00:46:50 --> 00:46:54 So before we go we'll try to see if we can do a clicker 895 00:46:54 --> 00:46:58 question for you on this, and we can, very good. 896 00:46:58 --> 00:47:01 So, everyone take those clickers back out and tell me, 897 00:47:01 --> 00:47:03 if a beam of light with a certain energy, and we're going 898 00:47:03 --> 00:47:08 to say four electron volts strikes a gold surface, and 899 00:47:08 --> 00:47:11 here we're saying that the gold surface has a work function of 900 00:47:11 --> 00:47:17 5.1 electron volts, what is the maximum kinetic energy of the 901 00:47:17 --> 00:47:31 electron that is ejected? 902 00:47:31 --> 00:47:37 So why don't you go ahead and take ten seconds on that. 903 00:47:37 --> 00:47:40 And if you don't know, that's okay, just type in an answer 904 00:47:40 --> 00:47:44 and give it your best shot. 905 00:47:44 --> 00:47:47 And let's see what we come up with here. 906 00:47:47 --> 00:47:47 Alright. 907 00:47:47 --> 00:47:52 So, it looks like some of you were tricked, but many of you 908 00:47:52 --> 00:47:55 were not, so no electrons will be ejected. 909 00:47:55 --> 00:47:59 The reason for that is because this is the minimum amount of 910 00:47:59 --> 00:48:02 energy -- hold off a sec on the packing up, so in case someone 911 00:48:02 --> 00:48:04 doesn't understand -- this is the minimum amount of energy 912 00:48:04 --> 00:48:07 that's required from the energy going in in order to 913 00:48:07 --> 00:48:08 eject an electron. 914 00:48:08 --> 00:48:12 So if the incident energy is less than the energy that's 915 00:48:12 --> 00:48:14 required, absolutely nothing will happen. 916 00:48:14 --> 00:48:16 That's the same thing we were talking about with 917 00:48:16 --> 00:48:17 threshold frequency. 918 00:48:17 --> 00:48:20 All right, now you can pack up and we'll see you on Wednesday. 919 00:48:20 --> 00:48:21