1 00:00:00,090 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT Open Courseware 4 00:00:06,330 --> 00:00:10,690 continue to offer high quality educational resources for free. 5 00:00:10,690 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,250 from hundreds of MIT courses, visit MIT Open Courseware 7 00:00:17,250 --> 00:00:18,200 at ocw.mit.edu. 8 00:00:22,090 --> 00:00:23,250 ROBERT FIELD: OK. 9 00:00:23,250 --> 00:00:26,580 So this is going to be a fun lecture because it's mostly 10 00:00:26,580 --> 00:00:29,130 pictures and language. 11 00:00:29,130 --> 00:00:33,780 And it's also an area of physical chemistry, 12 00:00:33,780 --> 00:00:35,880 which is pretty much-- 13 00:00:35,880 --> 00:00:39,780 I mean, there's three areas of physical chemistry 14 00:00:39,780 --> 00:00:42,150 where people have trouble talking to each other. 15 00:00:42,150 --> 00:00:44,007 There's statistical mechanics. 16 00:00:44,007 --> 00:00:45,090 There's quantum mechanics. 17 00:00:45,090 --> 00:00:48,000 And there's the time dependent, time independent forms 18 00:00:48,000 --> 00:00:49,470 of quantum mechanics. 19 00:00:49,470 --> 00:00:51,990 And these three communities have to learn 20 00:00:51,990 --> 00:00:53,520 how to talk to each other. 21 00:00:53,520 --> 00:00:57,370 And I'm hoping that I will help to bridge that gap. 22 00:00:57,370 --> 00:00:57,870 OK. 23 00:00:57,870 --> 00:01:01,721 So last time, we talked about the time dependent Schrodinger 24 00:01:01,721 --> 00:01:02,220 equation. 25 00:01:02,220 --> 00:01:05,060 It's a simple looking little thing. 26 00:01:05,060 --> 00:01:07,610 Remember though, it's an extra level of complexity 27 00:01:07,610 --> 00:01:10,670 on what you already understand. 28 00:01:10,670 --> 00:01:15,020 Now, in the special case where the Hamiltonian is time 29 00:01:15,020 --> 00:01:21,650 independent, then, if you know all of the energy levels 30 00:01:21,650 --> 00:01:26,360 and wave functions for the time independent Hamiltonian, 31 00:01:26,360 --> 00:01:29,180 you can immediately write down the solution 32 00:01:29,180 --> 00:01:33,470 to the time dependent Hamiltonian. 33 00:01:33,470 --> 00:01:37,970 And so you have a complete set of wave functions. 34 00:01:37,970 --> 00:01:40,910 And you have a complete set of energy levels. 35 00:01:40,910 --> 00:01:44,900 And with that, you can basically describe something 36 00:01:44,900 --> 00:01:47,480 that satisfies this thing. 37 00:01:47,480 --> 00:01:49,730 Another set of coefficients is often something 38 00:01:49,730 --> 00:01:56,090 that you arrange for simplicity or for insight. 39 00:01:56,090 --> 00:01:57,260 And I want to understand-- 40 00:01:57,260 --> 00:01:59,240 I want to talk about that. 41 00:01:59,240 --> 00:02:00,020 OK. 42 00:02:00,020 --> 00:02:02,180 And so in the previous lecture, we 43 00:02:02,180 --> 00:02:10,250 talked about the probability density, which is psi star psi. 44 00:02:10,250 --> 00:02:12,300 And that can move. 45 00:02:12,300 --> 00:02:16,230 And that can move in two ways. 46 00:02:19,800 --> 00:02:24,270 In the wave function, if there are two or more states 47 00:02:24,270 --> 00:02:29,140 belonging to two or more different energies, 48 00:02:29,140 --> 00:02:30,780 then you can have motion. 49 00:02:30,780 --> 00:02:36,700 And you can have motion like breathing, where probability 50 00:02:36,700 --> 00:02:41,890 moves towards the extremes, or motion 51 00:02:41,890 --> 00:02:44,920 where there is actually a wave packet going from one side 52 00:02:44,920 --> 00:02:47,580 to another. 53 00:02:47,580 --> 00:02:50,600 So for anything interesting to happen, 54 00:02:50,600 --> 00:02:53,870 this probability density has to include 55 00:02:53,870 --> 00:02:56,945 at least two different eigenstates belonging 56 00:02:56,945 --> 00:02:59,300 to two different energy levels. 57 00:02:59,300 --> 00:03:02,540 Now, the total probability is just the integral 58 00:03:02,540 --> 00:03:08,540 over all coordinates of the probability density. 59 00:03:08,540 --> 00:03:12,180 And probability is conserved. 60 00:03:12,180 --> 00:03:14,000 So one of the things that happens-- 61 00:03:14,000 --> 00:03:16,550 when you do an integral, the wave functions go away. 62 00:03:16,550 --> 00:03:20,660 And in this particular case, integral psi star psi, 63 00:03:20,660 --> 00:03:22,550 all you get is one. 64 00:03:22,550 --> 00:03:25,800 Or you get a constant, depending on how you normalize things. 65 00:03:25,800 --> 00:03:28,930 But it's not time dependent. 66 00:03:28,930 --> 00:03:34,640 Now, in understanding motion from classical mechanics, 67 00:03:34,640 --> 00:03:36,310 we know Newton's equations. 68 00:03:36,310 --> 00:03:42,880 We know how the coordinate and the momentum depend on time. 69 00:03:42,880 --> 00:03:47,290 And so if we calculate the expectation 70 00:03:47,290 --> 00:03:49,900 value of the coordinate or the expectation 71 00:03:49,900 --> 00:03:53,650 value of the momentum, we get some results. 72 00:03:53,650 --> 00:03:58,180 And it turns out that these things describe the motion 73 00:03:58,180 --> 00:04:00,490 of the center of a wave packet. 74 00:04:00,490 --> 00:04:04,240 A wave packet is anything that's not just a single eigenstate, 75 00:04:04,240 --> 00:04:05,740 so it moves. 76 00:04:05,740 --> 00:04:10,560 And so the motion of a single eigenstate-- 77 00:04:10,560 --> 00:04:15,130 the motion of a wave packet is described by Newton's laws. 78 00:04:15,130 --> 00:04:17,050 And this is Ehrenfest theorem. 79 00:04:17,050 --> 00:04:18,940 And it can easily be proven. 80 00:04:18,940 --> 00:04:20,709 But we're not going to prove it. 81 00:04:20,709 --> 00:04:22,240 We're just going to use it. 82 00:04:22,240 --> 00:04:27,010 So we are very interested in expectation values 83 00:04:27,010 --> 00:04:29,410 of the coordinate and the momentum. 84 00:04:29,410 --> 00:04:33,700 And for a harmonic oscillator, this is duck soup. 85 00:04:33,700 --> 00:04:38,500 Then we have nothing but trivial integrals involving 86 00:04:38,500 --> 00:04:40,870 the a's and a-daggers. 87 00:04:40,870 --> 00:04:45,580 And so even though in some ways the harmonic oscillator 88 00:04:45,580 --> 00:04:49,880 is a more complicated problem than the particle in a box, 89 00:04:49,880 --> 00:04:51,610 the harmonic oscillator is the problem 90 00:04:51,610 --> 00:05:00,190 of choice for dealing with motion and developing insight. 91 00:05:00,190 --> 00:05:02,110 There's another useful-- 92 00:05:02,110 --> 00:05:02,990 OK. 93 00:05:02,990 --> 00:05:05,780 We have this wave function here. 94 00:05:05,780 --> 00:05:09,290 There is a huge amount of information embedded in it, 95 00:05:09,290 --> 00:05:12,050 too much to just look at. 96 00:05:12,050 --> 00:05:16,230 We need to have ways of reducing that information. 97 00:05:16,230 --> 00:05:19,880 And one of the nice ways is the survival probability. 98 00:05:19,880 --> 00:05:22,010 The survival probability tells you 99 00:05:22,010 --> 00:05:27,600 how fast does the initial state move away from itself. 100 00:05:27,600 --> 00:05:30,240 And this is very revealing. 101 00:05:30,240 --> 00:05:34,766 Another very revealing thing is if this survival probability, 102 00:05:34,766 --> 00:05:36,390 which is a function of time and nothing 103 00:05:36,390 --> 00:05:40,350 else, because we've integrated over the wave functions-- 104 00:05:40,350 --> 00:05:42,840 if this survival probability goes up and down 105 00:05:42,840 --> 00:05:47,160 and up and down, there are recurrences. 106 00:05:47,160 --> 00:05:49,860 So at a maximum, which is usually 107 00:05:49,860 --> 00:05:53,520 a maximum at t equals 0 because the wave function is it 108 00:05:53,520 --> 00:05:57,060 at its birthplace, the survival probability 109 00:05:57,060 --> 00:06:00,630 will start high and go down, and come up, and go down. 110 00:06:00,630 --> 00:06:03,780 And there's all sorts of information about recurrences. 111 00:06:03,780 --> 00:06:07,800 As the wave function returns to its birthplace, 112 00:06:07,800 --> 00:06:11,040 it might not return completely. 113 00:06:11,040 --> 00:06:13,420 And so we get partial recurrences. 114 00:06:13,420 --> 00:06:15,910 And these tell us something. 115 00:06:15,910 --> 00:06:20,830 And the times at which the maxima in the survival 116 00:06:20,830 --> 00:06:27,120 probability occur tell us a lot about the potential. 117 00:06:27,120 --> 00:06:29,280 And there are some grand recurrences 118 00:06:29,280 --> 00:06:33,390 where, because all of the energy level differences 119 00:06:33,390 --> 00:06:36,130 are an integer multiple of a common factor, 120 00:06:36,130 --> 00:06:39,120 then you get a perfect recurrence. 121 00:06:39,120 --> 00:06:43,150 And these are wonderful for drawing pictures. 122 00:06:43,150 --> 00:06:50,590 Now, you really want to understand the concepts of time 123 00:06:50,590 --> 00:06:53,900 dependent quantum mechanics pictorially 124 00:06:53,900 --> 00:06:56,720 and to develop a language that describes it. 125 00:06:56,720 --> 00:06:58,220 And there are a lot of little pieces 126 00:06:58,220 --> 00:07:02,904 here that you need to convince yourself you understand 127 00:07:02,904 --> 00:07:04,070 so you can use the language. 128 00:07:21,230 --> 00:07:22,080 OK. 129 00:07:22,080 --> 00:07:24,840 One of the things I said at the beginning of the course 130 00:07:24,840 --> 00:07:32,310 is the central object in quantum mechanics is the wave function. 131 00:07:32,310 --> 00:07:36,900 Or we could generalize to this. 132 00:07:36,900 --> 00:07:39,760 This is actually the truth. 133 00:07:39,760 --> 00:07:42,360 This is a partial truth. 134 00:07:42,360 --> 00:07:46,050 This is everything that we can possibly know. 135 00:07:46,050 --> 00:07:47,910 But we can't know this. 136 00:07:47,910 --> 00:07:50,560 We can never observe the wave function. 137 00:07:50,560 --> 00:07:53,490 However, we do observations. 138 00:07:53,490 --> 00:07:58,980 And we construct a picture, which 139 00:07:58,980 --> 00:08:03,040 is what we call the effective Hamiltonian. 140 00:08:03,040 --> 00:08:06,660 This effective Hamiltonian describes everything 141 00:08:06,660 --> 00:08:09,930 we know, mostly energy levels. 142 00:08:09,930 --> 00:08:12,640 We have formulas for the energy levels. 143 00:08:12,640 --> 00:08:14,490 And we determine the constants. 144 00:08:14,490 --> 00:08:18,130 There's all sorts of stuff that we collect by experiment. 145 00:08:18,130 --> 00:08:20,820 And so we create an object which we think 146 00:08:20,820 --> 00:08:23,730 is like the true Hamiltonian. 147 00:08:23,730 --> 00:08:27,660 And by having the effective Hamiltonian, 148 00:08:27,660 --> 00:08:32,309 we can get these things. 149 00:08:32,309 --> 00:08:36,330 So we observe we make some observations. 150 00:08:36,330 --> 00:08:40,320 And then this is a reduced version of the truth. 151 00:08:40,320 --> 00:08:43,770 And we can get a pretty good representation 152 00:08:43,770 --> 00:08:46,440 of the thing that is supposedly hidden from us. 153 00:08:50,201 --> 00:08:50,700 OK. 154 00:08:53,352 --> 00:08:58,085 So let's start now at t equals 0. 155 00:08:58,085 --> 00:09:01,430 At t equals 0, which I like to call the pluck-- 156 00:09:01,430 --> 00:09:04,430 and it really makes it contact with those of you who 157 00:09:04,430 --> 00:09:08,330 are musicians, that you know how to create 158 00:09:08,330 --> 00:09:12,700 a particular kind of sound, which will evolve in time. 159 00:09:12,700 --> 00:09:18,200 And it depends on the details of how you handle the instrument. 160 00:09:18,200 --> 00:09:24,080 So at t equals 0, we have this thing 161 00:09:24,080 --> 00:09:29,030 which, if we have a complete set of eigenfunctions 162 00:09:29,030 --> 00:09:34,160 of the Hamiltonian, of the time independent Hamiltonian, 163 00:09:34,160 --> 00:09:39,146 we know we can write this as cj psi j of x. 164 00:09:45,930 --> 00:09:48,960 Now, often we don't need an infinite number of terms. 165 00:09:48,960 --> 00:09:53,190 But we know that there are lots of different kinds of plucks. 166 00:09:53,190 --> 00:10:00,390 And you can find out what these coefficients are 167 00:10:00,390 --> 00:10:05,190 by just calculating overlap integral of these functions 168 00:10:05,190 --> 00:10:06,570 with this initial state. 169 00:10:10,961 --> 00:10:11,460 OK. 170 00:10:11,460 --> 00:10:14,730 So this is telling you what happens at t equals 0. 171 00:10:14,730 --> 00:10:18,630 And if you can write the initial state 172 00:10:18,630 --> 00:10:20,970 as a superposition of eigenstates, 173 00:10:20,970 --> 00:10:24,030 then you can immediately write the time 174 00:10:24,030 --> 00:10:46,620 evolving object this way. 175 00:10:46,620 --> 00:10:52,190 Now, we have a minus i. 176 00:10:52,190 --> 00:10:55,190 Well, why do we need a minus i? 177 00:10:55,190 --> 00:10:56,700 Well, because we're going to take 178 00:10:56,700 --> 00:10:59,760 the time derivative of the Hamiltonian 179 00:10:59,760 --> 00:11:02,430 and multiply it by iH bar. 180 00:11:02,430 --> 00:11:06,780 We're going to want to get the energy, not minus the energy. 181 00:11:09,380 --> 00:11:12,440 And so we need a minus i here. 182 00:11:12,440 --> 00:11:15,970 And we need a divided by H bar here 183 00:11:15,970 --> 00:11:20,240 in order to satisfy the time independence or time 184 00:11:20,240 --> 00:11:22,130 dependence. 185 00:11:22,130 --> 00:11:24,681 So this always bothers people. 186 00:11:24,681 --> 00:11:25,930 Why should there be an i here? 187 00:11:25,930 --> 00:11:28,210 And why should it be minus i? 188 00:11:28,210 --> 00:11:31,330 And I recommend not trying to memorize it, 189 00:11:31,330 --> 00:11:32,890 but to just convince yourself. 190 00:11:32,890 --> 00:11:37,060 I need the minus i because I'm going to bring down a minus i 191 00:11:37,060 --> 00:11:39,880 over H bar times e. 192 00:11:39,880 --> 00:11:42,310 And we have this iH bar. 193 00:11:42,310 --> 00:11:48,170 The minus i times i gives plus 1. 194 00:11:48,170 --> 00:11:49,600 OK. 195 00:11:49,600 --> 00:11:59,570 So now, in understanding how the time dependent Hamiltonian can 196 00:11:59,570 --> 00:12:05,660 be sampled, we build the superposition states 197 00:12:05,660 --> 00:12:10,550 with a minimum number of terms, like two or three. 198 00:12:10,550 --> 00:12:15,530 Even though, in order to create a very, very sharply localized 199 00:12:15,530 --> 00:12:20,420 state, it needs a huge number of terms here. 200 00:12:20,420 --> 00:12:23,300 But you always build your insight 201 00:12:23,300 --> 00:12:25,890 with something you can do in your head. 202 00:12:25,890 --> 00:12:32,010 And so for some problems, you need only two states. 203 00:12:32,010 --> 00:12:34,440 And for others, you need three. 204 00:12:34,440 --> 00:12:37,440 And those are the ones you want to kill. 205 00:12:37,440 --> 00:12:39,330 You want to understand everything 206 00:12:39,330 --> 00:12:43,166 you can build with two and three state superpositions. 207 00:12:46,290 --> 00:12:47,130 OK. 208 00:12:47,130 --> 00:12:53,505 So there's two classes of problems. 209 00:13:07,500 --> 00:13:12,740 OK so for a half harmonic oscillator-- 210 00:13:24,710 --> 00:13:28,400 so for a half harmonic oscillator, 211 00:13:28,400 --> 00:13:30,440 we start with harmonic oscillator 212 00:13:30,440 --> 00:13:35,670 and divide it in half and say this goes to infinity. 213 00:13:35,670 --> 00:13:39,840 And so this side is not accessible. 214 00:13:39,840 --> 00:13:43,640 And so if the potential is half of a harmonic oscillator 215 00:13:43,640 --> 00:13:46,490 potential, then we know immediately what 216 00:13:46,490 --> 00:13:50,540 the eigenvalues and eigenfunctions are. 217 00:13:50,540 --> 00:13:52,519 They are the harmonic oscillator functions 218 00:13:52,519 --> 00:13:53,810 that have a node in the middle. 219 00:13:56,760 --> 00:14:07,230 So we know that we have v equals 1, v equals 3. 220 00:14:07,230 --> 00:14:10,600 No v equals 0 or 2. 221 00:14:10,600 --> 00:14:13,220 I mean, this corresponds to the lowest level, v 222 00:14:13,220 --> 00:14:14,740 equals 0 for the half oscillator. 223 00:14:14,740 --> 00:14:16,630 But this is basically-- now, what 224 00:14:16,630 --> 00:14:20,740 we're going to do is we create some state of a half 225 00:14:20,740 --> 00:14:23,250 oscillator. 226 00:14:23,250 --> 00:14:26,050 And then we take away the barrier. 227 00:14:26,050 --> 00:14:29,250 So that's a cheap way of creating localization. 228 00:14:29,250 --> 00:14:31,760 Of course, we can't do this experimentally. 229 00:14:31,760 --> 00:14:34,840 But you could, in principle, do it. 230 00:14:34,840 --> 00:14:46,650 And so basically, you want an initial state, 231 00:14:46,650 --> 00:14:50,885 which is a superposition of vibrational levels. 232 00:14:56,290 --> 00:14:56,970 OK. 233 00:14:56,970 --> 00:15:02,660 And this initial state needs to have the wave 234 00:15:02,660 --> 00:15:04,525 function be 0 at the barrier. 235 00:15:07,860 --> 00:15:13,740 So since we're going to be looking at the full potential, 236 00:15:13,740 --> 00:15:15,480 we're taking away the barrier. 237 00:15:15,480 --> 00:15:20,460 We're allowed to use the even and odd states. 238 00:15:20,460 --> 00:15:24,200 And so let's take the three lowest states. 239 00:15:24,200 --> 00:15:32,390 And so we have c0 psi 0 plus c1 psi 1 plus c2 psi 2. 240 00:15:34,811 --> 00:15:35,310 OK. 241 00:15:35,310 --> 00:15:37,140 This guy is OK. 242 00:15:37,140 --> 00:15:38,310 Yes? 243 00:15:38,310 --> 00:15:42,260 AUDIENCE: When you create the states of the half harmonic 244 00:15:42,260 --> 00:15:45,360 oscillator, do you have to re-scale them 245 00:15:45,360 --> 00:15:46,600 so that they normalize? 246 00:15:46,600 --> 00:15:47,350 ROBERT FIELD: Yes. 247 00:15:50,400 --> 00:15:52,980 We play fast and loose with normalization constants. 248 00:15:52,980 --> 00:15:55,530 We already always know that, whenever 249 00:15:55,530 --> 00:15:57,480 you want to calculate anything, you're 250 00:15:57,480 --> 00:16:02,130 going to divide by the normalization integral. 251 00:16:02,130 --> 00:16:04,890 However, it's OK. 252 00:16:04,890 --> 00:16:08,010 Because we're going to be using the full harmonic oscillator 253 00:16:08,010 --> 00:16:12,390 functions, which do exist over here. 254 00:16:12,390 --> 00:16:15,960 And it's fine once we remove the barrier. 255 00:16:15,960 --> 00:16:21,810 But we want to choose a problem which is as simple as possible. 256 00:16:21,810 --> 00:16:26,460 OK, now, this guy is not 0 at x equals 0. 257 00:16:26,460 --> 00:16:28,530 And this guy is not 0 at x equals 0. 258 00:16:28,530 --> 00:16:40,710 But you can say 0 is equal to c0 psi 0 of 0 plus c2 psi 2 of 0. 259 00:16:40,710 --> 00:16:46,590 We choose the coefficients so that these two guys together 260 00:16:46,590 --> 00:16:49,890 make 0 at the boundary. 261 00:16:49,890 --> 00:16:50,850 And why do we do this? 262 00:16:50,850 --> 00:16:54,720 We're going to be calculating expectation values of x and p. 263 00:16:54,720 --> 00:16:59,970 And if we only have half of the energy levels, 264 00:16:59,970 --> 00:17:03,940 all of the intricacies are going to be 0. 265 00:17:03,940 --> 00:17:05,950 So we have to have three consecutive levels 266 00:17:05,950 --> 00:17:07,511 to have anything interesting. 267 00:17:10,220 --> 00:17:14,640 OK, so the artifice of making these two-- 268 00:17:14,640 --> 00:17:20,190 arranging the coefficients so that you get a temporary node 269 00:17:20,190 --> 00:17:24,170 at x equals 0-- 270 00:17:24,170 --> 00:17:25,210 it won't stay a node. 271 00:17:25,210 --> 00:17:27,760 Because when we let things oscillate with time, 272 00:17:27,760 --> 00:17:30,670 these two will oscillate differently. 273 00:17:30,670 --> 00:17:33,409 And the node will go away. 274 00:17:33,409 --> 00:17:34,450 But that's how you do it. 275 00:17:34,450 --> 00:17:37,571 That's how you build a superposition. 276 00:17:37,571 --> 00:17:38,070 OK. 277 00:17:46,111 --> 00:17:46,610 OK. 278 00:17:46,610 --> 00:17:51,710 Now, what we want to do is to be able to draw pictures 279 00:17:51,710 --> 00:17:57,240 of what's happening and also to calculate what's going on. 280 00:17:57,240 --> 00:18:00,230 And one of the things we use for our pictures 281 00:18:00,230 --> 00:18:05,990 is the expectation value of x and the expectation value of p. 282 00:18:09,100 --> 00:18:12,460 So you know how to calculate the expectation value of x. 283 00:18:12,460 --> 00:18:18,650 We have this capital psi star x capital psi star capital 284 00:18:18,650 --> 00:18:21,610 psi integrate over x. 285 00:18:21,610 --> 00:18:35,535 And so we get to c0 c1 x01 cosine omega t plus 2 286 00:18:35,535 --> 00:18:43,710 c1 c2 x12 cosine omega t. 287 00:18:43,710 --> 00:18:45,000 Now, how did I do that? 288 00:18:48,784 --> 00:18:49,290 Oh, come on. 289 00:18:49,290 --> 00:18:51,330 You can do it, too. 290 00:18:51,330 --> 00:18:52,550 So what is this x01? 291 00:18:52,550 --> 00:19:03,020 Well, x01 is the integral psi 0 star x psi 1 dx. 292 00:19:03,020 --> 00:19:06,050 And we know we can replace x by a plus a-dagger 293 00:19:06,050 --> 00:19:08,370 times the constant. 294 00:19:08,370 --> 00:19:10,050 We always like to forget that concept. 295 00:19:10,050 --> 00:19:13,080 We only bring it in at the end anyway. 296 00:19:13,080 --> 00:19:16,700 And so, well, this has a value-- 297 00:19:16,700 --> 00:19:22,370 it's a constant, the constant that we're forgetting-- 298 00:19:22,370 --> 00:19:26,990 times 1 square root, right? 299 00:19:30,150 --> 00:19:34,080 And x12 is the same sort of thing. 300 00:19:34,080 --> 00:19:41,520 It's going to be the same constant times 2 square root. 301 00:19:41,520 --> 00:19:43,380 This is easy. 302 00:19:43,380 --> 00:19:48,450 So getting from this symbol to this, 303 00:19:48,450 --> 00:19:52,640 that just requires a little practice, 304 00:19:52,640 --> 00:19:54,730 and then simplifying further to know 305 00:19:54,730 --> 00:20:00,110 what these x's are, and then the constraints on c2 and c0. 306 00:20:00,110 --> 00:20:03,030 We have all that stuff. 307 00:20:03,030 --> 00:20:03,530 OK. 308 00:20:03,530 --> 00:20:10,230 And now, we can draw a picture of what's going to happen. 309 00:20:10,230 --> 00:20:12,380 So here we have the full oscillator. 310 00:20:12,380 --> 00:20:15,950 And let's just draw some energy which is-- 311 00:20:15,950 --> 00:20:17,070 now that's complicated. 312 00:20:17,070 --> 00:20:20,790 We have a time dependent wave function, 313 00:20:20,790 --> 00:20:25,910 which is composed of several different energy eigenstates. 314 00:20:25,910 --> 00:20:27,470 So what is its energy? 315 00:20:27,470 --> 00:20:29,210 Well, you can evaluate what the energy 316 00:20:29,210 --> 00:20:34,380 is by taking the expectation value of the Hamiltonian. 317 00:20:34,380 --> 00:20:38,300 And so you can do that. 318 00:20:38,300 --> 00:20:40,149 So what we've made at t equals 0 is 319 00:20:40,149 --> 00:20:41,440 something that looks like this. 320 00:20:47,060 --> 00:20:48,904 It's localized on the left side. 321 00:20:48,904 --> 00:20:50,570 Or it's more localized on the left side. 322 00:20:50,570 --> 00:20:54,655 Now, sometimes, you're going to worry about phase. 323 00:20:57,360 --> 00:21:02,280 And so many times when you're working symbolically 324 00:21:02,280 --> 00:21:05,160 rather than actually evaluating integrals, 325 00:21:05,160 --> 00:21:12,860 there are symbolic phase choices that what you're using 326 00:21:12,860 --> 00:21:13,990 has made. 327 00:21:13,990 --> 00:21:15,980 And for example, for the harmonic oscillator, 328 00:21:15,980 --> 00:21:18,740 if you look in the book, you'll see 329 00:21:18,740 --> 00:21:24,410 that, for all of the harmonic oscillator functions, 330 00:21:24,410 --> 00:21:26,780 the outer lobe is always positive. 331 00:21:26,780 --> 00:21:30,080 And the inner lobe is alternating 332 00:21:30,080 --> 00:21:33,330 with the quantum number. 333 00:21:33,330 --> 00:21:38,090 So that's a phase convention that's implicit in everything 334 00:21:38,090 --> 00:21:41,150 that people have derived. 335 00:21:41,150 --> 00:21:44,180 And as long as the different things 336 00:21:44,180 --> 00:21:48,560 you combine in doing a calculation involve 337 00:21:48,560 --> 00:21:52,190 the same phase convention, which is implicit-- we don't 338 00:21:52,190 --> 00:21:54,330 want to look at what functions. 339 00:21:54,330 --> 00:21:57,731 We want to look at these xij's. 340 00:21:57,731 --> 00:21:58,230 OK. 341 00:21:58,230 --> 00:22:03,570 And so those things that we're manipulating 342 00:22:03,570 --> 00:22:07,020 have a phase implicit in how you define them. 343 00:22:07,020 --> 00:22:09,350 But that's gone in your manipulation. 344 00:22:09,350 --> 00:22:11,340 So you want to be a little careful. 345 00:22:11,340 --> 00:22:13,350 OK. 346 00:22:13,350 --> 00:22:21,090 So we start out and the expectation value 347 00:22:21,090 --> 00:22:23,460 is on this side. 348 00:22:23,460 --> 00:22:30,370 And the psi star psi is localized on this side mostly. 349 00:22:30,370 --> 00:22:33,330 And at a later time-- 350 00:22:33,330 --> 00:22:35,950 so this is a later time. 351 00:22:35,950 --> 00:22:39,320 I shouldn't make it bigger. 352 00:22:39,320 --> 00:22:42,380 At a later time, this thing has moved to the other turning 353 00:22:42,380 --> 00:22:44,260 point. 354 00:22:44,260 --> 00:22:46,360 Back and forth, back and forth. 355 00:22:49,690 --> 00:22:55,610 Now, since we have three energy levels-- 356 00:22:55,610 --> 00:22:59,780 and well, actually, we have a coherence term 357 00:22:59,780 --> 00:23:04,970 which involves the product of psi 0 and psi 1 358 00:23:04,970 --> 00:23:07,610 and psi 1 and psi 2-- 359 00:23:07,610 --> 00:23:11,420 they differ in energy both by omega. 360 00:23:11,420 --> 00:23:15,680 So these two things have the same oscillation frequency. 361 00:23:15,680 --> 00:23:18,950 And so what's going to happen is the wave function, the wave 362 00:23:18,950 --> 00:23:21,620 packet, is going to move from this side to that side, 363 00:23:21,620 --> 00:23:26,690 back and forth always forever at the same frequency, 364 00:23:26,690 --> 00:23:28,660 no dephasing. 365 00:23:28,660 --> 00:23:32,470 In the middle, I can't tell you what it's going to look like. 366 00:23:32,470 --> 00:23:33,670 I don't want to tell you. 367 00:23:33,670 --> 00:23:34,650 Because you don't care. 368 00:23:37,320 --> 00:23:41,580 You care mostly about what's it going to look like at a turning 369 00:23:41,580 --> 00:23:42,870 point. 370 00:23:42,870 --> 00:23:47,920 Or what is it going to look like when it returns to home base? 371 00:23:47,920 --> 00:23:50,050 And all sorts of insights come from that. 372 00:23:52,600 --> 00:23:54,340 OK. 373 00:23:54,340 --> 00:23:59,940 Now, I said, well, if we want to know the energy of this wave 374 00:23:59,940 --> 00:24:03,300 packet, well, we take the expectation value 375 00:24:03,300 --> 00:24:04,890 of the Hamiltonian. 376 00:24:04,890 --> 00:24:07,020 And the expectation of the Hamiltonian 377 00:24:07,020 --> 00:24:29,450 is c0 squared e0 plus c1 squared e1 plus c2 squared e2. 378 00:24:35,240 --> 00:24:37,010 These things are easy once you've gone 379 00:24:37,010 --> 00:24:38,930 through it a couple of times. 380 00:24:38,930 --> 00:24:42,760 Because what's happening is you have these factors, 381 00:24:42,760 --> 00:24:50,310 e to the minus i ej t over H bar. 382 00:24:50,310 --> 00:24:54,960 And they cancel when you do a psi star psi. 383 00:24:54,960 --> 00:24:58,830 Or they generate a difference, omega H bar omega, 384 00:24:58,830 --> 00:25:01,750 when you have different values of the energy. 385 00:25:01,750 --> 00:25:05,700 So this is something that you can derive really quickly. 386 00:25:05,700 --> 00:25:09,090 And the fact that your eigenfunction-- 387 00:25:09,090 --> 00:25:13,780 the psi's in your linear combination 388 00:25:13,780 --> 00:25:16,530 up there are eigenfunctions of the Hamiltonian. 389 00:25:16,530 --> 00:25:21,460 So every time the Hamiltonian operates in a wave function, 390 00:25:21,460 --> 00:25:24,510 it gives the energy times that wave function. 391 00:25:24,510 --> 00:25:27,930 And then you're taking the expectation value. 392 00:25:27,930 --> 00:25:32,010 And so we have the wave function time itself integrated. 393 00:25:32,010 --> 00:25:33,490 And that goes away. 394 00:25:33,490 --> 00:25:36,870 So after doing this a few times, you 395 00:25:36,870 --> 00:25:40,000 don't need to write the intermediate steps. 396 00:25:40,000 --> 00:25:41,760 And you shouldn't. 397 00:25:41,760 --> 00:25:45,397 Because you'll get lost in the forest of notation. 398 00:25:45,397 --> 00:25:46,980 Because one of the things you probably 399 00:25:46,980 --> 00:25:50,790 noticed in the last lecture is the equations got really big. 400 00:25:50,790 --> 00:25:52,500 And then we calculate something else. 401 00:25:52,500 --> 00:25:54,660 And it gets twice as big. 402 00:25:54,660 --> 00:25:56,750 And then all of a sudden, it all goes away. 403 00:25:56,750 --> 00:25:59,588 And that's what you want to be able to anticipate. 404 00:26:09,160 --> 00:26:09,660 OK. 405 00:26:09,660 --> 00:26:14,260 So you can really kill this half oscillator problem. 406 00:26:14,260 --> 00:26:16,210 There's nothing much happening except you 407 00:26:16,210 --> 00:26:18,430 create a localization. 408 00:26:18,430 --> 00:26:20,080 And when you take away the other-- 409 00:26:20,080 --> 00:26:23,410 when you restore the full oscillator, 410 00:26:23,410 --> 00:26:28,320 everything oscillates at omega. 411 00:26:28,320 --> 00:26:30,350 And so you have a whole bunch of terms 412 00:26:30,350 --> 00:26:35,240 contributing to the motion of the wave packet. 413 00:26:35,240 --> 00:26:36,410 And they're all very simple. 414 00:26:36,410 --> 00:26:38,659 Because they're all oscillating at the same frequency. 415 00:26:42,780 --> 00:26:47,420 Now, as an aside, I want to say there's a huge number of stuff 416 00:26:47,420 --> 00:26:49,590 that's up in this lecture that's going to be 417 00:26:49,590 --> 00:26:53,850 on the exam, a huge amount. 418 00:26:53,850 --> 00:26:58,260 So if for example you wanted to calculate something 419 00:26:58,260 --> 00:27:02,770 like x squared, well, fine. 420 00:27:02,770 --> 00:27:06,750 You know what the selection rule for x squared is. 421 00:27:06,750 --> 00:27:10,560 It's delta v of 0 plus or minus 2. 422 00:27:10,560 --> 00:27:14,820 And so this thing is going to generate some constant terms 423 00:27:14,820 --> 00:27:16,320 and some terms at 2 omega. 424 00:27:21,940 --> 00:27:22,620 OK. 425 00:27:22,620 --> 00:27:26,750 So there are a lot of things about the harmonic oscillator 426 00:27:26,750 --> 00:27:31,370 that make it really wonderful to consider a problem, even 427 00:27:31,370 --> 00:27:34,100 a complicated problem, which is not explicitly 428 00:27:34,100 --> 00:27:35,690 a harmonic oscillator problem. 429 00:27:35,690 --> 00:27:39,740 Because you can get everything so quickly without any 430 00:27:39,740 --> 00:27:42,215 thought after a little bit of investment. 431 00:27:44,830 --> 00:27:46,640 OK. 432 00:27:46,640 --> 00:27:50,120 Now, we haven't talked about electronic transitions 433 00:27:50,120 --> 00:27:51,590 and potential energy curves. 434 00:27:51,590 --> 00:27:52,910 But we will. 435 00:27:52,910 --> 00:27:55,130 And I think you know about them. 436 00:27:55,130 --> 00:27:57,850 And so we have some electronic ground state. 437 00:27:57,850 --> 00:28:01,020 And we have some electronically excited state. 438 00:28:01,020 --> 00:28:07,790 And so each of these states has a vibrational coordinate. 439 00:28:07,790 --> 00:28:13,420 And we can pretend that it's harmonic even if it's not. 440 00:28:13,420 --> 00:28:15,520 Because we build a framework treating 441 00:28:15,520 --> 00:28:17,260 them as harmonic, and then discover 442 00:28:17,260 --> 00:28:23,320 that there's discrepancies which we can fit to a model. 443 00:28:23,320 --> 00:28:25,780 And we can determine from the time 444 00:28:25,780 --> 00:28:27,340 dependence what that model is. 445 00:28:27,340 --> 00:28:28,090 OK. 446 00:28:28,090 --> 00:28:34,220 So we start out with a molecule in v equals 0. 447 00:28:37,890 --> 00:28:45,420 And there is a much repeated truism. 448 00:28:45,420 --> 00:28:48,870 Electrons move fast, nuclei slow. 449 00:28:48,870 --> 00:28:52,450 Transition is instantaneous, or nearly instantaneous, 450 00:28:52,450 --> 00:28:54,400 because it involves the electrons. 451 00:28:54,400 --> 00:28:59,280 So what ends up happening is you draw a vertical line. 452 00:28:59,280 --> 00:29:05,750 And now, you transfer this wave function to the upper state. 453 00:29:05,750 --> 00:29:08,930 You just move-- this is the probability amplitude 454 00:29:08,930 --> 00:29:15,170 distribution of the vibration. 455 00:29:15,170 --> 00:29:17,360 We transfer that to the excited state. 456 00:29:17,360 --> 00:29:19,880 And that's not an eigenstate, the excited state. 457 00:29:19,880 --> 00:29:23,770 It's a localized state, localized at a turning point. 458 00:29:23,770 --> 00:29:26,560 Now, you want to know-- 459 00:29:26,560 --> 00:29:34,330 so there's the Franck-Condon principle, 460 00:29:34,330 --> 00:29:36,930 that is just another way of saying electrons 461 00:29:36,930 --> 00:29:38,950 move fast, nuclei slow. 462 00:29:38,950 --> 00:29:44,460 And there is delta x equals 0 delta p equals 0. 463 00:29:47,980 --> 00:29:51,160 If the nuclear state can't change while the electron is 464 00:29:51,160 --> 00:29:58,810 jumping, then the coordinate and the momentum are both constant. 465 00:29:58,810 --> 00:30:01,480 So you're creating a wave packet. 466 00:30:01,480 --> 00:30:04,600 And the best place to create it is near a turning point. 467 00:30:04,600 --> 00:30:08,080 Because then, you can match the momentum of this guy 468 00:30:08,080 --> 00:30:11,890 to that zero point momentum, which you can calculate. 469 00:30:11,890 --> 00:30:13,660 You know how to calculate this. 470 00:30:13,660 --> 00:30:25,540 Because the potential energy curve is 1/2 k x squared. 471 00:30:25,540 --> 00:30:34,100 And the momentum is given by this energy difference here. 472 00:30:34,100 --> 00:30:35,530 And so you can work it out. 473 00:30:35,530 --> 00:30:36,280 It's in the notes. 474 00:30:36,280 --> 00:30:38,290 I don't want to write it down. 475 00:30:38,290 --> 00:30:40,330 So you're going to create something 476 00:30:40,330 --> 00:30:43,960 which is vertical and is not quite exactly 477 00:30:43,960 --> 00:30:45,550 at the turning point. 478 00:30:45,550 --> 00:30:48,130 Because you have to have a little bit of momentum 479 00:30:48,130 --> 00:30:51,270 to match the zero point momentum here. 480 00:30:51,270 --> 00:30:55,220 So you know everything about the initial state. 481 00:30:55,220 --> 00:30:59,035 And so you can calculate what it is by taking the overlap of v 482 00:30:59,035 --> 00:31:03,550 equals 0 of the ground state with all 483 00:31:03,550 --> 00:31:06,620 of the vibrational levels of the excited state. 484 00:31:06,620 --> 00:31:09,730 And so we have the coefficients of each 485 00:31:09,730 --> 00:31:12,760 of those vibrational levels in the excited state that 486 00:31:12,760 --> 00:31:15,530 makes this wave packet. 487 00:31:15,530 --> 00:31:17,690 Now, of course, this wave packet is 488 00:31:17,690 --> 00:31:20,660 going to move back and forth, back and forth. 489 00:31:23,450 --> 00:31:25,570 And if this were a harmonic oscillator, 490 00:31:25,570 --> 00:31:28,070 it would move harmonically. 491 00:31:28,070 --> 00:31:30,400 And so the only thing that would appear 492 00:31:30,400 --> 00:31:33,700 in the expectation value of x is going 493 00:31:33,700 --> 00:31:38,450 to be this motion at omega. 494 00:31:38,450 --> 00:31:41,000 Now, this is usually a relatively high 495 00:31:41,000 --> 00:31:42,110 vibrational level. 496 00:31:42,110 --> 00:31:45,860 And the molecule is not being harmonic here. 497 00:31:45,860 --> 00:31:49,460 So there are correction terms called anharmonicity terms. 498 00:31:52,550 --> 00:32:05,320 So we have the energy level expression plus H bar omega e 499 00:32:05,320 --> 00:32:06,620 xe-- 500 00:32:06,620 --> 00:32:13,600 that's one number-- plus 1/2 squared. 501 00:32:13,600 --> 00:32:15,960 So we have a linear term and a quadratic term. 502 00:32:15,960 --> 00:32:24,180 And this omega e xe is on the order of 0.02 times omega 503 00:32:24,180 --> 00:32:26,550 e, 2%. 504 00:32:26,550 --> 00:32:28,870 So it's a small thing. 505 00:32:28,870 --> 00:32:31,800 But if you're going to be allowing a wave 506 00:32:31,800 --> 00:32:36,520 packet to be built out of many vibrational levels, 507 00:32:36,520 --> 00:32:38,500 this guy is going to de-phase a little bit. 508 00:32:38,500 --> 00:32:40,300 So you go around and come back. 509 00:32:40,300 --> 00:32:42,460 And you can't quite have everybody 510 00:32:42,460 --> 00:32:45,230 back where they started. 511 00:32:45,230 --> 00:32:53,900 And so you'll see a decreasing amplitude here. 512 00:32:53,900 --> 00:33:00,480 And that's best looked at by the survival probability. 513 00:33:00,480 --> 00:33:03,151 And you'll see characteristic behavior in the survival 514 00:33:03,151 --> 00:33:03,650 probably. 515 00:33:10,570 --> 00:33:12,010 OK. 516 00:33:12,010 --> 00:33:14,620 But let's go a little bit deeper before I 517 00:33:14,620 --> 00:33:17,240 move on to-- what time is it? 518 00:33:17,240 --> 00:33:20,860 Oh, I'm doing OK. 519 00:33:20,860 --> 00:33:23,380 So let's just say our superposition 520 00:33:23,380 --> 00:33:30,790 state, the electronically excited state, 521 00:33:30,790 --> 00:33:35,410 is a combination of v equals 10 and v equals 11. 522 00:33:51,710 --> 00:34:02,660 So we know immediately that psi star t psi t-- 523 00:34:02,660 --> 00:34:07,370 this probability amplitude-- probability, yes, 524 00:34:07,370 --> 00:34:16,389 this probability is c10 squared psi 10 525 00:34:16,389 --> 00:34:31,380 squared plus c11 squared psi 11 squared plus 2 c10 c11 526 00:34:31,380 --> 00:34:38,510 psi 10 psi 11 cosine omega t. 527 00:34:38,510 --> 00:34:40,100 That's not very legible. 528 00:34:45,540 --> 00:34:48,670 Now, I'm playing fast and loose here. 529 00:34:48,670 --> 00:34:50,050 Because I've done this before. 530 00:34:50,050 --> 00:34:52,000 I know what disappears. 531 00:34:52,000 --> 00:34:55,030 And I know, if it's harmonic, you just get this. 532 00:34:55,030 --> 00:34:57,250 Well, if there's two states, this thing 533 00:34:57,250 --> 00:35:02,380 is really just the frequency associated 534 00:35:02,380 --> 00:35:05,220 with these two levels. 535 00:35:05,220 --> 00:35:07,440 OK. 536 00:35:07,440 --> 00:35:13,560 Now, once we have this, we can also generate the survival 537 00:35:13,560 --> 00:35:15,240 probability. 538 00:35:15,240 --> 00:35:17,570 Well, the survival probability, capital P 539 00:35:17,570 --> 00:35:30,390 of t, that's defined as the square modulus of psi star xt 540 00:35:30,390 --> 00:35:34,768 psi x0 dx. 541 00:35:38,920 --> 00:35:39,420 OK. 542 00:35:39,420 --> 00:35:42,100 And you can immediately write what that is going to be. 543 00:35:42,100 --> 00:35:47,680 It's going to be c10 squared. 544 00:35:47,680 --> 00:35:53,010 And we've integrated so the wave functions go away. 545 00:35:53,010 --> 00:36:08,220 And so we have c10 e to the i H bar 10.5 omega 546 00:36:08,220 --> 00:36:29,970 t divided by H bar plus c11 squared e to the minus i plus H 547 00:36:29,970 --> 00:36:37,001 bar 11.5 omega e over H bar. 548 00:36:37,001 --> 00:36:37,500 OK. 549 00:36:37,500 --> 00:36:38,980 Why did I made the mistake here? 550 00:36:42,010 --> 00:36:43,350 Well, we have a psi star. 551 00:36:45,960 --> 00:36:53,490 And so we're going to get the complex conjugate of e 552 00:36:53,490 --> 00:36:56,660 to the minus i omega. 553 00:36:56,660 --> 00:36:59,730 So you end up getting this. 554 00:36:59,730 --> 00:37:00,755 Practice that. 555 00:37:00,755 --> 00:37:02,380 AUDIENCE: That whole thing is [? the ?] 556 00:37:02,380 --> 00:37:02,700 modulus squared, right? 557 00:37:02,700 --> 00:37:03,450 ROBERT FIELD: Yes. 558 00:37:03,450 --> 00:37:04,721 I have that right. 559 00:37:04,721 --> 00:37:05,220 Exactly. 560 00:37:08,100 --> 00:37:16,320 So and now, lo and behold, we know what to do with this too. 561 00:37:16,320 --> 00:37:23,790 And so we're going to get c1 0 to the 4 plus c-- 562 00:37:23,790 --> 00:37:33,700 not 1-0, 10, c11 to the 4 plus 2 c10 563 00:37:33,700 --> 00:37:39,430 squared c11 squared cosine omega t. 564 00:37:43,690 --> 00:37:45,200 Isn't that neat? 565 00:37:45,200 --> 00:37:48,872 So we've got a whole bunch of constant terms. 566 00:37:48,872 --> 00:37:49,580 This is constant. 567 00:37:49,580 --> 00:37:53,660 Because it's square modulus and it's to the fourth power. 568 00:37:53,660 --> 00:37:59,480 And so all of these coefficients are positive. 569 00:37:59,480 --> 00:38:02,640 At t equals 0, this is 1. 570 00:38:02,640 --> 00:38:09,110 And so at t equals 0, the survival probability 571 00:38:09,110 --> 00:38:09,800 is at a maximum. 572 00:38:13,680 --> 00:38:16,775 And at some later time, that survival probability 573 00:38:16,775 --> 00:38:17,650 will be at a minimum. 574 00:38:24,530 --> 00:38:25,030 OK. 575 00:38:25,030 --> 00:38:31,540 And so you can say the maximum will occur at integer when 576 00:38:31,540 --> 00:38:43,500 omega t is equal to 2 n pi. 577 00:38:48,290 --> 00:38:52,100 So then the exponential factor is always 1. 578 00:38:55,690 --> 00:39:04,210 And we have a minimum when omega t is an odd multiple of pi. 579 00:39:04,210 --> 00:39:06,900 And all the exponential factors are minus 1. 580 00:39:14,660 --> 00:39:19,120 So we can also now look at the expectation 581 00:39:19,120 --> 00:39:24,960 value for x of t and p of t. 582 00:39:24,960 --> 00:39:26,860 And I'm going to just draw sketches. 583 00:39:33,950 --> 00:39:41,690 So for x of t, we have something that looks like this. 584 00:39:41,690 --> 00:39:45,387 This is at the left turning point. 585 00:39:45,387 --> 00:39:47,470 This is the right turning point, or near the right 586 00:39:47,470 --> 00:39:48,970 turning point. 587 00:39:48,970 --> 00:39:56,250 And this is at pi over 2 omega. 588 00:39:56,250 --> 00:39:59,920 This is at pi over omega. 589 00:39:59,920 --> 00:40:01,950 So that's the half oscillator point. 590 00:40:01,950 --> 00:40:07,110 Now, it starts out and the expectation value 591 00:40:07,110 --> 00:40:08,730 is at the left turning point. 592 00:40:08,730 --> 00:40:11,210 And it's not changing. 593 00:40:11,210 --> 00:40:13,970 The derivative of the expectation value 594 00:40:13,970 --> 00:40:17,138 with respect to t is 0. 595 00:40:17,138 --> 00:40:24,970 The momentum, which has a different phase-- 596 00:40:24,970 --> 00:40:28,940 the momentum starts out at 0 also. 597 00:40:28,940 --> 00:40:35,270 And at pi over 2 omega, it reaches a maximum. 598 00:40:35,270 --> 00:40:40,460 And at pi over omega, it reaches 9 again. 599 00:40:40,460 --> 00:40:44,270 But the important thing here is at t 600 00:40:44,270 --> 00:40:47,420 equals 0 the derivative of the momentum 601 00:40:47,420 --> 00:40:48,770 is as big as it can get. 602 00:40:52,510 --> 00:40:56,600 So what's that telling you? 603 00:40:56,600 --> 00:40:58,660 It's telling you this wave packet, 604 00:40:58,660 --> 00:41:02,320 as far as coordinate space is concerned, it's not moving at t 605 00:41:02,320 --> 00:41:04,120 equals 0. 606 00:41:04,120 --> 00:41:09,470 As far as momentum is concerned, it's moving like crazy. 607 00:41:09,470 --> 00:41:16,990 And so this survival probability is changing entirely dominated 608 00:41:16,990 --> 00:41:19,420 by the change in momentum, which is encoded in the wave 609 00:41:19,420 --> 00:41:23,780 function, which is neat. 610 00:41:23,780 --> 00:41:31,340 Because Newton's equation says that the time derivative 611 00:41:31,340 --> 00:41:39,200 of the momentum is equal to-- 612 00:41:39,200 --> 00:41:44,330 that's the mass times acceleration-- 613 00:41:44,330 --> 00:41:47,660 is equal to the force, which is minus 614 00:41:47,660 --> 00:41:55,912 the gradient of the potential-- 615 00:41:55,912 --> 00:41:57,860 the function of time. 616 00:41:57,860 --> 00:42:04,530 And for a harmonic oscillator, the gradient potential is x kx. 617 00:42:04,530 --> 00:42:15,760 And so it's telling you that we have this relationship 618 00:42:15,760 --> 00:42:19,060 between x and t and p. 619 00:42:19,060 --> 00:42:23,350 And we know what the momentum is. 620 00:42:23,350 --> 00:42:24,760 And I'm sorry. 621 00:42:24,760 --> 00:42:27,320 And so the change in the momentum, 622 00:42:27,320 --> 00:42:30,760 which is responsible for the change in the survival 623 00:42:30,760 --> 00:42:35,890 probability at t equals 0, is due to the gradient 624 00:42:35,890 --> 00:42:37,550 of the potential. 625 00:42:37,550 --> 00:42:40,390 So we're actually sampling what is 626 00:42:40,390 --> 00:42:44,320 the gradient of the potential at the left turning point. 627 00:42:44,320 --> 00:42:46,150 And often, for any kind of a problem, 628 00:42:46,150 --> 00:42:49,060 we want to know what kind of force 629 00:42:49,060 --> 00:42:53,470 is acting on the wave packet. 630 00:42:53,470 --> 00:42:57,540 There it is, classic mechanics embedded in quantum mechanics. 631 00:42:57,540 --> 00:43:00,640 It's really amazing. 632 00:43:00,640 --> 00:43:04,110 So if you have some way of measuring 633 00:43:04,110 --> 00:43:07,560 the survival probability near t equals 0, 634 00:43:07,560 --> 00:43:11,460 it's telling you what the slope of the potential 635 00:43:11,460 --> 00:43:12,660 is at the turning point. 636 00:43:22,575 --> 00:43:23,075 OK. 637 00:43:39,990 --> 00:43:44,490 Well, if we have an initial state which 638 00:43:44,490 --> 00:43:49,830 involves many eigenstates, the Hamiltonian, 639 00:43:49,830 --> 00:43:54,840 we still know that there's going to be some complicated behavior 640 00:43:54,840 --> 00:43:58,540 which is modulated by omega. 641 00:43:58,540 --> 00:44:03,620 So we have a cosine omega t always. 642 00:44:03,620 --> 00:44:05,610 And so no matter what kind of a wave 643 00:44:05,610 --> 00:44:09,750 function we make, if we start out at one turning point, 644 00:44:09,750 --> 00:44:13,230 it's going to go to the other turning point. 645 00:44:13,230 --> 00:44:16,230 And it'll keep coming back, and back, and back. 646 00:44:16,230 --> 00:44:20,610 And so one could imagine doing an experiment where-- 647 00:44:20,610 --> 00:44:24,881 let's just draw the excited state and some other repulsive 648 00:44:24,881 --> 00:44:25,380 state. 649 00:44:29,980 --> 00:44:33,520 So at this turning point, vertical transition 650 00:44:33,520 --> 00:44:37,690 to that repulsive state is within the range of your laser. 651 00:44:37,690 --> 00:44:41,230 And at this turning point, it's way high. 652 00:44:41,230 --> 00:44:43,810 And so what you can imagine doing 653 00:44:43,810 --> 00:44:52,830 is probing where this wave packet is as a function of time 654 00:44:52,830 --> 00:44:59,570 by having a probe pulse which creates dissociating fragments. 655 00:44:59,570 --> 00:45:05,040 And now, that is what Ahmed Zewail did. 656 00:45:05,040 --> 00:45:09,500 He talked about real dynamics in real time. 657 00:45:09,500 --> 00:45:12,620 So when the wave packet is here, it can't be dissociated. 658 00:45:12,620 --> 00:45:14,380 When it's here, it can. 659 00:45:14,380 --> 00:45:16,650 And you look at the fragments. 660 00:45:16,650 --> 00:45:17,890 Very simple. 661 00:45:17,890 --> 00:45:21,810 So there are lots of ways of taking these simple pictures 662 00:45:21,810 --> 00:45:26,080 of wave packets moving and saying, 663 00:45:26,080 --> 00:45:29,120 OK, I can use them to set up an experiment where 664 00:45:29,120 --> 00:45:32,490 I ask a very specific question. 665 00:45:32,490 --> 00:45:33,270 And I get answers. 666 00:45:33,270 --> 00:45:34,894 And I know what to do with the answers. 667 00:45:37,530 --> 00:45:47,560 Now, if it's not harmonic, you can still 668 00:45:47,560 --> 00:45:49,930 do Zewail's experiment or some other experiment. 669 00:45:49,930 --> 00:45:54,970 And you can ask, OK, the wave packet moved over here. 670 00:45:54,970 --> 00:45:56,960 And there's some dynamics that occurs. 671 00:45:56,960 --> 00:45:59,230 It dissociates or you do something. 672 00:45:59,230 --> 00:46:04,240 And when it comes back, it's not at the same amplitude. 673 00:46:04,240 --> 00:46:13,600 And so you can use the time history of these recurrences 674 00:46:13,600 --> 00:46:14,500 if it's harmonic. 675 00:46:14,500 --> 00:46:18,700 If it's not harmonic, then the recurrences-- even if nothing 676 00:46:18,700 --> 00:46:21,250 happens over here, the recurrences 677 00:46:21,250 --> 00:46:24,930 will be increasingly less perfect. 678 00:46:24,930 --> 00:46:27,202 And so you measure the anharmonicity. 679 00:46:35,240 --> 00:46:36,290 OK. 680 00:46:36,290 --> 00:46:36,990 Five minutes. 681 00:46:50,980 --> 00:46:54,460 Tunneling is a quantum mechanical phenomenon. 682 00:46:54,460 --> 00:46:58,450 And again, you want to use the simplest possible picture 683 00:46:58,450 --> 00:47:01,450 to understand the signature of tunneling. 684 00:47:01,450 --> 00:47:02,450 And so what do you do? 685 00:47:02,450 --> 00:47:04,300 Well, the simplest thing you can do 686 00:47:04,300 --> 00:47:07,460 is start with a harmonic oscillator. 687 00:47:07,460 --> 00:47:11,740 The next thing you do is you put a barrier in the middle. 688 00:47:11,740 --> 00:47:14,400 And you make it really thin. 689 00:47:14,400 --> 00:47:17,100 Thin is because you don't want to calculate the integral. 690 00:47:17,100 --> 00:47:22,410 You want to just say, oh yeah, I know that the wave function-- 691 00:47:22,410 --> 00:47:23,640 forget about the barrier. 692 00:47:23,640 --> 00:47:31,230 We know the magnitude of the wave function at the barrier, 693 00:47:31,230 --> 00:47:31,770 OK? 694 00:47:31,770 --> 00:47:41,040 And so if you have a state which has a maximum here, 695 00:47:41,040 --> 00:47:44,720 well, that means that it's feeling the barrier. 696 00:47:44,720 --> 00:47:47,480 And so you know what to do. 697 00:47:47,480 --> 00:47:51,760 If there is a node here, it doesn't know about the barrier. 698 00:47:51,760 --> 00:47:53,961 So you can do all sorts of things really fast. 699 00:47:53,961 --> 00:47:54,460 OK. 700 00:47:54,460 --> 00:47:58,520 So and it's harmonic because you're going to want to use 701 00:47:58,520 --> 00:47:59,835 a's and a-dagger's. 702 00:48:04,950 --> 00:48:09,930 So v equals 1, 3, 5. 703 00:48:09,930 --> 00:48:12,790 They have nodes here. 704 00:48:12,790 --> 00:48:14,620 They don't know about the barrier. 705 00:48:14,620 --> 00:48:19,460 Or if they do, it's a very modest effect. 706 00:48:19,460 --> 00:48:21,920 If you make this thin enough, you won't know about it all. 707 00:48:25,110 --> 00:48:36,650 And then v equals 0, 2, 2 4, et cetera-- 708 00:48:36,650 --> 00:48:37,850 well, they're maxima. 709 00:48:37,850 --> 00:48:40,857 Well, they have a local maximum here. 710 00:48:40,857 --> 00:48:42,690 And remember, this is a harmonic oscillator. 711 00:48:42,690 --> 00:48:45,770 So the big lobes are on the extremes. 712 00:48:45,770 --> 00:48:49,900 The smallest lobe is in the middle, but it's not 0. 713 00:48:49,900 --> 00:48:50,400 OK. 714 00:48:50,400 --> 00:48:52,600 There are also symmetry restrictions. 715 00:48:52,600 --> 00:48:56,230 This is a problem that has symmetry. 716 00:48:56,230 --> 00:48:59,280 We have even functions and odd functions. 717 00:48:59,280 --> 00:49:11,970 And for the even functions, d psi dx is equal to 0. 718 00:49:11,970 --> 00:49:18,970 And for the odd functions, psi of 0 is equal to 0. 719 00:49:24,080 --> 00:49:24,730 OK. 720 00:49:24,730 --> 00:49:28,121 So this is something that-- you choose the simplest problem. 721 00:49:28,121 --> 00:49:28,995 And you use symmetry. 722 00:49:31,670 --> 00:49:34,040 And bang, all of a sudden, you get fantastic insights. 723 00:49:45,500 --> 00:49:48,390 So let's look at what happens to the even functions. 724 00:49:48,390 --> 00:49:53,100 So let's just draw a picture of the harmonic oscillator energy 725 00:49:53,100 --> 00:49:53,600 levels. 726 00:49:59,940 --> 00:50:04,120 And we have a barrier, which we say maybe goes that high. 727 00:50:07,170 --> 00:50:10,180 And so we have 1. 728 00:50:10,180 --> 00:50:12,380 And we have 3. 729 00:50:12,380 --> 00:50:14,180 And we have 5. 730 00:50:14,180 --> 00:50:16,804 They're basically not affected by the barrier. 731 00:50:16,804 --> 00:50:18,470 Because they have a node at the barrier. 732 00:50:21,300 --> 00:50:25,060 And then we have v equals 0. 733 00:50:25,060 --> 00:50:26,250 What happens to v equals 0? 734 00:50:26,250 --> 00:50:29,000 Well, it hits this barrier. 735 00:50:29,000 --> 00:50:33,340 And it's got a large amplitude. 736 00:50:33,340 --> 00:50:38,890 And it can't accrue more phase as you go through the barrier. 737 00:50:38,890 --> 00:50:41,050 So when it hits the other turning point, 738 00:50:41,050 --> 00:50:43,130 it doesn't satisfy the boundary conditions. 739 00:50:43,130 --> 00:50:47,420 It'll go to infinity at x equals positive infinity. 740 00:50:47,420 --> 00:50:54,010 So it has to be shifted so that the boundary conditions 741 00:50:54,010 --> 00:50:56,230 at the turning points are met. 742 00:50:56,230 --> 00:50:58,450 And the only way that can happen is 743 00:50:58,450 --> 00:51:01,420 it gets shifted up in energy a lot. 744 00:51:01,420 --> 00:51:05,520 So v equals 0 is here. 745 00:51:05,520 --> 00:51:08,025 Now, it can't be shifted above v equals 1. 746 00:51:08,025 --> 00:51:10,710 Because that would violate the node rule. 747 00:51:10,710 --> 00:51:13,650 And if you draw a picture of the wave function for v 748 00:51:13,650 --> 00:51:17,400 equals 0 in the boundary region, the wave function 749 00:51:17,400 --> 00:51:19,710 has to look something-- well, it has 750 00:51:19,710 --> 00:51:23,110 to look something like that. 751 00:51:23,110 --> 00:51:25,830 So we have two lobes. 752 00:51:25,830 --> 00:51:29,550 And it's trying to make a node, but it can't. 753 00:51:29,550 --> 00:51:32,040 It's a decreasing exponential. 754 00:51:32,040 --> 00:51:36,180 I mean, if you have a wave going into this region 755 00:51:36,180 --> 00:51:39,300 of the barrier, you can have an increasing exponential 756 00:51:39,300 --> 00:51:41,770 or a decreasing exponential. 757 00:51:41,770 --> 00:51:44,400 You can never satisfy continuity of the wave function 758 00:51:44,400 --> 00:51:46,560 with the increasing exponential. 759 00:51:46,560 --> 00:51:49,440 But you can with the decreasing exponential. 760 00:51:49,440 --> 00:51:53,550 And so the height of the barrier determines how close this guy 761 00:51:53,550 --> 00:51:56,520 comes to having a node. 762 00:51:56,520 --> 00:51:57,360 It never will. 763 00:52:00,080 --> 00:52:03,890 And how does this compare to the wave function for v equals 1? 764 00:52:03,890 --> 00:52:07,670 Well, the wave function for v equals 1 looks like this. 765 00:52:07,670 --> 00:52:11,300 The amplitude in this lobe and the amplitude in that lobe 766 00:52:11,300 --> 00:52:12,170 are the same. 767 00:52:12,170 --> 00:52:14,360 But the sign is reversed. 768 00:52:14,360 --> 00:52:17,750 And so basically, this picture tells you 769 00:52:17,750 --> 00:52:20,900 that these two levels have almost exactly the same energy. 770 00:52:23,470 --> 00:52:27,010 So v equals 0 is shifted up. 771 00:52:27,010 --> 00:52:30,940 v equals 2 is shifted up, but not quite so much. 772 00:52:30,940 --> 00:52:36,410 And v 4 is shifted up hardly at all. 773 00:52:36,410 --> 00:52:40,070 And so you get what's called level staggering. 774 00:52:40,070 --> 00:52:45,360 And this level staggering is the signature of tunneling. 775 00:52:45,360 --> 00:52:50,070 This is how we know about tunneling, the only way we 776 00:52:50,070 --> 00:52:52,510 know about tunneling. 777 00:52:52,510 --> 00:52:54,210 So we measure the energy levels. 778 00:52:54,210 --> 00:52:59,040 And it tells us how are the wave functions sampling 779 00:52:59,040 --> 00:53:00,750 this barrier. 780 00:53:00,750 --> 00:53:03,710 Now, this is a childish barrier. 781 00:53:03,710 --> 00:53:05,820 And there is a real barrier-- 782 00:53:05,820 --> 00:53:08,610 molecules isomerize. 783 00:53:08,610 --> 00:53:10,770 And they isomerize over a barrier. 784 00:53:10,770 --> 00:53:13,410 And the barrier isn't necessarily at x equals 0. 785 00:53:13,410 --> 00:53:15,030 But if you understand this problem, 786 00:53:15,030 --> 00:53:17,650 you can deal with isomerization. 787 00:53:17,650 --> 00:53:20,400 Now, I'm an author of a paper that's 788 00:53:20,400 --> 00:53:23,570 just appearing in Science in the next few weeks 789 00:53:23,570 --> 00:53:28,510 on the isomerization from vinylidene to acetylene. 790 00:53:28,510 --> 00:53:29,635 There's a barrier involved. 791 00:53:32,470 --> 00:53:37,510 So these sorts of pictures are important for understanding 792 00:53:37,510 --> 00:53:39,650 those sorts of phenomena. 793 00:53:39,650 --> 00:53:43,210 Now, so I'm done really. 794 00:53:43,210 --> 00:53:48,340 So what we are now encountering with the time 795 00:53:48,340 --> 00:53:52,300 dependent Schrodinger equation is discovering 796 00:53:52,300 --> 00:53:58,680 how dynamics, like tunneling, is encoded 797 00:53:58,680 --> 00:54:01,060 in an eigenstate spectrum. 798 00:54:01,060 --> 00:54:06,850 Because the encoding is level staggering eigenstates. 799 00:54:06,850 --> 00:54:11,080 So people normally have this naive idea 800 00:54:11,080 --> 00:54:19,420 that eigenstate time independent Hamiltonian spectroscopy 801 00:54:19,420 --> 00:54:21,820 does not sample dynamics because it's only 802 00:54:21,820 --> 00:54:23,620 measuring energy levels. 803 00:54:23,620 --> 00:54:26,890 Real dynamical processes are time dependent. 804 00:54:26,890 --> 00:54:31,000 But the dynamics is encoded in energy level patterns. 805 00:54:31,000 --> 00:54:36,650 And that is actually my signature 806 00:54:36,650 --> 00:54:39,440 in experimental spectroscopy. 807 00:54:39,440 --> 00:54:42,050 I've looked for ways in which dynamics is 808 00:54:42,050 --> 00:54:45,140 encoded in eigenstate spectra. 809 00:54:45,140 --> 00:54:49,340 And chemists are interested in dynamics much more than 810 00:54:49,340 --> 00:54:51,520 in structure. 811 00:54:51,520 --> 00:54:53,540 So that's it for today. 812 00:54:53,540 --> 00:54:57,660 I will see you on Wednesday.