1 00:00:03,040 --> 00:00:06,388 Hello everyone, welcome back to Atomic Structure and Spectra. 2 00:00:06,388 --> 00:00:12,568 I'm Charles Clark and I'll be concluding with part 5 of this lecture. 3 00:00:12,568 --> 00:00:19,001 And then in the subsequent lecture we'll turn to the old quantum theory. 4 00:00:19,001 --> 00:00:26,620 And the practical calculations using the wave functions to address atomic systems. 5 00:00:27,700 --> 00:00:31,663 So let's get started. Before proceeding, let me make a 6 00:00:31,663 --> 00:00:39,090 procedural announcement concerning the homework for this week. 7 00:00:39,090 --> 00:00:46,900 It's not mathematically demanding, in most cases but it is conceptually demand, 8 00:00:46,900 --> 00:00:52,088 demanding. And one part of it requires you to read a 9 00:00:52,088 --> 00:00:58,110 section of Albert Einstein's 1905 paper on the photoelectric effect. 10 00:00:58,110 --> 00:01:03,234 That is, can be found here in this document which is the additional 11 00:01:03,234 --> 00:01:08,120 materials linked from the course home page. 12 00:01:09,430 --> 00:01:13,588 Now we have in there some commentary by Cornelius Cilanchos, it's quite 13 00:01:13,588 --> 00:01:18,318 informative. The original German paper of Einstein, 14 00:01:18,318 --> 00:01:23,120 and then a link to the English translation that's on Wikimedia. 15 00:01:23,120 --> 00:01:28,374 I should mention that in preparing this material, I found that there was a very 16 00:01:28,374 --> 00:01:34,664 considerable error in the existing English translation on Wikimedia. 17 00:01:34,664 --> 00:01:39,739 And I posted that information to the German student study group. 18 00:01:39,739 --> 00:01:44,993 And I'm really grateful to Stephan Leger for stepping in, correcting that error, 19 00:01:44,993 --> 00:01:51,580 and I think a few others as well. So, this is really, for me, a positive 20 00:01:51,580 --> 00:01:57,970 outcome of the course that it's, it's giving rise to an improved version of, of 21 00:01:57,970 --> 00:02:05,361 this wonderful paper. That is potentially the most the widely 22 00:02:05,361 --> 00:02:11,521 accessed source of access to it given the fact that its in the public domain on 23 00:02:11,521 --> 00:02:17,391 Wikimedia. Okay, so now lets proceed and I will 24 00:02:17,391 --> 00:02:23,963 remind you were we left off. Again, we're on the trail of the spectrum 25 00:02:23,963 --> 00:02:29,387 of the hydrogen atom. and the real focus we're going to start 26 00:02:29,387 --> 00:02:34,633 up in earnest in the next, in the next lecture is on this famous Bulma Alpha 27 00:02:34,633 --> 00:02:41,048 Line at 656 nanometers. again, very very close to our, our red 28 00:02:41,048 --> 00:02:45,880 reference laser. this is as you may recall. 29 00:02:45,880 --> 00:02:55,262 Up here, we have the Emission spectrum. And then, this is Absorption spectrum of 30 00:02:55,262 --> 00:03:04,030 the sun, as seen by the 4A transform spectrometer at Kit Peak Observatory. 31 00:03:06,070 --> 00:03:12,622 Now in the last lecture we began studying this electron nucleus system, the Hy, 32 00:03:12,622 --> 00:03:18,160 Hydrogen-like, systems. Hydrogen, Helium plus, Lithium plus, and 33 00:03:18,160 --> 00:03:22,286 some other more exotic things. And from the standpoint of, of classical 34 00:03:22,286 --> 00:03:26,865 mechanics. And well, we didn't, we didn't get that 35 00:03:26,865 --> 00:03:29,976 far. But there was, there's quite a number of 36 00:03:29,976 --> 00:03:34,599 things associated with separating out the relative from the center of mass motion 37 00:03:34,599 --> 00:03:40,114 that are generic value. And we ended up finding that the due to 38 00:03:40,114 --> 00:03:45,190 the particular force law. which is the same as the, as the Coulomb 39 00:03:45,190 --> 00:03:48,620 force law, which is the same as the gravitational force law. 40 00:03:49,800 --> 00:03:53,290 We have a conserved quantity, the angular momentum. 41 00:03:53,290 --> 00:03:58,345 So that angular momentum. Let's, let's pretend we have, actually we 42 00:03:58,345 --> 00:04:01,974 do have, we have god like powers over this system. 43 00:04:01,974 --> 00:04:05,943 So we can, we have the ability to prepare this system with arbitrary initial 44 00:04:05,943 --> 00:04:09,900 conditions for the positions and the velocities. 45 00:04:09,900 --> 00:04:13,546 So think about it that way. we're trying to understand globally 46 00:04:13,546 --> 00:04:18,652 everything that can possibly happen if we fix a an initial position, an initial 47 00:04:18,652 --> 00:04:23,015 velocity. So if we just set the thing up so that we 48 00:04:23,015 --> 00:04:26,315 determine, we determine, we have a, we have a separation between the two 49 00:04:26,315 --> 00:04:30,704 particles. We determine that at time t equals 0 and 50 00:04:30,704 --> 00:04:35,247 then we also determine the relative velocities of the electron and the 51 00:04:35,247 --> 00:04:40,176 nucleus. Then that, those choices determine the 52 00:04:40,176 --> 00:04:44,320 value of the energy and they also determine the value of the angular 53 00:04:44,320 --> 00:04:49,580 momentum. For example, if I, I, I just take the two 54 00:04:49,580 --> 00:04:58,585 particles and hold them still at time t equals 0, separated by some distance, r. 55 00:04:58,585 --> 00:05:09,193 then I release them, well then we have r of 0 is equal to r not, whatever value 56 00:05:09,193 --> 00:05:18,763 that was. And r, sorry p of 0 vector r 0, here is 57 00:05:18,763 --> 00:05:29,324 just r not P of 0 is equal to 0. So tat means in a case like that, the 58 00:05:29,324 --> 00:05:38,030 energy is just now equal to minus ZE squared over r 0. 59 00:05:38,030 --> 00:05:43,860 And you might, you might just think about what is the angular momentum if we have. 60 00:05:43,860 --> 00:05:48,468 we have the 6 value r now we just put that in there. 61 00:05:48,468 --> 00:05:54,372 Well if initially the the momentum was 0 then the angular momentum is going to be 62 00:05:54,372 --> 00:05:59,046 0 and it remains that way for the entire remaining time, the system is 63 00:05:59,046 --> 00:06:05,378 interacting. Now is there anything else that we can do 64 00:06:05,378 --> 00:06:10,980 to get, to further, get a under command of this system? 65 00:06:10,980 --> 00:06:16,870 And the answer is yes. We're heading for that answer. 66 00:06:16,870 --> 00:06:23,870 And a good way to start, is for example, we found a constant of the motion. 67 00:06:23,870 --> 00:06:28,872 that was made by say squaring the momentum and then another one the vector 68 00:06:28,872 --> 00:06:36,397 cross of the motion with using a cross product of the position the momentum. 69 00:06:36,397 --> 00:06:41,941 And let me remind you that the angular momentum defines a plane, this is in the 70 00:06:41,941 --> 00:06:46,895 previous thing. defines a plane in which the, the 71 00:06:46,895 --> 00:06:51,150 position and the momentum remain for all times. 72 00:06:51,150 --> 00:06:54,867 So as soon as we set up our initial condition, for positional momentum, we 73 00:06:54,867 --> 00:06:59,549 define an orbital plane, and the particle moves in that plane. 74 00:06:59,549 --> 00:07:07,400 Now It's always a good idea to look at alternative coordinates for that plane. 75 00:07:07,400 --> 00:07:12,704 So, we don't really know right now how the momentum and how the momentum and the 76 00:07:12,704 --> 00:07:21,010 the position are going to evolve in time. But we do have 2 other variables that 77 00:07:21,010 --> 00:07:26,246 arise quite naturally. So we're interested in vectors that are 78 00:07:26,246 --> 00:07:32,075 in the same plane as the momentum and the the position. 79 00:07:32,075 --> 00:07:36,234 And here are 2 choices. one is r cross l. 80 00:07:36,234 --> 00:07:41,482 Now, I hope I've done this correctly, so here I'm taking L to be out of the plane 81 00:07:41,482 --> 00:07:46,508 of the image. So L is pointing toward you, from your 82 00:07:46,508 --> 00:07:51,578 display, and now, if we use the right hand rule to calculate r cross l from r, 83 00:07:51,578 --> 00:07:57,852 my result is that this is r cross L in that direction. 84 00:07:59,080 --> 00:08:03,859 And then there's another, another another vector independent of the previous one P 85 00:08:03,859 --> 00:08:06,876 cross l. And as you can see I've, I have again l 86 00:08:06,876 --> 00:08:11,020 out of the board. I do a cross product with P cross l. 87 00:08:11,020 --> 00:08:22,510 Now that, that turns in the other direction so this is P cross l. 88 00:08:24,360 --> 00:08:31,640 So these are 2, 2, 2 vectors that sort of have a parallelism with the position, the 89 00:08:31,640 --> 00:08:36,180 momentum. That is their, their basically, you're 90 00:08:36,180 --> 00:08:39,380 taking a, a constant vector, the angular momentum, and performing the cross 91 00:08:39,380 --> 00:08:43,395 product. So in some sense they're really just, a 92 00:08:43,395 --> 00:08:47,775 simple linear functions of the position momentum. 93 00:08:47,775 --> 00:08:53,100 But they well let's, let's just see if they give us any additional information 94 00:08:53,100 --> 00:08:58,155 on the problem. So the first thing we'll do is work out 95 00:08:58,155 --> 00:09:03,864 explicitly what r cross L is. So L is r cross p, so r is r cross r 96 00:09:03,864 --> 00:09:09,324 cross p, or just to simplify the just, just to put everything in terms of r in 97 00:09:09,324 --> 00:09:15,770 its time derivative, it has this simple form. 98 00:09:17,100 --> 00:09:22,898 Now this gives us an occasion to use a famous vector identity. 99 00:09:22,898 --> 00:09:28,988 Which some of you may have encountered in in electrodynamics or other mechanics 100 00:09:28,988 --> 00:09:32,844 courses. Very useful. 101 00:09:32,844 --> 00:09:39,280 The back cab rule is the pneumonic given to it, among people like me. 102 00:09:39,280 --> 00:09:42,350 So it's, it's actually, its relatively easier to prove. 103 00:09:42,350 --> 00:09:44,680 But its a good idea just to be aware of it. 104 00:09:44,680 --> 00:09:52,744 a cross a quantity b cross c is b a dot c minus c a dot b, that pronounce batcab in 105 00:09:52,744 --> 00:10:00,951 American. Just for those with a mathematical bent, 106 00:10:00,951 --> 00:10:08,247 this is not the same as a cross b cross c, so the cross product is not 107 00:10:08,247 --> 00:10:15,241 associative. It actually you, you have to be quite 108 00:10:15,241 --> 00:10:19,529 explicit in the in the, the grouping of operations in order to get the same 109 00:10:19,529 --> 00:10:23,599 result. And you can, you can see this directly 110 00:10:23,599 --> 00:10:27,648 from from applica, changing, changing the 111 00:10:27,648 --> 00:10:34,661 order around and applying the back cap. So this is a very simple calculation when 112 00:10:34,661 --> 00:10:39,794 you know how to do it. I said there's nothing very much of 113 00:10:39,794 --> 00:10:43,090 intrinsic of interest in using the back cab rule. 114 00:10:43,090 --> 00:10:47,490 I'm just showing the details, so you see how straight forward it is. 115 00:10:47,490 --> 00:10:52,021 And so when you when you do evaluate this identity. 116 00:10:52,021 --> 00:10:57,515 you get this form and then, remarkably enough, it turns out that, this r cross 117 00:10:57,515 --> 00:11:03,637 L, which we just took as a basically another basis vector. 118 00:11:03,637 --> 00:11:09,049 That's natural to used to describe the dynamics of the system because it lies in 119 00:11:09,049 --> 00:11:13,900 the, in the same. It, it points to locations where the 120 00:11:13,900 --> 00:11:17,624 particle can. Where the electron nucleus system can be 121 00:11:17,624 --> 00:11:21,801 in the plane. We find that it is proportional to a, a 122 00:11:21,801 --> 00:11:27,581 time derivative. just, which is the, it's the minus r 123 00:11:27,581 --> 00:11:35,130 cubed times the time derivative of the unit vector of r. 124 00:11:35,130 --> 00:11:41,114 So this is the unit vector of r is just the if, if the if the particle is moving 125 00:11:41,114 --> 00:11:49,325 in some trajectory, then the unit vector is always the, the same length. 126 00:11:49,325 --> 00:11:55,045 And it, it keeps, it keeps a track of the, the direction of the the direction 127 00:11:55,045 --> 00:12:01,884 of the electron nucleus separation. But not, it's the, its independent of the 128 00:12:01,884 --> 00:12:08,137 magnitude. Okay, so now let's see let's see if we 129 00:12:08,137 --> 00:12:15,324 can infer anything useful. About from this addition term, addition 130 00:12:15,324 --> 00:12:23,822 vector p cross l. So, what we do here first, is just to 131 00:12:23,822 --> 00:12:28,430 calculate the time dirivitive of P cross l. 132 00:12:28,430 --> 00:12:34,472 that's pretty easy because. p cross L dot would be p dot cross l plus 133 00:12:34,472 --> 00:12:41,592 p cross l dot. But l dot vanishes, because l is constant 134 00:12:41,592 --> 00:12:47,847 of motion. And so now we have we can just take the 135 00:12:47,847 --> 00:12:56,210 Newton's law and plug into p dot, a substitute for p dot. 136 00:12:56,210 --> 00:13:11,139 And then we get this this identity here. So, I guess you can see now that 137 00:13:12,690 --> 00:13:21,790 That this, these 2 can combine to give something that's independent of time. 138 00:13:21,790 --> 00:13:29,638 So you see, that p dot cross l ended up giving a something that's proportional to 139 00:13:29,638 --> 00:13:34,766 the Time derivative of the, of the unit 140 00:13:34,766 --> 00:13:38,960 vector. And so, here is our central result. 141 00:13:38,960 --> 00:13:42,816 this is a, called the Runge-Lenz vector. It's been known under different names, 142 00:13:42,816 --> 00:13:45,864 for a long period of time. There's a very informative Wikipedia 143 00:13:45,864 --> 00:13:48,770 article about it. And it is, 144 00:13:48,770 --> 00:13:53,320 Well, as you see, that's just as it is here. 145 00:13:53,320 --> 00:14:01,280 p cross l minus ze square, e squared mu times the unit vector in r. 146 00:14:01,280 --> 00:14:06,707 And it's a constant of the motion and we are going to now look at its properties 147 00:14:06,707 --> 00:14:12,750 and it turns out a very important Interpretative value. 148 00:14:12,750 --> 00:14:18,244 In fact, it makes it possible for us to concisely express all possible solutions 149 00:14:18,244 --> 00:14:24,310 of this equation of motion for the electron nucleus system. 150 00:14:25,630 --> 00:14:29,820 This is very unusual for most problems of classical mechanics. 151 00:14:29,820 --> 00:14:35,772 There is no such global Solution. In this case it's what we have, what we 152 00:14:35,772 --> 00:14:40,764 have is an integral system where the, the constants of the motion completely 153 00:14:40,764 --> 00:14:47,550 specify the properties up to the choice of the origin of time. 154 00:14:52,210 --> 00:14:57,202 So, the first thing to note, about this Runge-Lenz vector, is it enables us to 155 00:14:57,202 --> 00:15:02,310 define the orbit in a remarkably simple way. 156 00:15:02,310 --> 00:15:05,650 So let's say, we have the Runge-Lenz vector. 157 00:15:05,650 --> 00:15:10,609 Which is a constant, sits in space. And now let's just, we'll just refer the, 158 00:15:10,609 --> 00:15:16,070 electron orbit to it by the most straightforward, manner. 159 00:15:16,070 --> 00:15:22,360 We know that, A is in the plane of the, of the electron 160 00:15:22,360 --> 00:15:27,172 orbit. Because it was, we, we chose it, by using 161 00:15:27,172 --> 00:15:31,502 from vectors that were always in that plane. 162 00:15:31,502 --> 00:15:37,904 and so theres this angle phi between the position vector and the run lens vector 163 00:15:37,904 --> 00:15:44,016 completely defines where the position vector is. 164 00:15:44,016 --> 00:15:50,418 So, we all know how to take the scale or product r.a Is r the, this this is, this 165 00:15:50,418 --> 00:15:57,224 is the vector, sorry this the vector product R.A. 166 00:15:57,224 --> 00:16:01,479 R A cosign theta. And so now we just, we just work it 167 00:16:01,479 --> 00:16:09,580 through. We work it through and this, this gives 168 00:16:09,580 --> 00:16:27,280 us opportunity to Invoke another common vector identity a dot b cross c. 169 00:16:27,280 --> 00:16:33,973 So this is something also widely used throughout physics and I suggest that you 170 00:16:33,973 --> 00:16:42,223 think about what this implies for the evaluation of r dot p cross l, and 171 00:16:42,223 --> 00:16:48,964 that's the subject of an in-video quiz. Well I hope that was a straightforward 172 00:16:48,964 --> 00:16:53,874 exercise for some of you. And that for those who, whom it wasn't 173 00:16:53,874 --> 00:16:59,300 you learned something. I mean this is for example, that identity 174 00:16:59,300 --> 00:17:04,305 a.b cross c is a way of computing the volume of a tetrahedron that is defined 175 00:17:04,305 --> 00:17:10,890 by the, the vectors, three vectors, a, b, and c. 176 00:17:10,890 --> 00:17:20,115 Anyway, here's the result. this is just l squared and that leads to 177 00:17:20,115 --> 00:17:32,882 this simple formula for the, the radius as a function of the angle phi. 178 00:17:32,882 --> 00:17:36,899 So, another, another way of looking at this is that the. 179 00:17:39,420 --> 00:17:43,500 Dependence of one over R. If we just invert that equation. 180 00:17:43,500 --> 00:17:46,908 If we say one over r. As a function of fee. 181 00:17:46,908 --> 00:17:53,862 It just, it just oscillates sinusoidally. So there's a, there's a very simple, very 182 00:17:53,862 --> 00:17:57,705 simple dependence on the inverse of r so which is proportional to the, the 183 00:17:57,705 --> 00:18:02,322 potential energy. So, that shows, in some sense, there's a, 184 00:18:02,322 --> 00:18:06,795 there's a continual harmonic oscillating interchange of energy between the 185 00:18:06,795 --> 00:18:12,780 potential and the kinetic parts, kinetic contributions of the energy. 186 00:18:17,550 --> 00:18:20,640 Okay, so there's one last calculation to do. 187 00:18:20,640 --> 00:18:26,488 to illustrate the essential points of this system and that is to calculate 188 00:18:26,488 --> 00:18:34,240 theses, these, the, the magnitude of A is terms of of other quantities. 189 00:18:34,240 --> 00:18:39,553 And so I mean, this, this just involves some, some vector very straight forward 190 00:18:39,553 --> 00:18:43,740 vector. calculations who which you. 191 00:18:43,740 --> 00:18:47,732 We've done, we've done those already so I'm not I just show you the result. 192 00:18:47,732 --> 00:18:52,832 And so this shows us a significant thing in that. 193 00:18:52,832 --> 00:18:58,560 The, the run a lens vector. Its magnitude depends both upon the 194 00:18:58,560 --> 00:19:07,570 energy and the angular momentum. In a sense the run the lens factor. 195 00:19:07,570 --> 00:19:10,608 It looks like we have a large number of constants in motion here. 196 00:19:10,608 --> 00:19:17,046 The energy, the anglement of that vector, and the run of lens factor. 197 00:19:17,046 --> 00:19:24,096 So 0[INAUDIBLE] these are 7 constants of motion that are defined on the surface 198 00:19:24,096 --> 00:19:28,790 independently. Well, that's far too many. 199 00:19:28,790 --> 00:19:35,380 In fact the the number of independent ones is more like 5. 200 00:19:35,380 --> 00:19:41,890 That is the energy, the angular momentum, vector, and then the run of lens vector 201 00:19:41,890 --> 00:19:48,399 must be perpendicular to the anglementum vectors. 202 00:19:48,399 --> 00:19:51,819 So it has, it has 2 components, but in fact there's then this consistency 203 00:19:51,819 --> 00:19:56,419 equation so it reduces it to 1. Okay, so now you can see some. 204 00:20:01,511 --> 00:20:08,205 some possibilities here. For example, this r can go to well, it 205 00:20:08,205 --> 00:20:20,593 has a possibility of going to infinity if, if A is greater than this, Ze squared 206 00:20:20,593 --> 00:20:28,444 mill. Because the cosine phi can, can become 207 00:20:28,444 --> 00:20:33,745 negative. So let's just let's think about what we 208 00:20:33,745 --> 00:20:42,950 can, what we know about the, the, the values of these various. 209 00:20:42,950 --> 00:20:49,682 these various quantities and see how that determines the form of the possible 210 00:20:49,682 --> 00:20:55,715 orbits of this system. So that's a basic, test of that, of your 211 00:20:55,715 --> 00:21:00,087 understanding of that, is in the next in-video quiz. 212 00:21:01,550 --> 00:21:06,552 So, I hope it's clear to you that in classical mechanics, we can in fact 213 00:21:06,552 --> 00:21:13,837 choose initial conditions that correspond to any possib, any energy. 214 00:21:13,837 --> 00:21:18,877 Negative or positive, independent of the value the the physical parameters of the 215 00:21:18,877 --> 00:21:23,488 problem the charge of the, the charge of the mass. 216 00:21:23,488 --> 00:21:27,180 And that there's a very sharp dividing line. 217 00:21:27,180 --> 00:21:31,182 And this choice at this, this determination of the dividing line will 218 00:21:31,182 --> 00:21:36,297 remain to be important. So for energies less than 0, what do 219 00:21:36,297 --> 00:21:41,320 those correspond to? Well, for example, if we start the 220 00:21:41,320 --> 00:21:46,420 system, we start the system with the electron and nucleus at rest at some 221 00:21:46,420 --> 00:21:51,660 fixed distance. Then we saw early in the lecture, and 222 00:21:51,660 --> 00:21:55,062 it's very easy to, to see from this, that this is the only contribution to the 223 00:21:55,062 --> 00:21:58,850 energy so the energy isn't, is negative there. 224 00:22:00,270 --> 00:22:07,110 So, the indeed if we start the system at rest, what happens is as you'll probably 225 00:22:07,110 --> 00:22:13,932 be able to figure out. It represents, it, it corresponds to 226 00:22:13,932 --> 00:22:21,532 extreme of this of this denominator, and you, you get you get an elliptical 227 00:22:21,532 --> 00:22:30,840 motion. well extreme case-full elliptical motion. 228 00:22:30,840 --> 00:22:36,912 Now, on the other hand, if the energy is greater than 0, then the the result is 229 00:22:36,912 --> 00:22:43,847 that the electron nucleus system completely dissociates. 230 00:22:43,847 --> 00:22:50,563 That is the the value of the radius eventually the value of the separation 231 00:22:50,563 --> 00:22:57,874 eventually gets as large as you please. So the, the, case is e less than 0, 232 00:22:57,874 --> 00:23:02,164 correspond to the bound states of atoms and molecules that respond to the sharp 233 00:23:02,164 --> 00:23:06,761 emission lines. And absorption lines in which we're 234 00:23:06,761 --> 00:23:11,296 interested. Now for E greater than 0, we have at 235 00:23:11,296 --> 00:23:18,280 least discussed some of those, those such states. 236 00:23:18,280 --> 00:23:22,624 more or less in passing. So, for example the photoelectric effect, 237 00:23:22,624 --> 00:23:27,316 when an atom, an electron is ejected from an atom, correspond to a state of, E, E 238 00:23:27,316 --> 00:23:32,278 greater than 0. And the, is the, that's basically the 239 00:23:32,278 --> 00:23:38,140 same process as the ion, ionization of gases by ultraviolet radiation. 240 00:23:38,140 --> 00:23:42,492 Which is the section of Einstein's paper that's, it dealt with in the homework for 241 00:23:42,492 --> 00:23:46,740 this week, and Rutherford's Scattering experiment. 242 00:23:46,740 --> 00:23:51,765 So we will see in the next lecture, in the next set of lectures, how this run 243 00:23:51,765 --> 00:23:56,724 the lens vector enters into quantum mechanics. 244 00:23:56,724 --> 00:24:02,850 And just to be very give, give you some guidances to where we're heading. 245 00:24:02,850 --> 00:24:10,555 these, these statements are modified. one significantly, the other not very 246 00:24:10,555 --> 00:24:16,098 much at all. So we will see that the states of energy 247 00:24:16,098 --> 00:24:25,184 less than 0 are now discrete. So quantum mechanics indicates that only 248 00:24:25,184 --> 00:24:34,634 certain values of energies are possible. And these have to do with basically the 249 00:24:34,634 --> 00:24:42,080 interference of the quantum waves to form. 250 00:24:42,080 --> 00:24:46,170 Sort of echoing bound states. This is sort of implicit in the Feynman 251 00:24:46,170 --> 00:24:49,800 path integral picture that Victor showed you. 252 00:24:50,880 --> 00:24:56,688 For the continuum on the other hand, I mean there still are significant effects 253 00:24:56,688 --> 00:25:02,855 of quantum mechanics, but all energies remain possible. 254 00:25:05,330 --> 00:25:07,810 And we'll look at the details of that in the next lecture, so, hope to see you 255 00:25:07,810 --> 00:25:09,040 return then.