1 00:00:01,440 --> 00:00:06,215 Welcome back everyone, I'm Charles Clark. And now we're can continue our discussion 2 00:00:06,215 --> 00:00:09,625 of the hydrogen atom as seen from the perspective of classical mechanics, 3 00:00:09,625 --> 00:00:14,922 picking up where we left off. Now this is going to be the most detailed 4 00:00:14,922 --> 00:00:19,962 mathematical derivation so far, but I emphasize that it's simple vector algebra 5 00:00:19,962 --> 00:00:24,747 and calculus just taking time to relatives. 6 00:00:24,747 --> 00:00:29,149 So I hope it'll be accessible in any of then I have a number of inline quizzes 7 00:00:29,149 --> 00:00:33,906 that will pause during the derivation and give you some simple questions to help 8 00:00:33,906 --> 00:00:39,979 you keep pace. So good luck and let's see how it goes. 9 00:00:41,710 --> 00:00:44,430 Just to remind you where we left off last time. 10 00:00:44,430 --> 00:00:49,191 So we started, we're working on equation of motion for a hydrogen-like system that 11 00:00:49,191 --> 00:00:53,262 is a nucleus with a positive charge, a mass of capital M, and electron of 12 00:00:53,262 --> 00:01:01,425 negative charge, a mass of little m. And then we, we got, we just wrote down 13 00:01:01,425 --> 00:01:07,311 the Newtonian equations. Using the Coulomb interactions we found 14 00:01:07,311 --> 00:01:13,779 that we are able to get a cut an equation that involved only, the relative 15 00:01:13,779 --> 00:01:21,990 separation of the two particles. So that's a big step forward. 16 00:01:21,990 --> 00:01:28,272 here is that equation in a simpler form. So it's a, just a motion of a, of a 17 00:01:28,272 --> 00:01:32,178 particle in the Coulomb field but with a reduced mass, and with a, a relative 18 00:01:32,178 --> 00:01:35,820 co-ordinate. Here's the expression for the reduced 19 00:01:35,820 --> 00:01:38,899 mass. We'll be using this later on, in order to 20 00:01:38,899 --> 00:01:44,384 replicate the. The investigations that led to the 21 00:01:44,384 --> 00:01:49,280 discovery of deuterium which is a heavier isotope of the hydrogen atom. 22 00:01:49,280 --> 00:01:54,510 It's the, the, neutrium nucleus consisted of proton and a neutron. 23 00:01:54,510 --> 00:01:59,264 So it has the same charge as hydrogen but about twice the mass. 24 00:01:59,264 --> 00:02:04,906 And then by denoting this momentum in terms of the, the velo, the mass times 25 00:02:04,906 --> 00:02:10,912 the velocity, the particle, very much the conventional way, we got to these two 26 00:02:10,912 --> 00:02:17,372 first order equations. And I put emphasis on this because this 27 00:02:17,372 --> 00:02:20,301 is really. The general type of evolution equation 28 00:02:20,301 --> 00:02:24,772 that one wants to have. In other words if we have initial values 29 00:02:24,772 --> 00:02:33,360 of the position, sorry, the position and the momentum, we have initial values. 30 00:02:33,360 --> 00:02:36,898 Then we can calculate the first derivative of those, because we have, 31 00:02:36,898 --> 00:02:42,430 basically we just plug in the initial values to these equations. 32 00:02:42,430 --> 00:02:45,898 That first derivative, let's say we have t equals 0, that first derivative allows 33 00:02:45,898 --> 00:02:49,009 us to propagate a little way to, a next high step and so on, so, with just the 34 00:02:49,009 --> 00:02:51,916 initial values of position of the momentum, we can integrate these, 35 00:02:51,916 --> 00:02:56,429 equations of motion and time indefinitely. 36 00:02:57,570 --> 00:03:01,389 Now, whether we can do that in a practical sense with a simple looking 37 00:03:01,389 --> 00:03:07,417 solution or not remains to be determined. The answer is yes, but the hydrogen atom 38 00:03:07,417 --> 00:03:11,833 is quite unusual in this degree and it shows aspects that are not necessarily 39 00:03:11,833 --> 00:03:16,760 encountered in most other dynamic systems. 40 00:03:16,760 --> 00:03:21,558 But more about that later. So now we continue by seeing, we're 41 00:03:21,558 --> 00:03:26,598 going to try to integrate this equation to the best degree by finding so called 42 00:03:26,598 --> 00:03:32,225 constants of the motion. That is, things that stay constant in 43 00:03:32,225 --> 00:03:34,855 time. So, so one of these, one such constant in 44 00:03:34,855 --> 00:03:39,699 the motion which characterizes. They were really all dynamical systems 45 00:03:39,699 --> 00:03:43,983 where the, the the, the, the, the, the potentials that govern our system are 46 00:03:43,983 --> 00:03:48,190 independent of time. Is the total energy. 47 00:03:48,190 --> 00:03:51,260 We put a bunch of particles together and they just have neutral interactions. 48 00:03:51,260 --> 00:03:54,150 The, and they're not influenced by an external environment. 49 00:03:54,150 --> 00:03:58,040 Their energy will be the same. So lets 50 00:03:58,040 --> 00:04:00,540 Let's see how that's done for this current system. 51 00:04:00,540 --> 00:04:05,508 And, very natural procedure is to start from the dynamical variables themselves, 52 00:04:05,508 --> 00:04:10,188 form simple functions of those variables and see to what degree that those 53 00:04:10,188 --> 00:04:16,635 functions are conserved in time. So if we take, if we just think about 54 00:04:16,635 --> 00:04:23,575 products of p and r, here's, here's a, rather obvious candidate to look at. 55 00:04:23,575 --> 00:04:27,420 p squared over 2 mu. This is the expression for a kinetic 56 00:04:27,420 --> 00:04:32,050 energy operator. So does this correspond to a kinetic 57 00:04:32,050 --> 00:04:38,420 energy, operator in this system. And does it lead to anything useful? 58 00:04:38,420 --> 00:04:43,284 So I'm now going to give you a little in line quiz just to see how well you're 59 00:04:43,284 --> 00:04:47,864 tracking that. So I hope that you did well on that 60 00:04:47,864 --> 00:04:52,804 question and the, the message is that this t that we've identified here is a 61 00:04:52,804 --> 00:04:59,730 kinetic energy operator. It's the, the net kinetic energy of the 62 00:04:59,730 --> 00:05:06,408 system in a frame in which the center of mass is not moving. 63 00:05:06,408 --> 00:05:10,495 So that's basically if you get on board a frame that's you when, both these 64 00:05:10,495 --> 00:05:16,733 electron and the nucleus could be moving along at a million miles an hour. 65 00:05:16,733 --> 00:05:20,693 If you have a fast enough car you can catch up with that and then you're in, 66 00:05:20,693 --> 00:05:24,587 then you're in, in a, basically in a system, in a, in a, in a frame in which 67 00:05:24,587 --> 00:05:29,207 you're only seeing the relative motion of the of the coordinate, the separation 68 00:05:29,207 --> 00:05:37,920 seems to, the weighted, mass weighted separation seems to be constant time. 69 00:05:40,260 --> 00:05:43,354 Well, now how does the kinetic energy depend upon time? 70 00:05:43,354 --> 00:05:48,452 Well, let's take it's time derivative. here we have the first derivative of time 71 00:05:48,452 --> 00:05:53,173 in, of the kinetic energy. Well it's a time derivative of p.p, 72 00:05:53,173 --> 00:05:58,090 that's just over two mu, that's just clearly equal to p. 73 00:05:59,190 --> 00:06:10,258 Dot P dot, it's the over mu. And now so for for P, this term P here we 74 00:06:10,258 --> 00:06:19,670 can, we can use, we can use this identity. 75 00:06:19,670 --> 00:06:24,908 To relate it to r dot. And now, let's, maybe I better choose a 76 00:06:24,908 --> 00:06:29,636 different color for the next one. For the p dot, you see we have the 77 00:06:29,636 --> 00:06:37,280 equation of motion up here. So we substitute that in. 78 00:06:38,540 --> 00:06:43,360 So here's, here's, here's what this. P dot, dot, p dot. 79 00:06:43,360 --> 00:06:46,410 I'm sorry for all the dots, but that's you know, our way of talking. 80 00:06:47,480 --> 00:06:50,105 You can pass for a physicist if you pick some of this lingo up. 81 00:06:50,105 --> 00:06:57,350 you see, it's now reduces from this equation, it's easy to reduce to this, so 82 00:06:57,350 --> 00:07:05,646 that r dot, r dot is equal to the scalar r times scalar r dot. 83 00:07:05,646 --> 00:07:12,513 You can use, basically, maybe you need to go here, if you want to follow that in 84 00:07:12,513 --> 00:07:17,820 detail. Some of you will see this self evidently. 85 00:07:17,820 --> 00:07:20,070 But, I mean, you know, I have done this before. 86 00:07:20,070 --> 00:07:23,100 And, it always pays to be careful-,[INAUDIBLE] carefully. 87 00:07:23,100 --> 00:07:27,577 And now, here is the whole point. Is that, you can recognize that this, 88 00:07:27,577 --> 00:07:35,340 this whole term here, is actually the time derivative of something else. 89 00:07:35,340 --> 00:07:38,855 It's basically, let's, you know? It's got this minus ze squared. 90 00:07:38,855 --> 00:07:43,273 Drdt divide by r squared. Is, so, the rdt times 1 over r squared is 91 00:07:43,273 --> 00:07:49,373 equal to minus ddt 1 over r. If you know how to, how to differentiate, 92 00:07:49,373 --> 00:07:58,223 if you know how that ddt of x to the n is equal to n x dot x to the n minus 1, you 93 00:07:58,223 --> 00:08:08,412 can work all this. This is sort of the, kind of the 94 00:08:08,412 --> 00:08:15,276 fundamental identity of functional differentiate, how to differentiate any 95 00:08:15,276 --> 00:08:22,225 power of a quantity. So this is just a, this general rule 96 00:08:22,225 --> 00:08:29,386 gives you the example here. So in other words the time derivative of 97 00:08:29,386 --> 00:08:36,382 the potential energy and, can use another color, time derivative of potential 98 00:08:36,382 --> 00:08:43,166 energy is minus the time derivative of this thing that we've denoted by V of r, 99 00:08:43,166 --> 00:08:50,411 okay. So now I am going to give you another set 100 00:08:50,411 --> 00:08:56,047 of questions to ponder. Okay I hope that you've grasped the 101 00:08:56,047 --> 00:09:01,712 difference and that second question. So now V of r, is the total potential 102 00:09:01,712 --> 00:09:06,752 the, energy of the system. The potential energy doesn't actually 103 00:09:06,752 --> 00:09:12,359 depend upon the reference frame, at least in, in non-relativistic quantum 104 00:09:12,359 --> 00:09:16,823 mechanics. So, let's, let's go on and buy a vowel 105 00:09:16,823 --> 00:09:23,516 and complete the puzzle. I've decided to buy the vowel E, okay? 106 00:09:23,516 --> 00:09:29,660 Because from, from this, from this relationship that the time derivative of 107 00:09:29,660 --> 00:09:38,650 t is minus the time derivative of the potential v we find that t plus v. 108 00:09:38,650 --> 00:09:41,770 The sum of these two, is a constant of motion. 109 00:09:41,770 --> 00:09:45,840 And that is, in fact, from the yo, your previous analyses. 110 00:09:45,840 --> 00:09:49,100 an energy. Well now like, what energy is it, and 111 00:09:49,100 --> 00:09:53,974 that's going to be sorry, just another little brief. 112 00:09:53,974 --> 00:09:59,254 I hope that was straightforward and that you understand that the energy here is 113 00:09:59,254 --> 00:10:04,648 the energy of the system. In the frame that moves with the center 114 00:10:04,648 --> 00:10:07,056 of mass. Once again, the fact that the center of 115 00:10:07,056 --> 00:10:10,395 mass strength could be moving at any speed whatever, we would still have the 116 00:10:10,395 --> 00:10:13,416 same equation of motion to solve, to describe this, this relative in 117 00:10:13,416 --> 00:10:17,230 coordinate. Which is the important thing. 118 00:10:17,230 --> 00:10:20,530 So it's, this is one of these initial condition problems. 119 00:10:20,530 --> 00:10:23,415 You're, we're not really. If we're talking about an atom, we 120 00:10:23,415 --> 00:10:26,871 recognize, it could be moving at any speed, but what we're really interested 121 00:10:26,871 --> 00:10:31,020 in is the apparent constancy of its internal structure. 122 00:10:31,020 --> 00:10:35,812 This is really the important issue. and this is, you know, this is consistent 123 00:10:35,812 --> 00:10:41,370 with the picture of quantum mechanics. Okay. 124 00:10:41,370 --> 00:10:45,470 So now, just to recapitulate what we've done. 125 00:10:45,470 --> 00:10:56,299 we've got ourselves a nice, first order equations of motion. 126 00:10:56,299 --> 00:11:01,320 And we found the conserved energy. Okay. 127 00:11:01,320 --> 00:11:08,652 Now are there other constants in motion this system well what about a something 128 00:11:08,652 --> 00:11:17,890 having to do with who, we, we started with p.p as a, to see whether if. 129 00:11:17,890 --> 00:11:20,659 When you took its time derivative, anything interesting? 130 00:11:20,659 --> 00:11:24,628 Well this would just be r squared. So that's, that's not likely to offer 131 00:11:24,628 --> 00:11:29,530 anything more because, you see, we already have an equation of motion for r. 132 00:11:29,530 --> 00:11:34,384 how about r dot p? Well that's, that's not without interest, 133 00:11:34,384 --> 00:11:41,820 but it doesn't really lead to. lead to much. 134 00:11:41,820 --> 00:11:46,670 Okay, now, how about this, I think many of you will have seen this before. 135 00:11:46,670 --> 00:11:49,523 Here's a, you know, we're sort of going in order. 136 00:11:49,523 --> 00:11:56,882 We really sort of, we did the dot products, r-dot-r, r-dot-p, p-dot-p. 137 00:11:56,882 --> 00:12:04,652 Now, here's the here's the cross product. What about this l, r cross p? 138 00:12:04,652 --> 00:12:08,390 What's the time derivative? Well that's easy. 139 00:12:08,390 --> 00:12:15,334 It's r dot cross p plus r cross p dot. That's a basic rule of differentiation. 140 00:12:15,334 --> 00:12:22,650 And now for this term, we look up here. Okay. 141 00:12:22,650 --> 00:12:26,010 And we see that r dot is just proportional to p. 142 00:12:27,360 --> 00:12:33,940 So that means we get a p cross p, term. That's automatically equal to zero. 143 00:12:33,940 --> 00:12:37,390 We don't even have to worry about the, whether the mu is there or not. 144 00:12:37,390 --> 00:12:40,755 Cross product of a vector with itself, of a vector with itself is zero. 145 00:12:40,755 --> 00:12:45,230 Okay, and now for p dot. Once again, a very nice, you know, we 146 00:12:45,230 --> 00:12:48,648 have a very nice equation. Let's see, I'm going to switch colors 147 00:12:48,648 --> 00:12:56,060 here so it doesn't get too confusing. Apologize to the color blind, 148 00:12:58,470 --> 00:13:01,876 So p dot you see is just proportional to r. 149 00:13:01,876 --> 00:13:05,838 So when we bring that back that's also equal to 0. 150 00:13:05,838 --> 00:13:15,160 So, we have the result that L is a constant of the motion. 151 00:13:15,160 --> 00:13:18,845 Very nice. And you might say it's actually three 152 00:13:18,845 --> 00:13:24,556 constants of the motion because it's a vector quantity. 153 00:13:24,556 --> 00:13:29,022 Now I have a, and I have a question for you about this vector L that we've 154 00:13:29,022 --> 00:13:34,104 discovered. So L is in fact an angular momentum and 155 00:13:34,104 --> 00:13:39,900 I, in the, in the quiz that you just took it, it defines, you know, the, the 156 00:13:39,900 --> 00:13:49,231 location of the coordinate origin which that hanging[UNKNOWN] is defined. 157 00:13:49,231 --> 00:13:55,011 And it's very significant, you see, that that the issue of the momentum and the 158 00:13:55,011 --> 00:14:00,840 position always being in a plane perpendicular to l. 159 00:14:00,840 --> 00:14:07,522 So this is a, this is actually a, it's a property of any, force law. 160 00:14:07,522 --> 00:14:13,250 That, that turns out to be proportional. that is proportional. 161 00:14:13,250 --> 00:14:15,710 Where the force is proportional to the, the vector r. 162 00:14:15,710 --> 00:14:19,900 So, This Coloumb force is a 1 over r squared 163 00:14:19,900 --> 00:14:25,610 law. One over r law 1 over and it's a linear 164 00:14:25,610 --> 00:14:31,721 law in r and so on. All those yield the same result that the 165 00:14:31,721 --> 00:14:35,255 angular minimum is conserved and that means I guess you can, you can now see 166 00:14:35,255 --> 00:14:42,800 with this. r dot l equals 0 always. 167 00:14:45,950 --> 00:14:54,928 And p dot l equals 0 always. So r and p are always perpendicular to l, 168 00:14:54,928 --> 00:14:59,672 and there's, there's only one plane that's perpendicular to l. 169 00:14:59,672 --> 00:15:03,578 So the position the momentum evolve in that plane, so this is, you might say 170 00:15:03,578 --> 00:15:09,074 this is a very important simplification. We started out with two particles they 171 00:15:09,074 --> 00:15:12,918 can each move in 3 dimensions so we started out with a 6 dimensional problem 172 00:15:12,918 --> 00:15:17,061 in some sense and by now we've reduced it. 173 00:15:17,061 --> 00:15:21,379 To, Well, how many dimensions is it? 174 00:15:21,379 --> 00:15:27,229 I mean, the, we have two particles that move in a plane. 175 00:15:27,229 --> 00:15:29,310 So, we, we, we know. We know. 176 00:15:29,310 --> 00:15:34,850 We've, we've really located the orbit. we, we know that, that, that, basically 177 00:15:34,850 --> 00:15:41,050 the, the system has gotta be evolving. Somehow in this plane. 178 00:15:41,050 --> 00:15:46,055 And in the next lecture, we're going to take, and we're going to actually find, 179 00:15:46,055 --> 00:15:50,983 the specific origin orbit, and that is made possible by the discovery of a 180 00:15:50,983 --> 00:15:54,563 hidden symmetry.