1 00:00:00,400 --> 00:00:03,250 Welcome back everyone. I'm Charles Clark. 2 00:00:03,250 --> 00:00:07,000 We're now going to start to look at the spectrum of the hydrogen atom in detail. 3 00:00:07,000 --> 00:00:11,146 We'll review the historical background. we'll see what classical mechanics 4 00:00:11,146 --> 00:00:14,948 yields. And we'll go into the old quantum theory, 5 00:00:14,948 --> 00:00:19,340 always trying to understand the world and take steps to reproduce actual 6 00:00:19,340 --> 00:00:26,285 experimental data. Now just to remind you, our focus is on 7 00:00:26,285 --> 00:00:31,190 the spectrum of hydrogen. Here's the emission spectrum. 8 00:00:31,190 --> 00:00:36,860 and you can see I'm sorry, [SOUND] you can see here this red line at 656 9 00:00:36,860 --> 00:00:42,980 nanometers, which I'll remind you, is the same as this line seen in the Fraunhofer 10 00:00:42,980 --> 00:00:49,160 spectrum. this is one of the first things to be 11 00:00:49,160 --> 00:00:53,629 explained by Bohr's theory. so we're going to look at the things 12 00:00:53,629 --> 00:00:58,045 that, that govern the, the spectrum of hydrogen in this region and we'll, we'll 13 00:00:58,045 --> 00:01:03,522 show how they pertain to others. Again, just keep in mind, we're we're 14 00:01:03,522 --> 00:01:10,050 staying within the visible spectrum here. Here are our, our three reference lasers. 15 00:01:10,050 --> 00:01:13,263 So, I'd like to, you know, to just keep those the, the wavelengths and 16 00:01:13,263 --> 00:01:22,240 frequencies in mind. Now, just a reminder of why this system 17 00:01:22,240 --> 00:01:25,910 was very hard to understand by classical mechanics. 18 00:01:25,910 --> 00:01:28,990 So, it seems that atoms have these invariable properties that are 19 00:01:28,990 --> 00:01:33,400 demonstrated by the spectral lines. They don't change at all. 20 00:01:33,400 --> 00:01:37,264 one sees the same state of an atom on the surface of the, in the, in the atmosphere 21 00:01:37,264 --> 00:01:43,640 of the sun as one does for something in gas in a, in a terrestrial laboratory. 22 00:01:43,640 --> 00:01:47,727 And it's hard to understand how that can be if you think of things from standpoint 23 00:01:47,727 --> 00:01:53,301 of simple Newtonian, Newtonian mechanics. So, in classical mechanics you can assign 24 00:01:53,301 --> 00:01:58,020 any initial value of position velocity. Let's say, one particle class mechanics. 25 00:01:58,020 --> 00:02:01,772 A particle can have any position or velocity at an initial time. 26 00:02:01,772 --> 00:02:07,936 And so the interaction with its environment can also change the position 27 00:02:07,936 --> 00:02:14,517 and velocity in some random way. It's very difficult to see how you could 28 00:02:14,517 --> 00:02:18,450 have permanent structures. There are a lot of very interesting ideas 29 00:02:18,450 --> 00:02:23,376 that were put forward in those days. Here's, here's one that appeals to me the 30 00:02:23,376 --> 00:02:29,920 idea that atoms could be vortex rings in some kind of pervasive fluid. 31 00:02:29,920 --> 00:02:35,007 This was worked on quite a bit by Lord Kelvin William Thompson. 32 00:02:35,007 --> 00:02:39,831 And the, the rationale for that was that it was if Helmholtz had found that there 33 00:02:39,831 --> 00:02:44,186 was a conservation law in effect for vortices, and that you could get actually 34 00:02:44,186 --> 00:02:50,880 fairly complex structures that would last for long periods of time. 35 00:02:50,880 --> 00:02:54,900 So maybe an atom was some sort of vortex structure in a fluid. 36 00:02:54,900 --> 00:02:59,636 In fact the, the appearance of vortex rings in atomic gases has returned to be 37 00:02:59,636 --> 00:03:04,520 of interest in recent years due to the the development of the Bose-Einstein 38 00:03:04,520 --> 00:03:11,420 condensations which which are subject of a later talk. 39 00:03:11,420 --> 00:03:16,316 This is a theoretical paper from last year that shows the the generation of 40 00:03:16,316 --> 00:03:20,780 vortex knots in in a Bose-Einstein condensate and shows how they, they 41 00:03:20,780 --> 00:03:27,410 preserve their form to some extent as time evolves. 42 00:03:27,410 --> 00:03:31,826 We won't discuss that in detail now, but it turns out that this is not, this is 43 00:03:31,826 --> 00:03:38,460 not the solution to the problem of atomic structure for quite a fundamental reason. 44 00:03:41,090 --> 00:03:45,238 So, the problem with continuum models of the structure of atoms, the structure of 45 00:03:45,238 --> 00:03:51,532 the, the, the spectra that we're seeing. Is that the the electron and the protons 46 00:03:51,532 --> 00:03:55,237 were discovered and they seem to behave very much like particles that behave, 47 00:03:55,237 --> 00:04:01,192 behave as classical mechanics suggests. They're subject to electrical magnetic 48 00:04:01,192 --> 00:04:04,665 forces. Now, a good understanding of the size of 49 00:04:04,665 --> 00:04:10,340 atom was obtained in the 19th century. And it's pretty much consistent with what 50 00:04:10,340 --> 00:04:12,900 we know now. This was, this was developed on the 51 00:04:12,900 --> 00:04:15,920 understanding of the kinetic theory of gases. 52 00:04:15,920 --> 00:04:21,398 Even experiments like a lack of spreading of, of a drop of oil on a big pond allows 53 00:04:21,398 --> 00:04:27,562 you to estimate the upper bound or atomic sizes. 54 00:04:27,562 --> 00:04:33,455 And the experiment report, analysis of an experiment reported in 1911 by Rutherford 55 00:04:33,455 --> 00:04:40,600 was quite shocking at the time. Rutherford showed that almost all of the 56 00:04:40,600 --> 00:04:44,800 mass in an, in an atom was concentrated in a very small part of its volume, 57 00:04:44,800 --> 00:04:51,050 almost, almost in a point compared to the size of the atom itself. 58 00:04:52,190 --> 00:04:56,740 And the, the only possible conclusion was that atoms are mostly empty space with 59 00:04:56,740 --> 00:05:00,770 electrons orbiting the nucleus under the influence of long range Coulomb 60 00:05:00,770 --> 00:05:05,483 interactions. So indeed, that's not such a strange 61 00:05:05,483 --> 00:05:10,340 system compared to others we now of. For example, if you think the solar 62 00:05:10,340 --> 00:05:14,695 system as just containing the sun and the earth, the sun contains most of the mass 63 00:05:14,695 --> 00:05:18,660 and the orbit the earth is revolving around the sun in a regular way in a 64 00:05:18,660 --> 00:05:24,517 regular period. And and, and most, most of the space in 65 00:05:24,517 --> 00:05:29,428 between them is empty. However, it's rather difficult from the 66 00:05:29,428 --> 00:05:34,051 standpoint of classical mechanics to reconcile that, that idea with the atoms 67 00:05:34,051 --> 00:05:38,272 that are sort of thought to be like little, little balls that are, you put 68 00:05:38,272 --> 00:05:45,370 together in solids and close packing format, or something like that. 69 00:05:45,370 --> 00:05:50,206 So we're going to start now to see, since, since it appears to be the case 70 00:05:50,206 --> 00:05:55,588 that the, the Coulomb interaction between particles is the dominating feature of 71 00:05:55,588 --> 00:06:00,814 atomic structure. We're going to, we're going to see what 72 00:06:00,814 --> 00:06:04,486 classical mechanics actually says about that in detail because lessons learned 73 00:06:04,486 --> 00:06:07,810 there are applicable to the quantum system. 74 00:06:07,810 --> 00:06:11,482 So in particular, we're going to look at hydrogen-like systems, by which I mean to 75 00:06:11,482 --> 00:06:14,900 say, a system of one electron and one nucleus. 76 00:06:14,900 --> 00:06:24,570 Now, the nucleus could be hydrogen helium plus. 77 00:06:24,570 --> 00:06:32,260 So, hydrogen has a has a nuclear charge of one unit of electronic charge. 78 00:06:32,260 --> 00:06:36,816 Helium, the neutral helium atom has two units and nor, normally has two electrons 79 00:06:36,816 --> 00:06:40,370 about it. So, this would be the spectrum of the 80 00:06:40,370 --> 00:06:44,000 structure of helium plus, the first ion of helium which is something that is 81 00:06:44,000 --> 00:06:49,590 present in the sun, very important object in, in astrophysics. 82 00:06:49,590 --> 00:06:54,146 And in fact there's been quite a number of studies of these one electron ions, 83 00:06:54,146 --> 00:06:59,300 all the way up to Uranium 91 plus. that sounds like a lot. 84 00:06:59,300 --> 00:07:03,070 But that, that, in fact, has been observed experimentally in at Livermore 85 00:07:03,070 --> 00:07:07,695 Laboratory. Another system that has been studied 86 00:07:07,695 --> 00:07:12,810 extensively in recent years that's, that's of this type is Positronium. 87 00:07:12,810 --> 00:07:15,960 That's the bound state of the positron, the antielectron, and the ordinary 88 00:07:15,960 --> 00:07:22,002 electron. that's a group at the University of 89 00:07:22,002 --> 00:07:26,190 California at Riverside as one of the world leaders in that area. 90 00:07:26,190 --> 00:07:34,582 And even more remarkable to me the system called antiprotonic helium. 91 00:07:34,582 --> 00:07:39,826 This is a system where you have a, an antiproton, which is is like the proton, 92 00:07:39,826 --> 00:07:46,290 but with a negative charge bound to the nucleus of the helium atom. 93 00:07:46,290 --> 00:07:50,637 This has been studied in some detail at the CERN Laboratory near Geneva, 94 00:07:50,637 --> 00:07:55,024 Switzerland. Now, just to remind you about the, I was 95 00:07:55,024 --> 00:07:59,700 talking about all those elements. Here's a periodic table. 96 00:07:59,700 --> 00:08:04,663 You can get a free high resolution copy of this at this address. 97 00:08:04,663 --> 00:08:09,902 So, here's hydrogen in the upper corner. Here would be helium with, with one 98 00:08:09,902 --> 00:08:15,650 electron, lithium two plus, going all the way down, where is it, to uranium. 99 00:08:15,650 --> 00:08:18,290 So many, many, I don't, I can't name them all. 100 00:08:18,290 --> 00:08:23,603 But many of the the atoms that you see here have been stripped down to have all 101 00:08:23,603 --> 00:08:32,057 their electrons stripped off but one. the reason for doing that for the case of 102 00:08:32,057 --> 00:08:36,656 uranium, for example, is to understand the effects of relativity in atomic 103 00:08:36,656 --> 00:08:42,060 structure, relativity in quantum electrodynamics. 104 00:08:42,060 --> 00:08:45,354 Okay, so let's see what classical mechanics tells us about these 105 00:08:45,354 --> 00:08:51,328 hydrogen-like systems. So, we have, a single nucleus with a 106 00:08:51,328 --> 00:08:59,850 position denoted by r sub n the mass, capital M, a charge of plus Ze. 107 00:08:59,850 --> 00:09:03,405 Z is the atomic number. So, it would be one for hydrogen two for 108 00:09:03,405 --> 00:09:08,660 helium, three for lithium and so on. And then e is what we call the specific 109 00:09:08,660 --> 00:09:12,019 charge. That's the the absolute value of the 110 00:09:12,019 --> 00:09:16,542 charge of the electron. Then there's the electron with a position 111 00:09:16,542 --> 00:09:20,142 vector, described by position vector r sub e little mass, little m, and a 112 00:09:20,142 --> 00:09:24,998 negative charge of minus e. We're going to keep, we're going to keep 113 00:09:24,998 --> 00:09:29,258 the masses in play represented explicitly in this treatment because there's not 114 00:09:29,258 --> 00:09:34,220 such a big simplification from changing it to something else. 115 00:09:34,220 --> 00:09:38,189 And one of the exercises that we want to do is to go through the process that led 116 00:09:38,189 --> 00:09:44,250 to the discovery of deuterium. It's a very important event both from the 117 00:09:44,250 --> 00:09:50,550 standpoint of scientific development of nuclear physics and also the development 118 00:09:50,550 --> 00:09:58,697 of nuclear energy and nuclear weapons. And it was a discovery that was made by 119 00:09:58,697 --> 00:10:07,444 looking at the Balmer series of hydrogen. Okay, now here are the Newton's equation. 120 00:10:07,444 --> 00:10:12,754 It's Ma, mass times the second derivative, the acceleration of the 121 00:10:12,754 --> 00:10:20,082 nuclear coordinate. And then, these are the, this is 122 00:10:20,082 --> 00:10:25,980 Coulomb's law. So, there's two class of the mechanical 123 00:10:25,980 --> 00:10:30,530 class of, there is a equation of motion for the nuclear and coordinate which is 124 00:10:30,530 --> 00:10:35,430 just inversely propose, it's, the force on it is in the direction of the electron 125 00:10:35,430 --> 00:10:41,828 nuclear distance. And it's inversely proportional to the 126 00:10:41,828 --> 00:10:46,095 square of that distance. You see, there is a factor of r from the 127 00:10:46,095 --> 00:10:52,210 vector and a factor of of r cubed below. So, that's 1 over r squared Force Law. 128 00:10:52,210 --> 00:10:56,890 And then, according to Newton's what is it, third law, there's an equal but 129 00:10:56,890 --> 00:11:01,806 opposite force on the electron. So, you have the same thing there. 130 00:11:01,806 --> 00:11:07,694 Now, I, the I'm using this, this coordinate r here which is the separation 131 00:11:07,694 --> 00:11:13,582 between the nucleus and the electron coordinates because that is in fact the 132 00:11:13,582 --> 00:11:22,512 relevant length for, that describes the electrical Force Law. 133 00:11:22,512 --> 00:11:28,016 What you can do now is you can get, you divide this, you divide, okay, take this 134 00:11:28,016 --> 00:11:33,004 equation and divide it by 1 over M, multiply it by 1 over M, you multiply 135 00:11:33,004 --> 00:11:40,076 this equation by 1 over little, 1 over little m. 136 00:11:40,076 --> 00:11:46,776 And you subtract them and so you get an equation of motion for the the electron 137 00:11:46,776 --> 00:11:52,850 nucleus separation alone. So, you see on this side, you have the 138 00:11:52,850 --> 00:11:57,220 second derivative of that is just a function of itself. 139 00:11:57,220 --> 00:12:03,000 So, we've now managed to get an equation that can be solved directly for the the 140 00:12:03,000 --> 00:12:10,395 separation between the two particles. So, from, from this equation, we go 141 00:12:10,395 --> 00:12:15,825 directly here. You see, we're just taking the mass, this 142 00:12:15,825 --> 00:12:20,640 is an, in, this, this term here has, is the dimension. 143 00:12:20,640 --> 00:12:26,240 This term here has the dimensions of 1 over mass. 144 00:12:26,240 --> 00:12:29,130 So, we just invert it to get what's called the reduced mass. 145 00:12:29,130 --> 00:12:34,028 And now, we have a simple Newtonian equation for the motion of the of the 146 00:12:34,028 --> 00:12:40,050 relative distance between the, the electron and the nucleus. 147 00:12:41,850 --> 00:12:47,510 Now, we define the momentum, p, associated with this coordinate. 148 00:12:47,510 --> 00:12:51,350 It's just the mass times the velocity. Usual, the usual definition. 149 00:12:53,030 --> 00:12:58,458 And this then allows us to rewrite the equations of motion as a, a pair of 150 00:12:58,458 --> 00:13:06,755 couple first order equations in time. So, here, here's, here's that equation we 151 00:13:06,755 --> 00:13:13,340 just made up. That's that r dot is 1 over mu times p. 152 00:13:13,340 --> 00:13:22,430 And then p dot is just taken from, from this equation. 153 00:13:22,430 --> 00:13:28,766 because it, P dot is our double dot and so it's just minus Ze squared r over r 154 00:13:28,766 --> 00:13:33,902 cubed. Now, why do we reduce this to first-order 155 00:13:33,902 --> 00:13:39,107 equations, you might ask. This is actually something that's come 156 00:13:39,107 --> 00:13:43,305 up, in the discussion forums. And so, I want to make a statement here 157 00:13:43,305 --> 00:13:46,095 about why it's essential to have equations of motion that are first-order 158 00:13:46,095 --> 00:13:51,358 in time. So, when we say, as physicists, that we 159 00:13:51,358 --> 00:13:55,798 understand a system, that we can determine it's future from what we know 160 00:13:55,798 --> 00:14:00,898 at the present. That means, if we know the state of a 161 00:14:00,898 --> 00:14:06,106 system arbitrary system of s at some given time t, where we want a framework 162 00:14:06,106 --> 00:14:11,734 that gives us the ability to calculate or predict what the state would be like at a 163 00:14:11,734 --> 00:14:20,598 short time later, t plus delta t. So, that seems to me to be a very basic 164 00:14:20,598 --> 00:14:26,575 idea of a dynamical theory. Now well, if you have a, you have, you 165 00:14:26,575 --> 00:14:31,237 have a system at state t its state at a sub, a slightly later time, delta t, 166 00:14:31,237 --> 00:14:36,269 delta t being very small, is going to be approximately given by the original state 167 00:14:36,269 --> 00:14:40,709 here plus the time interval times the derivative of the system, the time 168 00:14:40,709 --> 00:14:49,360 derivative of the system. Now, this is a loose way of speaking. 169 00:14:49,360 --> 00:14:54,509 Let's, let's show explicitly how that's represented in quantum mechanics. 170 00:14:54,509 --> 00:14:58,326 And this is what you saw in Victor's lectures. 171 00:14:58,326 --> 00:15:03,760 Schrodinger's equation must be first-order in time. 172 00:15:03,760 --> 00:15:09,246 Why? Because quantum mechanics tells us that 173 00:15:09,246 --> 00:15:18,500 all the information about the system that we can have is containance wavefunction. 174 00:15:18,500 --> 00:15:23,530 So, in other, the, the wave function fully describes a system in a given time. 175 00:15:23,530 --> 00:15:28,414 Therefore, if we want to find the, the wavefunction at a slightly greater time, 176 00:15:28,414 --> 00:15:35,287 we have to be, you know, we have to be able to take its, its time derivative. 177 00:15:35,287 --> 00:15:39,875 We have to have some operator that advances it in time and experiment tells 178 00:15:39,875 --> 00:15:45,550 us that this is the operator. It's one minus i times delta t times the 179 00:15:45,550 --> 00:15:51,244 Hamiltonian operator divided by h bar. So, if you, if we apply this, and delta t 180 00:15:51,244 --> 00:15:55,800 is, is, is sufficiently small, then we'll get the state would, to apply that to 181 00:15:55,800 --> 00:16:03,200 science, state of time t will get it to get it at a, at a time t plus delta t. 182 00:16:03,200 --> 00:16:10,330 In fact, this is an approximation of e to minus i delta t H over h bar to the 183 00:16:10,330 --> 00:16:16,825 first-order. So that, this would be the, this would be 184 00:16:16,825 --> 00:16:21,196 the full propagator. But that's, that's the lowest order 185 00:16:21,196 --> 00:16:24,930 approximation. Now, to make sure that you grasp that 186 00:16:24,930 --> 00:16:31,670 idea here's a question for you to answer. Now, I hope some of you are able to 187 00:16:31,670 --> 00:16:38,284 answer the previous question correctly. And this is a, is a subject that came up 188 00:16:38,284 --> 00:16:42,035 in the forum on several occasions. And it is a very good question. 189 00:16:42,035 --> 00:16:46,707 Why isn't Schrodinger's equation second-order in time, like the other wave 190 00:16:46,707 --> 00:16:50,412 equations? So, the traditional wave equations that 191 00:16:50,412 --> 00:16:53,625 you, you're taught in school for a vibrating string or for the 192 00:16:53,625 --> 00:16:57,909 electromagnetic field, if you, if you take college level physics, are generally 193 00:16:57,909 --> 00:17:01,815 packed for your convenience as second-order equations, but began life as 194 00:17:01,815 --> 00:17:09,873 first-order equations. So, you know, the the old gag Abbot asked 195 00:17:09,873 --> 00:17:15,763 Costello, where does milk come from? Costello says, oh, it comes from a 196 00:17:15,763 --> 00:17:19,298 bottle. Which Abbott says, oh, you're silly, the 197 00:17:19,298 --> 00:17:22,901 bottle comes from a shop. When you consider, consider first 198 00:17:22,901 --> 00:17:25,630 principles, milk actually comes from a shop. 199 00:17:25,630 --> 00:17:30,153 So the real underlying equa, equations that we deal with. 200 00:17:30,153 --> 00:17:35,185 Sometimes it's most convenient to think of them in terms of a second-order 201 00:17:35,185 --> 00:17:40,291 equations, there, they can, they can be decoupled and solved more conveniently in 202 00:17:40,291 --> 00:17:48,120 second order form, but they're always derived from first-order equations. 203 00:17:48,120 --> 00:17:52,856 And that is the, that is the form that we're going to use enough to treat the 204 00:17:52,856 --> 00:17:57,290 hydrogen atom in classical mechanics now.