1 00:00:01,390 --> 00:00:03,400 Welcome back everyone. I'm Charles Clark. 2 00:00:03,400 --> 00:00:08,182 This is exploring quantum physics. And we're going to, finish up for this 3 00:00:08,182 --> 00:00:12,010 week with a discussion of angular momentum. 4 00:00:12,010 --> 00:00:17,990 Next week, you'll have a set of lectures by Ian Applebaum that introduce spin. 5 00:00:17,990 --> 00:00:24,460 And then we'll return to bring those parts together in the following week. 6 00:00:24,460 --> 00:00:27,971 So let's begin. We're going to say hello again to an old 7 00:00:27,971 --> 00:00:34,165 friend the hydrogen atom. where as we saw in the classical 8 00:00:34,165 --> 00:00:42,020 treatment, there was this remarkable unusual conserved quantity. 9 00:00:42,020 --> 00:00:44,213 The run that described in terms of the Runge-Lenz vector. 10 00:00:45,340 --> 00:00:49,308 and that as you will recall helps us completely define the orbit of the 11 00:00:49,308 --> 00:00:53,785 classical particle. It makes possible a classification of all 12 00:00:53,785 --> 00:00:58,140 possible, orbits of the electron in the hydrogen atom, in, in terms of the 13 00:00:58,140 --> 00:01:04,374 single, in terms of the angular momentum and the Runge-Lenz vector. 14 00:01:04,374 --> 00:01:13,010 Now as defined in classical mechanics this has some issues because, in 15 00:01:13,010 --> 00:01:22,154 classical mechanics P cross L is equal to L cross P and as we saw the, sorry, minus 16 00:01:22,154 --> 00:01:30,981 L cross B. There's no, there's no issue about 17 00:01:30,981 --> 00:01:37,033 commutation so in quantum mechanics we have to, we have to use a form that is 18 00:01:37,033 --> 00:01:44,393 symmetric, it makes sense. And so Wolfgang Pauli derived, this 19 00:01:44,393 --> 00:01:50,069 formula for the quantum mechanical counterpart of the Runge-Lenz vector, 20 00:01:50,069 --> 00:01:57,901 it's as you can see, it's symmetric. So P cross L minus L cross P, that's 21 00:01:57,901 --> 00:02:05,380 something which, it looks like P cross L in classical mechanics. 22 00:02:05,380 --> 00:02:09,868 So actually, why don't you just take a quick look at a comparison, and give me 23 00:02:09,868 --> 00:02:14,940 your opinion about the result in this in the video quiz. 24 00:02:17,300 --> 00:02:23,670 Well I hope it was clear to you that the, the form that Pauli derived is the same 25 00:02:23,670 --> 00:02:29,858 the same as this, this classical conventional classical form in the limit 26 00:02:29,858 --> 00:02:39,450 of Planck's constant going to zero, except that it involves a division. 27 00:02:39,450 --> 00:02:52,654 So basically M, M goes to A divide by Mu. So it's, it's just multiplied by one over 28 00:02:52,654 --> 00:02:56,380 mass, so it's changed only by a constance, that means it's still a 29 00:02:56,380 --> 00:03:06,469 constant emotion. Now we can actually find out a lot by 30 00:03:06,469 --> 00:03:11,815 working out the commutators of L and M, where M is this this operator this 31 00:03:11,815 --> 00:03:16,837 produced by Pauli that is the quantum mechanical analog of the run and lens 32 00:03:16,837 --> 00:03:23,548 vector. So, some of these you've already done, so 33 00:03:23,548 --> 00:03:27,708 you have already proved, I think, if you followed my advice this important 34 00:03:27,708 --> 00:03:32,670 identity for the angular momentum operator. 35 00:03:32,670 --> 00:03:40,090 Now, it doesn't take much of a stretch to see that this commutator must be the same 36 00:03:40,090 --> 00:03:46,556 as the commutator of any other truly vector quantity in the hydrogen atom 37 00:03:46,556 --> 00:03:53,000 problem. In other words, the commutator of L with 38 00:03:53,000 --> 00:03:58,738 and vector, gives, gives the relationship, that's described here. 39 00:03:58,738 --> 00:04:06,534 What is novel however, is the commutator of the Runge-Lenz vector itself. 40 00:04:06,534 --> 00:04:14,164 And the interesting thing about this is that it comes back, Mu, Mv is, Ih bar, a 41 00:04:14,164 --> 00:04:23,920 constant, which is minus twice the energy divided by the reduced mass. 42 00:04:23,920 --> 00:04:29,125 Epsilon U nu, mu V, UVWLW. So, now in some sense you see that the L 43 00:04:29,125 --> 00:04:37,627 and the M together are a closed set, with respect to the commutation relations. 44 00:04:37,627 --> 00:04:42,919 So this means they actually define [UNKNOWN] algebra which is, turns out to 45 00:04:42,919 --> 00:04:48,164 be a rotation group in a four dimensional space. 46 00:04:48,164 --> 00:04:53,519 And we're not going to go into detail on this, but just to say that by an 47 00:04:53,519 --> 00:05:00,149 appropriate choice of of scaling factors, basically compressing, compressing this 48 00:05:00,149 --> 00:05:08,378 factor into the definition of M. We can find, we can find all the bound 49 00:05:08,378 --> 00:05:14,318 state energies of the hydrogen atom without solving the Schrodinger equation 50 00:05:14,318 --> 00:05:20,715 via, completely algebraic approach. So were first going to look at the 51 00:05:20,715 --> 00:05:23,770 spectrum of the angular minimum operator itself. 52 00:05:23,770 --> 00:05:28,054 That is something that's of generic importance [INAUDIBLE] describes the 53 00:05:28,054 --> 00:05:32,474 motion of the particle of any central potential, not just that of the hydrogen 54 00:05:32,474 --> 00:05:36,340 atom. And it's widely used in scattering 55 00:05:36,340 --> 00:05:39,220 theory, and the calculation of bound states. 56 00:05:39,220 --> 00:05:42,972 And the expansion of wave functions and eigenfunctions of the angular momentum 57 00:05:42,972 --> 00:05:47,830 operator for a sort of general solution of vanisotropic problems as well. 58 00:05:49,170 --> 00:05:55,126 So, let's start. We're going to choose eigenfunctions of, 59 00:05:55,126 --> 00:06:00,433 both the L3, the projections of the anglar minimum onto the axis E3, or the 60 00:06:00,433 --> 00:06:05,392 Lz as it's often called, and of L squared, the total, the square of the 61 00:06:05,392 --> 00:06:13,858 angular minimum operator. So, we're just going to designate them 62 00:06:13,858 --> 00:06:23,440 for the moment in this form. Lm where L3 operating on Lm gives Mh bar 63 00:06:23,440 --> 00:06:28,798 times Lm. Now this, we haven't yet said anything 64 00:06:28,798 --> 00:06:32,092 about what the value of M is, except it's a pure number, and this comes from 65 00:06:32,092 --> 00:06:38,160 dimensional analysis. So Recall that L is equal to H bar over 66 00:06:38,160 --> 00:06:45,920 I, by definition H bar over I. R cross gram, this part, the operational 67 00:06:45,920 --> 00:06:51,200 part of the Hamiltonian contributes nothing in terms of its dimensionality so 68 00:06:51,200 --> 00:06:58,140 all the, all the units in the system come from Planck's constant. 69 00:06:58,140 --> 00:06:59,526 This is basically Bohr's idea, that the angular momentum is, is valued in units 70 00:06:59,526 --> 00:07:01,038 of Planck's constant, rather than, than that is, that's a fundamental property of 71 00:07:01,038 --> 00:07:07,526 quantum physics. So this just means that at the moment M 72 00:07:07,526 --> 00:07:18,580 is a value, a pure number that's going to be determined. 73 00:07:18,580 --> 00:07:20,660 Now how does that, how is that determination made? 74 00:07:20,660 --> 00:07:26,780 Well, it's handy to define two new operators, L plus, L1, which is L1 plus 75 00:07:26,780 --> 00:07:36,393 il two and L minus, L1 minus i L two, and it's straightforward to see that. 76 00:07:36,393 --> 00:07:43,261 When, you take the commutator of L3 with either of those, you get plus or minus H 77 00:07:43,261 --> 00:07:51,206 bar, times the, the thing that you started with. 78 00:07:51,206 --> 00:07:58,438 So, this is. You might say, this is suggestive that 79 00:07:58,438 --> 00:08:02,391 somehow, if you think about this, our previous geometric interpretation of the 80 00:08:02,391 --> 00:08:07,270 commutator with the angular momentum, it involves a rotation. 81 00:08:07,270 --> 00:08:13,501 So L plus is rotated a little bit in the positive direction about the the L, the 82 00:08:13,501 --> 00:08:18,990 L3 axis. And L minus in a negative direction. 83 00:08:20,700 --> 00:08:29,752 So that gives some sense of what we're going to find next, and that is, we're 84 00:08:29,752 --> 00:08:42,128 trying to see if, LM is an eigenvector of L3 is L plus of Lm. 85 00:08:42,128 --> 00:08:51,670 Also an eigenvector. And the answer is, it must be. 86 00:08:51,670 --> 00:08:55,895 And that we can see from the commutator because what we're going to do now is 87 00:08:55,895 --> 00:09:00,435 apply L3 to this new vector, L plus. Let's just use the plus sign here for a 88 00:09:00,435 --> 00:09:02,884 moment. L plus of LM. 89 00:09:02,884 --> 00:09:07,650 now we can do that. We can pass through the L3, if we add the 90 00:09:07,650 --> 00:09:15,826 commentator on the other side. So that's equal to L plus L3of Lm, plus H 91 00:09:15,826 --> 00:09:24,950 bar, l plus Lm, because this H bar is the commutator you see. 92 00:09:24,950 --> 00:09:30,902 So, then that means that when we apply L3 to L plus Lm, we find that we get a new 93 00:09:30,902 --> 00:09:37,134 eigenvector. With the eigenvalue increased by H bar, 94 00:09:37,134 --> 00:09:42,145 so we'd write that as H bar times M plus one. 95 00:09:42,145 --> 00:09:48,725 And we get the same the same description for the L minus, except the the new 96 00:09:48,725 --> 00:09:55,490 eigenvector has a, has an eigenvalue of M minus one. 97 00:09:55,490 --> 00:10:03,426 So in some sense, if we start with, we start with we start with M then by, say 98 00:10:03,426 --> 00:10:10,618 Mh bar as an [UNKNOWN], then by sequential application of, of L plus we 99 00:10:10,618 --> 00:10:19,982 can increase, we create a new eigenvector. 100 00:10:19,982 --> 00:10:25,370 With an eigenvalue of, H bar greater each time. 101 00:10:25,370 --> 00:10:29,658 We just get a ladder of equally spaced rungs going up, and the same for a ladder 102 00:10:29,658 --> 00:10:39,696 of equally space going, rungs going down. And you can find that with fairly simple 103 00:10:39,696 --> 00:10:49,020 arguments the the the permitted range of M depends upon the value, the eigenvalue 104 00:10:49,020 --> 00:11:02,120 of L squared, and you get, so if the Eigenvalue of L squared of this form. 105 00:11:02,120 --> 00:11:09,690 L times L plus one H bar squared. And the the allowed values of this M are 106 00:11:09,690 --> 00:11:15,184 then, so these are integers, and the allowed values of M just range from minus 107 00:11:15,184 --> 00:11:21,876 L to plus L or 2l plus one, 2l plus one possible value. 108 00:11:23,690 --> 00:11:28,216 Now I'm not going to go into that argument in detail, but you might think 109 00:11:28,216 --> 00:11:32,815 about it and this is again, this is a similar type of approach that was used in 110 00:11:32,815 --> 00:11:37,268 the, in the harmonic oscillator to generate, using these raising and 111 00:11:37,268 --> 00:11:45,320 lowering operators to generate all possible eigenfunctions. 112 00:11:45,320 --> 00:11:49,810 So the harmonic oscillator of the ark. The argument is similar here. 113 00:11:49,810 --> 00:11:55,066 But there's a termination because the let's just say, because we've seen there 114 00:11:55,066 --> 00:12:00,249 is an energy associated with the value of an angular minimum, and you can't exceed 115 00:12:00,249 --> 00:12:09,985 the, that amount of energy. without increasing the toal angular 116 00:12:09,985 --> 00:12:14,176 momentum. So I've described to you how the, the the 117 00:12:14,176 --> 00:12:17,644 spectrum is generated now what about the eigenfunctions of the angular momentum 118 00:12:17,644 --> 00:12:22,040 operator. Now for those we need to find specific 119 00:12:22,040 --> 00:12:26,663 representations and So a well documented basis, called 120 00:12:26,663 --> 00:12:31,583 spherical harmonics, of which you can find details in the mathematical 121 00:12:31,583 --> 00:12:36,238 references. And the, the major thing to know, and I 122 00:12:36,238 --> 00:12:40,999 alluded to this previously, is that these are functions of angles only in the polar 123 00:12:40,999 --> 00:12:46,794 coordinate system. So in other words, the eugenfunctions of 124 00:12:46,794 --> 00:12:52,626 the angular minimum operator do not depend upon the, the distance from the, 125 00:12:52,626 --> 00:12:59,656 the coordinate origin. They're just functions of angles and so 126 00:12:59,656 --> 00:13:04,528 again we have, we have this Lm designation and so here are portraits of 127 00:13:04,528 --> 00:13:10,240 some of the spherical harmonics in the usual coordinate system so there's one 128 00:13:10,240 --> 00:13:17,848 for L equals zero. So for L equals zero we have M equals 129 00:13:17,848 --> 00:13:24,731 zero only because we have this minus L, minus L, plus one. 130 00:13:24,731 --> 00:13:32,223 Up to L, provides the only choice of the M values. 131 00:13:32,223 --> 00:13:38,453 And now for L equal one there are three such, terms, and these can be also be 132 00:13:38,453 --> 00:13:44,940 thought of in, in alternative representation. 133 00:13:44,940 --> 00:13:50,960 Is to think of these, so we, think of these as angular momentum oriented along 134 00:13:50,960 --> 00:13:56,028 the several axis'. So we have an X, a Y, and Z axis, so 135 00:13:56,028 --> 00:14:01,254 what's actually portrayed here, the M equals zero state is, is a line that 136 00:14:01,254 --> 00:14:08,115 looks like a cosine of theta. About the zed axis, and then, what I'm 137 00:14:08,115 --> 00:14:12,717 going to describe now as an alternative representation to the, the usual 138 00:14:12,717 --> 00:14:18,458 spherical harmonics. You can also, equally well have the, the 139 00:14:18,458 --> 00:14:24,625 same distribution along the Y axis, or indeed any other axis in space. 140 00:14:24,625 --> 00:14:29,918 And then a third, a third function which provides a complete basis for L equal one 141 00:14:29,918 --> 00:14:35,137 is to have that type of function along the X axis. 142 00:14:35,137 --> 00:14:41,762 and as, as the, as the value of L gets larger. 143 00:14:41,762 --> 00:14:48,902 There's a, you know, a more, a richer set of eigenfunctions, that are given by the 144 00:14:48,902 --> 00:14:56,717 various, values of M that are possible. So the angular momentum and the 145 00:14:56,717 --> 00:15:00,438 Runge-Lenz lens vector, or the poly symmetrized version of the run a lens 146 00:15:00,438 --> 00:15:03,915 vector, can be combined to form generators of rotation group of four 147 00:15:03,915 --> 00:15:09,388 dimensions. And as it turns out the same sort of 148 00:15:09,388 --> 00:15:15,240 ladder operator approach is used to get the, define the angular momentum. 149 00:15:15,240 --> 00:15:20,153 I can vouch the angular momentum. Or is also used to find the spectrum of 150 00:15:20,153 --> 00:15:26,793 the harmonic oscillator. Gives the following spectrum for the 151 00:15:26,793 --> 00:15:34,950 bound state of the hydrogen atom. So it's we, we recover exactly the Bohr 152 00:15:34,950 --> 00:15:43,320 formula for the energy levels, but with a big difference. 153 00:15:43,320 --> 00:15:49,961 so again it's this universal constant. The [UNKNOWN] times Hc times the ratio of 154 00:15:49,961 --> 00:15:54,268 the reduced mass to the mass of the electron, divided by the square of an 155 00:15:54,268 --> 00:16:00,152 integer. But this integer has well, is usually 156 00:16:00,152 --> 00:16:05,450 thought of have, of having two separate contributions. 157 00:16:05,450 --> 00:16:08,390 one is the so called radial quantum number. 158 00:16:08,390 --> 00:16:32,057 N, and this, this describes increasing orbit size, roughly speaking. 159 00:16:32,057 --> 00:16:41,000 and then this is the angular momentum. the angular momentum, quantum number that 160 00:16:41,000 --> 00:16:49,384 we just discussed previously. And you see it enters in a very, in a 161 00:16:49,384 --> 00:16:53,150 very equivalent way to the radial quantum number. 162 00:16:53,150 --> 00:16:58,130 So in some sense, the angular on the radio motions of the electron in a 163 00:16:58,130 --> 00:17:06,100 hydrogen, in a partition of the energy in a, in more or less equivalent way. 164 00:17:07,940 --> 00:17:10,956 And a major difference so the, the spectrum is the same as predicted by 165 00:17:10,956 --> 00:17:15,872 Bohr, but a major difference is the total numbers of states that are available. 166 00:17:15,872 --> 00:17:23,080 So there are when you be, because so also you see for each one of these values of L 167 00:17:23,080 --> 00:17:31,653 we have an M equal minus L, up to L. It turns out that for given energy, we 168 00:17:31,653 --> 00:17:37,570 have N plus L plus one squared states, all at the same energy. 169 00:17:37,570 --> 00:17:44,880 It's a very high degree of degeneracy. So for, N equals zero, we have one. 170 00:17:44,880 --> 00:17:55,568 For N equals, N equals one, [INAUDIBLE] rewrite differently, N plus L equals 171 00:17:55,568 --> 00:18:02,410 zero. We have one state, which is N equals 172 00:18:02,410 --> 00:18:07,980 zero. So for N plus L equals one, we have four 173 00:18:07,980 --> 00:18:12,129 states and so on. So increases roughly is the 174 00:18:12,129 --> 00:18:18,620 square,exactly is the square. And this means that, there, there's a, 175 00:18:18,620 --> 00:18:24,124 for large values of this integer there are quite a large number of degenerate 176 00:18:24,124 --> 00:18:29,947 item states. What I'd like to remind you of this 177 00:18:29,947 --> 00:18:36,935 amazing data that we, we saw earlier in aprevious lecture. 178 00:18:36,935 --> 00:18:43,259 the radio recombination lines in molecular clouds around the supernova, 179 00:18:43,259 --> 00:18:51,562 around the casiopa a. And this showed, this integer that we 180 00:18:51,562 --> 00:18:58,090 refereed to, it showed transitions of value with that integer being up around 181 00:18:58,090 --> 00:19:06,040 1000. So 1000 squared is equal to a million. 182 00:19:08,650 --> 00:19:16,309 So that means that these, these lines that you see, in the, in the absorption 183 00:19:16,309 --> 00:19:23,857 spectrum are actually, they are combine, they're sums of transitions involving 184 00:19:23,857 --> 00:19:32,630 millions of states, all of the same energy. 185 00:19:32,630 --> 00:19:37,220 And you see them in this one sharp line. So this is a very, to me a very, poignant 186 00:19:37,220 --> 00:19:41,124 demonstration of the high degree of degeneracy of all these states of the 187 00:19:41,124 --> 00:19:47,725 hydrogen atom in quantum mechanics. since there are about a million of these 188 00:19:47,725 --> 00:19:52,900 states that are presumably formed, they're being formed by recombination of, 189 00:19:52,900 --> 00:20:00,970 a very slow, electron in the interstellar medium and an ionized carbon atom. 190 00:20:00,970 --> 00:20:05,938 It's almost certain that these states are populated basically, completely randomly 191 00:20:05,938 --> 00:20:11,610 so the probability of any state being popular is one in a million. 192 00:20:11,610 --> 00:20:15,228 So if there was a significant difference in, difference in energy between them, 193 00:20:15,228 --> 00:20:19,370 you wouldn't see a sharp line, but you'd see something blurred. 194 00:20:19,370 --> 00:20:26,104 So again, the the, the power of quantum mechanics to describe atomic transitions 195 00:20:26,104 --> 00:20:33,730 over such a wide range of frequencies I think is really impressive. 196 00:20:33,730 --> 00:20:37,450 And, it, it validates Bohr's basic idea that there's a, you know, sharp 197 00:20:37,450 --> 00:20:41,590 quantization of atomic motion associated with the, with the, the quantization of 198 00:20:41,590 --> 00:20:47,040 angular minimum . Okay, so we'll conclude for now. 199 00:20:47,040 --> 00:20:51,066 next week will be a set of lectures on spin, then we'll return and, and wrap up 200 00:20:51,066 --> 00:20:52,880 after that.