1 00:00:00,540 --> 00:00:02,810 Hello, everyone. Welcome back. 2 00:00:02,810 --> 00:00:06,066 I'm Charles Clark. And in this lecture, we're gong to look 3 00:00:06,066 --> 00:00:10,148 at quantum theory old and new. we'll conclude our discussion of the Bohr 4 00:00:10,148 --> 00:00:14,693 model, show what it predicted. give a demonstration of its mnemonic 5 00:00:14,693 --> 00:00:20,021 value for when you have to infer things about atomic spectra or quantum mechanics 6 00:00:20,021 --> 00:00:27,115 from the imperfect knowledge. And then deal with actual solution of the 7 00:00:27,115 --> 00:00:31,145 Schrodinger equation for atomic molecular systems and look at some of the 8 00:00:31,145 --> 00:00:35,760 approximation methods, and the analytic results that are needed to give a modern 9 00:00:35,760 --> 00:00:40,600 account of atomic structure. So, let's proceed. 10 00:00:43,300 --> 00:00:47,460 Now as before, we have some original scientific literature available with a 11 00:00:47,460 --> 00:00:52,205 commentary that you might look at the one thing that will be quite helpful I think 12 00:00:52,205 --> 00:01:00,000 is, of course, original paper here. then another topic that we'll treat is 13 00:01:00,000 --> 00:01:06,205 the application of the Bohr theory, early important application of the Bohr theory 14 00:01:06,205 --> 00:01:12,485 to the discovery of deuterium. There's actually two things two items 15 00:01:12,485 --> 00:01:15,962 there. The second one is written for popular 16 00:01:15,962 --> 00:01:20,446 audience. And so it's either more accessible of the 17 00:01:20,446 --> 00:01:23,593 two. The first one contains some of the 18 00:01:23,593 --> 00:01:29,433 original literatures original papers from the physical review and then a memoir 19 00:01:29,433 --> 00:01:37,988 written by one of the co-discoverers. So, do have a look at that if you're 20 00:01:37,988 --> 00:01:43,688 interested. So, returning to where we left off at the 21 00:01:43,688 --> 00:01:50,310 end of the previous lecture we had solved by rather elementary techniques for the 22 00:01:50,310 --> 00:01:55,814 really complete description of the classical mechanics of an electron and a 23 00:01:55,814 --> 00:02:04,283 nucleus interacting through the Coulomb interaction. 24 00:02:04,283 --> 00:02:09,833 And then we found these constant motion angle atom which is a characteristic of 25 00:02:09,833 --> 00:02:15,235 any central, any force that depends any potential, any potential interaction that 26 00:02:15,235 --> 00:02:21,484 depends only upon the distance between the particles. 27 00:02:21,484 --> 00:02:25,670 And then this an additional Runge-Lenz vector. 28 00:02:25,670 --> 00:02:32,202 And with that Runge-Lenz vector, we're able to write a very simple expression 29 00:02:32,202 --> 00:02:37,998 for the, the orbit of the of the, the orbit executed by the interparticle 30 00:02:37,998 --> 00:02:43,160 separations. So, it's an, an orbit of the distance 31 00:02:43,160 --> 00:02:52,335 between the electron and the nucleus. And these are for, for energies less than 32 00:02:52,335 --> 00:03:03,796 0, which means the the, the system never escapes to infinity. 33 00:03:03,796 --> 00:03:12,421 the the orbit is a, is a generalized ellipse, including the special case of a 34 00:03:12,421 --> 00:03:19,830 circle, which occurs when A is equal to 0. 35 00:03:19,830 --> 00:03:26,652 then for A greater than 0 the the orbit clearly escapes to infinity. 36 00:03:26,652 --> 00:03:33,113 and that's that can be seen well that, that, that can be shown by look, using 37 00:03:33,113 --> 00:03:41,037 this identify and, and replacing replacing a denominator there. 38 00:03:41,037 --> 00:03:46,143 Right, but there are no restrictions whatever imposed by classical mechanics 39 00:03:46,143 --> 00:03:52,125 on the possible values of the energy. It can be any number, any, any real 40 00:03:52,125 --> 00:03:56,787 number of, you know, a large, a negative number very large magnitude, which 41 00:03:56,787 --> 00:04:03,840 corresponds to the electron orbiting very closely to the nucleus. 42 00:04:03,840 --> 00:04:08,354 Or a very large positive number, in which case, the electron, the nucleus separate 43 00:04:08,354 --> 00:04:14,570 to infinite seperationals or they go to very large separation of the large times. 44 00:04:14,570 --> 00:04:20,180 It was about a hundred years ago to the day at the time I'm recording this that 45 00:04:20,180 --> 00:04:25,535 Neils Bohr conceived of a solution to this problem on application of the 46 00:04:25,535 --> 00:04:31,145 quantization idea to the motion of a particle in orbit about you know, in 47 00:04:31,145 --> 00:04:40,090 particular, the motion of the electron and hydrogen atom. 48 00:04:41,650 --> 00:04:47,120 So in, in this case he states it in terms of the motion of the electron. 49 00:04:47,120 --> 00:04:51,193 And it really has to do with the, the motion of the center of mass. 50 00:04:51,193 --> 00:04:55,323 But since the nucleus of the the hydrogen atom is nearly 2,000 times as massive as 51 00:04:55,323 --> 00:04:58,981 the electron it, it makes sense to just think of , of it of be, as being 52 00:04:58,981 --> 00:05:04,286 infinitely massive. And all the kinetic energy and angular 53 00:05:04,286 --> 00:05:07,974 momentum is associated with the motion of an electron. 54 00:05:07,974 --> 00:05:16,494 And so, Bohr proposed that the electrons in the hydrogen atoms or the the, the, 55 00:05:16,494 --> 00:05:24,534 the stationary state of a hydrogen atom that were inferred from atomic spectra, 56 00:05:24,534 --> 00:05:35,800 corresponded to circular orbits, Runge-Lenz vector equals zero. 57 00:05:37,100 --> 00:05:42,476 And that in, in such an orbit, the angular momentum, L, would be equal to an 58 00:05:42,476 --> 00:05:47,670 integer times the reduced Planck's constant. 59 00:05:47,670 --> 00:05:53,600 And he, he expresses that condition here. You can read it on page 15 of his paper. 60 00:05:53,600 --> 00:05:58,730 Now as we'll see, this had an enormous effect. 61 00:05:58,730 --> 00:06:06,250 it was accepted very quickly. it led to Bohr's getting the Nobel Prize 62 00:06:06,250 --> 00:06:13,150 in Physics in 1922 explicitly for his understanding the structure of atoms and 63 00:06:13,150 --> 00:06:21,202 their radiative properties. And I have to say before going much 64 00:06:21,202 --> 00:06:27,180 further, that Bohr's idea is not, no longer accepted now. 65 00:06:27,180 --> 00:06:32,052 It's not consistent with the Schrodinger equation as we know it. 66 00:06:32,052 --> 00:06:38,322 But it, it contains the germ of a, of an important idea, which is the the use of 67 00:06:38,322 --> 00:06:45,922 the Planck constant which, was originally devised to explain the thermodynamics of 68 00:06:45,922 --> 00:06:54,178 the electromagnetic field. The, this constant which had the units of 69 00:06:54,178 --> 00:06:58,930 angular momentum, should actually be associated with quantization of the 70 00:06:58,930 --> 00:07:04,770 motion of of elementary systems. And that idea still is correct. 71 00:07:04,770 --> 00:07:10,510 So this, this relationship appropriately understood, is still valid. 72 00:07:10,510 --> 00:07:15,487 And furthermore, the, the Bohr model in its for, own form gave an incredibly 73 00:07:15,487 --> 00:07:20,430 precise explanation of many different facts. 74 00:07:26,440 --> 00:07:30,565 Now, the value of the Bohr model, at least to me, is that it's easy to 75 00:07:30,565 --> 00:07:36,198 understand and to reproduce. All you have to remember is the orbit's 76 00:07:36,198 --> 00:07:40,510 are circular and they're quantized in units of Planck's constant. 77 00:07:40,510 --> 00:07:45,988 And then, this, this gives an amazing number of useful results, which actually 78 00:07:45,988 --> 00:07:53,690 scale in the same way as the, as the results of modern quantum theory. 79 00:07:53,690 --> 00:07:58,230 So, for example here's, here's the here's a restatement. 80 00:07:58,230 --> 00:08:01,480 We have circular orbits with quantized angular momentum, that means that A is 81 00:08:01,480 --> 00:08:06,088 equal to 0. So instantly, from this equation here 82 00:08:06,088 --> 00:08:13,543 that relates the, the magnitudes of the Runge-Lenz vector, the angular momentum 83 00:08:13,543 --> 00:08:20,790 and the energy. we can see that E is negative and it just 84 00:08:20,790 --> 00:08:27,615 by sub, substitution here by substituting, L is nh bar, or L squared 85 00:08:27,615 --> 00:08:36,530 is nh bar squared. we get, we get this simple form which in 86 00:08:36,530 --> 00:08:43,733 modern terms it's convenient to write in, in this way. 87 00:08:43,733 --> 00:08:50,433 So Z is the, the charge, Alpha is the so-called fine structure constant given 88 00:08:50,433 --> 00:08:58,112 by E squared over h bar Z. And this is, this just, and the mnemonic 89 00:08:58,112 --> 00:09:05,456 for this is E is equal to mc squared, this is the Einsteinian expression for 90 00:09:05,456 --> 00:09:13,300 rest mass energy of a particle reduced mass mu. 91 00:09:13,300 --> 00:09:17,992 And, in fact, this very type of the way that I've written this here, is something 92 00:09:17,992 --> 00:09:22,548 that falls out of a Dirac equation from a hydrogen atom, which we may discuss very 93 00:09:22,548 --> 00:09:29,032 briefly later. Okay, now just to get your own sense of 94 00:09:29,032 --> 00:09:36,880 familiarity with this, this equation, I'm going to pose a brief question to you. 95 00:09:40,570 --> 00:09:44,730 Another use of the Bohr model is to help in clarifying the definition of some of 96 00:09:44,730 --> 00:09:50,523 the fundamental constants. so, in particular, what, we, we replaced 97 00:09:50,523 --> 00:09:56,930 the, the, you know, arbitrary reduced mass by the mass of the electron. 98 00:09:56,930 --> 00:10:00,258 So, in other words, we're, we're now using the Bohr model to describe a system 99 00:10:00,258 --> 00:10:04,890 where the, the nucleus has infinite mass, and only the electron is moving. 100 00:10:06,600 --> 00:10:13,490 So, with that choice the energy of the nth Bohr orbit, which is given by this 101 00:10:13,490 --> 00:10:21,016 expression we've seen previously takes the form of minus 1 over n squared R 102 00:10:21,016 --> 00:10:30,103 infinity hc. Where R infinity is the Rydberg constant, 103 00:10:30,103 --> 00:10:36,834 which is about 10 million inverse meters. This is one of the most precisely 104 00:10:36,834 --> 00:10:41,360 measured of the fundamental constants and subject to very active interest today. 105 00:10:41,360 --> 00:10:49,796 the quantity R infinity h bar c has the dimensions of energy and so the this says 106 00:10:49,796 --> 00:10:59,060 E, En is minus 13.606 approximately, eV divided by n squared. 107 00:10:59,060 --> 00:11:05,160 So again, with the with the choices Z equal 1 and mu equal the mass of the 108 00:11:05,160 --> 00:11:14,550 electron, we have an expression for the radius f the nth Bohr orbit. 109 00:11:14,550 --> 00:11:16,360 So, this is something that works for you generally. 110 00:11:16,360 --> 00:11:20,310 The radius goes as the square of the principal quantum number. 111 00:11:20,310 --> 00:11:24,864 This is a relationship that enables you to make pretty good estimates of the 112 00:11:24,864 --> 00:11:33,294 actual size of excited states of atoms. And the constant of proportionality is 113 00:11:33,294 --> 00:11:39,960 the Bohr radius a0 which is 53 picometers approximately. 114 00:11:39,960 --> 00:11:46,890 this is the or in older terminology, we would say is about half an Angstrom. 115 00:11:46,890 --> 00:11:53,590 and it's a good measure of the general size of atoms in their ground states.