1 00:00:00,070 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,800 under a Creative Commons license. 3 00:00:03,800 --> 00:00:06,530 Your support will help MIT OpenCourseWare continue 4 00:00:06,530 --> 00:00:10,130 to offer high-quality educational resources for free. 5 00:00:10,130 --> 00:00:12,700 To make a donation, or to view additional materials 6 00:00:12,700 --> 00:00:17,575 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,575 --> 00:00:18,730 at ocw.mit.edu. 8 00:00:23,377 --> 00:00:25,210 PROFESSOR: OK, so let me just quickly remind 9 00:00:25,210 --> 00:00:28,120 you of what we've done what we did last time. 10 00:00:28,120 --> 00:00:31,931 So unfortunately, today is going to be a little bit of what I 11 00:00:31,931 --> 00:00:34,430 wanted to do this time and a little bit of what I would have 12 00:00:34,430 --> 00:00:36,050 wanted to have been done in recitation yesterday, 13 00:00:36,050 --> 00:00:37,980 because all recitations were cancelled. 14 00:00:37,980 --> 00:00:41,900 So today is going to include a few steps along the way 15 00:00:41,900 --> 00:00:44,010 that I skipped over last time. 16 00:00:44,010 --> 00:00:46,157 OK, so to review from last time, we 17 00:00:46,157 --> 00:00:47,490 talked about central potentials. 18 00:00:47,490 --> 00:00:50,110 The energy operator had a radial derivative term, an angular 19 00:00:50,110 --> 00:00:51,822 momentum term, and a potential term. 20 00:00:51,822 --> 00:00:54,280 So this would be true regardless of whether we were central 21 00:00:54,280 --> 00:00:54,880 or not. 22 00:00:54,880 --> 00:00:57,004 But if we have a central potential, where this only 23 00:00:57,004 --> 00:00:59,890 depends on the magnitude of the radius vector, 24 00:00:59,890 --> 00:01:01,970 then we can use separation of variables, 25 00:01:01,970 --> 00:01:05,560 and write an energy eigenstate as a radial part-- which 26 00:01:05,560 --> 00:01:09,300 we will sneakily pull out a factor of 1 over the radius-- 27 00:01:09,300 --> 00:01:12,350 as we saw last time, that was particularly convenient. 28 00:01:12,350 --> 00:01:15,870 It took care of this r and made everything nice and simple. 29 00:01:15,870 --> 00:01:19,102 So a radial part times a spherical harmonic. 30 00:01:19,102 --> 00:01:20,810 The reason we use a spherical harmonic is 31 00:01:20,810 --> 00:01:23,940 that neither of these two terms depend on the angular 32 00:01:23,940 --> 00:01:24,610 coordinates. 33 00:01:24,610 --> 00:01:27,240 The angular momentum, we know what its eigenfunctions are, 34 00:01:27,240 --> 00:01:28,656 so if we use a spherical harmonic, 35 00:01:28,656 --> 00:01:32,060 this angular momentum becomes just a constant. 36 00:01:32,060 --> 00:01:34,320 And having separated in this form, 37 00:01:34,320 --> 00:01:37,110 and pulling out that sneaky factor of 1 over r, 38 00:01:37,110 --> 00:01:41,100 the energy eigenvalue equation reduces to a simple 1-D 39 00:01:41,100 --> 00:01:44,560 effective energy eigenvalue equation 40 00:01:44,560 --> 00:01:49,060 with a simple dr squared, and an effective potential, 41 00:01:49,060 --> 00:01:53,490 where the effective potential is the original central potential 42 00:01:53,490 --> 00:01:55,320 plus an angular momentum term, often 43 00:01:55,320 --> 00:01:59,870 referred to the angular momentum barrier-- which 44 00:01:59,870 --> 00:02:03,400 is proportional to-- as Matt says, 45 00:02:03,400 --> 00:02:06,800 as Professor Evans says, in 804, one 46 00:02:06,800 --> 00:02:09,680 is forced to say h bar squared upon 2m repeatedly. 47 00:02:09,680 --> 00:02:12,180 And he pointed out that it would be much more efficient if I 48 00:02:12,180 --> 00:02:14,620 would just come up with a simple variable for saying it 49 00:02:14,620 --> 00:02:15,750 or sound. 50 00:02:15,750 --> 00:02:19,188 So I will attempt to henceforth call it pfft. 51 00:02:19,188 --> 00:02:20,650 [LAUGHTER] 52 00:02:20,650 --> 00:02:23,000 So the effective potential is pfft, 53 00:02:23,000 --> 00:02:26,410 l, l plus 1 upon r squared plus the central part 54 00:02:26,410 --> 00:02:29,870 of the potential-- props to Professor Evans. 55 00:02:29,870 --> 00:02:32,160 And this is true in general for central potential. 56 00:02:32,160 --> 00:02:35,330 The last thing we notice of this is that since the energy 57 00:02:35,330 --> 00:02:37,920 eigenvalue equation has been reduced to this form, which 58 00:02:37,920 --> 00:02:40,120 depends on l, but is independent of m-- m 59 00:02:40,120 --> 00:02:43,020 appears nowhere here-- the energy can't possibly 60 00:02:43,020 --> 00:02:52,360 depend on m, although it can depend on l. 61 00:02:52,360 --> 00:02:53,500 It's just e of l. 62 00:02:53,500 --> 00:02:55,420 And thus, for every different value of m, 63 00:02:55,420 --> 00:02:59,410 I will get the same energy-- fixed l, different m. 64 00:02:59,410 --> 00:03:03,390 And there are how many values of m for a fixed l? 65 00:03:03,390 --> 00:03:04,110 2l plus 1. 66 00:03:04,110 --> 00:03:15,110 So the degeneracy of El is 2l plus 1. 67 00:03:15,110 --> 00:03:15,610 Right? 68 00:03:15,610 --> 00:03:17,290 That's general for any central potential. 69 00:03:17,290 --> 00:03:18,790 It only requires spherical symmetry. 70 00:03:22,990 --> 00:03:26,020 So then we say the example of the cooling potential, where 71 00:03:26,020 --> 00:03:27,920 the potential is minus E squared upon r. 72 00:03:30,670 --> 00:03:33,465 And we solved-- or I quickly reviewed the solution 73 00:03:33,465 --> 00:03:39,710 of the energy eigenvalue equation. 74 00:03:39,710 --> 00:03:42,870 I'll write it that way-- where r of what 75 00:03:42,870 --> 00:03:46,440 I will call-- in place of E I, will simply write n. 76 00:03:46,440 --> 00:03:56,240 Rnl is equal to e to the minus r over 2 77 00:03:56,240 --> 00:04:09,300 r naught n times r over r naught to the l plus 1 times 78 00:04:09,300 --> 00:04:18,200 some simple function v sub n of l of r over r naught, 79 00:04:18,200 --> 00:04:24,920 with the energy nlm being equal to E naught-- 80 00:04:24,920 --> 00:04:27,730 or I should say e Rydberg in our notation of last time-- 81 00:04:27,730 --> 00:04:30,750 upon n squared minus bound state. 82 00:04:30,750 --> 00:04:33,100 So let's talk about this quickly. 83 00:04:33,100 --> 00:04:37,560 This term-- the fact that it goes exponentially-- falls 84 00:04:37,560 --> 00:04:39,850 exponentially at large distances-- 85 00:04:39,850 --> 00:04:46,359 came from the asymptotic analysis at r goes to infinity. 86 00:04:46,359 --> 00:04:48,150 And this is just saying it's a bound state. 87 00:04:48,150 --> 00:04:50,410 The probability density falls off exponentially 88 00:04:50,410 --> 00:04:53,050 as we get to large radius. 89 00:04:53,050 --> 00:04:59,470 This term came from the asymptotic analysis 90 00:04:59,470 --> 00:05:01,180 near r goes to zero. 91 00:05:01,180 --> 00:05:03,620 And near r goes to zero, the only term 92 00:05:03,620 --> 00:05:07,850 that matters is in the effective-- this is one upon r. 93 00:05:07,850 --> 00:05:10,920 This is 1 over r squared, so this gets large more rapidly. 94 00:05:10,920 --> 00:05:13,110 This term is important, and that l, l plus 1 95 00:05:13,110 --> 00:05:15,660 gave us r to the l plus 1. 96 00:05:15,660 --> 00:05:18,080 We work in dimensionless variables, rho. 97 00:05:18,080 --> 00:05:20,270 And here, I replace rho with r over r naught, 98 00:05:20,270 --> 00:05:22,061 where r naught was the characteristic scale 99 00:05:22,061 --> 00:05:23,640 we computed last time. 100 00:05:23,640 --> 00:05:26,650 And finally, this function V sub nl 101 00:05:26,650 --> 00:05:31,710 satisfied-- it again satisfied the energy eigenvalue equation, 102 00:05:31,710 --> 00:05:33,500 but having pulled out these factors 103 00:05:33,500 --> 00:05:36,130 such that asymptotically, its regular. 104 00:05:36,130 --> 00:05:38,820 So this function must be regular, 105 00:05:38,820 --> 00:05:41,890 smooth-- no divergences, no poles-- 106 00:05:41,890 --> 00:05:46,610 as r goes to infinity and to 0. 107 00:05:46,610 --> 00:05:47,250 OK. 108 00:05:47,250 --> 00:05:50,380 And then we solved this by series expansion. 109 00:05:50,380 --> 00:05:52,340 And from termination of the series expansion, 110 00:05:52,340 --> 00:05:54,310 we got that the energy-- we got a relationship 111 00:05:54,310 --> 00:05:56,100 between the energy and l and n, and 112 00:05:56,100 --> 00:05:58,240 in particular the relationship-- this 113 00:05:58,240 --> 00:06:04,000 came from termination of the series, just 114 00:06:04,000 --> 00:06:05,985 like it did for the harmonic oscillator. 115 00:06:08,870 --> 00:06:10,300 And the termination series gave us 116 00:06:10,300 --> 00:06:12,721 that the energy was equal to minus a constant over n 117 00:06:12,721 --> 00:06:13,220 squared. 118 00:06:17,610 --> 00:06:19,580 With one additional condition, which 119 00:06:19,580 --> 00:06:22,800 is that n had to be greater than l. 120 00:06:22,800 --> 00:06:24,849 n had to be an integer greater than l. 121 00:06:24,849 --> 00:06:27,140 And l, of course, has to be greater than or equal to 0. 122 00:06:32,580 --> 00:06:35,250 So let me just quickly tell you how 123 00:06:35,250 --> 00:06:38,840 this arises, because this is an important fact about hydrogen 124 00:06:38,840 --> 00:06:41,860 which I want you to understand. 125 00:06:41,860 --> 00:06:45,430 So the way that arises is we got the energy from a series 126 00:06:45,430 --> 00:06:50,096 expansion for this remaining function of rv and l of r. 127 00:06:50,096 --> 00:06:51,900 And that series expansion started out 128 00:06:51,900 --> 00:07:00,020 by saying let v be of the form sum over j of a sub j, r 129 00:07:00,020 --> 00:07:05,235 over r naught, the dimensionless variable rho, to the j, 130 00:07:05,235 --> 00:07:07,660 to the j-th power, so it's a power series. 131 00:07:07,660 --> 00:07:09,890 And when we plug this whole form, together 132 00:07:09,890 --> 00:07:12,660 with this series expansion, into that differential equation, 133 00:07:12,660 --> 00:07:16,460 we get-- just like we did before for a harmonic oscillator-- 134 00:07:16,460 --> 00:07:18,870 we get a set of relations between 135 00:07:18,870 --> 00:07:21,120 the various coefficients in that series. 136 00:07:21,120 --> 00:07:24,240 And we solve that by saying a sub j plus 1-- 137 00:07:24,240 --> 00:07:26,530 by deriving a recursion relation-- is equal to a sub 138 00:07:26,530 --> 00:07:34,350 j times 2, square root of epsilon times 139 00:07:34,350 --> 00:07:53,930 j plus l plus 1 minus 1 upon j plus 1, j plus 2l plus 2. 140 00:07:58,210 --> 00:07:58,860 OK, fine. 141 00:07:58,860 --> 00:08:00,151 We get this recursion relation. 142 00:08:00,151 --> 00:08:02,410 It looks absolutely awful, but it tells you 143 00:08:02,410 --> 00:08:07,010 that if you know a0, say, is a constant which is not 0-- 144 00:08:07,010 --> 00:08:08,635 if it's zero, then they're all 0-- then 145 00:08:08,635 --> 00:08:11,170 this function is identically 0, because a j plus 1 146 00:08:11,170 --> 00:08:12,940 is a j times some constant. 147 00:08:12,940 --> 00:08:15,321 So this had better not vanish-- a sub 0. 148 00:08:15,321 --> 00:08:16,820 So the first term, the constant term 149 00:08:16,820 --> 00:08:18,774 doesn't vanish, which is good. 150 00:08:18,774 --> 00:08:20,690 It tells us V doesn't vanish, so we've already 151 00:08:20,690 --> 00:08:21,439 learned something. 152 00:08:21,439 --> 00:08:25,250 If V doesn't vanish identically at the origin, 153 00:08:25,250 --> 00:08:29,085 then we can determine a1 from a0 by multiplying by this. 154 00:08:29,085 --> 00:08:32,030 j is 0 in this, and l is whatever number 155 00:08:32,030 --> 00:08:33,919 we fix from ylm. 156 00:08:33,919 --> 00:08:34,460 And so on. 157 00:08:34,460 --> 00:08:35,799 We can deduce all the higher a sub l's. 158 00:08:35,799 --> 00:08:37,000 It's a series expansion. 159 00:08:37,000 --> 00:08:40,260 This is the recursion relation for it. 160 00:08:40,260 --> 00:08:44,920 However, note the following thing-- as j gets large, 161 00:08:44,920 --> 00:08:49,690 so we go to higher and higher orders in the series, 162 00:08:49,690 --> 00:08:53,100 then j is much larger than whatever fixed number l is, 163 00:08:53,100 --> 00:08:55,410 and much larger than whatever fixed number 2 is. 164 00:08:55,410 --> 00:09:01,610 So this asymptotes to aj times roughly 2 epsilon j upon j 165 00:09:01,610 --> 00:09:03,410 plus 1 j. 166 00:09:03,410 --> 00:09:06,915 At very large j, this just goes to 2 epsilon over j. 167 00:09:06,915 --> 00:09:08,540 Now, let's think about what that means. 168 00:09:08,540 --> 00:09:11,530 If I have that aj plus 1 is roughly equal 169 00:09:11,530 --> 00:09:18,720 to aj times root epsilon times 2 over j plus 1, 170 00:09:18,720 --> 00:09:24,900 I guess, then this tells me that every time I increase j, 171 00:09:24,900 --> 00:09:28,120 I get a factor of j plus 1 and another factor of 2 root 172 00:09:28,120 --> 00:09:28,810 epsilon. 173 00:09:28,810 --> 00:09:33,450 This tells me that a sub j is equal to a sub 0 times 174 00:09:33,450 --> 00:09:40,174 2 root epsilon to the j over j factorial. 175 00:09:40,174 --> 00:09:41,840 But those are the expansion coefficients 176 00:09:41,840 --> 00:09:48,230 of an exponential-- quantity to the n over n factorial. 177 00:09:48,230 --> 00:09:51,080 But that's bad, because if this series is 178 00:09:51,080 --> 00:09:55,220 the exponential series with argument 2 root epsilon, 179 00:09:55,220 --> 00:10:03,840 then this series becomes E to the 2 root epsilon r over r0. 180 00:10:03,840 --> 00:10:08,090 But that's bad, because that would exactly swamp this term. 181 00:10:08,090 --> 00:10:09,810 That would defeat the asymptotic analysis 182 00:10:09,810 --> 00:10:10,810 we did at the beginning. 183 00:10:10,810 --> 00:10:13,980 If this term, V, is growing so large twice 184 00:10:13,980 --> 00:10:17,310 as rapidly as this is going to 0, 185 00:10:17,310 --> 00:10:19,470 then the net is diverging at infinity. 186 00:10:19,470 --> 00:10:20,820 That's bad. 187 00:10:20,820 --> 00:10:23,790 So in order for this-- for the series expansion-- 188 00:10:23,790 --> 00:10:27,330 to describe a good function that really does vanish at infinity, 189 00:10:27,330 --> 00:10:30,150 we need that this doesn't happen. 190 00:10:30,150 --> 00:10:31,750 So it must be true that j doesn't 191 00:10:31,750 --> 00:10:35,350 get large enough that the exponential in V 192 00:10:35,350 --> 00:10:37,800 overwhelms the decaying exponential 193 00:10:37,800 --> 00:10:39,240 from the asymptotic analysis. 194 00:10:39,240 --> 00:10:40,450 Everyone cool with that? 195 00:10:40,450 --> 00:10:42,044 So it has to terminate. 196 00:10:42,044 --> 00:10:44,460 But if it terminates-- let's look at the coefficient here. 197 00:10:44,460 --> 00:10:51,210 If it terminates, this says that 2 root epsilon times j plus-- 198 00:10:51,210 --> 00:10:53,810 and if it terminates, that says that there's some j maximum. 199 00:10:56,437 --> 00:10:57,770 Well, let me write it over here. 200 00:10:57,770 --> 00:10:59,270 If it terminates, that means there's 201 00:10:59,270 --> 00:11:02,850 some value j such that aj plus 1 is 0. 202 00:11:02,850 --> 00:11:05,830 But if aj plus 1 is zero, how can that possibly be? 203 00:11:05,830 --> 00:11:08,180 It must be that for that maximum value of j, 204 00:11:08,180 --> 00:11:10,850 the numerator vanishes-- specifically 205 00:11:10,850 --> 00:11:12,900 for that maximum value of j. 206 00:11:12,900 --> 00:11:13,700 Yeah? 207 00:11:13,700 --> 00:11:15,690 Well, what's the condition for that numerator 208 00:11:15,690 --> 00:11:17,390 to vanish for a specific value of j? 209 00:11:17,390 --> 00:11:21,360 I'll j sub m, or j max, the maximum value, 210 00:11:21,360 --> 00:11:25,470 such that a sub j max plus 1 vanishes. 211 00:11:25,470 --> 00:11:30,080 So j plus l plus 1 must be equal to 1. 212 00:11:30,080 --> 00:11:33,300 Or said differently, epsilon-- which 213 00:11:33,300 --> 00:11:37,060 is the energy in units of e0, must be equal to 1 214 00:11:37,060 --> 00:11:45,080 over 4 g max plus l plus 1, quantity squared. 215 00:11:45,080 --> 00:11:46,510 Cool? 216 00:11:46,510 --> 00:11:50,070 So that means that our solutions, or states psi, 217 00:11:50,070 --> 00:11:52,700 are now labeled by three integers. 218 00:11:52,700 --> 00:11:56,440 they're labeled by l And by m from spherical harmonics, 219 00:11:56,440 --> 00:11:58,440 but they're also labeled by the value j max. 220 00:12:02,710 --> 00:12:07,452 And j max can be any number from 0 to infinity, right? 221 00:12:07,452 --> 00:12:08,910 Strictly, it shouldn't be infinite, 222 00:12:08,910 --> 00:12:12,930 but any finite number, any countable, any integer-- 223 00:12:12,930 --> 00:12:16,570 because eventually the series terminates at j max. 224 00:12:16,570 --> 00:12:18,230 So it's labeled by these three numbers, 225 00:12:18,230 --> 00:12:28,680 and the energy of lmj max is equal to e0 minus over 4. 226 00:12:28,680 --> 00:12:32,520 And this e0 over 4 we already saw was e Rydberg, 13.6 dv, 227 00:12:32,520 --> 00:12:33,680 times what value? 228 00:12:33,680 --> 00:12:39,722 Well, here's the energy in units of e0-- jm plus l plus 1, 229 00:12:39,722 --> 00:12:40,430 quantity squared. 230 00:12:43,890 --> 00:12:46,650 Where L is greater than or equal to 0, 231 00:12:46,650 --> 00:12:51,090 j sub m is greater than or equal to 0, and 1 is 1. 232 00:12:51,090 --> 00:12:52,990 But this is sort of cumbersome. 233 00:12:52,990 --> 00:12:56,620 We can just as well call this quantity n, 234 00:12:56,620 --> 00:12:59,370 where n is greater than l as an integer. 235 00:12:59,370 --> 00:13:01,210 It can't be equal to l, but it could 236 00:13:01,210 --> 00:13:03,900 be 1 greater than l, or any greater integer. 237 00:13:03,900 --> 00:13:04,930 Is that cool? 238 00:13:04,930 --> 00:13:07,420 So instead of labeling it by jm, the point 239 00:13:07,420 --> 00:13:11,370 at which the series terminates, we'll label it by n, 240 00:13:11,370 --> 00:13:15,230 and you can deduce what jm is by subtracting I plus 1 from n. 241 00:13:15,230 --> 00:13:17,200 That tells you where the series terminates. 242 00:13:17,200 --> 00:13:19,510 Notice that this tells you something nice. 243 00:13:19,510 --> 00:13:25,640 This power series for V, if n is equal to l plus 1, 244 00:13:25,640 --> 00:13:26,850 has how many terms in it? 245 00:13:29,389 --> 00:13:31,430 So say it out loud, just think about it a second. 246 00:13:31,430 --> 00:13:35,834 If n is equal to l plus 1, how many terms 247 00:13:35,834 --> 00:13:37,750 are there, that are non-vanishing in the power 248 00:13:37,750 --> 00:13:38,250 series? 249 00:13:40,874 --> 00:13:42,540 I won't call on you, but raise your hand 250 00:13:42,540 --> 00:13:43,540 when you have an answer. 251 00:13:50,690 --> 00:13:55,480 What's the value of j max if n is equal to l plus 1? 252 00:13:55,480 --> 00:13:58,060 So what term is the first one to vanish in the series? 253 00:14:01,254 --> 00:14:02,920 OK, so how many terms are non-vanishing? 254 00:14:05,540 --> 00:14:07,560 Yeah, just the one-- just the first term. 255 00:14:07,560 --> 00:14:11,320 So when l is as large as it can possibly 256 00:14:11,320 --> 00:14:18,480 be-- when l is equal to n minus 1, V is a constant function. 257 00:14:18,480 --> 00:14:20,451 It's just constant. 258 00:14:20,451 --> 00:14:27,340 When l is one less, when l is n minus 1, or n is l plus 2, 259 00:14:27,340 --> 00:14:31,559 then V has two terms in it-- a constant, and a linear. 260 00:14:31,559 --> 00:14:33,475 So the greater the difference between l and n, 261 00:14:33,475 --> 00:14:37,140 the more terms there are in the series. 262 00:14:37,140 --> 00:14:39,380 OK? 263 00:14:39,380 --> 00:14:44,360 So anyway, this leads to this form of the energy. 264 00:14:44,360 --> 00:14:48,720 But as is usually the case, this sort of a brute force analysis 265 00:14:48,720 --> 00:14:50,570 doesn't give us a whole lot of insight 266 00:14:50,570 --> 00:14:53,770 into why we get the qualitative features we do. 267 00:14:53,770 --> 00:14:55,990 In particular, the qualitative feature 268 00:14:55,990 --> 00:14:58,870 that stands out most obviously to me-- I don't know about you 269 00:14:58,870 --> 00:15:02,830 guys-- is the fact that the degeneracy is now much, 270 00:15:02,830 --> 00:15:03,980 much larger. 271 00:15:03,980 --> 00:15:07,200 In particular, we know from basic principles 272 00:15:07,200 --> 00:15:09,220 of a central potential-- as we just reviewed-- 273 00:15:09,220 --> 00:15:12,980 that anytime a potential is symmetrically invariant, 274 00:15:12,980 --> 00:15:16,930 is spherically symmetric-- any time 275 00:15:16,930 --> 00:15:20,390 the potential is spherically symmetric, rotationally 276 00:15:20,390 --> 00:15:22,720 invariant-- then there must be degeneracy 277 00:15:22,720 --> 00:15:26,400 of 2l plus 1 for every energy eigenstate-- there must be. 278 00:15:26,400 --> 00:15:29,420 The l could be zero, but if there's any angular momentum, 279 00:15:29,420 --> 00:15:31,750 then that degeneracy must be 2l plus 1. 280 00:15:31,750 --> 00:15:33,315 But this is much more degenerate. 281 00:15:37,200 --> 00:15:39,480 And in fact, how degenerate is it? 282 00:15:39,480 --> 00:15:45,430 Well, for any given En, the degeneracy 283 00:15:45,430 --> 00:15:50,210 is equal to-- well, what are the possible values of l? 284 00:15:50,210 --> 00:15:54,390 So we have some state for every different value of l. 285 00:15:54,390 --> 00:15:58,580 l could be equal to 0, and it can go up to n minus 1, right? 286 00:15:58,580 --> 00:16:07,290 Because, again, n is some integer 287 00:16:07,290 --> 00:16:09,640 which could vanish, l plus 1. 288 00:16:09,640 --> 00:16:12,360 So as small as this can be is 0. 289 00:16:12,360 --> 00:16:14,520 So n can be l plus 1, or anything greater. 290 00:16:14,520 --> 00:16:20,300 I.e.: l can go up to n minus 1, but it can't be any greater. 291 00:16:20,300 --> 00:16:24,806 Then for every value of l, there is a state n-- 292 00:16:24,806 --> 00:16:27,222 there's a value of n which could be minus l all the way up 293 00:16:27,222 --> 00:16:28,202 to l in integer steps. 294 00:16:28,202 --> 00:16:29,410 So how many states are there? 295 00:16:29,410 --> 00:16:32,830 Well there's one state for every value of n, l, and m. 296 00:16:32,830 --> 00:16:34,490 So it's this sum. 297 00:16:34,490 --> 00:16:37,140 The sum on m for minus l to l is 2l plus 1. 298 00:16:37,140 --> 00:16:43,240 So this is the sum n minus 1, l equals 0, of 2l plus 1. 299 00:16:43,240 --> 00:16:45,600 And this is equal to, kind of beautifully, n 300 00:16:45,600 --> 00:16:52,150 squared-- just arithmetic series. 301 00:16:52,150 --> 00:16:57,440 So we have this huge degeneracy, which 302 00:16:57,440 --> 00:17:00,340 is much larger than 2l plus 1. 303 00:17:00,340 --> 00:17:03,910 It's the sum over 2l plus 1 from l to 0 to n minus 1. 304 00:17:03,910 --> 00:17:05,615 Where is this degeneracy coming from? 305 00:17:05,615 --> 00:17:06,115 Why? 306 00:17:11,220 --> 00:17:16,160 So quantum mechanically, we know the following-- 307 00:17:16,160 --> 00:17:17,880 we've learned the following-- and I 308 00:17:17,880 --> 00:17:20,940 hope it's under your fingernails at this point-- 309 00:17:20,940 --> 00:17:22,890 that when you see a degeneracy, you 310 00:17:22,890 --> 00:17:26,619 should expect that degeneracy to follow from some symmetry. 311 00:17:26,619 --> 00:17:28,980 There's some symmetry of the system that 312 00:17:28,980 --> 00:17:31,160 is leading to a degeneracy by virtue of the fact 313 00:17:31,160 --> 00:17:33,285 that the generators of that symmetry-- for example, 314 00:17:33,285 --> 00:17:34,970 for rotation it's angular momentum-- 315 00:17:34,970 --> 00:17:36,380 commute with the energy. 316 00:17:36,380 --> 00:17:38,860 And when you have an operator that commutes with the energy 317 00:17:38,860 --> 00:17:41,350 operator, as we've seen, you can generally 318 00:17:41,350 --> 00:17:44,300 construct new states given a single energy eigenstate 319 00:17:44,300 --> 00:17:48,230 by acting suitably with that operator. 320 00:17:48,230 --> 00:17:54,814 So we should expect there to be some new symmetry. 321 00:17:54,814 --> 00:17:56,480 But it's very hard to see what symmetry. 322 00:17:56,480 --> 00:17:58,287 I mean, this is hydrogen. 323 00:17:58,287 --> 00:17:59,370 This is central potential. 324 00:17:59,370 --> 00:18:01,100 It's just some stupid central potential. 325 00:18:01,100 --> 00:18:03,300 What's so special about the hydrogen system as 326 00:18:03,300 --> 00:18:05,650 opposed to, for example, the harmonic oscillator? 327 00:18:05,650 --> 00:18:08,830 Which hopefully you will see in recitation at some point. 328 00:18:08,830 --> 00:18:11,360 The harmonic oscillator in three dimensions 329 00:18:11,360 --> 00:18:13,590 has degeneracy 2l plus 1. 330 00:18:13,590 --> 00:18:14,650 It's a central potential. 331 00:18:14,650 --> 00:18:19,320 It has to have degeneracy 2l plus 1-- but not n squared. 332 00:18:19,320 --> 00:18:20,370 Where did that come from? 333 00:18:20,370 --> 00:18:22,290 What's so special about the Kepler problem? 334 00:18:22,290 --> 00:18:23,870 So there must be some symmetry, but it's not obvious 335 00:18:23,870 --> 00:18:24,640 what it is. 336 00:18:24,640 --> 00:18:28,415 But now let's use a note from classical mechanics. 337 00:18:28,415 --> 00:18:30,540 Noether's Theorem tells us when we have a symmetry, 338 00:18:30,540 --> 00:18:32,617 we have a conserved quantity. 339 00:18:32,617 --> 00:18:34,700 We also saw this in the quantum mechanical version 340 00:18:34,700 --> 00:18:36,022 for expectation values. 341 00:18:36,022 --> 00:18:37,230 We have a conserved quantity. 342 00:18:41,632 --> 00:18:43,840 And in quantum mechanics, having a conserved quantity 343 00:18:43,840 --> 00:18:47,520 means that the energy commutes with some quantity. 344 00:18:50,734 --> 00:18:52,400 Because that controls the time evolution 345 00:18:52,400 --> 00:18:53,441 of the expectation value. 346 00:18:55,739 --> 00:18:57,280 So there must be some quantity, which 347 00:18:57,280 --> 00:19:00,440 I've written by a question mark, that commutes with the energy 348 00:19:00,440 --> 00:19:02,290 operator, specifically in the case 349 00:19:02,290 --> 00:19:04,150 of the harmonic oscillator. 350 00:19:04,150 --> 00:19:05,529 So what is that quantity? 351 00:19:05,529 --> 00:19:07,570 So actually, this quantity in classical mechanics 352 00:19:07,570 --> 00:19:09,816 was studied, because the same thing 353 00:19:09,816 --> 00:19:10,940 is true in the Kepler case. 354 00:19:10,940 --> 00:19:13,230 In the Kepler case, the orbits close, 355 00:19:13,230 --> 00:19:15,006 and they have simple ellipses, and they 356 00:19:15,006 --> 00:19:16,380 have all sorts of nice properties 357 00:19:16,380 --> 00:19:18,869 that beg for an explanation in terms of symmetry. 358 00:19:18,869 --> 00:19:21,160 And it was pointed out by a number of people-- Laplace, 359 00:19:21,160 --> 00:19:23,560 Runge, Lenz, various people-- and this is often 360 00:19:23,560 --> 00:19:25,020 called the Runge-Lenz vector, but I 361 00:19:25,020 --> 00:19:27,630 think you can blame it on any number of people-- 362 00:19:27,630 --> 00:19:30,740 that there's a vector that in classical mechanics, 363 00:19:30,740 --> 00:19:33,830 for the Kepler problem-- or for the Coulomb problem, 364 00:19:33,830 --> 00:19:35,200 is conserved. 365 00:19:35,200 --> 00:19:40,100 And that quantity is p cross l-- momentum cross angular 366 00:19:40,100 --> 00:19:46,825 momentum-- minus m e squared-- and let 367 00:19:46,825 --> 00:19:49,290 me write m sub electrons-- it's the mass rather 368 00:19:49,290 --> 00:19:55,800 than the quantum number little m-- e squared, r vector. 369 00:19:55,800 --> 00:19:58,940 So it turns out this quantity is classically conserved, 370 00:19:58,940 --> 00:20:01,280 and if you make a quantum operator out of it, 371 00:20:01,280 --> 00:20:05,260 then e with A is equal to 0. 372 00:20:05,260 --> 00:20:07,582 Add hat. 373 00:20:07,582 --> 00:20:09,700 This was enormously non-obvious. 374 00:20:09,700 --> 00:20:12,800 If it's obvious to you, you're a freak. 375 00:20:12,800 --> 00:20:14,230 It's really, really not obvious. 376 00:20:14,230 --> 00:20:16,310 You can go through and do the calculation. 377 00:20:16,310 --> 00:20:20,320 And when I say really not obvious, I mean-- well, 378 00:20:20,320 --> 00:20:22,310 I'll show you what I mean in just a second. 379 00:20:22,310 --> 00:20:24,825 So you can go through and do the calculation classically. 380 00:20:24,825 --> 00:20:26,880 Is this quantity conserved in the Kepler problem 381 00:20:26,880 --> 00:20:27,520 using Newton's law? 382 00:20:27,520 --> 00:20:28,110 And the answer is yes. 383 00:20:28,110 --> 00:20:28,920 It's amazing. 384 00:20:28,920 --> 00:20:30,210 Quantum mechanically, you can do the same thing. 385 00:20:30,210 --> 00:20:32,120 But I'm going to warn you for anyone who has the chutzpah 386 00:20:32,120 --> 00:20:33,620 to try-- because I encourage you to, 387 00:20:33,620 --> 00:20:36,280 but it takes a little bit of brawn-- 388 00:20:36,280 --> 00:20:38,570 think about this operator for second. 389 00:20:38,570 --> 00:20:42,019 Does this classical quantity have an obvious interpretation 390 00:20:42,019 --> 00:20:42,935 as a quantum operator? 391 00:20:45,980 --> 00:20:49,230 No, because l is r cross p. 392 00:20:49,230 --> 00:20:50,940 But now we have prp. 393 00:20:50,940 --> 00:20:52,770 There's an ordering ambiguity. 394 00:20:52,770 --> 00:20:54,730 So you have to decide which order do you 395 00:20:54,730 --> 00:20:58,020 put that p and that r, and the other p-- ppr, rpr, 396 00:20:58,020 --> 00:20:59,469 prr-- right? 397 00:20:59,469 --> 00:21:01,510 You can write out which components you mean here. 398 00:21:01,510 --> 00:21:04,140 For example, you mean for the z component of a, 399 00:21:04,140 --> 00:21:08,090 you're going to get px ly, and ly px. 400 00:21:08,090 --> 00:21:12,460 But within that, how do you order the p's and the r's, 401 00:21:12,460 --> 00:21:19,510 because ly contains px, and in particular, it contains x. 402 00:21:19,510 --> 00:21:21,860 So there's an ordering ambiguity. 403 00:21:21,860 --> 00:21:25,130 So just be a little bit careful about this as a quantum 404 00:21:25,130 --> 00:21:27,470 mechanical operator if you do play with this. 405 00:21:27,470 --> 00:21:30,050 But if you do it, and you're thoughtful about it, 406 00:21:30,050 --> 00:21:32,850 it's pretty easy to see that it can be computed and checked 407 00:21:32,850 --> 00:21:35,090 that, in fact, the commutator vanishes. 408 00:21:35,090 --> 00:21:37,311 So at this point, it's tempting to say, aha! 409 00:21:37,311 --> 00:21:38,810 There's an extra conserved quantity. 410 00:21:38,810 --> 00:21:40,180 We declare victory. 411 00:21:40,180 --> 00:21:42,980 The problem is, this didn't give us the symmetry. 412 00:21:42,980 --> 00:21:44,960 This just told us there's a conserved quantity. 413 00:21:44,960 --> 00:21:48,361 What's the symmetry behind this? 414 00:21:48,361 --> 00:21:50,110 What's the symmetry that insures that this 415 00:21:50,110 --> 00:21:52,070 is a conserved quantity? 416 00:21:52,070 --> 00:21:53,630 It's obviously not rotations. 417 00:21:53,630 --> 00:21:56,240 It's something else. 418 00:21:56,240 --> 00:21:58,400 And so there are many answers to this question. 419 00:21:58,400 --> 00:22:02,840 One answer involves an explicit expression for in phase space, 420 00:22:02,840 --> 00:22:05,930 the change of variables dx-- and this 421 00:22:05,930 --> 00:22:15,730 is a variation that depends on f of x and p. 422 00:22:15,730 --> 00:22:17,560 So there's a little vector parameter, 423 00:22:17,560 --> 00:22:22,060 and the change of the physical coordinates 424 00:22:22,060 --> 00:22:24,796 depends on both the coordinates and the momenta. 425 00:22:24,796 --> 00:22:26,170 This is a slightly strange thing, 426 00:22:26,170 --> 00:22:27,450 because it's like a change of variables 427 00:22:27,450 --> 00:22:29,949 where the position gets mapped into some non-linear function 428 00:22:29,949 --> 00:22:31,590 of the positions and momenta. 429 00:22:31,590 --> 00:22:34,580 That's a little bit weird, but you can do this. 430 00:22:34,580 --> 00:22:36,622 But I don't find it a terribly satisfying answer. 431 00:22:36,622 --> 00:22:38,913 And here's the answer I find most satisfying-- it's not 432 00:22:38,913 --> 00:22:40,700 terribly useful for our present purposes, 433 00:22:40,700 --> 00:22:43,520 but you're going to run into this again in 805, when 434 00:22:43,520 --> 00:22:46,717 you do what's called the operator method for the Coulomb 435 00:22:46,717 --> 00:22:48,092 potential, and it turns out to be 436 00:22:48,092 --> 00:22:49,686 an enormously useful machine. 437 00:22:49,686 --> 00:22:51,060 We're not going to use it in 804, 438 00:22:51,060 --> 00:22:52,610 but I want to advertise it for you. 439 00:22:52,610 --> 00:22:53,590 I want to give you a description, though, 440 00:22:53,590 --> 00:22:55,950 that isn't the description you're going to get in 805. 441 00:22:55,950 --> 00:22:59,280 The description is the following-- 1935, 442 00:22:59,280 --> 00:23:01,880 guy named Fock, crazy guy, amazing, 443 00:23:01,880 --> 00:23:03,820 amazing physicist-- played a huge role 444 00:23:03,820 --> 00:23:05,660 in the development of quantum field theory. 445 00:23:05,660 --> 00:23:08,290 I'm not exactly sure why he was thinking about this problem, 446 00:23:08,290 --> 00:23:12,500 but he was, and this was a very sort of Soviet thing 447 00:23:12,500 --> 00:23:15,150 to do at the time, I guess, to be hard core and mathematical. 448 00:23:15,150 --> 00:23:18,200 So off he was going, computing things, 449 00:23:18,200 --> 00:23:19,870 and he observed the following fact-- 450 00:23:19,870 --> 00:23:23,260 how he observed this, again, crazy long story. 451 00:23:23,260 --> 00:23:26,170 He observed the following fact-- if you take a Kepler problem 452 00:23:26,170 --> 00:23:29,010 and you work in momentum space-- so you write everything 453 00:23:29,010 --> 00:23:32,430 in terms of the momentum-- it turns out that the Kepler 454 00:23:32,430 --> 00:23:36,370 problem, or he referred to as the Kepler problem, 455 00:23:36,370 --> 00:23:41,986 but I'll refer to it as the Coulomb problem in 3D, 456 00:23:41,986 --> 00:23:45,610 three spatial dimensions-- this is really crazy-- 457 00:23:45,610 --> 00:23:47,060 is exactly equivalent. 458 00:23:47,060 --> 00:23:50,202 There's a change of variables-- it's kind of complicated, 459 00:23:50,202 --> 00:23:51,910 but it's just a projection, it turns out. 460 00:23:51,910 --> 00:23:58,980 It's exactly equivalent to the problem of a free particle 461 00:23:58,980 --> 00:24:07,320 in four dimensions, constrained to the surface 462 00:24:07,320 --> 00:24:11,274 of a sphere-- to a three sphere in four dimensions. 463 00:24:11,274 --> 00:24:13,190 So you take a three sphere in four dimensions. 464 00:24:13,190 --> 00:24:14,090 You take a marble. 465 00:24:14,090 --> 00:24:17,060 You kick it, but make sure that it's stuck to the sphere. 466 00:24:17,060 --> 00:24:18,900 It will move in perfect circles, right? 467 00:24:18,900 --> 00:24:20,400 Obviously-- it's moving on a sphere, 468 00:24:20,400 --> 00:24:22,110 it's just going to move in great circles. 469 00:24:22,110 --> 00:24:23,910 OK, that's nice and simple. 470 00:24:23,910 --> 00:24:26,950 What's the symmetry group of that system? 471 00:24:26,950 --> 00:24:29,060 Well, the sphere is invariant under any rotation 472 00:24:29,060 --> 00:24:31,780 in four dimensions. 473 00:24:31,780 --> 00:24:32,980 STUDENT: [INAUDIBLE]. 474 00:24:32,980 --> 00:24:36,924 PROFESSOR: Ah, SO4-- there's a symmetry SO4-- 475 00:24:36,924 --> 00:24:38,340 special orthogonal transformation. 476 00:24:38,340 --> 00:24:39,965 Special means it doesn't change length. 477 00:24:39,965 --> 00:24:41,730 Orthogonal means that it's a rotation. 478 00:24:41,730 --> 00:24:44,930 So it's orthogonal rotations in four dimensions, 479 00:24:44,930 --> 00:24:49,060 and so now, this should sound familiar, 480 00:24:49,060 --> 00:24:57,780 because the angular momentum group in 3D is SO3. 481 00:24:57,780 --> 00:25:01,580 It's rotations that don't change the length in three dimensions. 482 00:25:01,580 --> 00:25:02,080 OK? 483 00:25:02,080 --> 00:25:04,050 How many elements does it have? 484 00:25:04,050 --> 00:25:06,587 How many conserved quantities go along with this? 485 00:25:06,587 --> 00:25:08,420 Well, how many conserved quantities go along 486 00:25:08,420 --> 00:25:09,878 with rotations in three dimensions? 487 00:25:12,010 --> 00:25:14,510 What's conserved by virtue of rotational invariance in three 488 00:25:14,510 --> 00:25:15,350 dimensions? 489 00:25:15,350 --> 00:25:17,766 Angular momentum-- how many components of angular momentum 490 00:25:17,766 --> 00:25:18,500 are there? 491 00:25:18,500 --> 00:25:19,880 Three, right, there are three. 492 00:25:19,880 --> 00:25:23,420 So it turns out that if you calculate in SO4 493 00:25:23,420 --> 00:25:27,690 how many conserved quantities are there, there are six. 494 00:25:27,690 --> 00:25:29,430 And when you do this mapping back down 495 00:25:29,430 --> 00:25:32,510 from the free particle in 4D to the Coulomb problem in 3D, 496 00:25:32,510 --> 00:25:36,780 three of them are the l's, and the other three of them 497 00:25:36,780 --> 00:25:41,280 are this A vector-- comes out spot on. 498 00:25:41,280 --> 00:25:44,447 So what it's telling you is that the spectrum of hydrogen-- 499 00:25:44,447 --> 00:25:46,280 or at least the I shouldn't say of hydrogen, 500 00:25:46,280 --> 00:25:48,370 the spectrum of the Coulomb system-- 501 00:25:48,370 --> 00:25:50,350 has an enhanced symmetry. 502 00:25:50,350 --> 00:25:54,060 It's the symmetry group of rotations in four dimensions. 503 00:25:54,060 --> 00:25:55,990 Why? 504 00:25:55,990 --> 00:25:57,790 I don't know, it's just kind awesome. 505 00:25:57,790 --> 00:26:01,220 But I think that's about the best that can be said. 506 00:26:01,220 --> 00:26:02,510 It's just true. 507 00:26:02,510 --> 00:26:04,170 So for certain very special potentials, 508 00:26:04,170 --> 00:26:05,836 we find that this sort of thing happens. 509 00:26:05,836 --> 00:26:07,230 It happens in other systems, too. 510 00:26:07,230 --> 00:26:09,500 You discover that there are accidental symmetries 511 00:26:09,500 --> 00:26:10,379 in your system. 512 00:26:10,379 --> 00:26:12,670 They shouldn't be referred to as accidental symmetries. 513 00:26:12,670 --> 00:26:14,970 They are enormously nontrivial functions 514 00:26:14,970 --> 00:26:16,774 of the rest of the system, and we're 515 00:26:16,774 --> 00:26:18,440 going to break the symmetry in a minute, 516 00:26:18,440 --> 00:26:20,856 and you'll see why we shouldn't refer to it as accidental. 517 00:26:20,856 --> 00:26:23,080 But in any case, the answer to the question, 518 00:26:23,080 --> 00:26:26,240 why do we have this giant degeneracy in the Coulomb 519 00:26:26,240 --> 00:26:28,900 problem-- why is it n squared rather than 2l plus 1-- 520 00:26:28,900 --> 00:26:31,407 is that there is an enhanced symmetry group. 521 00:26:31,407 --> 00:26:33,490 There's more symmetry then you would have thought. 522 00:26:33,490 --> 00:26:35,980 And there's a conserved quantity associated with it. 523 00:26:35,980 --> 00:26:37,790 And that conserved quantity corresponds 524 00:26:37,790 --> 00:26:39,910 quantum mechanically to an operator 525 00:26:39,910 --> 00:26:43,070 that commutes with the energy operator. 526 00:26:43,070 --> 00:26:44,018 OK? 527 00:26:44,018 --> 00:26:46,030 Yeah. 528 00:26:46,030 --> 00:26:46,946 AUDIENCE: [INAUDIBLE]. 529 00:26:52,330 --> 00:26:53,521 PROFESSOR: Indeed, indeed. 530 00:26:53,521 --> 00:26:55,770 AUDIENCE: So how do you get three conserved components 531 00:26:55,770 --> 00:26:57,995 [INAUDIBLE]. 532 00:26:57,995 --> 00:26:58,870 PROFESSOR: Excellent. 533 00:26:58,870 --> 00:27:01,770 What do I mean in classical mechanics for a quantity 534 00:27:01,770 --> 00:27:02,932 to be conserved? 535 00:27:02,932 --> 00:27:05,140 I mean that the time rate of change of that quantity, 536 00:27:05,140 --> 00:27:07,620 under the classical dynamics, under classical evolution, 537 00:27:07,620 --> 00:27:09,990 is preserved. 538 00:27:09,990 --> 00:27:13,600 So what's the quantum mechanical version of that? 539 00:27:13,600 --> 00:27:15,520 So the mechanical version of that is, well, 540 00:27:15,520 --> 00:27:21,310 can we say that d dt of some operator-- 541 00:27:21,310 --> 00:27:23,740 or of some observable a is 0. 542 00:27:23,740 --> 00:27:25,210 Does this makes sense? 543 00:27:25,210 --> 00:27:28,740 No, because it turns out, in the language we've 544 00:27:28,740 --> 00:27:31,410 been using-- in the Schrodinger evolution description 545 00:27:31,410 --> 00:27:32,550 that we've been using-- this doesn't really 546 00:27:32,550 --> 00:27:33,840 make sense, because this thing is an operator. 547 00:27:33,840 --> 00:27:34,790 That's not a quantity. 548 00:27:34,790 --> 00:27:36,000 Quantity is something you measure. 549 00:27:36,000 --> 00:27:36,958 So what do you measure? 550 00:27:36,958 --> 00:27:39,130 Do you measure the operator? 551 00:27:39,130 --> 00:27:42,394 In a state, you measure its expectation values. 552 00:27:42,394 --> 00:27:43,810 You might measure its eigenvalues, 553 00:27:43,810 --> 00:27:48,450 but the more general thing is expectation values. 554 00:27:48,450 --> 00:27:50,640 So this is something that depends on time, 555 00:27:50,640 --> 00:27:53,660 because the wave function depends on time-- psi of t. 556 00:27:53,660 --> 00:27:56,390 OK, so the best to say something is conserved, 557 00:27:56,390 --> 00:27:58,863 is to say d dt of A is 0. 558 00:28:01,790 --> 00:28:03,660 But we've seen what the condition 559 00:28:03,660 --> 00:28:08,200 is for the expectation value to not change in time. 560 00:28:08,200 --> 00:28:11,420 In general, for any operator A, the time rate 561 00:28:11,420 --> 00:28:14,430 of change of its expectation value 562 00:28:14,430 --> 00:28:27,840 is given by the commutator of E with A. 563 00:28:27,840 --> 00:28:30,690 I can do the symbols-- good. 564 00:28:30,690 --> 00:28:35,880 OK, so in particular, if this commutator vanishes, 565 00:28:35,880 --> 00:28:41,660 if the commutator vanishes-- equals 0--, this is equal to-- 566 00:28:41,660 --> 00:28:43,960 AUDIENCE: [INAUDIBLE]. 567 00:28:43,960 --> 00:28:46,217 PROFESSOR: Sorry? 568 00:28:46,217 --> 00:28:48,050 Well, the wave function can be time variant, 569 00:28:48,050 --> 00:28:49,030 and I'm going to assume for the moment 570 00:28:49,030 --> 00:28:50,650 that the energy operator and the A operator are not, 571 00:28:50,650 --> 00:28:52,600 themselves, explicitly time dependent. 572 00:28:52,600 --> 00:28:54,150 Yes, that's an important assumption. 573 00:28:54,150 --> 00:28:55,649 So let me just assume for simplicity 574 00:28:55,649 --> 00:28:58,281 that neither the energy nor the operator A that I'm looking at 575 00:28:58,281 --> 00:28:59,530 are explicitly time dependent. 576 00:28:59,530 --> 00:29:02,332 Like, it shouldn't be, like, x plus tp. 577 00:29:02,332 --> 00:29:03,290 That would be annoying. 578 00:29:03,290 --> 00:29:05,634 So let me assume that that's not the case. 579 00:29:05,634 --> 00:29:07,300 You do have to deal with that sometimes. 580 00:29:07,300 --> 00:29:08,990 For example, if you are dealing with a system that 581 00:29:08,990 --> 00:29:11,495 changes in time, where there's a background magnetic field, 582 00:29:11,495 --> 00:29:13,370 or something like that, that changes in time. 583 00:29:13,370 --> 00:29:14,661 But module that sort of detail. 584 00:29:14,661 --> 00:29:17,310 And also, there's an i h bar, so where do the i h bars go? 585 00:29:20,010 --> 00:29:21,760 Do the units make sense here? 586 00:29:25,560 --> 00:29:26,550 A, A, good. 587 00:29:26,550 --> 00:29:28,100 E and d dt-- whoops. 588 00:29:28,100 --> 00:29:32,169 So where do the i h bars go? 589 00:29:32,169 --> 00:29:33,460 Well, it can go on either side. 590 00:29:33,460 --> 00:29:35,209 You have to tell me if they go up or down. 591 00:29:37,216 --> 00:29:38,590 What is the Schrodinger equation? 592 00:29:45,760 --> 00:29:49,980 Yeah, exactly, i h bar, OK so we need an i h bar. 593 00:29:49,980 --> 00:29:55,460 [LAUGHTER] 594 00:29:55,460 --> 00:29:59,470 OK, this spring has got to us all. 595 00:29:59,470 --> 00:30:03,060 So now, in the case that e with A as 0, 596 00:30:03,060 --> 00:30:06,670 imagine we have an operator a that commutes equal to 0. 597 00:30:06,670 --> 00:30:12,190 Then d dt of the expectation value of A in any state 598 00:30:12,190 --> 00:30:13,210 is equal to 0. 599 00:30:13,210 --> 00:30:15,080 And that's what I'll call conserved. 600 00:30:15,080 --> 00:30:17,466 So classically, to be a conserved quantity 601 00:30:17,466 --> 00:30:18,840 means that it's time independent. 602 00:30:18,840 --> 00:30:20,940 Quantum mechanically, to be a conserved quantity 603 00:30:20,940 --> 00:30:22,635 means that it commutes with the energy. 604 00:30:22,635 --> 00:30:24,760 Now, going back to the question-- the question was, 605 00:30:24,760 --> 00:30:26,150 look for angular momentum. 606 00:30:26,150 --> 00:30:29,555 There are only two numbers you can know at any moment in time. 607 00:30:29,555 --> 00:30:30,430 You can't know three. 608 00:30:30,430 --> 00:30:32,559 You can know the total angular momentum, little l. 609 00:30:32,559 --> 00:30:34,100 And you can know the angular momentum 610 00:30:34,100 --> 00:30:36,183 in a particular direction, which we conventionally 611 00:30:36,183 --> 00:30:37,862 call z-- little n, right? 612 00:30:37,862 --> 00:30:39,820 But you can't know some other angular momentum, 613 00:30:39,820 --> 00:30:41,770 like angular momentum in x direction simultaneously. 614 00:30:41,770 --> 00:30:42,720 So what does it mean to say that there 615 00:30:42,720 --> 00:30:43,960 are three conserved quantities? 616 00:30:43,960 --> 00:30:46,335 I can't know three conserved quantities at the same time. 617 00:30:46,335 --> 00:30:48,640 They're not even well defined at some moment in time. 618 00:30:48,640 --> 00:30:51,400 The answer is there are three operators-- lx, 619 00:30:51,400 --> 00:30:54,870 and ly, and lz-- which I will simply 620 00:30:54,870 --> 00:31:00,200 write as l vector-- all of which commute with E. So 621 00:31:00,200 --> 00:31:04,510 while it's true that systems don't have 622 00:31:04,510 --> 00:31:07,460 definite values of lx, ly and lz simultaneously, 623 00:31:07,460 --> 00:31:09,730 nonetheless lx, ly and lz all commute 624 00:31:09,730 --> 00:31:11,245 with the energy operator. 625 00:31:11,245 --> 00:31:13,370 And as a consequence, their time expectation values 626 00:31:13,370 --> 00:31:15,657 are all 0 in a central potential. 627 00:31:15,657 --> 00:31:17,490 So the time derivatives of their expectation 628 00:31:17,490 --> 00:31:18,890 are all 0s in a central potential. 629 00:31:18,890 --> 00:31:20,723 They're all constant in a central potential. 630 00:31:20,723 --> 00:31:21,854 Yeah. 631 00:31:21,854 --> 00:31:22,770 AUDIENCE: [INAUDIBLE]. 632 00:31:29,296 --> 00:31:31,654 PROFESSOR: You mean 1 over r? 633 00:31:31,654 --> 00:31:32,654 STUDENT: Yeah, 1 over r. 634 00:31:35,880 --> 00:31:38,451 PROFESSOR: So the question is, if you don't have 1 over r, 635 00:31:38,451 --> 00:31:39,950 do you get extra-special things that 636 00:31:39,950 --> 00:31:41,575 happen for other potentials that aren't 637 00:31:41,575 --> 00:31:43,756 1 over r, but something else? 638 00:31:43,756 --> 00:31:44,672 AUDIENCE: [INAUDIBLE]. 639 00:31:48,514 --> 00:31:50,180 PROFESSOR: It's not quite a unique case. 640 00:31:50,180 --> 00:31:54,120 So the question is-- this is a very good question. 641 00:31:54,120 --> 00:31:56,931 So the question is this-- this is a great question-- look, 642 00:31:56,931 --> 00:31:59,180 there's something peculiar about 1 over r classically. 643 00:31:59,180 --> 00:32:01,040 The orbit's closed. 644 00:32:01,040 --> 00:32:01,760 You get ellipses. 645 00:32:01,760 --> 00:32:03,220 You don't get progressing ellipses. 646 00:32:03,220 --> 00:32:05,250 That's what you get if you have an n harmonic. 647 00:32:05,250 --> 00:32:07,920 If you have an extra 1 over r to the sixth term, or something, 648 00:32:07,920 --> 00:32:11,040 a small correction to 1 over r, which we 649 00:32:11,040 --> 00:32:12,270 do in the real solar system. 650 00:32:12,270 --> 00:32:14,040 It's a many body system instead of a two-body system, 651 00:32:14,040 --> 00:32:15,235 so r orbit doesn't close. 652 00:32:15,235 --> 00:32:19,207 It's close, but-- so that's special. 653 00:32:19,207 --> 00:32:21,290 And here we see that there's an enhanced symmetry. 654 00:32:21,290 --> 00:32:23,300 Are these things related to each other? 655 00:32:23,300 --> 00:32:27,260 So the answer is almost, in the following sense-- 656 00:32:27,260 --> 00:32:30,200 for the harmonic oscillator, we also have it close, 657 00:32:30,200 --> 00:32:33,560 but there isn't a Runge-Lenz vector. 658 00:32:33,560 --> 00:32:35,102 However, the orbits close, so there's 659 00:32:35,102 --> 00:32:36,685 something special about that one, too. 660 00:32:36,685 --> 00:32:38,210 Is there a conserved quantity there? 661 00:32:38,210 --> 00:32:39,600 And it turns out the answer to that is yes. 662 00:32:39,600 --> 00:32:41,391 There's an enhanced symmetry there as well. 663 00:32:41,391 --> 00:32:43,040 But it's not like this. 664 00:32:43,040 --> 00:32:45,210 It doesn't lead to this gigantic degeneracy. 665 00:32:45,210 --> 00:32:46,780 There is just extra structure that's 666 00:32:46,780 --> 00:32:48,238 nice about the harmonic oscillator. 667 00:32:48,238 --> 00:32:51,400 But for the harmonic oscillator, of course, it also closes. 668 00:32:51,400 --> 00:32:54,684 It's not this orbits this. 669 00:32:54,684 --> 00:32:56,100 So there's a nice structure there, 670 00:32:56,100 --> 00:32:58,450 but it's not quite so simple. 671 00:33:01,260 --> 00:33:05,520 OK, yeah, one last question. 672 00:33:05,520 --> 00:33:12,511 AUDIENCE: [INAUDIBLE] Kepler's problem. 673 00:33:12,511 --> 00:33:17,441 I was wondering if it's at all correct thinking [INAUDIBLE] 674 00:33:17,441 --> 00:33:21,385 symmetry or degeneracy [INAUDIBLE] 675 00:33:21,385 --> 00:33:22,890 equal time instead of equal area? 676 00:33:22,890 --> 00:33:24,662 PROFESSOR: Yeah, it's not unrelated. 677 00:33:24,662 --> 00:33:26,620 I don't know how to make that connection sharp, 678 00:33:26,620 --> 00:33:29,391 but the thing I can say sharp-- so the question is, look, 679 00:33:29,391 --> 00:33:31,640 is that conserved quantity somehow related to the fact 680 00:33:31,640 --> 00:33:34,880 that there's equal time in equal-- equal areas swept out 681 00:33:34,880 --> 00:33:37,035 in equal time at all points in the orbit? 682 00:33:37,035 --> 00:33:38,910 Which is a true thing about a Kepler problem. 683 00:33:38,910 --> 00:33:40,220 That's a good observation. 684 00:33:40,220 --> 00:33:41,340 I don't know how to make that connection, 685 00:33:41,340 --> 00:33:42,990 but it is true in the following sense-- 686 00:33:42,990 --> 00:33:45,114 that both of those facts are guaranteed by the fact 687 00:33:45,114 --> 00:33:48,170 that you have closed orbits with a particular ratio. 688 00:33:48,170 --> 00:33:50,270 And that is, indeed, can be blamed 689 00:33:50,270 --> 00:33:52,939 on the conservation of the Runge-Lenz vector. 690 00:33:52,939 --> 00:33:55,480 I don't know how to make that connection direct and explicit, 691 00:33:55,480 --> 00:33:57,460 but indeed it is. 692 00:33:57,460 --> 00:33:58,550 So that's fun question. 693 00:33:58,550 --> 00:34:00,010 I'll have to think about that. 694 00:34:00,010 --> 00:34:00,792 That's a good one. 695 00:34:03,624 --> 00:34:05,790 One of the totally awesome things about teaching 804 696 00:34:05,790 --> 00:34:07,895 is that every single time, someone 697 00:34:07,895 --> 00:34:09,520 asks a question I've never even thought 698 00:34:09,520 --> 00:34:11,070 of before-- forget never answered. 699 00:34:11,070 --> 00:34:11,655 Never even thought of. 700 00:34:11,655 --> 00:34:12,655 I never thought of that. 701 00:34:12,655 --> 00:34:14,909 It's a great question. 702 00:34:14,909 --> 00:34:18,679 OK, so everyone cool with the degeneracy? 703 00:34:18,679 --> 00:34:20,110 We have this gigantic degeneracy. 704 00:34:20,110 --> 00:34:23,409 So knowing what the degeneracy is, or where it's coming from, 705 00:34:23,409 --> 00:34:26,199 at least more or less-- is progress. 706 00:34:26,199 --> 00:34:29,630 But if you really understand why something is conserved, 707 00:34:29,630 --> 00:34:31,989 then you can break that conservation. 708 00:34:31,989 --> 00:34:33,590 You can break that degeneracy. 709 00:34:33,590 --> 00:34:37,190 So here's my question, is it possible-- can 710 00:34:37,190 --> 00:34:46,240 we, or indeed must we, can we break or lift this degeneracy? 711 00:34:46,240 --> 00:34:48,239 So first, let me tell you what those words mean. 712 00:34:51,350 --> 00:35:00,130 So we have this gigantic degeneracy 713 00:35:00,130 --> 00:35:04,504 of the n squared of the Coulomb potential. 714 00:35:04,504 --> 00:35:06,170 Can we break that degeneracy or lift it? 715 00:35:06,170 --> 00:35:08,461 What I mean by lift the degeneracy, what I mean by lift 716 00:35:08,461 --> 00:35:09,837 the degeneracy is the following-- 717 00:35:09,837 --> 00:35:11,586 suppose I have a system that's degenerate, 718 00:35:11,586 --> 00:35:13,090 so it's got two energy eigenstates 719 00:35:13,090 --> 00:35:14,440 with the same energy eigenvalue. 720 00:35:14,440 --> 00:35:16,330 So this is energy. 721 00:35:16,330 --> 00:35:18,256 Then if I can kick the system in some way, 722 00:35:18,256 --> 00:35:20,830 or modify it, or bend it such that the energy levels are 723 00:35:20,830 --> 00:35:23,580 no longer to degenerate, one state 724 00:35:23,580 --> 00:35:25,137 has been lifted above the other. 725 00:35:25,137 --> 00:35:26,970 And so we use the word to lift a degeneracy. 726 00:35:26,970 --> 00:35:29,011 Now you can see that they're two different states 727 00:35:29,011 --> 00:35:30,300 by measuring the energies. 728 00:35:30,300 --> 00:35:32,330 It's also said to split a degeneracy, 729 00:35:32,330 --> 00:35:33,790 or to break a degeneracy. 730 00:35:33,790 --> 00:35:35,117 So those are just words. 731 00:35:35,117 --> 00:35:36,950 OK, so can we break or lift this degeneracy? 732 00:35:40,170 --> 00:35:40,792 So sure. 733 00:35:40,792 --> 00:35:42,000 How do we break a degeneracy? 734 00:35:42,000 --> 00:35:43,210 How do we lift a degeneracy? 735 00:35:46,610 --> 00:35:47,860 We need to break the symmetry. 736 00:35:54,825 --> 00:35:56,200 So if we don't have the symmetry, 737 00:35:56,200 --> 00:35:57,366 we won't get the degeneracy. 738 00:35:57,366 --> 00:35:58,630 We've seen this before. 739 00:35:58,630 --> 00:36:01,100 So first off, let's think about what 740 00:36:01,100 --> 00:36:02,600 are the two symmetries that we want? 741 00:36:02,600 --> 00:36:04,660 Well, we got this degeneracy of 2l plus 1 742 00:36:04,660 --> 00:36:06,650 from spherical symmetry. 743 00:36:06,650 --> 00:36:10,210 And we got this extra degeneracy of n squared from conservation 744 00:36:10,210 --> 00:36:11,550 of the Runge-Lenz vector. 745 00:36:11,550 --> 00:36:13,300 And conservation of the Runge-Lenz vector 746 00:36:13,300 --> 00:36:16,400 came from the fact that we had exactly the Coulomb system. 747 00:36:16,400 --> 00:36:18,940 So anything that isn't exactly the Coulomb system, 748 00:36:18,940 --> 00:36:21,470 but is still rotationally invariant, 749 00:36:21,470 --> 00:36:23,280 should still have the 2l plus 1 degeneracy, 750 00:36:23,280 --> 00:36:25,260 but not the n squared degeneracy-- 751 00:36:25,260 --> 00:36:28,060 like the harmonic oscillator, for example. 752 00:36:28,060 --> 00:36:29,800 So let's change the system. 753 00:36:29,800 --> 00:36:33,330 Let's find a small correction to the system that leaves it 754 00:36:33,330 --> 00:36:36,080 mostly the Coulomb problem, but with some small correction. 755 00:36:36,080 --> 00:36:40,830 So what comes to mind? 756 00:36:40,830 --> 00:36:45,290 So one good way to think about a physical, natural way 757 00:36:45,290 --> 00:36:47,380 to break this degeneracy, to break the symmetry, 758 00:36:47,380 --> 00:36:49,555 is to think a real system that's described by the Coulomb 759 00:36:49,555 --> 00:36:50,055 potential. 760 00:36:50,055 --> 00:36:52,330 So the system that we usually talk about is hydrogen. 761 00:36:52,330 --> 00:36:53,444 Hydrogen is the proton. 762 00:36:53,444 --> 00:36:55,360 It's got a plus charge, and it's got electron. 763 00:36:55,360 --> 00:36:58,510 And it's attractive with a minus E squared upon r. 764 00:36:58,510 --> 00:37:03,239 So in hydrogen, is this the energy operator for hydrogen? 765 00:37:03,239 --> 00:37:05,280 Is this a good description of the energy operator 766 00:37:05,280 --> 00:37:08,120 for hydrogen? 767 00:37:08,120 --> 00:37:10,632 It's a good start, but it's not exactly right, 768 00:37:10,632 --> 00:37:11,590 for a bunch of reasons. 769 00:37:11,590 --> 00:37:13,620 And in fact, in your first problem set, 770 00:37:13,620 --> 00:37:15,980 when you computed the time it takes for hydrogen, 771 00:37:15,980 --> 00:37:23,564 classically, to decay from the classical Bohr radius, 772 00:37:23,564 --> 00:37:26,230 one of the things you discovered is that the electron starts out 773 00:37:26,230 --> 00:37:30,330 being non-relativistic, but very rapidly becomes relativistic. 774 00:37:30,330 --> 00:37:34,030 And rapidly is 10 to the minus 15 seconds. 775 00:37:34,030 --> 00:37:37,850 This is a non-relativistic, p squared upon 2m kinetic energy. 776 00:37:37,850 --> 00:37:40,820 That's probably not a perfect approximation. 777 00:37:40,820 --> 00:37:48,020 So the first example is going to be relativistic corrections. 778 00:37:48,020 --> 00:37:49,510 Now, will relativistic corrections 779 00:37:49,510 --> 00:37:50,595 break rotational symmetry? 780 00:37:54,260 --> 00:37:56,370 They can't, because relativity doesn't say, 781 00:37:56,370 --> 00:37:57,580 this is the z direction. 782 00:37:57,580 --> 00:37:59,080 In fact, it does quite the opposite. 783 00:37:59,080 --> 00:38:00,621 Relativity, among other things, says, 784 00:38:00,621 --> 00:38:02,880 don't get too hung up on what's the z direction. 785 00:38:02,880 --> 00:38:05,920 But relativistic corrections will not break the symmetry. 786 00:38:05,920 --> 00:38:13,420 So if we include relativistic corrections 787 00:38:13,420 --> 00:38:20,270 to the kinetic energy, this will change E, 788 00:38:20,270 --> 00:38:21,910 change the energy operator. 789 00:38:21,910 --> 00:38:27,740 But it will preserve the rotational symmetry. 790 00:38:27,740 --> 00:38:30,899 E with l is equal to 0. 791 00:38:30,899 --> 00:38:32,690 So we won't break this rotational symmetry, 792 00:38:32,690 --> 00:38:34,160 so we'll preserve the 2l plus 1. 793 00:38:34,160 --> 00:38:36,890 So let's check to see that this is true. 794 00:38:36,890 --> 00:38:38,940 So to check to see that this is true, 795 00:38:38,940 --> 00:38:44,950 let's actually estimate out the correction to the energy 796 00:38:44,950 --> 00:38:48,640 due to the first nontrivial relativistic effect 797 00:38:48,640 --> 00:38:50,000 in the atom. 798 00:38:50,000 --> 00:38:51,970 The first nontrivial, relativistic effect 799 00:38:51,970 --> 00:38:54,220 can be calculated [INAUDIBLE] kinetic energy 800 00:38:54,220 --> 00:38:57,780 in special relativity, of a particle of mass m. 801 00:38:57,780 --> 00:38:58,650 So that's easy. 802 00:38:58,650 --> 00:39:06,130 It's the total energy is the square root of m squared 803 00:39:06,130 --> 00:39:12,842 c to the 4th plus p squared c squared minus mc squared. 804 00:39:12,842 --> 00:39:13,800 That's the rest energy. 805 00:39:13,800 --> 00:39:17,830 That's the total energy, so this is the kinetic energy. 806 00:39:17,830 --> 00:39:20,360 But this is equal to-- well, if we pull out 807 00:39:20,360 --> 00:39:21,925 a factor of m squared c squared, this 808 00:39:21,925 --> 00:39:31,550 is m c squared times the square root of 1 plus p squared over n 809 00:39:31,550 --> 00:39:36,054 squared c squared minus 1. 810 00:39:36,054 --> 00:39:37,470 And now if we Taylor expand this-- 811 00:39:37,470 --> 00:39:42,100 because if the momentum is small compared to mc squared-- or mc, 812 00:39:42,100 --> 00:39:44,850 I should say-- the momentum is small compared to mc which is, 813 00:39:44,850 --> 00:39:47,170 morally speaking, at low velocity is, 814 00:39:47,170 --> 00:39:48,870 the velocity is small compared to c, 815 00:39:48,870 --> 00:39:51,070 then this is 1 plus a small number, 816 00:39:51,070 --> 00:39:56,120 and we can Taylor expand this to get 1 plus 1/2 a small number. 817 00:39:56,120 --> 00:39:58,060 And the 1s will cancel, giving us only 1/2. 818 00:39:58,060 --> 00:40:01,630 So this is going to be equal to 1/2 of that small number, 819 00:40:01,630 --> 00:40:05,610 p squared over m squared c squared, 820 00:40:05,610 --> 00:40:07,820 times mc squared upstairs, giving us 821 00:40:07,820 --> 00:40:10,100 the first term, which is p squared upon 2m. 822 00:40:10,100 --> 00:40:12,360 That's the familiar classical kinetic energy. 823 00:40:12,360 --> 00:40:13,170 Rock on. 824 00:40:13,170 --> 00:40:14,320 What's next correction? 825 00:40:14,320 --> 00:40:16,400 The next correction is the next term for this Taylor series, 826 00:40:16,400 --> 00:40:18,110 which is going to involve this quantity squared, 827 00:40:18,110 --> 00:40:20,470 so it's going to be p to the 4th over m to the 4th, c 828 00:40:20,470 --> 00:40:22,830 the 4th times mc to the 4th. 829 00:40:22,830 --> 00:40:25,760 The coefficient is a minus 1/8, so we get minus p 830 00:40:25,760 --> 00:40:31,790 to the 4th over 8m cubed c squared. 831 00:40:36,380 --> 00:40:38,780 And that's down by a factor of m squared 832 00:40:38,780 --> 00:40:40,900 c squared from this guy, n of p squared 833 00:40:40,900 --> 00:40:42,750 upstairs-- which is what you'd get-- p 834 00:40:42,750 --> 00:40:45,080 squared m squared, over c squared, OK? 835 00:40:45,080 --> 00:40:47,890 So the correction to the kinetic energy for a given momentum p 836 00:40:47,890 --> 00:40:51,490 is to correct it down by p to the 4th term. 837 00:40:51,490 --> 00:40:52,580 Everyone cool with that? 838 00:40:52,580 --> 00:40:54,418 Yeah. 839 00:40:54,418 --> 00:40:58,150 AUDIENCE: Why do you make the assumption [INAUDIBLE]. 840 00:40:58,150 --> 00:41:00,390 PROFESSOR: Because you, like Schrodinger, 841 00:41:00,390 --> 00:41:02,580 did problem Set One. 842 00:41:02,580 --> 00:41:06,050 And problem Set One said, if you have an electron at the Bohr 843 00:41:06,050 --> 00:41:09,530 radius classically, its velocity is about 1/400 844 00:41:09,530 --> 00:41:10,910 of the speed of light. 845 00:41:10,910 --> 00:41:12,350 200? 846 00:41:12,350 --> 00:41:13,005 It's small. 847 00:41:13,005 --> 00:41:14,505 It's a couple of orders of magnitude 848 00:41:14,505 --> 00:41:15,838 smaller than the speed of light. 849 00:41:15,838 --> 00:41:19,890 So that number, that ratio there, is 1 upon 200, 850 00:41:19,890 --> 00:41:21,462 or 400-- I don't remember-- squared. 851 00:41:21,462 --> 00:41:22,920 That's a very small number, so it's 852 00:41:22,920 --> 00:41:24,210 a reasonable approximation. 853 00:41:24,210 --> 00:41:26,460 AUDIENCE: [INAUDIBLE]. 854 00:41:26,460 --> 00:41:28,510 PROFESSOR: Because what is the typical-- that's 855 00:41:28,510 --> 00:41:29,150 an excellent question. 856 00:41:29,150 --> 00:41:31,108 So why am I talking about it as if the electron 857 00:41:31,108 --> 00:41:32,600 is sitting at that radius? 858 00:41:32,600 --> 00:41:36,680 And the answer is, well, look-- what is the average momentum? 859 00:41:36,680 --> 00:41:39,299 What is the average value of p squared in, 860 00:41:39,299 --> 00:41:41,340 let's say, the ground state of the hydrogen atom? 861 00:41:41,340 --> 00:41:42,589 What's the average value of p? 862 00:41:45,650 --> 00:41:47,150 What must be the average value of p 863 00:41:47,150 --> 00:41:49,570 in the ground state of the hydrogen atom? 864 00:41:49,570 --> 00:41:52,090 Zero on two grounds-- first off, p is a vector, 865 00:41:52,090 --> 00:41:54,580 and the ground state of hydrogen is rotationally invariant. 866 00:41:54,580 --> 00:41:55,663 It doesn't break anything. 867 00:41:55,663 --> 00:41:57,630 Secondly, it's not going anywhere. 868 00:41:57,630 --> 00:41:59,588 It's a bound state, so it's just sitting there. 869 00:41:59,588 --> 00:42:01,640 It's got an average momentum of 0. 870 00:42:01,640 --> 00:42:03,980 If it carried momentum, it would be cruising. 871 00:42:03,980 --> 00:42:05,470 So it's got average momentum 0. 872 00:42:05,470 --> 00:42:07,553 OK, fine, but what's the average momentum squared? 873 00:42:07,553 --> 00:42:09,414 What's the average of p squared? 874 00:42:09,414 --> 00:42:11,080 We can actually do out that calculation. 875 00:42:11,080 --> 00:42:13,705 You're going to have to do that calculation on the problem set. 876 00:42:13,705 --> 00:42:15,315 And it's a small number. 877 00:42:15,315 --> 00:42:17,190 And it's, in fact, almost exactly what 878 00:42:17,190 --> 00:42:19,890 you'd guess classically from the Bohr radius. 879 00:42:19,890 --> 00:42:22,456 So you have to know the h bar to compute the Bohr 880 00:42:22,456 --> 00:42:24,040 radius in the first place. 881 00:42:24,040 --> 00:42:26,530 So the answer is, to the best of anyone's ability 882 00:42:26,530 --> 00:42:29,559 to define what's the momentum, or the typical scale 883 00:42:29,559 --> 00:42:31,350 of momentum, that typical scale of momentum 884 00:42:31,350 --> 00:42:32,850 for the electron in the ground state of hydrogen 885 00:42:32,850 --> 00:42:35,340 is extraordinarily small compared to the speed of light 886 00:42:35,340 --> 00:42:37,749 by a factor of 100, 200, or 400. 887 00:42:37,749 --> 00:42:39,040 Does that answer your question? 888 00:42:39,040 --> 00:42:39,270 AUDIENCE: Yes. 889 00:42:39,270 --> 00:42:40,811 PROFESSOR: It's a very good question. 890 00:42:40,811 --> 00:42:42,750 OK, so here's the kinetic energy. 891 00:42:42,750 --> 00:42:44,881 And by the way, another way to say 892 00:42:44,881 --> 00:42:46,380 that this approximation must be good 893 00:42:46,380 --> 00:42:48,080 is that we've already computed-- ignoring 894 00:42:48,080 --> 00:42:50,246 the relativistic correction-- we computed the ground 895 00:42:50,246 --> 00:42:52,070 state of energy, and it was 13.6 Ev, 896 00:42:52,070 --> 00:42:55,450 which is pretty good, since the binding energy of hydrogen 897 00:42:55,450 --> 00:42:57,370 is about 13.6 Ev. 898 00:42:57,370 --> 00:42:59,640 So apparently, that was a good approximation. 899 00:42:59,640 --> 00:43:02,160 And you might be disgusted by answer analysis like that, 900 00:43:02,160 --> 00:43:04,117 but that's what physics is. 901 00:43:04,117 --> 00:43:05,950 You write down some stupid, cockamamie model 902 00:43:05,950 --> 00:43:07,350 that you know is wrong, but if it 903 00:43:07,350 --> 00:43:10,522 does a good job of fitting the data, you declare triumph, 904 00:43:10,522 --> 00:43:11,480 and you call Stockholm. 905 00:43:11,480 --> 00:43:13,440 [LAUGHTER] 906 00:43:13,440 --> 00:43:17,300 So it depends on the problem you've just 907 00:43:17,300 --> 00:43:19,210 solved approximately. 908 00:43:19,210 --> 00:43:24,080 So here's the first correction. 909 00:43:24,080 --> 00:43:26,570 And now, what this tells us is therefore, 910 00:43:26,570 --> 00:43:29,675 the energy of hydrogen-- or at least 911 00:43:29,675 --> 00:43:31,050 a better model of hydrogen-- I'll 912 00:43:31,050 --> 00:43:37,070 call it hydrogen tilde-- is equal to E Coulomb 913 00:43:37,070 --> 00:43:43,450 minus p to the 4th over 8m cubed c squared. 914 00:43:53,540 --> 00:43:56,870 OK so this, however, we know is small compared 915 00:43:56,870 --> 00:44:00,812 to the kinetic energy, because 13.6 is pretty close to 13.6. 916 00:44:00,812 --> 00:44:03,270 Notice I didn't include the rest of the significant digits. 917 00:44:03,270 --> 00:44:07,010 So this must be in the insignificant digits. 918 00:44:07,010 --> 00:44:08,460 So this must be small. 919 00:44:08,460 --> 00:44:10,460 So here's what I want to ask-- we 920 00:44:10,460 --> 00:44:14,400 know that the energy of the Coulomb problem, nlm, 921 00:44:14,400 --> 00:44:17,660 is equal to e0 squared minus e0 squared 922 00:44:17,660 --> 00:44:21,400 over n squared, independent of l and m. 923 00:44:21,400 --> 00:44:25,090 What's the energy in this toy model of hydrogen 924 00:44:25,090 --> 00:44:29,580 that includes the first relativistic correction-- nlm. 925 00:44:29,580 --> 00:44:32,436 Well, to answer that question we have to resolve the energy 926 00:44:32,436 --> 00:44:33,560 eigenvalue equation, right? 927 00:44:33,560 --> 00:44:35,310 We have to refined the energy eigenstates, 928 00:44:35,310 --> 00:44:36,610 and we have to solve that. 929 00:44:36,610 --> 00:44:38,410 And this is going to be a much worse problem, because it's 930 00:44:38,410 --> 00:44:40,201 going to involve a p to the 4th term, which 931 00:44:40,201 --> 00:44:41,820 is going to involve four derivatives. 932 00:44:41,820 --> 00:44:44,580 That sounds horrible. 933 00:44:44,580 --> 00:44:47,040 Can anyone think of a better way to just approximate, 934 00:44:47,040 --> 00:44:50,640 estimate the magnitude and the value of the leading correction 935 00:44:50,640 --> 00:44:53,163 from this quantity? 936 00:44:53,163 --> 00:44:53,663 Yeah. 937 00:44:56,960 --> 00:45:00,865 AUDIENCE: Some variation on take an eigenvalue, 938 00:45:00,865 --> 00:45:03,933 plug it in, see what the error is, 939 00:45:03,933 --> 00:45:06,630 correct for that error-- something like that. 940 00:45:06,630 --> 00:45:08,880 PROFESSOR: There's an iterative method for doing that. 941 00:45:08,880 --> 00:45:10,880 It's a very good guess, but there's an even easier way 942 00:45:10,880 --> 00:45:11,505 to estimate it. 943 00:45:11,505 --> 00:45:15,320 That's something we will call perturbation theory when 944 00:45:15,320 --> 00:45:16,640 we do this in 805. 945 00:45:16,640 --> 00:45:19,419 And doing that systematically is exactly the right answer. 946 00:45:19,419 --> 00:45:21,710 However I want you to do just the leading part of that. 947 00:45:21,710 --> 00:45:22,110 Yeah. 948 00:45:22,110 --> 00:45:23,310 AUDIENCE: Dimensional analysis. 949 00:45:23,310 --> 00:45:25,518 PROFESSOR: Oh, dimensional analysis would be awesome. 950 00:45:25,518 --> 00:45:29,080 That's fantastic, brilliant. 951 00:45:29,080 --> 00:45:30,387 So how would you compute p? 952 00:45:34,123 --> 00:45:35,570 OK, so we didn't calculate p. 953 00:45:35,570 --> 00:45:36,560 You can do a dimensional analysis 954 00:45:36,560 --> 00:45:38,018 and get a p, because you have an e, 955 00:45:38,018 --> 00:45:39,630 you have an l, you can probably do it. 956 00:45:39,630 --> 00:45:41,004 But here's the problem with that, 957 00:45:41,004 --> 00:45:43,530 though-- this is the correct first answer to any question 958 00:45:43,530 --> 00:45:45,230 of this kind-- correct first answer 959 00:45:45,230 --> 00:45:47,826 is plug in dimensional analysis, and get an estimate 960 00:45:47,826 --> 00:45:48,950 for the order of magnitude. 961 00:45:48,950 --> 00:45:52,030 But we want more than the order of magnitude here. 962 00:45:52,030 --> 00:45:54,760 We're interested in splitting the degeneracy 963 00:45:54,760 --> 00:45:56,154 due to this interaction. 964 00:45:56,154 --> 00:45:58,570 So what that means is we want to see that different values 965 00:45:58,570 --> 00:46:01,330 use of l lead to different energy. 966 00:46:01,330 --> 00:46:03,560 What we care about is the l dependence 967 00:46:03,560 --> 00:46:04,980 that comes out of this correction. 968 00:46:04,980 --> 00:46:07,230 And we're not going to be able to get the l dependence 969 00:46:07,230 --> 00:46:10,390 from dimensional analysis, because l is an integer. 970 00:46:10,390 --> 00:46:12,300 It is dimensionless. 971 00:46:12,300 --> 00:46:14,750 Yeah, so that will give you the correct magnitude, 972 00:46:14,750 --> 00:46:17,220 and that is the first thing you do. 973 00:46:17,220 --> 00:46:18,934 What's the second thing you do? 974 00:46:18,934 --> 00:46:20,600 AUDIENCE: Stick in the expectation value 975 00:46:20,600 --> 00:46:21,365 of p to the 4th. 976 00:46:21,365 --> 00:46:22,240 PROFESSOR: Excellent. 977 00:46:22,240 --> 00:46:24,670 Let's just stick in the expectation value p to the 4th. 978 00:46:24,670 --> 00:46:26,420 That's not exactly the right thing to do, 979 00:46:26,420 --> 00:46:28,987 but if this is small, that's probably pretty reasonable. 980 00:46:28,987 --> 00:46:31,320 And what you'll find when you do it systematically using 981 00:46:31,320 --> 00:46:33,520 perturbation theory, as was pointed out earlier, 982 00:46:33,520 --> 00:46:36,780 we can take the entire system, the entire energy eigenvalue 983 00:46:36,780 --> 00:46:38,800 equation-- we can think of this 1 over m 984 00:46:38,800 --> 00:46:41,439 cubed c squared as a small number, 985 00:46:41,439 --> 00:46:43,730 and we can do perturbation theory on that small number. 986 00:46:43,730 --> 00:46:46,120 We can Taylor expand everything in that small number. 987 00:46:46,120 --> 00:46:48,690 And when you Taylor expand the exact equation, what 988 00:46:48,690 --> 00:46:52,470 you discover is the leading term in that Taylor expansion 989 00:46:52,470 --> 00:46:55,040 is the expectation value p to the 4th over 8m cubed 990 00:46:55,040 --> 00:46:56,260 c squared. 991 00:46:56,260 --> 00:46:57,970 OK, so this is a fantastic guess. 992 00:46:57,970 --> 00:46:59,490 OK, so let's compute it. 993 00:46:59,490 --> 00:47:02,930 So E hydrogen is going to be equal to E nlm, which I'm just 994 00:47:02,930 --> 00:47:06,020 to write as minus E0 over m squared. 995 00:47:23,600 --> 00:47:27,210 OK, so what is the correct answer here? 996 00:47:27,210 --> 00:47:30,450 So it's this quantity minus-- so this 997 00:47:30,450 --> 00:47:31,940 is making it more tightly bound. 998 00:47:31,940 --> 00:47:33,790 Notice the sign, because p to the 4th 999 00:47:33,790 --> 00:47:36,770 is strictly positive if p is real. 1000 00:47:36,770 --> 00:47:38,670 So minus E0 over m squared, and if you go out 1001 00:47:38,670 --> 00:47:40,711 and you do this expectation value, which I really 1002 00:47:40,711 --> 00:47:43,839 should have put on the problem set-- oh, I still can. 1003 00:47:43,839 --> 00:47:46,380 It hasn't been posted yet, so it'll be posted this afternoon. 1004 00:47:46,380 --> 00:47:48,600 But I won't. 1005 00:47:48,600 --> 00:47:51,450 It's not that bad, actually. 1006 00:47:51,450 --> 00:47:53,680 [LAUGHTER] 1007 00:47:53,680 --> 00:47:54,923 Matt, what do you think? 1008 00:47:54,923 --> 00:47:56,855 AUDIENCE: No! 1009 00:47:56,855 --> 00:48:00,240 [LAUGHS] 1010 00:48:00,240 --> 00:48:02,680 PROFESSOR: We'll discuss this afterwards, OK? 1011 00:48:02,680 --> 00:48:05,299 Which may or may not appear on your problem set. 1012 00:48:05,299 --> 00:48:07,840 So if you go ahead and you do this calculation, what you find 1013 00:48:07,840 --> 00:48:12,497 is the correction is En squared, so E0 squared over n 1014 00:48:12,497 --> 00:48:14,580 to the 4th-- this time it really is n to the 4th-- 1015 00:48:14,580 --> 00:48:16,370 and now you have to worry about dimensional analysis. 1016 00:48:16,370 --> 00:48:17,400 This is two energies. 1017 00:48:17,400 --> 00:48:20,806 We want one energy, divided by-- wait for it-- mc squared, 1018 00:48:20,806 --> 00:48:22,430 because it's a relativistic correction. 1019 00:48:22,430 --> 00:48:24,110 That shouldn't be too surprising. 1020 00:48:24,110 --> 00:48:27,765 Times the quantity 4n over-- and here 1021 00:48:27,765 --> 00:48:32,932 where life gets awesome-- l plus 1/2 minus 3. 1022 00:48:32,932 --> 00:48:37,550 [LAUGHTER] 1023 00:48:37,550 --> 00:48:40,990 PROFESSOR: You laugh, but I feel delighted 1024 00:48:40,990 --> 00:48:42,480 by this for so many reasons. 1025 00:48:42,480 --> 00:48:44,090 One is, that's beautiful. 1026 00:48:44,090 --> 00:48:45,650 What a crazy combination of symbols. 1027 00:48:45,650 --> 00:48:47,100 But the second and more important 1028 00:48:47,100 --> 00:48:52,390 one is, look it has an l in it, and the l is downstairs, right? 1029 00:48:52,390 --> 00:48:53,891 That's good. 1030 00:48:53,891 --> 00:48:55,140 It has more leverage that way. 1031 00:48:55,140 --> 00:48:57,960 But so we've got this l appearing, 1032 00:48:57,960 --> 00:49:01,780 so the energy depends on l. 1033 00:49:01,780 --> 00:49:03,200 Good, we've broken the degeneracy. 1034 00:49:03,200 --> 00:49:05,957 Different values of l give different energies. 1035 00:49:05,957 --> 00:49:07,165 So let's do that graphically. 1036 00:49:12,510 --> 00:49:17,130 So in particular, what this tells 1037 00:49:17,130 --> 00:49:23,780 us is that now while the Coulomb energy commuted 1038 00:49:23,780 --> 00:49:26,950 with the Runge-Lenz vector, it must be true 1039 00:49:26,950 --> 00:49:29,960 that the energy of this model of hydrogen, 1040 00:49:29,960 --> 00:49:33,620 which includes the first relativistic correction, 1041 00:49:33,620 --> 00:49:36,960 does not commute with the Runge-Lenz vector. 1042 00:49:36,960 --> 00:49:38,810 And indeed, if you check this-- which 1043 00:49:38,810 --> 00:49:42,160 is effortful, but not intellectually 1044 00:49:42,160 --> 00:49:45,870 difficult-- indeed, it doesn't vanish. 1045 00:49:45,870 --> 00:49:48,540 And the failure of this conservation equation, 1046 00:49:48,540 --> 00:49:52,430 the failure this commutational relation to vanish, 1047 00:49:52,430 --> 00:49:54,720 tell us that it's possible for the system 1048 00:49:54,720 --> 00:49:55,760 to not be degenerate. 1049 00:49:55,760 --> 00:49:58,430 And lo and behold, it is not degenerate. 1050 00:49:58,430 --> 00:49:59,680 OK? 1051 00:49:59,680 --> 00:50:00,500 Yeah. 1052 00:50:00,500 --> 00:50:03,120 AUDIENCE: So I just wanted to clarify something that-- 1053 00:50:03,120 --> 00:50:06,424 did you say before that the actual first term when you 1054 00:50:06,424 --> 00:50:08,532 encounter relativity just includes substituting 1055 00:50:08,532 --> 00:50:11,516 the non-relativistic p to the 4th? 1056 00:50:11,516 --> 00:50:12,140 PROFESSOR: Yes. 1057 00:50:12,140 --> 00:50:13,010 Sorry, I should have said that. 1058 00:50:13,010 --> 00:50:15,190 There are other things are important for relativity 1059 00:50:15,190 --> 00:50:16,648 that we'll come to in a little bit. 1060 00:50:16,648 --> 00:50:18,700 But if you just look at the kinetic energy, 1061 00:50:18,700 --> 00:50:21,172 this is the first non-trivial correction. 1062 00:50:21,172 --> 00:50:23,020 Yeah. 1063 00:50:23,020 --> 00:50:25,330 AUDIENCE: So we still have the 2l plus 1 degeneracy. 1064 00:50:25,330 --> 00:50:27,950 PROFESSOR: We still have the 2l plus 1 degeneracy, exactly. 1065 00:50:27,950 --> 00:50:31,796 Because does this correction violate rotational invariance? 1066 00:50:31,796 --> 00:50:32,670 No, that's p squared. 1067 00:50:32,670 --> 00:50:33,961 That's a scalar under rotation. 1068 00:50:33,961 --> 00:50:36,070 It's just the number, not a vector. 1069 00:50:36,070 --> 00:50:42,320 But E still commutes with l. 1070 00:50:42,320 --> 00:50:45,310 And you see that because this is independent of little m. 1071 00:50:45,310 --> 00:50:47,160 This is the mass of the electron. 1072 00:50:47,160 --> 00:50:49,460 So this is independent of the quantum number little m. 1073 00:50:49,460 --> 00:50:53,630 It only depends on the quantum number little l. 1074 00:50:53,630 --> 00:50:55,220 So indeed, it is still degenerate. 1075 00:50:55,220 --> 00:50:58,997 So let's go through this and see the degeneracy explicitly. 1076 00:50:58,997 --> 00:51:01,330 And let me do that by drawing you the following diagram. 1077 00:51:01,330 --> 00:51:08,780 So what happens if we have E1, which is the ground state? 1078 00:51:08,780 --> 00:51:13,670 It has some energy which is negative, which is minus E0. 1079 00:51:13,670 --> 00:51:17,700 And when we include this first correction, what does it do? 1080 00:51:17,700 --> 00:51:19,542 Well, it decreases it by some amount. 1081 00:51:22,640 --> 00:51:23,962 Everyone cool with that? 1082 00:51:23,962 --> 00:51:25,170 It doesn't change the number. 1083 00:51:25,170 --> 00:51:26,860 It doesn't split any degeneracy, because there was only 1084 00:51:26,860 --> 00:51:28,360 one ground state in the first place, 1085 00:51:28,360 --> 00:51:30,590 because l had to be 0 when n is 1. 1086 00:51:30,590 --> 00:51:33,690 l has to be one less than n or smaller. 1087 00:51:33,690 --> 00:51:35,780 And if n is 1, l has to be 0. 1088 00:51:35,780 --> 00:51:38,030 It can't be any smaller than 0, so it's 0. 1089 00:51:38,030 --> 00:51:40,530 There's just a one state, with degeneracy 2, 0 1090 00:51:40,530 --> 00:51:42,850 plus 1, which is 1. 1091 00:51:42,850 --> 00:51:43,760 OK so that's fine. 1092 00:51:43,760 --> 00:51:46,400 It just changes the ground state by some amount. 1093 00:51:46,400 --> 00:51:48,380 Let's consider the second excited level. 1094 00:51:48,380 --> 00:51:50,741 So in the second excited level, we have the states-- 1095 00:51:50,741 --> 00:51:52,490 and let me name this slightly differently, 1096 00:51:52,490 --> 00:51:54,170 so this has energy minus E0. 1097 00:51:54,170 --> 00:52:00,490 The ground state had l equals 0, and m equals 0. 1098 00:52:00,490 --> 00:52:02,370 It had to. 1099 00:52:02,370 --> 00:52:04,770 For the second excited state, n equals 2, 1100 00:52:04,770 --> 00:52:08,180 how many states are there? 1101 00:52:08,180 --> 00:52:12,470 2 squared is 4, so there are four states, two, three, four. 1102 00:52:12,470 --> 00:52:14,640 And what are their values of l and m? 1103 00:52:14,640 --> 00:52:18,660 Well, if n is 2, l can be either 0 or 1. 1104 00:52:18,660 --> 00:52:20,650 If it's 0, then m has to be 0. 1105 00:52:20,650 --> 00:52:23,455 If it's 1, m could be 1, or it could be 0, 1106 00:52:23,455 --> 00:52:27,170 or it could be minus 1. 1107 00:52:27,170 --> 00:52:29,700 So this, again, is lm, lm, lm. 1108 00:52:29,700 --> 00:52:33,280 So those are the four states with energy E2 in the hydrogen 1109 00:52:33,280 --> 00:52:33,986 atom. 1110 00:52:33,986 --> 00:52:36,360 And now when we ask, what are the energies of these three 1111 00:52:36,360 --> 00:52:40,320 states, when we include this so-called fine structure, what 1112 00:52:40,320 --> 00:52:41,660 is this going to do? 1113 00:52:41,660 --> 00:52:43,940 Well, the ground state is going to get pushed down 1114 00:52:43,940 --> 00:52:44,600 by some amount. 1115 00:52:47,820 --> 00:52:50,290 I should draw this differently to make it more vivid. 1116 00:52:52,957 --> 00:52:55,415 This gets pushed down by some amount, with some value of l. 1117 00:52:55,415 --> 00:52:57,330 l here is 0. 1118 00:52:57,330 --> 00:52:59,250 But for these three states, which 1119 00:52:59,250 --> 00:53:01,420 all have the same value of little l, 1, 1120 00:53:01,420 --> 00:53:03,360 the denominator is larger. 1121 00:53:03,360 --> 00:53:06,260 So the amount by which it gets pushed down is less. 1122 00:53:06,260 --> 00:53:08,384 So these three states get pushed down less, 1123 00:53:08,384 --> 00:53:09,800 but the same amount as each other. 1124 00:53:13,830 --> 00:53:15,730 That cool? 1125 00:53:15,730 --> 00:53:22,350 So these three states-- degeneracy is 3, 1126 00:53:22,350 --> 00:53:24,800 and this state degeneracy is 1. 1127 00:53:24,800 --> 00:53:29,410 Now, notice that 1 plus 3 is equal to 4 1128 00:53:29,410 --> 00:53:32,130 is equal to n squared. 1129 00:53:32,130 --> 00:53:37,470 So these states split into multiplets-- groups 1130 00:53:37,470 --> 00:53:40,135 of states-- which have to have the same energy 1131 00:53:40,135 --> 00:53:42,214 by rotational invariance. 1132 00:53:42,214 --> 00:53:43,630 If they have l equas 1, there must 1133 00:53:43,630 --> 00:53:46,260 be three states with the same energy-- there must be. 1134 00:53:46,260 --> 00:53:49,020 And states with l equals 0, there's just the one state. 1135 00:53:51,740 --> 00:53:54,520 And if we did the same thing for E3, 1136 00:53:54,520 --> 00:53:56,560 let's just think about what would happen. 1137 00:53:56,560 --> 00:53:58,140 So I'm just going to write the l. 1138 00:53:58,140 --> 00:53:59,431 I'm not going to include the m. 1139 00:53:59,431 --> 00:54:03,940 We have the l equals 0 times 1. 1140 00:54:03,940 --> 00:54:07,500 We have the l equals 1 state, times 3. 1141 00:54:07,500 --> 00:54:09,060 And we have l equals 2 state. 1142 00:54:09,060 --> 00:54:12,180 And how many states are there in the l equals 2 state? 1143 00:54:12,180 --> 00:54:14,180 Times 5, good. 1144 00:54:14,180 --> 00:54:16,860 And again, this gets pushed down to some value. 1145 00:54:16,860 --> 00:54:18,360 This gets pushed down to some value. 1146 00:54:18,360 --> 00:54:20,800 This gets pushed down to some value, 1147 00:54:20,800 --> 00:54:24,012 where this is one, three, and five. 1148 00:54:24,012 --> 00:54:25,470 And again, these add up to 9, which 1149 00:54:25,470 --> 00:54:27,180 is 3 squared-- the degeneracy. 1150 00:54:27,180 --> 00:54:29,930 So by turning on the relativistic corrections, 1151 00:54:29,930 --> 00:54:31,170 we break this degeneracy. 1152 00:54:31,170 --> 00:54:32,170 We lift this degeneracy. 1153 00:54:32,170 --> 00:54:38,104 You break the symmetry, you lift the degeneracy. 1154 00:54:38,104 --> 00:54:39,520 Now, unfortunately, this isn't one 1155 00:54:39,520 --> 00:54:41,700 we can control by turning a dial in the lab. 1156 00:54:41,700 --> 00:54:44,140 No experiment that I can do in the lab 1157 00:54:44,140 --> 00:54:46,160 will change the value of the speed of light. 1158 00:54:48,425 --> 00:54:49,925 Well, actually there are experiments 1159 00:54:49,925 --> 00:54:51,966 that will change the value of the speed of light. 1160 00:54:51,966 --> 00:54:55,269 If you work in medium, the effective speed of light 1161 00:54:55,269 --> 00:54:55,810 is different. 1162 00:54:55,810 --> 00:54:57,804 So that's going to change relativistic effects. 1163 00:54:57,804 --> 00:54:58,720 So that's interesting. 1164 00:54:58,720 --> 00:55:01,178 That tells you that you can actually vary this continuously 1165 00:55:01,178 --> 00:55:04,220 if you can vary continuously the index of refraction 1166 00:55:04,220 --> 00:55:06,569 of the ambient system. 1167 00:55:06,569 --> 00:55:08,610 I invite you to think about how you might do that 1168 00:55:08,610 --> 00:55:10,925 experimentally, because it's a fun question. 1169 00:55:13,940 --> 00:55:16,150 OK, however, what I'd really like 1170 00:55:16,150 --> 00:55:19,660 is something where Professor Evan in his lab 1171 00:55:19,660 --> 00:55:26,730 can actually turn a dial, and change the spectrum 1172 00:55:26,730 --> 00:55:28,870 in a way that explicitly breaks the symmetry. 1173 00:55:28,870 --> 00:55:30,350 Meanwhile, I also want an example 1174 00:55:30,350 --> 00:55:32,570 that breaks the rotational invariance. 1175 00:55:32,570 --> 00:55:34,910 I want to break the 2l plus 1 states up into 2l 1176 00:55:34,910 --> 00:55:36,550 plus 1 states with different energies. 1177 00:55:36,550 --> 00:55:38,876 I want to make them depend on the angular 1178 00:55:38,876 --> 00:55:40,000 momentum in some direction. 1179 00:55:40,000 --> 00:55:42,095 What must I do in order to lift that degeneracy? 1180 00:55:47,670 --> 00:55:49,120 AUDIENCE: Add a magnetic field. 1181 00:55:49,120 --> 00:55:49,510 PROFESSOR: Fantastic. 1182 00:55:49,510 --> 00:55:51,200 If you add a magnetic field, why is that 1183 00:55:51,200 --> 00:55:52,449 going to break the degeneracy? 1184 00:55:56,194 --> 00:55:57,110 AUDIENCE: [INAUDIBLE]. 1185 00:56:02,255 --> 00:56:03,630 PROFESSOR: Give me an answer that 1186 00:56:03,630 --> 00:56:05,610 depends, not on your knowledge of chemistry, 1187 00:56:05,610 --> 00:56:09,860 but give me an answer that depends on symmetries. 1188 00:56:09,860 --> 00:56:11,342 Why doe turning on a magnetic field 1189 00:56:11,342 --> 00:56:13,800 have the potential to split the degeneracy of the 2l plus 1 1190 00:56:13,800 --> 00:56:14,120 states? 1191 00:56:14,120 --> 00:56:15,840 AUDIENCE: You don't have rotational symmetry. 1192 00:56:15,840 --> 00:56:16,830 PROFESSOR: Thank you, exactly. 1193 00:56:16,830 --> 00:56:18,680 You have no rotational symmetry, exactly. 1194 00:56:18,680 --> 00:56:19,760 So that's pretty good. 1195 00:56:19,760 --> 00:56:29,330 So let's break the 2l plus 1 by breaking rotational symmetry. 1196 00:56:29,330 --> 00:56:32,672 There are many ways we could break rotational symmetry. 1197 00:56:32,672 --> 00:56:34,130 We could turn on an electric field. 1198 00:56:34,130 --> 00:56:35,910 We could turn on a magnetic field. 1199 00:56:35,910 --> 00:56:39,597 We could put a shovel in the room. 1200 00:56:39,597 --> 00:56:41,930 But anything that breaks rotational symmetry will do it, 1201 00:56:41,930 --> 00:56:44,250 but a very convenient way is to do it with a magnetic field. 1202 00:56:44,250 --> 00:56:45,958 Magnetic field is particularly beautiful. 1203 00:56:45,958 --> 00:56:50,060 So this is effect is now call the Zeeman effect. 1204 00:56:50,060 --> 00:56:54,190 I believe it was Faraday-- one of 1205 00:56:54,190 --> 00:56:59,580 the classic electromagnetists of the 19th century attempted 1206 00:56:59,580 --> 00:57:02,770 to look at the spectral lines coming off glowing gas, 1207 00:57:02,770 --> 00:57:04,790 and see if they changed by turning 1208 00:57:04,790 --> 00:57:06,407 on an electric or a magnetic field. 1209 00:57:06,407 --> 00:57:07,990 It turns out to be a tricky experiment 1210 00:57:07,990 --> 00:57:10,130 to do, because you need to have a very well-controlled 1211 00:57:10,130 --> 00:57:11,380 magnetic field, otherwise what you 1212 00:57:11,380 --> 00:57:12,790 get is just a bunch of schmutz. 1213 00:57:12,790 --> 00:57:15,380 So to get a clean spectrum, you have 1214 00:57:15,380 --> 00:57:18,550 to have a very uniform magnetic field, which is not trivial. 1215 00:57:18,550 --> 00:57:19,800 It has to be time independent. 1216 00:57:19,800 --> 00:57:23,477 So Zeeman, at the very end of the 19th century-- in fact, 1217 00:57:23,477 --> 00:57:25,310 this experiment was done, I believe, in '96. 1218 00:57:28,790 --> 00:57:32,170 So I think the paper came out in '97 1219 00:57:32,170 --> 00:57:34,455 but the experiment was done in '96. 1220 00:57:34,455 --> 00:57:35,580 Zeeman did this experiment. 1221 00:57:35,580 --> 00:57:37,210 He said look, if i turn on a magnetic field 1222 00:57:37,210 --> 00:57:39,418 next to a hot, glowing gas and look the spectrum that 1223 00:57:39,418 --> 00:57:43,060 comes off, anything happen to it? 1224 00:57:43,060 --> 00:57:46,931 Why would you think that anything might happen? 1225 00:57:46,931 --> 00:57:48,930 And the answer to this one is really quite nice. 1226 00:57:48,930 --> 00:57:51,130 Look, if we compute the energy, the energy 1227 00:57:51,130 --> 00:57:53,710 is going to be equal to the Coulomb 1228 00:57:53,710 --> 00:57:57,030 energy plus a correction, which is the following-- 1229 00:57:57,030 --> 00:58:00,540 if you think of the atom as an electron that carries 1230 00:58:00,540 --> 00:58:03,599 some angular momentum in the presence of a proton, 1231 00:58:03,599 --> 00:58:06,140 if you think of the electron as a thing carrying some angular 1232 00:58:06,140 --> 00:58:07,681 momentum, and a thing with a charge-- 1233 00:58:07,681 --> 00:58:09,940 it has a charge and angular momentum. 1234 00:58:09,940 --> 00:58:12,100 It's going to have magnetic moment, mu. 1235 00:58:12,100 --> 00:58:13,800 And the magnetic moment is going to be 1236 00:58:13,800 --> 00:58:21,000 equal to the current of the charge times the area swept out 1237 00:58:21,000 --> 00:58:24,220 divided by c, because we work with sensible units. 1238 00:58:24,220 --> 00:58:36,750 And this is equal to minus-- for the case of the atom-- 1239 00:58:36,750 --> 00:58:41,180 minus mu, the magnetic moment of the system 1240 00:58:41,180 --> 00:58:43,920 dotted into the magnetic field that you turn on 1241 00:58:43,920 --> 00:58:44,670 in the background. 1242 00:58:44,670 --> 00:58:46,420 So this is a background magnetic field. 1243 00:58:46,420 --> 00:58:48,049 I am turning on an electromagnet, which 1244 00:58:48,049 --> 00:58:50,090 induces a magnetic field in some known direction, 1245 00:58:50,090 --> 00:58:51,670 with some known magnitude. 1246 00:58:51,670 --> 00:58:53,630 So my dial is the magnitude of that B field. 1247 00:58:53,630 --> 00:58:55,920 I could make it large, or I could make it small. 1248 00:58:55,920 --> 00:58:58,010 And it's in a known direction. 1249 00:58:58,010 --> 00:59:02,034 And the point is, our electron because it's orbiting, 1250 00:59:02,034 --> 00:59:03,700 because it carries some angular momentum 1251 00:59:03,700 --> 00:59:06,506 and it carries some in charge, has some magnetic moment. 1252 00:59:06,506 --> 00:59:08,130 So it behaves like a little bar magnet. 1253 00:59:08,130 --> 00:59:10,200 And that bar magnet wants to anti-align with the background 1254 00:59:10,200 --> 00:59:11,800 magnetic field that I've turned on. 1255 00:59:11,800 --> 00:59:14,550 So there's an energy penalty to not being aligned. 1256 00:59:14,550 --> 00:59:21,550 This is the usual dipole-dipole potential from electrostatics. 1257 00:59:21,550 --> 00:59:29,160 OK and mu here is equal to minus E over 2mc-- angular momentum. 1258 00:59:32,920 --> 00:59:36,370 So this can be computed from a simple model of the atom. 1259 00:59:39,540 --> 00:59:40,930 Whatever, it's easy to compute. 1260 00:59:40,930 --> 00:59:44,201 So this is slightly annoying, because if E 1261 00:59:44,201 --> 00:59:46,700 is in some arbitrary direction, let's just simplify our life 1262 00:59:46,700 --> 00:59:50,392 and say that's equal to E Coulomb minus mu B. 1263 00:59:50,392 --> 00:59:52,350 And I want to turn on a magnetic field of known 1264 00:59:52,350 --> 00:59:53,650 amplitude in a known direction. 1265 00:59:53,650 --> 00:59:54,899 Let's call it the z direction. 1266 00:59:54,899 --> 00:59:57,480 So this is mu z. 1267 00:59:57,480 --> 01:00:03,040 But this mu z, that's just mu z is lz times minus E over 2mc. 1268 01:00:03,040 --> 01:00:09,190 So this is going to be plus E over 2mc, 1269 01:00:09,190 --> 01:00:11,874 Bz-- I'm going to put this all together, Bz Lz. 1270 01:00:16,440 --> 01:00:18,321 Everyone cool with that? 1271 01:00:18,321 --> 01:00:19,820 So where this is coming from, again, 1272 01:00:19,820 --> 01:00:23,000 is just the angular momentum is telling you the velocity. 1273 01:00:23,000 --> 01:00:25,380 The velocity is telling you the current. 1274 01:00:25,380 --> 01:00:27,790 And then the area is the radius. 1275 01:00:27,790 --> 01:00:33,581 So that's what's giving you the radius and the angular 1276 01:00:33,581 --> 01:00:34,080 momentum. 1277 01:00:36,820 --> 01:00:39,907 So this is our energy. 1278 01:00:39,907 --> 01:00:42,115 What does this tell you about the energy eigenvalues? 1279 01:00:45,340 --> 01:00:47,340 They depend on m, right? 1280 01:00:47,340 --> 01:00:49,250 And do we have to solve the energy eigenvalue 1281 01:00:49,250 --> 01:00:51,080 equation again? 1282 01:00:51,080 --> 01:00:54,060 No, because our energy eigenfunctions of Ec 1283 01:00:54,060 --> 01:00:55,950 are proportional to ylm, so they're already 1284 01:00:55,950 --> 01:00:59,300 eigenfunctions of lz. 1285 01:00:59,300 --> 01:01:03,050 So when I take this beast, and I act on the wave functions 1286 01:01:03,050 --> 01:01:08,440 I erased, but on the 1 upon little r, capital R, sub nl, y 1287 01:01:08,440 --> 01:01:10,810 sub lm-- when I act on the ylm's, this 1288 01:01:10,810 --> 01:01:14,880 is just going to give me h bar m, right? 1289 01:01:14,880 --> 01:01:18,877 So this is just a constant when acting on the Coulomb energy 1290 01:01:18,877 --> 01:01:19,460 eigenfunction. 1291 01:01:19,460 --> 01:01:20,460 So this is a much easier problem. 1292 01:01:20,460 --> 01:01:22,793 That way, we don't even have to make any approximations. 1293 01:01:22,793 --> 01:01:29,280 We can just say exactly energy Zeeman is equal to minus E0 1294 01:01:29,280 --> 01:01:40,960 upon n squared, nlm plus E h bar upon 2mc, 1295 01:01:40,960 --> 01:01:42,650 Bz-- the background magnetic field, 1296 01:01:42,650 --> 01:01:47,730 and this is the m of electron mass-- times little m. 1297 01:01:47,730 --> 01:01:49,720 And I'm going to write little m sub l, just 1298 01:01:49,720 --> 01:01:56,676 to distinguish little m sub l from m sub E. 1299 01:01:56,676 --> 01:01:58,050 So when we turn on the prediction 1300 01:01:58,050 --> 01:02:02,280 here, when we turn on the magnetic field a la Zeeman, 1301 01:02:02,280 --> 01:02:04,590 our energy levels will get split according 1302 01:02:04,590 --> 01:02:07,160 to the different values of m. 1303 01:02:07,160 --> 01:02:10,812 Now if instead of working with the pure Coulomb system, 1304 01:02:10,812 --> 01:02:12,270 if we were in fact a little clever, 1305 01:02:12,270 --> 01:02:14,490 and we'd already noticed that there's a fine structure-- 1306 01:02:14,490 --> 01:02:15,600 because they already knew that there 1307 01:02:15,600 --> 01:02:17,100 was a fine structure at this point-- 1308 01:02:17,100 --> 01:02:20,380 they could see the different spectra 1309 01:02:20,380 --> 01:02:24,936 in lines coming off of hot gas. 1310 01:02:24,936 --> 01:02:26,310 So it's just an experimental fact 1311 01:02:26,310 --> 01:02:27,893 you can't turn off the fine structure. 1312 01:02:27,893 --> 01:02:33,507 This is going to be equal to e nlm of our correction 1313 01:02:33,507 --> 01:02:35,840 to the hydrogen atom, with the fine structure corrected. 1314 01:02:35,840 --> 01:02:39,670 So the l equals 1 states and the l equals 0 states are split. 1315 01:02:39,670 --> 01:02:42,967 Let's focus, for example, on E2. 1316 01:02:42,967 --> 01:02:44,550 So these guys have a different energy. 1317 01:02:44,550 --> 01:02:46,341 The l equals 1 guys have a different energy 1318 01:02:46,341 --> 01:02:47,970 than the l equals 0 guys. 1319 01:02:47,970 --> 01:02:49,990 What does this predict is going to happen 1320 01:02:49,990 --> 01:02:52,240 if I look at my spectral line? 1321 01:02:52,240 --> 01:02:54,930 So what this predicts is the following-- 1322 01:02:54,930 --> 01:02:57,630 it says that if you have some spectrum that 1323 01:02:57,630 --> 01:03:08,660 looks like this-- so this is as a function of wavelength 1324 01:03:08,660 --> 01:03:12,650 and i'm looking literally at what comes off a prism. 1325 01:03:12,650 --> 01:03:14,050 I'm looking at spectrum lines. 1326 01:03:14,050 --> 01:03:15,740 And I'll have a line here. 1327 01:03:15,740 --> 01:03:17,900 And these are going to be the l equals 0 states. 1328 01:03:17,900 --> 01:03:20,770 This is n equals 2. 1329 01:03:20,770 --> 01:03:22,860 And then there's the l equals 1 states. 1330 01:03:22,860 --> 01:03:25,820 This is with zero magnetic field-- Bz equals 0. 1331 01:03:29,820 --> 01:03:38,040 If we turn on a magnetic field, Bz not equal to zero, 1332 01:03:38,040 --> 01:03:39,720 what should we see? 1333 01:03:39,720 --> 01:03:43,140 Well, what happens to this state? 1334 01:03:43,140 --> 01:03:46,610 Nothing-- it's the same thing, because the m is strictly 0, 1335 01:03:46,610 --> 01:03:48,050 so it's the same value. 1336 01:03:48,050 --> 01:03:50,850 And this guy, what happens to it? 1337 01:03:50,850 --> 01:03:53,242 The m equals 0 state is the same. 1338 01:03:53,242 --> 01:03:54,950 But then it breaks up-- as everyone said, 1339 01:03:54,950 --> 01:03:57,555 it breaks up into three lines that are equally spaced. 1340 01:04:01,260 --> 01:04:03,710 It breaks up into three lines, because one of them 1341 01:04:03,710 --> 01:04:06,750 has m which is plus 1, one of them has m which is 0, 1342 01:04:06,750 --> 01:04:08,540 and one is m is minus 1. 1343 01:04:08,540 --> 01:04:10,770 So energies will increase or decrease. 1344 01:04:10,770 --> 01:04:14,170 Let's say this is frequency. 1345 01:04:14,170 --> 01:04:15,290 So we get this splitting. 1346 01:04:15,290 --> 01:04:18,460 And another way to draw this is as a function of B, 1347 01:04:18,460 --> 01:04:21,570 let's draw the energy levels. 1348 01:04:21,570 --> 01:04:22,680 So there's the 1. 1349 01:04:22,680 --> 01:04:24,150 There's the 0. 1350 01:04:24,150 --> 01:04:25,960 And this stays constant. 1351 01:04:25,960 --> 01:04:30,035 And these guys-- one is constant, one increases, 1352 01:04:30,035 --> 01:04:30,785 and one decreases. 1353 01:04:34,006 --> 01:04:36,130 When you start getting into strong magnetic fields, 1354 01:04:36,130 --> 01:04:38,025 weird things happen, like these guys cross, 1355 01:04:38,025 --> 01:04:40,930 and all sorts of subtle things can happen. 1356 01:04:40,930 --> 01:04:42,930 But let's ignore the strong magnetic field case, 1357 01:04:42,930 --> 01:04:44,010 and just focus on this. 1358 01:04:44,010 --> 01:04:46,290 So we see that these guys split. 1359 01:04:46,290 --> 01:04:51,690 So this is what the Schrodinger theory, or the full story 1360 01:04:51,690 --> 01:04:54,000 would predict-- it came a little bit after this, 1361 01:04:54,000 --> 01:04:57,097 but this experiment was done in '96, 1362 01:04:57,097 --> 01:04:59,680 so there's actually kind of an entertaining story about people 1363 01:04:59,680 --> 01:05:01,560 trying to explain this effect classically. 1364 01:05:01,560 --> 01:05:03,010 Well, look you could have some angular 1365 01:05:03,010 --> 01:05:04,445 momentum in the z direction, some angular 1366 01:05:04,445 --> 01:05:06,030 momentum in the x direction, some angular momentum in the y 1367 01:05:06,030 --> 01:05:06,810 direction. 1368 01:05:06,810 --> 01:05:09,018 The part that has angular momentum in the x direction 1369 01:05:09,018 --> 01:05:11,170 doesn't do anything, or in the z direction. 1370 01:05:11,170 --> 01:05:12,795 The part with the x and part with the y 1371 01:05:12,795 --> 01:05:13,982 can have different polars. 1372 01:05:13,982 --> 01:05:15,690 There's a whole classical song and dance. 1373 01:05:15,690 --> 01:05:16,290 It's crazy. 1374 01:05:16,290 --> 01:05:17,230 It doesn't make any sense. 1375 01:05:17,230 --> 01:05:18,980 But that's because they were trying to classical mechanics 1376 01:05:18,980 --> 01:05:20,840 to do quantum mechanics, which doesn't work. 1377 01:05:20,840 --> 01:05:22,180 So I'm not even going to bother explaining 1378 01:05:22,180 --> 01:05:23,180 what they were thinking. 1379 01:05:23,180 --> 01:05:26,660 But anyway, so people came up with a sort of classical crutch 1380 01:05:26,660 --> 01:05:29,720 to say, well that's not totally insane. 1381 01:05:29,720 --> 01:05:31,870 So this is what Zeeman got, for example, 1382 01:05:31,870 --> 01:05:33,840 by looking at hydrogen. 1383 01:05:33,840 --> 01:05:37,550 But he did more than just look at hydrogen. 1384 01:05:37,550 --> 01:05:40,385 I can't remember exactly-- I think he did it with sodium. 1385 01:05:40,385 --> 01:05:42,010 Matt, do you remember? 1386 01:05:42,010 --> 01:05:44,280 I think he did sodium and iron. 1387 01:05:44,280 --> 01:05:45,755 But anyway, here's the funny thing 1388 01:05:45,755 --> 01:05:47,130 that he got with sodium and iron. 1389 01:05:47,130 --> 01:05:48,546 So this is a different experiment. 1390 01:05:48,546 --> 01:05:50,060 It's exactly the same set up, but he 1391 01:05:50,060 --> 01:05:52,030 was looking at spectral lines in sodium. 1392 01:05:52,030 --> 01:05:55,130 And spectral lines in sodium did a very similar thing. 1393 01:05:55,130 --> 01:05:57,450 There were some where 1 went to 1. 1394 01:06:01,400 --> 01:06:04,520 And there were others where 1 went 1395 01:06:04,520 --> 01:06:08,546 to-- why did we have three here? 1396 01:06:08,546 --> 01:06:10,920 Just as quick reminder, why do we have three states here? 1397 01:06:10,920 --> 01:06:12,200 AUDIENCE: [INAUDIBLE]. 1398 01:06:12,200 --> 01:06:14,840 PROFESSOR: Because we have three states for l equals 1. 1399 01:06:14,840 --> 01:06:25,600 This one state went to one, two, three, four states, 1400 01:06:25,600 --> 01:06:27,734 with none of them in the middle. 1401 01:06:27,734 --> 01:06:29,150 And in fact, it's worse than that, 1402 01:06:29,150 --> 01:06:31,108 because they didn't go to equally spaced states 1403 01:06:31,108 --> 01:06:31,700 in general. 1404 01:06:31,700 --> 01:06:33,650 In general, they did something slightly funny. 1405 01:06:33,650 --> 01:06:36,690 And the beautiful plot in his paper looks like this. 1406 01:06:40,130 --> 01:06:44,410 There are four states, and as you take the magnetic field 1407 01:06:44,410 --> 01:06:47,230 to 0, these four states all coalesce exactly 1408 01:06:47,230 --> 01:06:49,064 to this one state. 1409 01:06:49,064 --> 01:06:51,355 So this one line is representing four different states. 1410 01:06:54,000 --> 01:06:56,990 And this already included the fine structure, 1411 01:06:56,990 --> 01:07:00,535 so we know that this is all a single value of l. 1412 01:07:00,535 --> 01:07:02,035 But that apparently has four states. 1413 01:07:07,000 --> 01:07:08,810 That's should worry you. 1414 01:07:08,810 --> 01:07:12,926 What does that tell you about l? 1415 01:07:12,926 --> 01:07:14,550 What does l have to be in order to have 1416 01:07:14,550 --> 01:07:16,444 four states in your tower? 1417 01:07:16,444 --> 01:07:17,360 AUDIENCE: [INAUDIBLE]. 1418 01:07:17,360 --> 01:07:19,600 PROFESSOR: Half integer-- in particular, what value? 1419 01:07:19,600 --> 01:07:22,830 3/2, crap. 1420 01:07:22,830 --> 01:07:24,790 We already know that these wave functions 1421 01:07:24,790 --> 01:07:27,090 are described by the ylm's, right? 1422 01:07:27,090 --> 01:07:28,600 We solved that problem. 1423 01:07:28,600 --> 01:07:29,870 We explicitly solved it. 1424 01:07:29,870 --> 01:07:30,920 Here's the exact answer. 1425 01:07:30,920 --> 01:07:33,246 It's the ylm's. 1426 01:07:33,246 --> 01:07:34,620 But we've observed experimentally 1427 01:07:34,620 --> 01:07:36,210 that l has to be 3/2. 1428 01:07:36,210 --> 01:07:40,347 But if l is 3/2, ylm is equal to 0, 1429 01:07:40,347 --> 01:07:41,805 because it's equal to minus itself. 1430 01:07:45,070 --> 01:07:46,400 This is bad. 1431 01:07:46,400 --> 01:07:47,860 This is 0. 1432 01:07:47,860 --> 01:07:49,840 So this effect is called the Zeeman effect, 1433 01:07:49,840 --> 01:07:52,743 and this effect is called the anomalous Zeeman effect. 1434 01:07:52,743 --> 01:07:53,630 [LAUGHTER] 1435 01:07:53,630 --> 01:07:55,300 Which is strange for two reasons. 1436 01:07:55,300 --> 01:07:57,258 The first reason it's strange is they showed up 1437 01:07:57,258 --> 01:07:58,312 in the same paper. 1438 01:07:58,312 --> 01:08:00,770 And the second thing is, OK, you call it the Zeeman effect, 1439 01:08:00,770 --> 01:08:03,110 because there's some guy named Zeeman. 1440 01:08:03,110 --> 01:08:06,009 Is there some guy named Anomalous Zeeman? 1441 01:08:06,009 --> 01:08:07,050 It's a very strange name. 1442 01:08:07,050 --> 01:08:09,299 Anyway so this was called the anomalous Zeeman effect, 1443 01:08:09,299 --> 01:08:12,230 despite appearing in the same paper, because it's weird. 1444 01:08:12,230 --> 01:08:14,087 It was deeply disconcerting to people. 1445 01:08:14,087 --> 01:08:15,670 We now just call it the Zeeman effect, 1446 01:08:15,670 --> 01:08:17,964 but we have the bad habit-- for entertainment value-- 1447 01:08:17,964 --> 01:08:20,297 of referring to it still as the anamolous Zeeman effect. 1448 01:08:20,297 --> 01:08:23,292 I'm not the only one responsible for that bad habit. 1449 01:08:23,292 --> 01:08:25,750 I'm going to point out that at the end of this paper-- it's 1450 01:08:25,750 --> 01:08:27,500 a totally awesome paper, by the way-- it's 1451 01:08:27,500 --> 01:08:29,600 very readable and short-- he says-- I was looking 1452 01:08:29,600 --> 01:08:33,399 at it last night-- it says, quote possibly 1453 01:08:33,399 --> 01:08:36,580 the observed phenomenon will be regarded 1454 01:08:36,580 --> 01:08:39,810 as nothing of any consequence. 1455 01:08:39,810 --> 01:08:44,029 OK, so a few years later, Pauli says the following-- 1456 01:08:44,029 --> 01:08:46,319 so Pauli says this, actually, and this is also 1457 01:08:46,319 --> 01:08:51,410 totally lovely in Science from 1946 Pauli was in the US 1458 01:08:51,410 --> 01:08:52,170 during the war. 1459 01:08:52,170 --> 01:08:54,674 After the war, he was at the Institute for Advanced Study. 1460 01:08:54,674 --> 01:08:57,090 At the Institute for Advanced Studies, while he was there, 1461 01:08:57,090 --> 01:08:58,439 he got the Nobel Prize. 1462 01:08:58,439 --> 01:09:00,744 And he gave-- for the exclusion principle, which 1463 01:09:00,744 --> 01:09:02,910 we are about to get to-- and he gave a little spiel, 1464 01:09:02,910 --> 01:09:06,310 and it's written up in this edition of Science. 1465 01:09:06,310 --> 01:09:09,040 And he says, quote, a colleague who 1466 01:09:09,040 --> 01:09:11,080 met me strolling rather aimlessly 1467 01:09:11,080 --> 01:09:13,000 in the beautiful streets of Copenhagen 1468 01:09:13,000 --> 01:09:19,010 said to me in a friendly manner, "You look very unhappy." 1469 01:09:19,010 --> 01:09:22,910 Whereupon, I responded fiercely, "How can one look happy 1470 01:09:22,910 --> 01:09:25,686 when he is thinking about the anamolous Zeeman effect?" 1471 01:09:25,686 --> 01:09:32,500 [LAUGHTER] 1472 01:09:32,500 --> 01:09:33,830 So this troubled people. 1473 01:09:33,830 --> 01:09:34,374 Yeah. 1474 01:09:34,374 --> 01:09:35,290 AUDIENCE: [INAUDIBLE]. 1475 01:09:42,220 --> 01:09:44,389 PROFESSOR: Yeah he found all sorts of crazy states. 1476 01:09:44,389 --> 01:09:45,680 So this isn't the only example. 1477 01:09:45,680 --> 01:09:47,388 It's just the only one I'm going to draw. 1478 01:09:47,388 --> 01:09:49,479 But indeed, you can find five half states 1479 01:09:49,479 --> 01:09:50,967 you can find states with six. 1480 01:09:50,967 --> 01:09:52,050 AUDIENCE: Why [INAUDIBLE]? 1481 01:09:56,787 --> 01:09:58,620 PROFESSOR: Oh, I didn't draw all the states. 1482 01:09:58,620 --> 01:10:01,078 The spectrum is complicated and messy, because it's sodium. 1483 01:10:01,078 --> 01:10:02,970 And so there's all sorts of crap. 1484 01:10:02,970 --> 01:10:04,570 So I'm just not drawing it. 1485 01:10:04,570 --> 01:10:08,370 I'm just focusing on a specific set of spectral lines. 1486 01:10:08,370 --> 01:10:16,310 Maybe the best way to draw that is-- OK. 1487 01:10:16,310 --> 01:10:20,540 So this was really troubling. 1488 01:10:20,540 --> 01:10:24,679 So let's put this on pause for the moment, 1489 01:10:24,679 --> 01:10:26,470 and we'll come back to it in just a second. 1490 01:10:26,470 --> 01:10:28,010 So meanwhile, there's something else 1491 01:10:28,010 --> 01:10:29,760 that's sort of annoying about this system. 1492 01:10:29,760 --> 01:10:34,097 We've got the Coulomb potential, and the Coulomb potential 1493 01:10:34,097 --> 01:10:35,680 has the following beautiful property-- 1494 01:10:35,680 --> 01:10:41,710 the degeneracy of any energy state is equal to n squared. 1495 01:10:41,710 --> 01:10:44,410 Meanwhile, various people were fond 1496 01:10:44,410 --> 01:10:49,550 of observing the fact that the number of states, which 1497 01:10:49,550 --> 01:10:51,080 I will-- in an abuse of language-- 1498 01:10:51,080 --> 01:10:56,080 call the degeneracy of the level n in the periodic table, 1499 01:10:56,080 --> 01:10:58,770 is equal to-- well, the first one there's two. 1500 01:10:58,770 --> 01:11:01,736 And then there's eight, and then there's 18. 1501 01:11:05,930 --> 01:11:13,180 So two, 8, 8, 18, that's almost 2n squared. 1502 01:11:15,720 --> 01:11:20,040 Two four 8, 9, 18. 1503 01:11:20,040 --> 01:11:21,870 Cool, maybe there's a relationship, right? 1504 01:11:21,870 --> 01:11:23,245 And meanwhile, it's very tempting 1505 01:11:23,245 --> 01:11:25,244 to think there's a relationship, because imagine 1506 01:11:25,244 --> 01:11:27,012 you have four electrons. 1507 01:11:27,012 --> 01:11:28,970 If you have four electrons, what you do you do? 1508 01:11:28,970 --> 01:11:30,820 Well, you put an electron here, and then 1509 01:11:30,820 --> 01:11:32,730 you could put an electron here, or you 1510 01:11:32,730 --> 01:11:34,357 could put an other electron here. 1511 01:11:34,357 --> 01:11:36,440 You could also put another electron here and here. 1512 01:11:36,440 --> 01:11:38,356 Or you could put them in any combination-- two 1513 01:11:38,356 --> 01:11:40,410 here, put two here, two here, one, 1514 01:11:40,410 --> 01:11:42,450 one-- you could do all sorts of stuff, right? 1515 01:11:42,450 --> 01:11:45,990 But if, for the moment, you imagine 1516 01:11:45,990 --> 01:11:47,722 that electrons have this funny property. 1517 01:11:47,722 --> 01:11:49,180 Imagine for a moment that electrons 1518 01:11:49,180 --> 01:11:51,430 have the property that they can't be in the same state 1519 01:11:51,430 --> 01:11:52,315 at once. 1520 01:11:52,315 --> 01:11:54,440 Then if you have four electrons-- I don't know why. 1521 01:11:54,440 --> 01:11:55,130 I'm desperate. 1522 01:11:55,130 --> 01:11:56,490 I'm thinking about the anamolous Zeeman effect, 1523 01:11:56,490 --> 01:11:57,040 among other things. 1524 01:11:57,040 --> 01:11:58,206 I have nothing better to do. 1525 01:11:58,206 --> 01:12:00,560 So just for fun, let's imagine if I put an elector here, 1526 01:12:00,560 --> 01:12:01,560 I can't put another one. 1527 01:12:01,560 --> 01:12:02,910 So if I have a two electron atom, 1528 01:12:02,910 --> 01:12:04,360 the second one goes with one of these. 1529 01:12:04,360 --> 01:12:06,300 A three-electron atom, a four-electron atom, 1530 01:12:06,300 --> 01:12:07,810 and these guys are all related to each other, because they 1531 01:12:07,810 --> 01:12:09,212 live in the same energy level, but they 1532 01:12:09,212 --> 01:12:10,290 have different l values, so they'll 1533 01:12:10,290 --> 01:12:11,748 have slightly different properties. 1534 01:12:11,748 --> 01:12:14,950 That suggests that the one l equals 1535 01:12:14,950 --> 01:12:17,530 0 state should be different from the 3 l equals 1 states, 1536 01:12:17,530 --> 01:12:18,470 chemically. 1537 01:12:18,470 --> 01:12:21,630 But wait-- there aren't groupings of one and three. 1538 01:12:21,630 --> 01:12:25,500 They're groupings of two and six. 1539 01:12:25,500 --> 01:12:32,802 Ah, two-- well, so Pauli staring at this in 1925 1540 01:12:32,802 --> 01:12:34,260 said the following-- he said, look, 1541 01:12:34,260 --> 01:12:36,180 I'm going to conjecture two things. 1542 01:12:36,180 --> 01:12:41,000 The first is that no two electrons 1543 01:12:41,000 --> 01:12:48,190 can live in the same state-- inhabit the same quantum state. 1544 01:12:53,170 --> 01:12:55,570 That doesn't give us the periodic table, though. 1545 01:12:55,570 --> 01:12:58,060 It gives us half the periodic table. 1546 01:12:58,060 --> 01:13:12,240 There are twice as many states in hydrogen as you think. 1547 01:13:15,126 --> 01:13:17,000 And you might think that I'm being facetious, 1548 01:13:17,000 --> 01:13:19,315 that he said something a little more sophisticated. 1549 01:13:19,315 --> 01:13:20,190 But he really didn't. 1550 01:13:20,190 --> 01:13:22,020 What he said is, I posit of the existence 1551 01:13:22,020 --> 01:13:25,060 of an additional quantum number which the electron can have, 1552 01:13:25,060 --> 01:13:27,840 which takes one of two values. 1553 01:13:27,840 --> 01:13:30,460 Which translated, says, there are twice as many states 1554 01:13:30,460 --> 01:13:33,309 as you think. 1555 01:13:33,309 --> 01:13:34,600 So that's exactly what he said. 1556 01:13:34,600 --> 01:13:36,740 So this is called the Pauli exclusion principle. 1557 01:13:36,740 --> 01:13:40,262 And now this does a totally spectacular thing-- 1558 01:13:40,262 --> 01:13:42,720 and you've probably all seen this in high school chemistry, 1559 01:13:42,720 --> 01:13:44,370 or even college chemistry. 1560 01:13:44,370 --> 01:13:46,579 Now, first electron goes here, second electron 1561 01:13:46,579 --> 01:13:48,620 goes here, because there are twice as many states 1562 01:13:48,620 --> 01:13:50,880 as you think, but then I can't put another state in there, 1563 01:13:50,880 --> 01:13:52,520 because that's as many as you can have. 1564 01:13:52,520 --> 01:13:55,690 One can go here, and you can put up to six in here. 1565 01:13:55,690 --> 01:13:56,340 And that's it. 1566 01:13:56,340 --> 01:14:01,840 So the E2 states, the n squared, which is 4 times 2 states, 1567 01:14:01,840 --> 01:14:03,940 are somehow related to each other naturally. 1568 01:14:03,940 --> 01:14:05,920 And this gives you the structure of the periodic table. 1569 01:14:05,920 --> 01:14:07,336 In fact, an awful lot of chemistry 1570 01:14:07,336 --> 01:14:08,480 follows directly from this. 1571 01:14:08,480 --> 01:14:10,188 So this is the Pauli exclusion principle. 1572 01:14:10,188 --> 01:14:12,430 However, it's a little bit disappointing, 1573 01:14:12,430 --> 01:14:14,980 because while this is whatever-- some ridiculous rule, 1574 01:14:14,980 --> 01:14:16,354 but we're doing quantum mechanics 1575 01:14:16,354 --> 01:14:18,800 and they're all ridiculous rules. 1576 01:14:18,800 --> 01:14:20,730 On the other hand, this one is just stupid. 1577 01:14:20,730 --> 01:14:21,230 Right? 1578 01:14:21,230 --> 01:14:24,890 This is just, like, look, it wouldn't it be nice? 1579 01:14:24,890 --> 01:14:31,520 So in '25, a couple guys-- really interesting characters-- 1580 01:14:31,520 --> 01:14:37,739 named Goudsmit and Uhlenbeck-- so Uhlenbeck-- wow, 1581 01:14:37,739 --> 01:14:38,780 we're really low on time. 1582 01:14:38,780 --> 01:14:41,100 OK, so I'll just tell you this, and then I'll 1583 01:14:41,100 --> 01:14:43,320 get to the last of it later. 1584 01:14:43,320 --> 01:14:48,310 So Goudsmit and Uhlenbeck-- who were young, kind of naive-- 1585 01:14:48,310 --> 01:14:51,310 said, look we've got two effects here. 1586 01:14:51,310 --> 01:14:54,110 One effect is the anamolous Zeeman effect, 1587 01:14:54,110 --> 01:14:56,380 and there's this weird fact that states 1588 01:14:56,380 --> 01:14:58,335 have the wrong total angular momentum. 1589 01:14:58,335 --> 01:15:00,460 And we have this second thing, that there are twice 1590 01:15:00,460 --> 01:15:02,250 as many states for an electron as you 1591 01:15:02,250 --> 01:15:04,410 think in the Coulomb potential. 1592 01:15:04,410 --> 01:15:07,490 How can these two things fit together naturally? 1593 01:15:07,490 --> 01:15:10,060 Well here's a guess-- suppose that the angular 1594 01:15:10,060 --> 01:15:12,390 momentum that we calculated when we 1595 01:15:12,390 --> 01:15:17,569 did the estimate of the braking due to the Zeeman effect, 1596 01:15:17,569 --> 01:15:19,860 suppose that angular momentum was not the right angular 1597 01:15:19,860 --> 01:15:20,310 momentum. 1598 01:15:20,310 --> 01:15:22,130 Maybe there's more angular momentum in the system. 1599 01:15:22,130 --> 01:15:23,670 There's angular momentum in the system from the electron 1600 01:15:23,670 --> 01:15:25,878 orbiting around, but maybe like a little, tiny Earth, 1601 01:15:25,878 --> 01:15:28,180 the electron itself can have some intrinsic angular 1602 01:15:28,180 --> 01:15:29,347 momentum. 1603 01:15:29,347 --> 01:15:31,180 It turns out Pauli had had this idea before. 1604 01:15:31,180 --> 01:15:32,640 In fact, Kramers had suggested this 1605 01:15:32,640 --> 01:15:35,800 to him, who was a very young guy at the time. 1606 01:15:35,800 --> 01:15:37,660 And Pauli said, you're a blithering idiot, 1607 01:15:37,660 --> 01:15:39,490 because if you calculate how small an electron has 1608 01:15:39,490 --> 01:15:41,410 to be in order to fit all the other things we know, and you 1609 01:15:41,410 --> 01:15:43,370 figure out how fast it would have to rotate to explain 1610 01:15:43,370 --> 01:15:44,840 the anamolous Zeeman effect, the surface 1611 01:15:44,840 --> 01:15:46,990 would have to be moving faster the speed of light. 1612 01:15:46,990 --> 01:15:48,110 That's ridiculous. 1613 01:15:48,110 --> 01:15:49,630 Leave my office. 1614 01:15:49,630 --> 01:15:58,750 So then Uhlenbeck and Goudsmit-- these two guys-- write a paper, 1615 01:15:58,750 --> 01:15:59,440 and say aha! 1616 01:15:59,440 --> 01:16:01,350 Well, we can explain this, but let's just not 1617 01:16:01,350 --> 01:16:02,980 assume that the stupid thing is rotating. 1618 01:16:02,980 --> 01:16:04,810 Let's just say an electron has some intrinsic angular 1619 01:16:04,810 --> 01:16:05,630 momentum. 1620 01:16:05,630 --> 01:16:07,650 I don't know, why not? 1621 01:16:07,650 --> 01:16:09,400 Electrons have intrinsic angular momentum, 1622 01:16:09,400 --> 01:16:12,370 and if you assume that electrons have an intrinsic angular 1623 01:16:12,370 --> 01:16:16,880 momentum-- which I'll call S for the moment-- so then 1624 01:16:16,880 --> 01:16:18,390 the total angular momentum, j, is 1625 01:16:18,390 --> 01:16:21,320 equal to l, its orbital angular momentum, 1626 01:16:21,320 --> 01:16:23,420 plus some intrinsic angular momentum, which-- 1627 01:16:23,420 --> 01:16:25,795 I don't remember what symbol they used, but we'll call it 1628 01:16:25,795 --> 01:16:28,040 for the moment S-- where this has the property that S 1629 01:16:28,040 --> 01:16:34,480 squared, for an electron, sorry, the principle quantum 1630 01:16:34,480 --> 01:16:37,040 number, which I will call little s is equal to 1/2, 1631 01:16:37,040 --> 01:16:41,214 so m sub s is equal to plus or minus 1/2. 1632 01:16:41,214 --> 01:16:43,505 So this is like the l equals 1/2 state, but it's not l. 1633 01:16:43,505 --> 01:16:44,546 It's something intrinsic. 1634 01:16:44,546 --> 01:16:46,730 It has nothing to do with anything rotating. 1635 01:16:46,730 --> 01:16:48,357 It's just a fact about electrons. 1636 01:16:48,357 --> 01:16:50,190 Suppose electrons have a little bit of spin. 1637 01:16:50,190 --> 01:16:53,390 Then what you discover is if you have the l equals 1 state, 1638 01:16:53,390 --> 01:16:55,280 and the electron has spin 1/2, what's 1639 01:16:55,280 --> 01:16:57,720 the total angular momentum? 1640 01:16:57,720 --> 01:17:00,769 3/2, right? 1641 01:17:00,769 --> 01:17:01,560 So what do you get? 1642 01:17:01,560 --> 01:17:05,755 You get quadruplets, like that. 1643 01:17:05,755 --> 01:17:07,630 So this turns out to explain-- if you include 1644 01:17:07,630 --> 01:17:09,463 a small relativistic effect-- this turns out 1645 01:17:09,463 --> 01:17:14,720 to explain the anomalous Zeeman effect bang on. 1646 01:17:14,720 --> 01:17:16,640 Meanwhile, there was an experiment 1647 01:17:16,640 --> 01:17:19,210 done in 1922, which is the Stern-Gerlach experiment, 1648 01:17:19,210 --> 01:17:23,080 in which Stern and Gerlach discovered that nickel, when 1649 01:17:23,080 --> 01:17:25,315 sent through a magnetic field gradient, 1650 01:17:25,315 --> 01:17:28,600 bent into one of two different spots-- never three, 1651 01:17:28,600 --> 01:17:31,110 never zero, always two. 1652 01:17:31,110 --> 01:17:33,040 How can that be? 1653 01:17:33,040 --> 01:17:35,410 Electron spin-- that wasn't realized until 1929, 1654 01:17:35,410 --> 01:17:37,130 that the connection was there. 1655 01:17:37,130 --> 01:17:41,870 So these guys came up with this ridiculous postulate. 1656 01:17:41,870 --> 01:17:43,560 This was Uhlenbeck and Goudsmit. 1657 01:17:43,560 --> 01:17:49,460 And let me just quickly-- U-H-L-E-N-B-E-C-K-- 1658 01:17:49,460 --> 01:17:51,610 so Uhlenbeck-- amazing, amazing scientist, 1659 01:17:51,610 --> 01:17:55,390 also the father of the mathematician Uhlenbeck, 1660 01:17:55,390 --> 01:17:58,140 and she was a total badass, and has inspired an awful lot 1661 01:17:58,140 --> 01:18:02,259 of physics-- Karen Uhlenbeck, who's at UT. 1662 01:18:02,259 --> 01:18:04,300 So this is a pretty interesting and prolific guy. 1663 01:18:04,300 --> 01:18:06,350 But he was also prolific in the following way-- 1664 01:18:06,350 --> 01:18:10,540 he ran a program at the Rad Lab at MIT during World War II. 1665 01:18:10,540 --> 01:18:12,740 And Goudsmit worked with him. 1666 01:18:12,740 --> 01:18:15,476 And when they were finishing up, when the war was ending, 1667 01:18:15,476 --> 01:18:17,600 Goudsmit became the scientific adviser to something 1668 01:18:17,600 --> 01:18:20,140 called Project Alsos. 1669 01:18:20,140 --> 01:18:22,780 Project Alsos was a project where the military went 1670 01:18:22,780 --> 01:18:25,230 to the conquered territories in Germany 1671 01:18:25,230 --> 01:18:27,870 to catch the German atomic scientists, 1672 01:18:27,870 --> 01:18:30,710 and bring them to a place called Farm Hall 1673 01:18:30,710 --> 01:18:34,190 in England, where they listened and eavesdropped on them-- 1674 01:18:34,190 --> 01:18:37,820 Heisenberg, all the good guys. 1675 01:18:37,820 --> 01:18:40,420 Well, bad guys-- it depends on-- all the great physicist 1676 01:18:40,420 --> 01:18:42,000 in the German territory at the time 1677 01:18:42,000 --> 01:18:44,360 were deeply complicated people. 1678 01:18:44,360 --> 01:18:45,500 And they listened to them. 1679 01:18:45,500 --> 01:18:47,300 Something called the Farm Hall Transcripts 1680 01:18:47,300 --> 01:18:49,470 are the transcripts of those recordings. 1681 01:18:49,470 --> 01:18:52,684 They were written up in a book called "The Epsilon Project," 1682 01:18:52,684 --> 01:18:54,350 which is totally breathtakingly awesome. 1683 01:18:54,350 --> 01:18:57,580 And Goudsmit wrote a book called "Alsos" 1684 01:18:57,580 --> 01:19:01,240 about this process of hunting down the German scientists. 1685 01:19:01,240 --> 01:19:07,390 So Goudsmit, G-O-U-D-S-M-I-T, I think, 1686 01:19:07,390 --> 01:19:09,900 he wrote a book called "Alsos", which I heartily recommend 1687 01:19:09,900 --> 01:19:12,530 to you, because it's like a combination adventure story 1688 01:19:12,530 --> 01:19:15,020 and beautiful bit of physics history. 1689 01:19:15,020 --> 01:19:18,020 So these guys both were at MIT during the World War. 1690 01:19:18,020 --> 01:19:19,720 So there's a nice connection here. 1691 01:19:19,720 --> 01:19:21,360 So these guys-- fascinating characters, 1692 01:19:21,360 --> 01:19:23,050 and they came up with this idea of some intrinsic angular 1693 01:19:23,050 --> 01:19:23,740 momentum. 1694 01:19:23,740 --> 01:19:27,926 Pauli then calls it spin. 1695 01:19:27,926 --> 01:19:28,800 He gives it the name. 1696 01:19:28,800 --> 01:19:32,590 And he develops a mathematical theory of spin. 1697 01:19:32,590 --> 01:19:34,220 And the mathematical theory of spin 1698 01:19:34,220 --> 01:19:36,680 will lead to quantum field theory, relativistic quantum 1699 01:19:36,680 --> 01:19:38,380 mechanics, and will eventually lead 1700 01:19:38,380 --> 01:19:40,040 to quantum computation, which is going 1701 01:19:40,040 --> 01:19:42,074 to be the topic of the last week of our course. 1702 01:19:42,074 --> 01:19:43,490 So what we've done so far is we've 1703 01:19:43,490 --> 01:19:46,610 explained the discreteness of atomic spectra, 1704 01:19:46,610 --> 01:19:48,140 and we've explained the structure 1705 01:19:48,140 --> 01:19:49,422 of the periodic table. 1706 01:19:49,422 --> 01:19:50,880 What we haven't done, is we haven't 1707 01:19:50,880 --> 01:19:54,200 explained why atoms form molecules or solids. 1708 01:19:54,200 --> 01:19:56,890 And we also haven't explained what spin is at all. 1709 01:19:56,890 --> 01:19:59,150 Those are the topics in the next three weeks. 1710 01:19:59,150 --> 01:20:01,110 See you next time.