1 00:00:00,000 --> 00:00:03,280 PROFESSOR: What is a Rydberg atom? 2 00:00:03,280 --> 00:00:11,520 Well, it's a-- an atom can be a Rydberg atom if the outermost 3 00:00:11,520 --> 00:00:16,860 electron is in a very high principal quantum number. 4 00:00:16,860 --> 00:00:19,560 That's a definition of a Rydberg atom. 5 00:00:19,560 --> 00:00:24,270 The last electron is an n-- 6 00:00:24,270 --> 00:00:27,880 little n, very large. 7 00:00:27,880 --> 00:00:32,530 Now that is very interesting because when 8 00:00:32,530 --> 00:00:36,400 that happens, Rydberg atoms-- 9 00:00:40,550 --> 00:00:49,300 when that happens, you have a nucleus with charge ze 10 00:00:49,300 --> 00:00:51,640 and you have lots of electrons. 11 00:00:51,640 --> 00:00:55,600 And the last electron-- suppose the last electrode 12 00:00:55,600 --> 00:00:58,900 is in an orbit further out. 13 00:00:58,900 --> 00:01:04,379 You see the orbits do become further out in general. 14 00:01:04,379 --> 00:01:09,250 And suppose the last electron is in some large value 15 00:01:09,250 --> 00:01:11,680 of little n, so it's somewhat out. 16 00:01:11,680 --> 00:01:15,070 So here is the last electron. 17 00:01:15,070 --> 00:01:20,500 This electron sees a nucleus with charge z, 18 00:01:20,500 --> 00:01:26,050 but it also sees z minus 1 electrons, all 19 00:01:26,050 --> 00:01:27,370 the other electrons. 20 00:01:30,380 --> 00:01:35,060 The last electron is outside the nucleus and the cloud 21 00:01:35,060 --> 00:01:36,590 of the other electrons. 22 00:01:36,590 --> 00:01:45,740 So the last electron sees a charge plus 1 z from here 23 00:01:45,740 --> 00:01:54,210 and z minus 1 of the electrons. 24 00:01:54,210 --> 00:01:57,020 So the last electron sees charge 1. 25 00:01:57,020 --> 00:02:01,690 So in some sense, to a good approximation, 26 00:02:01,690 --> 00:02:07,250 the last electron says, oh, there's a hydrogen atom here. 27 00:02:07,250 --> 00:02:09,350 I'm part of a hydrogen atom. 28 00:02:09,350 --> 00:02:10,820 I don't see my friends. 29 00:02:10,820 --> 00:02:13,900 The other electrons are too close to the nucleus. 30 00:02:13,900 --> 00:02:20,370 And I'm out there going around as if I were hydrogen. 31 00:02:20,370 --> 00:02:24,540 So that's a very nice application of hydrogen atoms. 32 00:02:24,540 --> 00:02:32,340 So the first question that we want to understand, 33 00:02:32,340 --> 00:02:37,450 that it's, again conceptually, is what is the size. 34 00:02:37,450 --> 00:02:44,795 If n is large what is the size of the atom? 35 00:02:57,540 --> 00:03:00,770 Now I must say, I myself, when I look 36 00:03:00,770 --> 00:03:03,830 at these things after a few years 37 00:03:03,830 --> 00:03:07,460 that I don't teach quantum mechanics, I look at here 38 00:03:07,460 --> 00:03:10,910 and say, OK, this is the solution. 39 00:03:10,910 --> 00:03:14,820 Well, maybe the size is na0. 40 00:03:17,790 --> 00:03:22,150 Unfortunately, that's completely wrong. 41 00:03:22,150 --> 00:03:24,070 And we're going to try to explain what 42 00:03:24,070 --> 00:03:25,690 was wrong in looking there. 43 00:03:25,690 --> 00:03:29,600 You know, you see your hydrogen atom wave function. 44 00:03:29,600 --> 00:03:33,710 There's nothing like really the size, 45 00:03:33,710 --> 00:03:36,940 there's no such precise definition as the size. 46 00:03:36,940 --> 00:03:41,560 But you could say, what's the expected value of the radius. 47 00:03:41,560 --> 00:03:44,470 That's a reasonable definition of size. 48 00:03:44,470 --> 00:03:46,930 And you know from this wave function, 49 00:03:46,930 --> 00:03:56,440 is going to come out to a0, a0 over 2, 2a0, maybe pi over 2a0. 50 00:03:56,440 --> 00:03:58,520 Something like that. 51 00:03:58,520 --> 00:04:00,470 So from here you would say, well, 52 00:04:00,470 --> 00:04:04,780 it's going to come out to pi and na0 or something like that. 53 00:04:04,780 --> 00:04:06,490 That is not true. 54 00:04:06,490 --> 00:04:11,500 So how do we see that it's not true? 55 00:04:11,500 --> 00:04:14,500 We'll take a little time in a few minutes. 56 00:04:14,500 --> 00:04:18,670 But how can we get the size of that atom 57 00:04:18,670 --> 00:04:22,100 correctly in an intuitive way? 58 00:04:22,100 --> 00:04:24,700 Again, we want to just understand 59 00:04:24,700 --> 00:04:27,220 a few results about hydrogen atom 60 00:04:27,220 --> 00:04:31,750 that become part of your intuition. 61 00:04:31,750 --> 00:04:35,285 So the important result here is the virial theorem. 62 00:04:42,880 --> 00:04:45,450 Now whenever I think of the virial theorem I say, 63 00:04:45,450 --> 00:04:49,750 oh, there was a factor of 2 or a 1/2 there. 64 00:04:49,750 --> 00:04:51,560 How did it go? 65 00:04:51,560 --> 00:04:58,365 It takes me a few seconds to try to reconstruct that it's this. 66 00:05:02,430 --> 00:05:07,590 For the hydrogen system, for any 1 over r potential, 67 00:05:07,590 --> 00:05:10,230 there is this relation within the expectation 68 00:05:10,230 --> 00:05:13,160 value of the kinetic energy, the potential energy. 69 00:05:13,160 --> 00:05:16,200 You've seen that result probably a couple of times 70 00:05:16,200 --> 00:05:18,550 already in this course. 71 00:05:18,550 --> 00:05:21,070 Because it's a real important result. 72 00:05:21,070 --> 00:05:24,920 It looks like OK, just a theorem about these things. 73 00:05:24,920 --> 00:05:29,330 But the intuition is really important. 74 00:05:29,330 --> 00:05:34,280 So how does one remember or picture that? 75 00:05:34,280 --> 00:05:35,930 One way is the following. 76 00:05:35,930 --> 00:05:38,800 And that's the way I like it. 77 00:05:38,800 --> 00:05:41,620 I imagine the energy line here. 78 00:05:41,620 --> 00:05:43,570 And here is 0. 79 00:05:43,570 --> 00:05:46,980 And the one thing I remember is that, yes, 80 00:05:46,980 --> 00:05:52,930 there is a kinetic energy here. 81 00:05:52,930 --> 00:05:55,480 And actually the bound state energy 82 00:05:55,480 --> 00:05:58,390 is exactly the same, but the opposite. 83 00:06:02,140 --> 00:06:05,320 That's a way to remember that. 84 00:06:05,320 --> 00:06:07,240 That's really what is going on. 85 00:06:07,240 --> 00:06:08,500 That's the key thing. 86 00:06:08,500 --> 00:06:11,710 The kinetic energy and the bound state energy 87 00:06:11,710 --> 00:06:13,120 are just of opposite signs. 88 00:06:16,380 --> 00:06:17,730 Why do we see that? 89 00:06:17,730 --> 00:06:22,350 Because then we have that V, from this equation, 90 00:06:22,350 --> 00:06:23,880 is minus 2T. 91 00:06:27,380 --> 00:06:37,720 And if V is minus 2T, means that T plus V is T minus 2T. 92 00:06:37,720 --> 00:06:41,620 And T plus V is the total energy. 93 00:06:41,620 --> 00:06:49,610 So T plus V is equal to the total energy 94 00:06:49,610 --> 00:06:52,750 E of the bound state. 95 00:06:52,750 --> 00:06:56,340 For any stationary state, this is true. 96 00:06:56,340 --> 00:07:04,910 And this thing, given this condition, is minus T. 97 00:07:04,910 --> 00:07:09,630 So that's the way we think of it. 98 00:07:09,630 --> 00:07:16,350 So we then have a nice result because the expectation value 99 00:07:16,350 --> 00:07:27,510 of the potential would then be minus 2 times the expectation 100 00:07:27,510 --> 00:07:37,510 value of T, which is itself minus the energy. 101 00:07:37,510 --> 00:07:42,420 So you have a minus 2 times minus the energy. 102 00:07:42,420 --> 00:07:45,020 And it's equal to the energy. 103 00:07:45,020 --> 00:07:47,070 And this is correct. 104 00:07:47,070 --> 00:07:51,630 The expectation value of V is the energy 105 00:07:51,630 --> 00:07:55,140 and the expectation value of Eb is negative, 106 00:07:55,140 --> 00:07:57,180 and the energy's negative. 107 00:07:57,180 --> 00:07:58,140 So what is this? 108 00:07:58,140 --> 00:08:06,330 This is the expectation value of minus e squared over r. 109 00:08:06,330 --> 00:08:09,030 In fact, of-- yeah, I'll leave it like that. 110 00:08:12,320 --> 00:08:15,110 So I'm back to z equal 1, because we're 111 00:08:15,110 --> 00:08:18,710 talking about Rydberg atoms. 112 00:08:18,710 --> 00:08:21,370 Oh, 2Eb, I'm sorry, here. 113 00:08:21,370 --> 00:08:23,420 Thank you. 114 00:08:23,420 --> 00:08:27,590 OK, so expectation value of the potential we have here. 115 00:08:27,590 --> 00:08:30,590 And 2 times the energy-- 116 00:08:30,590 --> 00:08:32,240 yeah, I would have gotten this wrong. 117 00:08:32,240 --> 00:08:35,570 Thanks for correcting it before it 118 00:08:35,570 --> 00:08:39,280 did damage to the derivation. 119 00:08:39,280 --> 00:08:48,100 We have it there, e squared over 2a0, 1 over n squared. 120 00:08:50,660 --> 00:08:52,730 So what can we cancel? 121 00:08:52,730 --> 00:08:55,870 Well, the 2's cancel. 122 00:08:55,870 --> 00:08:57,670 The e squares cancel. 123 00:08:57,670 --> 00:08:59,920 The signs cancel. 124 00:08:59,920 --> 00:09:05,080 And we get expectation value of 1 over r 125 00:09:05,080 --> 00:09:12,680 is equal to 1 over n squared a0, which 126 00:09:12,680 --> 00:09:20,830 is suggesting very clearly that the typical radius is not na0. 127 00:09:20,830 --> 00:09:24,220 It's n square a0. 128 00:09:24,220 --> 00:09:28,870 So this is exact. 129 00:09:28,870 --> 00:09:31,000 The virial theorem is exact. 130 00:09:31,000 --> 00:09:32,200 The energy is exact. 131 00:09:32,200 --> 00:09:34,570 This is exact. 132 00:09:34,570 --> 00:09:37,720 That's not quite the expectation value of r. 133 00:09:37,720 --> 00:09:41,260 The expectation value of r is not 134 00:09:41,260 --> 00:09:45,410 the inverse of the expectation value of 1 over r. 135 00:09:45,410 --> 00:09:47,260 It's somewhat related. 136 00:09:47,260 --> 00:09:52,330 But there's no theorem, because it would be false, 137 00:09:52,330 --> 00:09:56,110 that the expectation value of 1 over a random variable 138 00:09:56,110 --> 00:09:59,644 is the expectation value of-- 139 00:09:59,644 --> 00:10:03,750 is 1 over the expectation value of the random variable. 140 00:10:03,750 --> 00:10:06,240 It's just not true. 141 00:10:06,240 --> 00:10:11,380 This nice result for 1 over r is exactly true. 142 00:10:11,380 --> 00:10:13,480 And it's l independent. 143 00:10:13,480 --> 00:10:22,650 On the other hand, the expectation value of r 144 00:10:22,650 --> 00:10:27,630 can be calculated with a bit more effort-- 145 00:10:27,630 --> 00:10:29,700 a lot more effort. 146 00:10:29,700 --> 00:10:31,660 And it's equal to this. 147 00:10:31,660 --> 00:10:36,720 It's just the same thing and 1, with a little correction, 148 00:10:36,720 --> 00:10:43,635 which is 1/2 1 minus l times l plus 1 over n squared. 149 00:10:47,980 --> 00:10:53,550 So actually, the expectation value of r in the hydrogen atom 150 00:10:53,550 --> 00:10:56,010 is l dependent. 151 00:10:56,010 --> 00:11:00,290 Not terribly strongly l dependent, but somewhat l 152 00:11:00,290 --> 00:11:02,460 dependent. 153 00:11:02,460 --> 00:11:10,180 To get an idea, this is equal to n squared a0 times 3/2 154 00:11:10,180 --> 00:11:13,790 for l equals 0. 155 00:11:13,790 --> 00:11:18,680 When l is equal to 0, you see this whole bracket becomes 3/2. 156 00:11:18,680 --> 00:11:30,920 And for the maximum l, l equals n minus 1, this is roughly 1. 157 00:11:30,920 --> 00:11:35,780 This is roughly 0, to a good approximation but not exactly. 158 00:11:35,780 --> 00:11:39,920 It becomes n squared a0 with corrections 159 00:11:39,920 --> 00:11:41,790 that are very, very small. 160 00:11:41,790 --> 00:11:43,364 It's pretty accurate. 161 00:11:47,160 --> 00:11:47,880 All right. 162 00:11:47,880 --> 00:11:53,360 So, first thing we've learned is that we 163 00:11:53,360 --> 00:11:59,250 got a radius, expected radius, that goes back n squared a0. 164 00:11:59,250 --> 00:12:01,680 So it's kind of interesting to see 165 00:12:01,680 --> 00:12:07,830 what went wrong if you would have thought 166 00:12:07,830 --> 00:12:11,470 with a form of the solution. 167 00:12:11,470 --> 00:12:20,395 So psi nlm equal Ar to the l Wnl, a polynomial, 168 00:12:20,395 --> 00:12:28,360 e to the minus r over na0 Ylm of theta and phi 169 00:12:28,360 --> 00:12:37,840 or fnl of r times Ylm of the solid angle. 170 00:12:37,840 --> 00:12:41,490 And what do we know about this polynomial? 171 00:12:41,490 --> 00:12:49,430 It's of degree n minus l plus 1. 172 00:12:49,430 --> 00:12:52,130 That's it. 173 00:12:52,130 --> 00:12:53,705 And it depends on r. 174 00:12:57,370 --> 00:13:01,840 So what we're looking at is what was the error in thinking 175 00:13:01,840 --> 00:13:06,130 that the typical r was na0. 176 00:13:06,130 --> 00:13:08,560 And you see, when you make a mistake like this-- 177 00:13:08,560 --> 00:13:13,630 the mistake I made of saying oh, na0 must be right-- 178 00:13:13,630 --> 00:13:17,350 and you find that it's wrong, it's very important 179 00:13:17,350 --> 00:13:22,420 to go back and learn why did you get the wrong answer. 180 00:13:22,420 --> 00:13:25,240 That's what we're doing now. 181 00:13:25,240 --> 00:13:29,170 So the one question I can ask to begin 182 00:13:29,170 --> 00:13:34,000 with is what is a probability density 183 00:13:34,000 --> 00:13:41,470 to find the electron between some radius r and a radius r 184 00:13:41,470 --> 00:13:49,850 plus dr. So, you know, this is a probability. 185 00:13:49,850 --> 00:13:52,290 And it depends on theta and phi. 186 00:13:52,290 --> 00:13:54,740 And it's very complicated. 187 00:13:54,740 --> 00:13:58,460 How about giving me a probability 188 00:13:58,460 --> 00:14:03,020 along r that they can integrate along r and visualize 189 00:14:03,020 --> 00:14:05,270 how this is dependant on r? 190 00:14:05,270 --> 00:14:09,980 So the probability to find the electron in this shell 191 00:14:09,980 --> 00:14:17,260 must be equal to the value of the wave function squared 192 00:14:17,260 --> 00:14:21,220 times the volume element. 193 00:14:21,220 --> 00:14:24,500 And the volume element here is psi 194 00:14:24,500 --> 00:14:29,480 squared r squared dr times-- 195 00:14:29,480 --> 00:14:33,580 you would say 4 pi, but it's not spherically symmetric. 196 00:14:33,580 --> 00:14:41,120 So you have to integrate over solid angle. 197 00:14:41,120 --> 00:14:43,130 That is the volume element. 198 00:14:43,130 --> 00:14:45,920 And since I have to integrate over solid angle, 199 00:14:45,920 --> 00:14:48,650 I must have psi squared here. 200 00:14:48,650 --> 00:14:50,120 So that's the right equation. 201 00:14:53,260 --> 00:14:56,510 If you want to make things look perfect, 202 00:14:56,510 --> 00:15:01,150 put the d cubed x before the psi squared here. 203 00:15:01,150 --> 00:15:03,700 And the problem is that the d cubed 204 00:15:03,700 --> 00:15:09,080 x is big enough that it, in some sense, has partial integrals. 205 00:15:09,080 --> 00:15:15,450 The So notation is not perfect, but somehow you 206 00:15:15,450 --> 00:15:18,090 must imagine this whole volume element 207 00:15:18,090 --> 00:15:24,660 that is still infinitesimal but involves some integral already. 208 00:15:24,660 --> 00:15:26,370 So you have this. 209 00:15:26,370 --> 00:15:37,030 And you get then r squared dr fnl squared 210 00:15:37,030 --> 00:15:43,158 and you have the integral, the omega of this Y star lm Ylm. 211 00:15:46,014 --> 00:15:52,120 And that integral is exactly 1. 212 00:15:52,120 --> 00:15:55,020 Spherical harmonics are normalized. 213 00:15:55,020 --> 00:15:58,330 So now I can cancel the dr. And I 214 00:15:58,330 --> 00:16:02,680 get that the radial probability distribution, which 215 00:16:02,680 --> 00:16:12,105 is a nice concept, is really r squared fnl of r squared. 216 00:16:16,440 --> 00:16:17,940 Radial probability. 217 00:16:23,530 --> 00:16:28,210 So our mistake-- my mistake must have 218 00:16:28,210 --> 00:16:34,780 been that I didn't include all that was relevant. 219 00:16:34,780 --> 00:16:36,760 The exponential is one part. 220 00:16:36,760 --> 00:16:38,350 But there is the polynomial. 221 00:16:38,350 --> 00:16:42,730 And the polynomial must be causing the trouble. 222 00:16:42,730 --> 00:16:50,780 Indeed, that's what is happening. 223 00:16:50,780 --> 00:16:58,300 Let's look at fnl from the top blackboard. 224 00:16:58,300 --> 00:17:05,579 That includes r to the l times that polynomial. 225 00:17:05,579 --> 00:17:08,200 And it's a polynomial of that degree. 226 00:17:08,200 --> 00:17:20,770 So it begins like a0 plus up to coefficient a prime r 227 00:17:20,770 --> 00:17:25,470 to the n minus l plus-- 228 00:17:25,470 --> 00:17:27,329 well, minus l minus 1. 229 00:17:32,140 --> 00:17:35,370 And then I have the exponential. 230 00:17:40,010 --> 00:17:41,920 OK. 231 00:17:41,920 --> 00:17:44,120 Now I cannot do this-- 232 00:17:44,120 --> 00:17:46,440 I don't want to do this calculation exactly. 233 00:17:46,440 --> 00:17:48,710 It's too complicated. 234 00:17:48,710 --> 00:17:57,580 So let's ignore the lower part of the polynomial. 235 00:17:57,580 --> 00:18:01,200 And we're thinking the radius is going to be reasonably big. 236 00:18:01,200 --> 00:18:04,090 So it's a reasonable idea to keep 237 00:18:04,090 --> 00:18:09,130 the power of the polynomial that is the largest. 238 00:18:09,130 --> 00:18:10,400 So what is the largest? 239 00:18:10,400 --> 00:18:13,540 And here you see a nice thing, actually r 240 00:18:13,540 --> 00:18:16,360 to the l times this polynomial is a polynomial that 241 00:18:16,360 --> 00:18:20,860 begins with r to the l and finishes 242 00:18:20,860 --> 00:18:24,136 with r to the n minus 1. 243 00:18:24,136 --> 00:18:31,980 So it has like equal number of terms as reaches 244 00:18:31,980 --> 00:18:34,020 a value of n minus 1. 245 00:18:34,020 --> 00:18:39,090 The last term in the polynomial, when you multiply it in, 246 00:18:39,090 --> 00:18:42,990 this begins like r to the l and the last term 247 00:18:42,990 --> 00:18:45,040 is r to the n minus 1. 248 00:18:45,040 --> 00:18:48,570 So let's take this to be proportional 249 00:18:48,570 --> 00:18:54,480 to r to the n minus 1 equal to minus r na0. 250 00:18:58,640 --> 00:19:03,410 And here is the fight that actually changes the answer. 251 00:19:03,410 --> 00:19:07,370 Because this is fnl. 252 00:19:07,370 --> 00:19:14,080 So actually I can write p of r is proportional up 253 00:19:14,080 --> 00:19:19,640 to a constant of normalization, and the approximation 254 00:19:19,640 --> 00:19:26,010 they have then r squared times this polynomial squared. 255 00:19:26,010 --> 00:19:28,490 But I'm taking the last term of the polynomial, 256 00:19:28,490 --> 00:19:34,820 so we get r to the 2n e to the minus 2r over na0. 257 00:19:39,400 --> 00:19:41,682 That's a probability distribution. 258 00:19:44,520 --> 00:19:46,990 And with this probability distribution, 259 00:19:46,990 --> 00:19:50,080 then you see what's happening is that there 260 00:19:50,080 --> 00:19:53,510 is a fight between an exponential that 261 00:19:53,510 --> 00:19:58,420 has a typical length where it decays to half its value, that 262 00:19:58,420 --> 00:20:03,700 is related to na0 but the maximum 263 00:20:03,700 --> 00:20:08,170 is delayed because it's multiplied by a function 264 00:20:08,170 --> 00:20:13,630 that the higher the value of n, the slower it is to take off. 265 00:20:13,630 --> 00:20:17,920 x squared or r squared takes off slower than r, 266 00:20:17,920 --> 00:20:21,340 takes of slower than r to the 4th. 267 00:20:21,340 --> 00:20:27,250 So here you have a thing that just grows like that, 268 00:20:27,250 --> 00:20:30,550 but it takes forever to take off. 269 00:20:30,550 --> 00:20:34,920 And the result of this is a function that just picks up 270 00:20:34,920 --> 00:20:37,200 at some point over here. 271 00:20:37,200 --> 00:20:42,300 And we want to find the maximum. 272 00:20:42,300 --> 00:20:47,520 So the maximum comes from taking a derivative. 273 00:20:47,520 --> 00:20:55,590 So the maximum of p of r is determined 274 00:20:55,590 --> 00:21:00,690 by setting the derivative of this equal to 0. 275 00:21:00,690 --> 00:21:09,900 So you get 2n over r times the same r to the 2n e 276 00:21:09,900 --> 00:21:13,830 to the minus 22 over na0. 277 00:21:13,830 --> 00:21:15,480 So the first term, the derivative 278 00:21:15,480 --> 00:21:20,460 is 2r and r to the 2n minus 1, which is divided 279 00:21:20,460 --> 00:21:23,220 by r and the same thing. 280 00:21:23,220 --> 00:21:28,470 The second derivative gives you minus 2 over na0. 281 00:21:31,280 --> 00:21:32,350 And that's it. 282 00:21:32,350 --> 00:21:41,670 So from these two you get that n over r is equal to 1 over na0. 283 00:21:44,530 --> 00:21:54,320 So r equals n squared a0, as we had predicted. 284 00:21:54,320 --> 00:21:56,690 So it's the interplay of the polynomial 285 00:21:56,690 --> 00:22:03,390 with the other thing that makes the radius go very high. 286 00:22:03,390 --> 00:22:10,170 So what about this Rydberg atoms in nature? 287 00:22:10,170 --> 00:22:14,930 Well, they've been observed in interstellar gases. 288 00:22:14,930 --> 00:22:18,500 You see, there was this phenomenon of recombination 289 00:22:18,500 --> 00:22:24,530 when protons captured electrons as the universe cooled off, 290 00:22:24,530 --> 00:22:27,800 and formed hydrogen atoms. 291 00:22:27,800 --> 00:22:32,360 And that recombination sometimes works in such a way 292 00:22:32,360 --> 00:22:36,380 that the proton can capture an electron. 293 00:22:36,380 --> 00:22:40,160 And it captures it in a very high quantum number. 294 00:22:40,160 --> 00:22:46,970 And it keeps happening as we observe this electron. 295 00:22:46,970 --> 00:22:56,030 So people have observed in astrophysics n equal 350. 296 00:22:56,030 --> 00:22:59,790 And [INAUDIBLE] here the electron being captured, 297 00:22:59,790 --> 00:23:05,170 just not in the usual size but almost a million times bigger. 298 00:23:05,170 --> 00:23:12,970 So what happens for this thing r is equal 0.53 times 299 00:23:12,970 --> 00:23:16,930 10 to the minus 10 meters. 300 00:23:16,930 --> 00:23:19,050 That's a0. 301 00:23:19,050 --> 00:23:23,150 And then you have 350 squared. 302 00:23:23,150 --> 00:23:27,023 And that gives you 6.5 microns. 303 00:23:30,250 --> 00:23:31,750 That's actually pretty big. 304 00:23:36,330 --> 00:23:38,340 A blood cell is 8 microns. 305 00:23:38,340 --> 00:23:45,660 A red blood cell is 8 microns. 306 00:23:45,660 --> 00:23:50,580 They are not stable because eventually they spiral in. 307 00:23:50,580 --> 00:23:55,090 But if you have an atom that, for example, has 308 00:23:55,090 --> 00:24:00,630 an n equal to 5, it jumps to n equals to 1 in 10 309 00:24:00,630 --> 00:24:02,320 to the minus 7 seconds. 310 00:24:02,320 --> 00:24:04,650 These atoms are rather stable. 311 00:24:04,650 --> 00:24:08,670 Instead of lasting 1/10 of a millionth of a second, 312 00:24:08,670 --> 00:24:12,450 they can last a millisecond, 1/10 of a second, 313 00:24:12,450 --> 00:24:15,300 sometimes even 1 second. 314 00:24:15,300 --> 00:24:18,900 It takes a long time to go down that spiral. 315 00:24:18,900 --> 00:24:25,890 The energy levels-- if you have energies that go like 1 over n 316 00:24:25,890 --> 00:24:31,590 squared, the energy levels, the separation between them, 317 00:24:31,590 --> 00:24:34,720 goes like 1 over n cubed. 318 00:24:34,720 --> 00:24:38,460 So lots and lots of states there. 319 00:24:38,460 --> 00:24:43,370 And it takes a long time for it to decay from one to another. 320 00:24:43,370 --> 00:24:47,680 So they observe them in the lab in different ways. 321 00:24:47,680 --> 00:24:49,720 They create them with lasers now. 322 00:24:49,720 --> 00:24:52,840 You have the outermost electron, you kick it 323 00:24:52,840 --> 00:24:54,980 to another orbit with one-- 324 00:24:54,980 --> 00:24:56,410 they use three lasers. 325 00:24:56,410 --> 00:24:58,780 In the lab at MIT, three lasers. 326 00:24:58,780 --> 00:25:02,200 One and two to kick it to an n equal 10, 327 00:25:02,200 --> 00:25:06,370 and then the third one to kick it to n equals 60. 328 00:25:06,370 --> 00:25:09,680 And they detect those atoms by ionization. 329 00:25:09,680 --> 00:25:12,590 A normal atom, you can't ionize. 330 00:25:12,590 --> 00:25:16,060 You would need millions of volts per centimeter 331 00:25:16,060 --> 00:25:18,570 to ionize it with an electric field. 332 00:25:18,570 --> 00:25:21,140 These atoms you can ionize very easily. 333 00:25:21,140 --> 00:25:25,910 So they can see that they've been created that way. 334 00:25:25,910 --> 00:25:37,290 So also in terms of sizes, the diameter of hair 335 00:25:37,290 --> 00:25:41,010 is about 50 microns. 336 00:25:41,010 --> 00:25:42,330 Hair is very thin. 337 00:25:42,330 --> 00:25:44,340 But you see it. 338 00:25:44,340 --> 00:25:48,240 So you're about a factor of 5 or 10 339 00:25:48,240 --> 00:25:52,140 to being able to see that atom with your naked eye. 340 00:25:52,140 --> 00:25:53,520 It's pretty impressive. 341 00:25:53,520 --> 00:25:54,775 Incredible, in fact. 342 00:25:57,300 --> 00:26:04,320 So a nice laboratory, those are almost semi-classical atoms. 343 00:26:04,320 --> 00:26:07,440 All what Bohr was doing of calculating, 344 00:26:07,440 --> 00:26:12,060 you can derive this law by assuming 345 00:26:12,060 --> 00:26:16,680 that the transitions between orbits in the hydrogen atom 346 00:26:16,680 --> 00:26:20,400 emit photons of the right frequency. 347 00:26:20,400 --> 00:26:23,180 It's all kinds of fun things.