1 00:00:00,060 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,800 Commons license. 3 00:00:03,800 --> 00:00:06,060 Your support will help MIT OpenCourseWare 4 00:00:06,060 --> 00:00:10,120 continue to offer high quality educational resources for free. 5 00:00:10,120 --> 00:00:12,690 To make a donation or view additional materials 6 00:00:12,690 --> 00:00:16,590 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,590 --> 00:00:17,297 at ocw.mit.edu. 8 00:00:23,489 --> 00:00:25,030 PROFESSOR: OK, so I want to start out 9 00:00:25,030 --> 00:00:27,420 by reviewing a few things and putting 10 00:00:27,420 --> 00:00:29,620 some machinery together. 11 00:00:29,620 --> 00:00:31,560 Unfortunately, this thing is sort of stuck. 12 00:00:31,560 --> 00:00:35,300 We're going to need a later, so I don't know. 13 00:00:35,300 --> 00:00:37,700 I'll put it up for now. 14 00:00:37,700 --> 00:00:40,410 So first just a bit of notation. 15 00:00:40,410 --> 00:00:42,250 This symbol, you should think of it 16 00:00:42,250 --> 00:00:44,520 like the dot product, or the inner product. 17 00:00:44,520 --> 00:00:49,160 It's just saying that bracket f g is the integral. 18 00:00:49,160 --> 00:00:50,450 It's a number that you get. 19 00:00:50,450 --> 00:00:52,780 So this is a number that you get from the function 20 00:00:52,780 --> 00:00:55,820 f and the function g by taking f, taking 21 00:00:55,820 --> 00:00:58,840 its complex conjugate, multiplying it by g, 22 00:00:58,840 --> 00:01:00,710 and then integrating overall positions. 23 00:01:00,710 --> 00:01:01,490 All right? 24 00:01:01,490 --> 00:01:02,740 So it's a way to get a number. 25 00:01:02,740 --> 00:01:05,310 And you should think about it as the analog for functions 26 00:01:05,310 --> 00:01:07,530 of the dot product for vectors. 27 00:01:07,530 --> 00:01:11,490 It's a way to get a number out of two vectors. 28 00:01:11,490 --> 00:01:15,860 And so, for example, with vectors we could do v dot w, 29 00:01:15,860 --> 00:01:16,940 and this is some number. 30 00:01:16,940 --> 00:01:20,712 And it has a nice property that v dot v, 31 00:01:20,712 --> 00:01:23,170 we can think it as v squared, it's something like a length. 32 00:01:23,170 --> 00:01:25,480 It's strictly positive, and it's something 33 00:01:25,480 --> 00:01:26,760 like the length of a vector. 34 00:01:26,760 --> 00:01:31,320 Similarly, if I take f and take its bracket with f, 35 00:01:31,320 --> 00:01:34,205 this is equal to the integral dx of f squared, 36 00:01:34,205 --> 00:01:36,580 and in particular, f could be complex, so f norm squared. 37 00:01:36,580 --> 00:01:38,000 This is strictly non-negative. 38 00:01:38,000 --> 00:01:41,550 It could vanish, but it's not negative at a point, 39 00:01:41,550 --> 00:01:42,740 hence the norm squared. 40 00:01:42,740 --> 00:01:46,990 So this will be zero if and only if what? 41 00:01:46,990 --> 00:01:49,010 f is 0, f is the 0 function, right. 42 00:01:49,010 --> 00:01:51,852 So the same way that if you take a vector, 43 00:01:51,852 --> 00:01:54,310 and you take its dot product with itself, take it the norm, 44 00:01:54,310 --> 00:01:56,470 it's 0 if an only if the vector is 0. 45 00:01:56,470 --> 00:01:59,270 So this beast satisfies a lot of the properties of a dot 46 00:01:59,270 --> 00:01:59,920 product. 47 00:01:59,920 --> 00:02:01,980 You should think about it as morally equivalent. 48 00:02:01,980 --> 00:02:04,190 We'll talk about that in more detail later. 49 00:02:04,190 --> 00:02:06,670 Second, basic postulate of quantum mechanics, 50 00:02:06,670 --> 00:02:08,972 to every observable is associated an operator, 51 00:02:08,972 --> 00:02:11,180 and it's an operator acting on the space of functions 52 00:02:11,180 --> 00:02:14,340 or on the space of wave functions. 53 00:02:14,340 --> 00:02:16,249 And to every operator corresponding 54 00:02:16,249 --> 00:02:17,790 to an observable in quantum mechanics 55 00:02:17,790 --> 00:02:20,060 are associated a special set of functions called 56 00:02:20,060 --> 00:02:22,640 the eigenfunctions, such that when the operator acts 57 00:02:22,640 --> 00:02:24,640 on that function, it gives you the same function 58 00:02:24,640 --> 00:02:26,560 back times a constant. 59 00:02:26,560 --> 00:02:29,100 What these functions mean, physically, 60 00:02:29,100 --> 00:02:32,860 is they are the wave functions describing configurations 61 00:02:32,860 --> 00:02:35,930 with a definite value of the corresponding observable. 62 00:02:35,930 --> 00:02:38,240 If I'm in an eigenfunction of position with 63 00:02:38,240 --> 00:02:41,610 eigenvalue x naught, awesome. 64 00:02:41,610 --> 00:02:44,740 Thank you, AV person, thank you. 65 00:02:44,740 --> 00:02:48,742 So if your system is described by a wave function which 66 00:02:48,742 --> 00:02:50,200 is an eigenfunction of the position 67 00:02:50,200 --> 00:02:52,470 operator with eigenvalue x naught, 68 00:02:52,470 --> 00:02:55,084 that means you can be confident that the system is 69 00:02:55,084 --> 00:02:56,500 in the configuration corresponding 70 00:02:56,500 --> 00:02:59,111 to having a definite position x naught. 71 00:02:59,111 --> 00:02:59,610 Right? 72 00:02:59,610 --> 00:03:02,360 It's not a superposition of different positions. 73 00:03:02,360 --> 00:03:04,980 It is at x naught. 74 00:03:04,980 --> 00:03:07,567 Similarly, momentum, momentum has eigenfunctions, 75 00:03:07,567 --> 00:03:08,900 and we know what these guys are. 76 00:03:08,900 --> 00:03:12,300 These are the exponentials, e to the iKX's. 77 00:03:12,300 --> 00:03:14,890 They're the eigenfunctions, and those are the wave functions 78 00:03:14,890 --> 00:03:18,130 describing states with definite value of the momentum, 79 00:03:18,130 --> 00:03:19,780 of the associated observable. 80 00:03:19,780 --> 00:03:26,640 Energy as an operator, energy is described by an operator, which 81 00:03:26,640 --> 00:03:34,340 has eigenfunctions which I'll call phi sub n, with energy 82 00:03:34,340 --> 00:03:37,910 as E sub n, those are the eigenvalues. 83 00:03:37,910 --> 00:03:40,920 And if I tell you that your wave function is the state phi 84 00:03:40,920 --> 00:03:43,330 sub 2, what that tells you is that the system has 85 00:03:43,330 --> 00:03:46,540 a definite energy, E sub 2, corresponding 86 00:03:46,540 --> 00:03:48,883 to that eigenvalue. 87 00:03:48,883 --> 00:03:51,166 Cool? 88 00:03:51,166 --> 00:03:53,040 And this is true for any physical observable. 89 00:03:53,040 --> 00:03:54,350 But these are sort of the basic ones 90 00:03:54,350 --> 00:03:56,849 that we'll keep focusing on, position, momentum, and energy, 91 00:03:56,849 --> 00:03:58,930 for the next while. 92 00:03:58,930 --> 00:04:01,360 Now a nice property about these eigenfunctions 93 00:04:01,360 --> 00:04:04,740 is that for different eigenvalues, 94 00:04:04,740 --> 00:04:07,220 the associated wave functions are different functions. 95 00:04:07,220 --> 00:04:08,320 And what I mean by saying they're different 96 00:04:08,320 --> 00:04:09,736 functions is that they're actually 97 00:04:09,736 --> 00:04:12,540 orthogonal functions in the sense of this dot product. 98 00:04:12,540 --> 00:04:16,050 If I have a state corresponding to be at x 0, 99 00:04:16,050 --> 00:04:17,579 definite position x 0, that means 100 00:04:17,579 --> 00:04:20,371 they're in eigenfunction of position with eigenvalue x 0, 101 00:04:20,371 --> 00:04:21,829 and I have another that corresponds 102 00:04:21,829 --> 00:04:24,630 to being at x1, an eigenfunction of the position operator 103 00:04:24,630 --> 00:04:28,240 or the eigenvalue x1, then these wave functions are 104 00:04:28,240 --> 00:04:29,920 orthogonal to each other. 105 00:04:29,920 --> 00:04:32,960 And we get 0 if x 0 is not equal to x1. 106 00:04:32,960 --> 00:04:35,250 Everyone cool with that? 107 00:04:35,250 --> 00:04:37,520 Now, meanwhile not only are they orthogonal 108 00:04:37,520 --> 00:04:39,837 but they're normalized in a particular way. 109 00:04:39,837 --> 00:04:41,670 The inner product gives me a delta function, 110 00:04:41,670 --> 00:04:44,490 which goes beep once, so that if I integrate against it 111 00:04:44,490 --> 00:04:45,319 I get a 1. 112 00:04:45,319 --> 00:04:46,360 Same thing with momentum. 113 00:04:46,360 --> 00:04:48,734 And you do this, this you're checking on the problem set. 114 00:04:48,734 --> 00:04:50,930 I don't remember if it was last one or this one. 115 00:04:50,930 --> 00:04:53,410 And for the energies, energy 1, if I 116 00:04:53,410 --> 00:04:58,436 know the system is in state energy 1, and let's say e sub n 117 00:04:58,436 --> 00:05:00,625 and e sub m, those are different states 118 00:05:00,625 --> 00:05:02,250 if n and m are not equal to each other. 119 00:05:02,250 --> 00:05:04,867 And this inner product is 0 if n and m are not 120 00:05:04,867 --> 00:05:06,450 equal to each other and 1 if they are. 121 00:05:06,450 --> 00:05:08,220 Their properly normalized. 122 00:05:08,220 --> 00:05:10,240 Everyone cool with that? 123 00:05:10,240 --> 00:05:11,769 Yeah. 124 00:05:11,769 --> 00:05:13,768 AUDIENCE: Is it possible that two eigenfunctions 125 00:05:13,768 --> 00:05:15,514 have the same eigenvalue? 126 00:05:15,514 --> 00:05:16,430 PROFESSOR: Absolutely. 127 00:05:16,430 --> 00:05:18,650 It is absolutely possible for two eigenfunctions 128 00:05:18,650 --> 00:05:19,996 to have the same eigenvalue. 129 00:05:19,996 --> 00:05:21,120 That is certainly possible. 130 00:05:21,120 --> 00:05:23,210 AUDIENCE: [INAUDIBLE] 131 00:05:23,210 --> 00:05:25,257 PROFESSOR: Yeah, good. 132 00:05:25,257 --> 00:05:26,840 Thank you, this is a good technicality 133 00:05:26,840 --> 00:05:28,970 that I didn't want to get into, but I'll go and get into it. 134 00:05:28,970 --> 00:05:30,053 It's a very good question. 135 00:05:30,053 --> 00:05:31,900 So the question is, is it possible for two 136 00:05:31,900 --> 00:05:34,620 different eigenfunctions to have the same eigenvalue. 137 00:05:34,620 --> 00:05:36,640 Could there be two states with the same energy , 138 00:05:36,640 --> 00:05:38,550 different states, same energy? 139 00:05:38,550 --> 00:05:40,080 Yeah, that's absolutely possible. 140 00:05:40,080 --> 00:05:41,280 And we'll run into that. 141 00:05:41,280 --> 00:05:43,550 And there's nice physics encoded in it. 142 00:05:43,550 --> 00:05:45,200 But let's think about what that means. 143 00:05:45,200 --> 00:05:47,400 The subsequent question is well, if that's the case, 144 00:05:47,400 --> 00:05:49,910 are they really still orthogonal? 145 00:05:49,910 --> 00:05:51,450 And here's the crucial thing. 146 00:05:51,450 --> 00:05:55,190 The crucial thing is, let's say I take one function, 147 00:05:55,190 --> 00:05:58,750 I'll call the function phi 1, consider the function phi 1. 148 00:05:58,750 --> 00:06:02,330 And let it have energy E1, so that E acting on phi 1 149 00:06:02,330 --> 00:06:05,020 is equal to E1 phi 1. 150 00:06:05,020 --> 00:06:08,600 And let there be another function, phi 2, 151 00:06:08,600 --> 00:06:10,810 such that the energy operator acting on phi 2 152 00:06:10,810 --> 00:06:14,050 is also equal to E1 phi 2. 153 00:06:14,050 --> 00:06:16,700 These are said to be degenerate. 154 00:06:16,700 --> 00:06:18,950 Degenerate doesn't mean you go out and trash your car, 155 00:06:18,950 --> 00:06:22,510 degenerate that the energies are the same. 156 00:06:22,510 --> 00:06:23,997 So what does this tell me? 157 00:06:23,997 --> 00:06:25,080 This tells me a cool fact. 158 00:06:25,080 --> 00:06:30,390 If I take a wave function phi, and I will call this phi star, 159 00:06:30,390 --> 00:06:32,560 in honor of Shri Kulkarni, so I've 160 00:06:32,560 --> 00:06:35,770 got this phi star, which is a linear combination alpha 161 00:06:35,770 --> 00:06:39,270 phi 1 plus beta phi 2, a linear combination 162 00:06:39,270 --> 00:06:42,700 of them, a superposition of those two states. 163 00:06:42,700 --> 00:06:46,060 Is this also an energy eigenfunction? 164 00:06:46,060 --> 00:06:52,630 Yeah, because if I act on phi star with E, then it's linear, 165 00:06:52,630 --> 00:06:55,960 so E acting on phi star is E acting on alpha phi 1, 166 00:06:55,960 --> 00:06:57,370 alpha's a constant, doesn't care. 167 00:06:57,370 --> 00:06:59,090 Phi 1 gives me an E1. 168 00:06:59,090 --> 00:07:02,060 Similarly, E acting on phi 2 gives me an E1. 169 00:07:02,060 --> 00:07:06,490 So if I act with E on this guy, this 170 00:07:06,490 --> 00:07:09,229 is equal to, from both of these I get an overall factor of E1. 171 00:07:09,229 --> 00:07:11,270 So notice that we get the same vector back, times 172 00:07:11,270 --> 00:07:16,260 a constant, a common constant. 173 00:07:16,260 --> 00:07:18,237 So when we have degenerate eigenfunctions, 174 00:07:18,237 --> 00:07:20,320 we can take arbitrary linear combinations to them, 175 00:07:20,320 --> 00:07:22,700 get another degenerate eigenfunction. 176 00:07:22,700 --> 00:07:23,522 Cool? 177 00:07:23,522 --> 00:07:25,230 So this is like, imagine I have a vector, 178 00:07:25,230 --> 00:07:26,460 and I have another vector. 179 00:07:26,460 --> 00:07:28,380 And they share the property that they're both eigenfunctions 180 00:07:28,380 --> 00:07:29,240 of some operator. 181 00:07:29,240 --> 00:07:30,948 That means any linear combination of them 182 00:07:30,948 --> 00:07:32,284 is also, right? 183 00:07:32,284 --> 00:07:33,950 So there's a whole vector space, there's 184 00:07:33,950 --> 00:07:36,850 a whole space of possible functions 185 00:07:36,850 --> 00:07:38,704 that all have the same eigenvalue. 186 00:07:38,704 --> 00:07:40,870 So now you say, well, look, are these two orthogonal 187 00:07:40,870 --> 00:07:41,360 to each other? 188 00:07:41,360 --> 00:07:41,640 No. 189 00:07:41,640 --> 00:07:41,820 These two? 190 00:07:41,820 --> 00:07:42,240 No. 191 00:07:42,240 --> 00:07:43,140 But here's the thing. 192 00:07:43,140 --> 00:07:45,306 If you have a vector space, if you have a the space, 193 00:07:45,306 --> 00:07:48,030 you can always find orthogonal guys and a basis 194 00:07:48,030 --> 00:07:50,190 for that space, yes? 195 00:07:50,190 --> 00:07:53,470 So while it's not true that the eigenfunctions are always 196 00:07:53,470 --> 00:07:56,460 orthogonal, it is true-- 197 00:07:56,460 --> 00:07:59,920 we will not prove this, but we will discuss the proof of it 198 00:07:59,920 --> 00:08:01,770 later by pulling the mathematician out 199 00:08:01,770 --> 00:08:03,930 of the closet-- 200 00:08:03,930 --> 00:08:06,370 the proof will say that it is possible to find 201 00:08:06,370 --> 00:08:09,620 a set of eigenfunctions which are orthogonal in precisely 202 00:08:09,620 --> 00:08:12,380 this fashion, even if there are degeneracies. 203 00:08:12,380 --> 00:08:12,880 OK? 204 00:08:12,880 --> 00:08:14,960 That theorem is called the spectral theorem. 205 00:08:14,960 --> 00:08:16,724 And we'll discuss it later. 206 00:08:16,724 --> 00:08:18,140 So it is always possible to do so. 207 00:08:18,140 --> 00:08:20,460 But you must be alert that there may be degeneracies. 208 00:08:20,460 --> 00:08:22,140 There aren't always degeneracies. 209 00:08:22,140 --> 00:08:24,040 In fact, degeneracies are very special. 210 00:08:24,040 --> 00:08:27,840 Why should two numbers happen to be the same? 211 00:08:27,840 --> 00:08:29,840 Something has to be forcing them to be the same. 212 00:08:29,840 --> 00:08:31,715 That's going to be an important theme for us. 213 00:08:31,715 --> 00:08:32,966 But it certainly is possible. 214 00:08:32,966 --> 00:08:33,549 Good question. 215 00:08:33,549 --> 00:08:36,085 Other questions? 216 00:08:36,085 --> 00:08:36,585 Yeah. 217 00:08:36,585 --> 00:08:42,340 AUDIENCE: [INAUDIBLE] 218 00:08:42,340 --> 00:08:46,110 PROFESSOR: Yeah, so using the triangular brackets-- 219 00:08:46,110 --> 00:08:49,140 so there's another notation for the same thing, which is f g, 220 00:08:49,140 --> 00:08:51,351 but this carries some slightly different weight. 221 00:08:51,351 --> 00:08:53,600 It mean something slightly-- you'll see this in books, 222 00:08:53,600 --> 00:08:55,500 and this means something very similar to this. 223 00:08:55,500 --> 00:08:56,680 But I'm not going to use this notation. 224 00:08:56,680 --> 00:08:58,010 It's called Dirac notation. 225 00:08:58,010 --> 00:08:59,360 We'll talk about it later in the semester, 226 00:08:59,360 --> 00:09:00,640 but we're not going to talk about it just yet. 227 00:09:00,640 --> 00:09:02,490 But when you see this, effectively it 228 00:09:02,490 --> 00:09:05,565 means the same thing as this. 229 00:09:05,565 --> 00:09:07,260 This is sort of like dialect. 230 00:09:07,260 --> 00:09:10,590 You know, it's like French and Quebecois. 231 00:09:10,590 --> 00:09:12,290 Other questions? 232 00:09:12,290 --> 00:09:14,190 My wife's Canadian. 233 00:09:14,190 --> 00:09:15,750 Other questions? 234 00:09:15,750 --> 00:09:17,760 OK. 235 00:09:17,760 --> 00:09:19,200 So given this fact, given the fact 236 00:09:19,200 --> 00:09:20,574 that we can associate observables 237 00:09:20,574 --> 00:09:23,210 to operators, operators come with special functions, 238 00:09:23,210 --> 00:09:25,460 the eigenfunctions, those eigenfunctions corresponding 239 00:09:25,460 --> 00:09:27,410 to have a definite value of the observable, 240 00:09:27,410 --> 00:09:29,320 and they're orthonormal. 241 00:09:29,320 --> 00:09:30,940 This tells us, and this is really 242 00:09:30,940 --> 00:09:32,523 the statement of the spectral theorem, 243 00:09:32,523 --> 00:09:36,340 that any function can be expanded in a basis of states 244 00:09:36,340 --> 00:09:38,550 with definite values of some observable. 245 00:09:38,550 --> 00:09:40,200 So for example, consider position. 246 00:09:40,200 --> 00:09:41,720 I claim that any wave function can 247 00:09:41,720 --> 00:09:44,660 be expanded as a superposition of states 248 00:09:44,660 --> 00:09:46,190 with definite position. 249 00:09:46,190 --> 00:09:48,510 So here's an arbitrary function, here's 250 00:09:48,510 --> 00:09:52,140 this set of states with definite position, the delta functions. 251 00:09:52,140 --> 00:09:55,290 And I can write any function as a superposition 252 00:09:55,290 --> 00:09:57,960 with some coefficients of states with definite position, 253 00:09:57,960 --> 00:10:01,814 integrating over all possible positions, x0. 254 00:10:01,814 --> 00:10:03,480 And this is also sort of trivially true, 255 00:10:03,480 --> 00:10:04,910 because what's this integral? 256 00:10:04,910 --> 00:10:07,650 Well, it's an integral, dx0 over all possible positions 257 00:10:07,650 --> 00:10:09,170 of this delta function. 258 00:10:09,170 --> 00:10:12,090 But we're evaluating at x, so this is 0 259 00:10:12,090 --> 00:10:13,550 unless x is equal to x0. 260 00:10:13,550 --> 00:10:15,125 So I can just put in x instead of x0, 261 00:10:15,125 --> 00:10:16,670 and that gives me psi of x. 262 00:10:16,670 --> 00:10:18,760 Sort of tautological We can do the same thing 263 00:10:18,760 --> 00:10:20,442 for momentum eigenfunctions. 264 00:10:20,442 --> 00:10:22,150 I claim that any function can be expanded 265 00:10:22,150 --> 00:10:24,340 in a superposition of momentum eigenfunctions, where 266 00:10:24,340 --> 00:10:27,160 I sum over all possible values in the momentum 267 00:10:27,160 --> 00:10:30,120 with some weight. 268 00:10:30,120 --> 00:10:31,670 This psi tilde of K is just telling 269 00:10:31,670 --> 00:10:34,480 me how much amplitude there is at that wave number. 270 00:10:34,480 --> 00:10:35,120 Cool? 271 00:10:35,120 --> 00:10:37,660 But this is the Fourier theorem, it's a Fourier expansion. 272 00:10:37,660 --> 00:10:40,864 So purely mathematically, we know that this is true. 273 00:10:40,864 --> 00:10:42,530 But there's also the physical statement. 274 00:10:42,530 --> 00:10:45,050 Any state can be expressed as a superposition of states 275 00:10:45,050 --> 00:10:47,950 with definite momentum. 276 00:10:47,950 --> 00:10:50,300 There's a math in here, but there's also physics in it. 277 00:10:50,300 --> 00:10:52,370 Finally, this is less obvious from a mathematical point 278 00:10:52,370 --> 00:10:54,036 of view, because I haven't even told you 279 00:10:54,036 --> 00:10:56,560 what energy is, any wave function can be expanded 280 00:10:56,560 --> 00:10:58,360 in states with definite energy. 281 00:10:58,360 --> 00:11:02,110 So this is a state, my state En, with definite energy, 282 00:11:02,110 --> 00:11:05,200 with some coefficient summed over all possible values 283 00:11:05,200 --> 00:11:07,790 of the energy. 284 00:11:07,790 --> 00:11:11,750 Given any physical observable, any physical observable, 285 00:11:11,750 --> 00:11:13,860 momentum, position, angular momentum, 286 00:11:13,860 --> 00:11:16,769 whatever, given any physical observable, 287 00:11:16,769 --> 00:11:18,310 a given wave function can be expanded 288 00:11:18,310 --> 00:11:21,920 as some superposition of having definite values of that. 289 00:11:21,920 --> 00:11:25,950 Will it in general have definite values of the observable? 290 00:11:25,950 --> 00:11:28,500 Well a general state be an energy eigenfunction? 291 00:11:28,500 --> 00:11:29,420 No. 292 00:11:29,420 --> 00:11:35,210 But any state is a superposition of energy eigenfunctions. 293 00:11:35,210 --> 00:11:38,240 Will a random state have definite position? 294 00:11:38,240 --> 00:11:38,870 Certainly not. 295 00:11:38,870 --> 00:11:41,010 You could have this wave function. 296 00:11:41,010 --> 00:11:43,110 Superposition. 297 00:11:43,110 --> 00:11:45,210 Yeah. 298 00:11:45,210 --> 00:11:50,710 AUDIENCE: Why is the energy special such 299 00:11:50,710 --> 00:11:54,680 that you can make an arbitrary state with a countable number 300 00:11:54,680 --> 00:11:56,954 of energy eigenfunctions rather than having 301 00:11:56,954 --> 00:11:58,120 to do a continuous spectrum? 302 00:11:58,120 --> 00:11:58,890 PROFESSOR: Excellent question. 303 00:11:58,890 --> 00:12:00,340 So I'm going to phrase that slightly differently. 304 00:12:00,340 --> 00:12:01,280 It's an excellent question, and we'll 305 00:12:01,280 --> 00:12:03,460 come to that at the end of today's lecture. 306 00:12:03,460 --> 00:12:06,190 So the question is, those are integrals, that 307 00:12:06,190 --> 00:12:08,115 is a sum over discrete things. 308 00:12:08,115 --> 00:12:08,615 Why? 309 00:12:08,615 --> 00:12:12,012 Why is the possible values of the position continuous, 310 00:12:12,012 --> 00:12:14,470 possible values of momentum continuous, and possible values 311 00:12:14,470 --> 00:12:16,690 of energy discrete? 312 00:12:16,690 --> 00:12:20,050 The answer to this will become apparent 313 00:12:20,050 --> 00:12:22,152 over the course of your next few problem sets. 314 00:12:22,152 --> 00:12:23,610 You have to do some problems to get 315 00:12:23,610 --> 00:12:25,190 your fingers dirty to really understand this. 316 00:12:25,190 --> 00:12:26,648 But here's the statement, and we'll 317 00:12:26,648 --> 00:12:30,700 see the first version of this at the end of today's lecture. 318 00:12:30,700 --> 00:12:33,420 Sometimes the allowed energies of a system, the energy 319 00:12:33,420 --> 00:12:35,240 eigenvalues, are discrete. 320 00:12:35,240 --> 00:12:37,170 Sometimes they are continuous. 321 00:12:37,170 --> 00:12:41,056 They will be discrete when you have bound states, states that 322 00:12:41,056 --> 00:12:42,930 are trapped in some region and aren't allowed 323 00:12:42,930 --> 00:12:44,410 to get arbitrarily far away. 324 00:12:44,410 --> 00:12:48,160 They'll be continuous when you have states that 325 00:12:48,160 --> 00:12:50,720 can get arbitrarily far away. 326 00:12:50,720 --> 00:12:54,090 Sometimes the momentum will be allowed to be discrete values, 327 00:12:54,090 --> 00:12:56,930 sometimes it will be allowed to be continuous values. 328 00:12:56,930 --> 00:12:59,416 And we'll see exactly why subsequently. 329 00:12:59,416 --> 00:13:00,790 But the thing I want to emphasize 330 00:13:00,790 --> 00:13:03,720 is that I'm writing this to emphasize that it's possible 331 00:13:03,720 --> 00:13:05,920 that each of these can be discrete or continuous. 332 00:13:05,920 --> 00:13:09,120 The important thing is that once you pick your physical system, 333 00:13:09,120 --> 00:13:11,572 you ask what are the allowed values of position, what 334 00:13:11,572 --> 00:13:13,030 are the allowed values of momentum, 335 00:13:13,030 --> 00:13:15,940 and what are the allowed values of energy. 336 00:13:15,940 --> 00:13:18,290 And then you sum over all possible values. 337 00:13:18,290 --> 00:13:21,450 Now, in the examples we looked at yesterday, or last lecture, 338 00:13:21,450 --> 00:13:23,070 the energy could have been discrete, 339 00:13:23,070 --> 00:13:25,070 as in the case of the infinite well, 340 00:13:25,070 --> 00:13:28,680 or continuous, as in the case of the free particle. 341 00:13:28,680 --> 00:13:30,180 In the case of a continuous particle 342 00:13:30,180 --> 00:13:31,810 this would have been an integral. 343 00:13:31,810 --> 00:13:35,890 In the case of the system such as a free particle, where 344 00:13:35,890 --> 00:13:37,890 the energy could take any of a continuous number 345 00:13:37,890 --> 00:13:40,424 of possible values, this would be a continuous integral. 346 00:13:40,424 --> 00:13:41,840 To deal with that, I'm often going 347 00:13:41,840 --> 00:13:45,660 to use the notation, just shorthand, integral sum. 348 00:13:45,660 --> 00:13:47,500 Which I know is a horrible bastardization 349 00:13:47,500 --> 00:13:50,120 of all that's good and just, but on the other hand, 350 00:13:50,120 --> 00:13:53,000 emphasizes the fact that in some systems 351 00:13:53,000 --> 00:13:55,200 you will get continuous, in some systems discrete, 352 00:13:55,200 --> 00:13:57,520 and sometimes you'll have both continuous and discrete. 353 00:13:57,520 --> 00:14:00,380 For example, in hydrogen, in hydrogen 354 00:14:00,380 --> 00:14:02,310 we'll find that there are bound states 355 00:14:02,310 --> 00:14:04,830 where the electron is stuck to the hydrogen nucleus, 356 00:14:04,830 --> 00:14:05,920 to the proton. 357 00:14:05,920 --> 00:14:08,510 And there are discrete allowed energy levels 358 00:14:08,510 --> 00:14:10,170 for that configuration. 359 00:14:10,170 --> 00:14:12,640 However, once you ionize the hydrogen, 360 00:14:12,640 --> 00:14:15,290 the electron can add any energy you want. 361 00:14:15,290 --> 00:14:16,170 It's no longer bound. 362 00:14:16,170 --> 00:14:18,090 It can just get arbitrarily far away. 363 00:14:18,090 --> 00:14:21,360 And there are an uncountable infinity, a continuous set 364 00:14:21,360 --> 00:14:22,744 of possible states. 365 00:14:22,744 --> 00:14:24,410 So in that situation, we'll find that we 366 00:14:24,410 --> 00:14:27,230 have both the discrete and continuous series 367 00:14:27,230 --> 00:14:28,582 of possible states. 368 00:14:28,582 --> 00:14:29,082 Yeah. 369 00:14:29,082 --> 00:14:32,420 AUDIENCE: [INAUDIBLE] 370 00:14:32,420 --> 00:14:34,620 PROFESSOR: Yeah, sure, if you work on a lattice. 371 00:14:34,620 --> 00:14:36,870 So for example, consider the following quantum system. 372 00:14:36,870 --> 00:14:38,550 I have an undergraduate. 373 00:14:38,550 --> 00:14:41,760 And that undergraduate has been placed in 1 of 12 boxes. 374 00:14:41,760 --> 00:14:42,450 OK? 375 00:14:42,450 --> 00:14:44,309 Now, what's the state of the undergraduate? 376 00:14:44,309 --> 00:14:44,850 I don't know. 377 00:14:44,850 --> 00:14:46,410 Is it a definite position state? 378 00:14:46,410 --> 00:14:47,290 It might be. 379 00:14:47,290 --> 00:14:49,660 But probably it's a superposition, an arbitrary 380 00:14:49,660 --> 00:14:51,360 superposition, right? 381 00:14:51,360 --> 00:14:54,912 Very impressive undergraduates at MIT. 382 00:14:54,912 --> 00:14:55,745 OK, other questions. 383 00:14:58,355 --> 00:14:59,491 Yeah. 384 00:14:59,491 --> 00:15:00,990 AUDIENCE: Do these three [INAUDIBLE] 385 00:15:00,990 --> 00:15:03,580 hold even if the probability changes over time? 386 00:15:03,580 --> 00:15:04,170 PROFESSOR: Excellent question. 387 00:15:04,170 --> 00:15:05,090 We'll come back to that. 388 00:15:05,090 --> 00:15:06,631 Very good question, leading question. 389 00:15:06,631 --> 00:15:09,340 OK, so we have this. 390 00:15:09,340 --> 00:15:12,130 The next thing is that energy eigenfunctions satisfy 391 00:15:12,130 --> 00:15:15,540 some very special properties. 392 00:15:15,540 --> 00:15:17,340 And in particular, energy eigenfunctions 393 00:15:17,340 --> 00:15:20,785 have the property from the Schrodinger equation i h 394 00:15:20,785 --> 00:15:26,150 bar d t on psi of x and t is equal to the energy operator 395 00:15:26,150 --> 00:15:28,680 acting on psi of x and t. 396 00:15:28,680 --> 00:15:30,150 This tells us that if we have psi 397 00:15:30,150 --> 00:15:37,000 x 0 time t 0 is equal to phi n of x, as we saw last time, 398 00:15:37,000 --> 00:15:41,080 then the wave function, psi at x at time t 399 00:15:41,080 --> 00:15:43,100 is equal to phi n of x. 400 00:15:43,100 --> 00:15:46,200 And it only changes by an overall phase, e to the minus i 401 00:15:46,200 --> 00:15:50,510 En t over h bar. 402 00:15:50,510 --> 00:15:52,760 And this ratio En upon h bar will often 403 00:15:52,760 --> 00:15:56,130 be written omega n is equal to En over h bar. 404 00:15:56,130 --> 00:15:59,252 This is just the Dupre relations. 405 00:15:59,252 --> 00:16:01,340 Everyone cool with that? 406 00:16:01,340 --> 00:16:02,600 So are energy eigenfunctions-- 407 00:16:07,020 --> 00:16:07,570 how to say. 408 00:16:07,570 --> 00:16:12,800 No wave function is more morally good than another. 409 00:16:12,800 --> 00:16:14,727 But some are particularly convenient. 410 00:16:14,727 --> 00:16:16,560 Energy eigenfunctions have the nice property 411 00:16:16,560 --> 00:16:17,990 that while they're not in a definite position 412 00:16:17,990 --> 00:16:20,156 and they don't necessarily have a definite momentum, 413 00:16:20,156 --> 00:16:22,950 they do evolve over time in a particularly simple way. 414 00:16:22,950 --> 00:16:25,690 And that and the superposition principle 415 00:16:25,690 --> 00:16:27,250 allow me to write the following. 416 00:16:27,250 --> 00:16:34,400 If I know that this is my wave function at psi at x at time 0, 417 00:16:34,400 --> 00:16:38,230 so let's say in all these cases, this is psi of x at time 0, 418 00:16:38,230 --> 00:16:40,590 how does this state evolve forward in time? 419 00:16:44,909 --> 00:16:45,950 It's kind of complicated. 420 00:16:45,950 --> 00:16:49,060 How does this description, how does psi tilde of k 421 00:16:49,060 --> 00:16:50,715 evolve forward in time? 422 00:16:50,715 --> 00:16:51,840 Again, kind of complicated. 423 00:16:51,840 --> 00:16:54,497 But when expressed in terms of the energy eigenstates, 424 00:16:54,497 --> 00:16:56,330 the answer to how it evolves forward in time 425 00:16:56,330 --> 00:16:58,460 is very simple, because I know that this 426 00:16:58,460 --> 00:17:01,440 is a superposition, a linear combination of states 427 00:17:01,440 --> 00:17:02,600 with definite energy. 428 00:17:02,600 --> 00:17:05,079 States with definite energy evolve with a phase. 429 00:17:05,079 --> 00:17:07,010 And the Schrodinger equation is linear, 430 00:17:07,010 --> 00:17:08,700 so solutions of the Schrodinger equation 431 00:17:08,700 --> 00:17:11,680 evolve to become solutions of the Schrodinger equation. 432 00:17:11,680 --> 00:17:13,950 So how does this state evolve forward in time? 433 00:17:13,950 --> 00:17:18,476 It evolves forward with a phase, e to the minus i omega n t. 434 00:17:18,476 --> 00:17:21,560 One for every different terms in this sum. 435 00:17:21,560 --> 00:17:23,020 Cool? 436 00:17:23,020 --> 00:17:25,089 So we are going to harp on energy functions, 437 00:17:25,089 --> 00:17:28,300 not because they're more moral, or more just, or more good, 438 00:17:28,300 --> 00:17:31,040 but because they're more convenient for solving the time 439 00:17:31,040 --> 00:17:33,680 evolution problem in quantum mechanics. 440 00:17:33,680 --> 00:17:37,020 So most of today is going to be about this expansion 441 00:17:37,020 --> 00:17:40,470 and qualitative features of energy eigenfunctions. 442 00:17:40,470 --> 00:17:42,690 Cool? 443 00:17:42,690 --> 00:17:43,510 OK. 444 00:17:43,510 --> 00:17:45,670 And just to close that out, I just 445 00:17:45,670 --> 00:17:47,420 want to remind you of a couple of examples 446 00:17:47,420 --> 00:17:50,022 that we did last time, just get them on board. 447 00:17:50,022 --> 00:17:51,355 So the first is a free particle. 448 00:17:54,440 --> 00:17:57,220 So for free particle, we have that our wave functions-- 449 00:18:00,380 --> 00:18:04,125 well, actually let me not write that down. 450 00:18:04,125 --> 00:18:06,000 Actually, let me skip over the free particle, 451 00:18:06,000 --> 00:18:06,999 because it's so trivial. 452 00:18:06,999 --> 00:18:10,417 Let me just talk about the infinite well. 453 00:18:10,417 --> 00:18:12,000 So the potential is infinite out here, 454 00:18:12,000 --> 00:18:13,750 and it's 0 inside the well, and it 455 00:18:13,750 --> 00:18:18,790 goes from 0 to L. This is just my choice of notation. 456 00:18:18,790 --> 00:18:22,300 And the energy operator, as usual, 457 00:18:22,300 --> 00:18:26,165 is p squared upon 2m plus u of x. 458 00:18:26,165 --> 00:18:27,790 You might say, where did I derive this, 459 00:18:27,790 --> 00:18:29,000 and the answer is I didn't derive this. 460 00:18:29,000 --> 00:18:30,030 I just wrote it down. 461 00:18:30,030 --> 00:18:32,690 It's like force in Newton's equations. 462 00:18:32,690 --> 00:18:34,730 You just declare some force and you ask, 463 00:18:34,730 --> 00:18:36,190 what system does is model. 464 00:18:36,190 --> 00:18:38,120 So here's my system. 465 00:18:38,120 --> 00:18:41,630 It has what looks like a classical kind of energy, 466 00:18:41,630 --> 00:18:43,450 except these are all operators. 467 00:18:43,450 --> 00:18:48,120 And the potential here is this guy, it's 0 between 0 and L, 468 00:18:48,120 --> 00:18:49,730 and it's infinite elsewhere. 469 00:18:49,730 --> 00:18:51,740 And as we saw last time, the solutions 470 00:18:51,740 --> 00:18:53,480 to the energy eigenvalue equation 471 00:18:53,480 --> 00:18:55,190 are particularly simple. 472 00:18:55,190 --> 00:18:59,500 Phi sub n of x is equal to root properly 473 00:18:59,500 --> 00:19:04,620 normalized 2 upon L sine of Kn x, 474 00:19:04,620 --> 00:19:08,790 where kn is equal to n plus 1 pi, where 475 00:19:08,790 --> 00:19:11,670 n is an integer upon L. 476 00:19:11,670 --> 00:19:14,070 And these were chosen to satisfy our boundary conditions, 477 00:19:14,070 --> 00:19:16,403 that the wave function must vanish here, hence the sine, 478 00:19:16,403 --> 00:19:19,690 and K was chosen so that it turned over and just hit 0 479 00:19:19,690 --> 00:19:25,210 as we got to L. And that gave us that the allowed energies were 480 00:19:25,210 --> 00:19:29,050 discrete, because the En, which you can get by just plugging 481 00:19:29,050 --> 00:19:31,200 into the energy eigenvalue equation, 482 00:19:31,200 --> 00:19:34,860 was equal to h bar squared Kn squared upon 2m. 483 00:19:37,550 --> 00:19:39,370 So this tells us a nice thing. 484 00:19:39,370 --> 00:19:41,430 First off, in this system, if I take a particle 485 00:19:41,430 --> 00:19:43,940 and I throw it in here in some arbitrary state 486 00:19:43,940 --> 00:19:46,070 so that at time t equals zero the wave 487 00:19:46,070 --> 00:19:54,170 function x 0 is equal to sum over n phi n of x Cn. 488 00:19:54,170 --> 00:19:56,440 OK? 489 00:19:56,440 --> 00:19:57,220 Can I do this? 490 00:19:57,220 --> 00:19:58,845 Can I just pick some arbitrary function 491 00:19:58,845 --> 00:20:01,320 which is a superposition of energy eigenstates? 492 00:20:01,320 --> 00:20:02,956 Sure, because any function is. 493 00:20:02,956 --> 00:20:04,955 Any function can be described as a superposition 494 00:20:04,955 --> 00:20:06,560 of energy eigenfunctions. 495 00:20:06,560 --> 00:20:09,350 And if I use the energy eigenfunctions, 496 00:20:09,350 --> 00:20:11,600 it will automatically satisfy the boundary conditions. 497 00:20:11,600 --> 00:20:13,190 All good things will happen. 498 00:20:13,190 --> 00:20:15,170 So this is perfectly fine initial condition. 499 00:20:15,170 --> 00:20:16,950 What is the system at time t? 500 00:20:20,986 --> 00:20:22,360 Yeah, we just pick up the phases. 501 00:20:22,360 --> 00:20:24,100 And what phase is this guy? 502 00:20:24,100 --> 00:20:28,420 It's this, e to the minus i omega n t. 503 00:20:28,420 --> 00:20:31,150 And when I write omega n, let me be more explicit about that, 504 00:20:31,150 --> 00:20:33,820 that's En over h bar. 505 00:20:33,820 --> 00:20:38,340 So that's h bar Kn squared upon 2m t. 506 00:20:41,420 --> 00:20:42,550 Cool? 507 00:20:42,550 --> 00:20:48,740 So there is our solution for arbitrary initial conditions 508 00:20:48,740 --> 00:20:54,069 to the infinite square well problem in quantum mechanics. 509 00:20:54,069 --> 00:20:56,610 And you're going to study this in some detail on your problem 510 00:20:56,610 --> 00:20:58,950 set. 511 00:20:58,950 --> 00:21:01,920 But just to start with a little bit of intuition, 512 00:21:01,920 --> 00:21:04,390 let's look at the wave functions and the probability 513 00:21:04,390 --> 00:21:06,177 distributions for the lowest lying states. 514 00:21:06,177 --> 00:21:07,760 So for example, let's look at the wave 515 00:21:07,760 --> 00:21:11,360 function for the ground state, what I will call psi sub 0. 516 00:21:11,360 --> 00:21:14,959 And this is from 0 to L. And I put these bars here 517 00:21:14,959 --> 00:21:16,750 not because we're looking at the potential. 518 00:21:16,750 --> 00:21:23,954 I'm going to be plotting the real part of the wave function. 519 00:21:23,954 --> 00:21:26,450 But I put these walls here just to emphasize 520 00:21:26,450 --> 00:21:29,617 that that's where the walls are, at x equals 0 and x equals L. 521 00:21:29,617 --> 00:21:30,700 So what does it look like? 522 00:21:30,700 --> 00:21:33,100 Well, the first one is going to sine of Kn x. 523 00:21:33,100 --> 00:21:34,010 n is 0. 524 00:21:34,010 --> 00:21:38,804 Kn is going to be pi upon L. So that's again just this guy. 525 00:21:38,804 --> 00:21:40,470 Now, what's the probability distribution 526 00:21:40,470 --> 00:21:43,734 associated with psi 0? 527 00:21:43,734 --> 00:21:45,025 Where do you find the particle? 528 00:21:47,920 --> 00:21:51,850 So we know that it's just the norm squared of this wave 529 00:21:51,850 --> 00:21:56,580 function and the norm squared is here at 0, it's 0 530 00:21:56,580 --> 00:21:58,040 and it rises linearly, because sine 531 00:21:58,040 --> 00:21:59,580 is linear for small values. 532 00:21:59,580 --> 00:22:02,980 That makes this quadratic, and a maximum, 533 00:22:02,980 --> 00:22:04,600 and then quadratic again. 534 00:22:04,600 --> 00:22:06,710 So there's our probability distribution. 535 00:22:06,710 --> 00:22:08,014 Now, here's a funny thing. 536 00:22:08,014 --> 00:22:09,930 Imagine I take a particle, classical particle, 537 00:22:09,930 --> 00:22:11,607 and I put it in a box. 538 00:22:11,607 --> 00:22:13,690 And you put it in a box, and you tell it, OK, it's 539 00:22:13,690 --> 00:22:14,390 got some energy. 540 00:22:14,390 --> 00:22:16,210 So classically it's got some momentum. 541 00:22:16,210 --> 00:22:17,834 So it's sort of bouncing back and forth 542 00:22:17,834 --> 00:22:20,000 and just bounces off the arbitrarily hard walls 543 00:22:20,000 --> 00:22:20,880 and moves around. 544 00:22:20,880 --> 00:22:22,880 Where are you most likely to find that particle? 545 00:22:27,850 --> 00:22:29,530 Where does it spend most of its time? 546 00:22:33,052 --> 00:22:35,010 It spends the same amount of time at any point. 547 00:22:35,010 --> 00:22:36,410 It's moving at constant velocity. 548 00:22:36,410 --> 00:22:39,247 It goes boo, boo, boo, boo, right? 549 00:22:39,247 --> 00:22:40,830 So what's the probability distribution 550 00:22:40,830 --> 00:22:44,020 for finding it at any point inside, classically? 551 00:22:44,020 --> 00:22:45,411 Constant. 552 00:22:45,411 --> 00:22:47,660 Classically, the probability distribution is constant. 553 00:22:47,660 --> 00:22:49,576 You're just as likely to find it near the wall 554 00:22:49,576 --> 00:22:51,615 as not near the wall. 555 00:22:51,615 --> 00:22:53,740 However, quantum mechanically, for the lowest lying 556 00:22:53,740 --> 00:22:55,090 state that is clearly not true. 557 00:22:55,090 --> 00:22:58,310 You're really likely to find it near the wall. 558 00:22:58,310 --> 00:23:01,060 What's up with that? 559 00:23:01,060 --> 00:23:03,630 So that's a question that I want to put in your head 560 00:23:03,630 --> 00:23:05,850 and have you think about. 561 00:23:05,850 --> 00:23:08,309 You're going to see a similar effect arising over and over. 562 00:23:08,309 --> 00:23:09,891 And we're going to see at the very end 563 00:23:09,891 --> 00:23:12,460 that that is directly related, the fact that this goes to 0, 564 00:23:12,460 --> 00:23:15,370 is directly related, and I'm not kidding, 565 00:23:15,370 --> 00:23:16,940 to the transparency of diamond. 566 00:23:22,385 --> 00:23:24,000 OK, I think it was pretty cool. 567 00:23:26,720 --> 00:23:27,470 They're expensive. 568 00:23:30,399 --> 00:23:31,940 It's also related to the transparency 569 00:23:31,940 --> 00:23:34,148 of cubic zirconium, which I guess is less impressive. 570 00:23:37,020 --> 00:23:39,090 So the first state, again, let's look 571 00:23:39,090 --> 00:23:43,790 at the real part of psi 1, the first excited state. 572 00:23:43,790 --> 00:23:46,240 Well, this is now a sine with one extra-- 573 00:23:46,240 --> 00:23:50,940 with a 2 here, 2 pi, so it goes through 0. 574 00:23:50,940 --> 00:23:53,740 So the probability distribution associated with psi 1, 575 00:23:53,740 --> 00:23:56,774 and I should say write this as a function of x, looks 576 00:23:56,774 --> 00:23:58,190 like, well, again, it's quadratic. 577 00:23:58,190 --> 00:24:01,310 But it has a 0 again in the middle. 578 00:24:01,310 --> 00:24:04,250 So it's going to look like-- 579 00:24:04,250 --> 00:24:07,670 oops, my bad art defeats me. 580 00:24:07,670 --> 00:24:08,580 OK, there we go. 581 00:24:08,580 --> 00:24:10,010 So now it's even worse. 582 00:24:10,010 --> 00:24:12,100 Not only is unlikely to be out here, 583 00:24:12,100 --> 00:24:14,183 it's also very unlikely to be found in the middle. 584 00:24:14,183 --> 00:24:16,750 In fact, there is 0 probability you'll find it in the middle. 585 00:24:16,750 --> 00:24:18,700 That's sort of surprising. 586 00:24:18,700 --> 00:24:21,290 But you can quickly guess what happens as you 587 00:24:21,290 --> 00:24:23,770 go to very high energies. 588 00:24:23,770 --> 00:24:30,010 The real part of psi n let's say 10,000, 10 to the 4, 589 00:24:30,010 --> 00:24:31,610 what is that going to look like? 590 00:24:31,610 --> 00:24:33,620 Well, this had no 0s, this had one 0, 591 00:24:33,620 --> 00:24:35,300 and every time you increase n by 1, 592 00:24:35,300 --> 00:24:37,570 you're just going to add one more 0 to the sign. 593 00:24:37,570 --> 00:24:39,570 That's an interesting and suggestive fact. 594 00:24:39,570 --> 00:24:42,650 So if it's size of 10,000, how many nodes 595 00:24:42,650 --> 00:24:45,411 are there going to be in the middle of the domain? 596 00:24:45,411 --> 00:24:45,910 10,000. 597 00:24:45,910 --> 00:24:48,690 And the amplitude is going to be the same. 598 00:24:48,690 --> 00:24:51,220 I'm not to be able to do this, but you get the idea. 599 00:24:51,220 --> 00:24:53,520 And now if I construct the probability distribution, 600 00:24:53,520 --> 00:24:55,780 what's the probability distribution going to be? 601 00:24:55,780 --> 00:25:02,030 Probability of the 10,000th psi sub 10 to the 4 of x. 602 00:25:02,030 --> 00:25:05,830 Well, it's again going to be strictly positive. 603 00:25:05,830 --> 00:25:10,040 And if you are not able to make measurements on the scale of L 604 00:25:10,040 --> 00:25:14,340 upon 10,000, but just say like L over 3, because you have 605 00:25:14,340 --> 00:25:17,331 a thumb and you don't have an infinitely accurate meter, what 606 00:25:17,331 --> 00:25:17,830 do you see? 607 00:25:17,830 --> 00:25:20,384 You see effectively a constant probability distribution. 608 00:25:20,384 --> 00:25:22,050 And actually, I shouldn't draw it there. 609 00:25:22,050 --> 00:25:23,834 I should draw it through the half, 610 00:25:23,834 --> 00:25:27,170 because sine squared over 2 averages to one half, 611 00:25:27,170 --> 00:25:30,500 or, sorry, sine squared averages to one half over many periods. 612 00:25:30,500 --> 00:25:33,660 So what we see is that the classical probability 613 00:25:33,660 --> 00:25:38,380 distribution constant does arise when we look 614 00:25:38,380 --> 00:25:41,590 at very high energy states. 615 00:25:41,590 --> 00:25:42,880 Cool? 616 00:25:42,880 --> 00:25:46,084 But it is manifestly not a good description. 617 00:25:46,084 --> 00:25:48,250 The classical description is not a good description. 618 00:25:48,250 --> 00:25:51,070 Your intuition is crappy at low energies, 619 00:25:51,070 --> 00:25:53,570 near the ground state, where quantum effects are dominating, 620 00:25:53,570 --> 00:25:55,980 because indeed, classically there was no minimum energy. 621 00:25:55,980 --> 00:25:57,930 Quantum effects have to be dominating there. 622 00:25:57,930 --> 00:25:59,780 And here we see that even the probability distribution's 623 00:25:59,780 --> 00:26:01,542 radically different than our intuition. 624 00:26:01,542 --> 00:26:02,042 Yeah. 625 00:26:02,042 --> 00:26:12,650 AUDIENCE: [INAUDIBLE] 626 00:26:12,650 --> 00:26:13,900 PROFESSOR: Keep working on it. 627 00:26:13,900 --> 00:26:18,820 So I want you all to think about what-- 628 00:26:18,820 --> 00:26:21,250 you're not, I promise you, unless you've already 629 00:26:21,250 --> 00:26:22,230 seen some quantum mechanics, you're 630 00:26:22,230 --> 00:26:24,271 not going to be able to answer this question now. 631 00:26:24,271 --> 00:26:27,010 But I want you to have it as an uncomfortable little piece 632 00:26:27,010 --> 00:26:31,420 of sand in the back of your oyster mind-- 633 00:26:31,420 --> 00:26:38,020 no offense-- what is causing that 0? 634 00:26:38,020 --> 00:26:39,300 Why are we getting 0? 635 00:26:39,300 --> 00:26:40,575 And I'll give you a hint. 636 00:26:40,575 --> 00:26:42,960 In quantum mechanics, anytime something interesting 637 00:26:42,960 --> 00:26:45,985 happens it's because of superposition and interference. 638 00:26:49,220 --> 00:26:50,850 All right. 639 00:26:50,850 --> 00:26:55,044 So with all that said, so any questions now over this story 640 00:26:55,044 --> 00:26:56,585 about energy eigenfunctions expanding 641 00:26:56,585 --> 00:27:01,136 in a basis, et cetera, before we get moving? 642 00:27:01,136 --> 00:27:03,370 No, OK. 643 00:27:03,370 --> 00:27:05,880 In that case, get out your clickers. 644 00:27:05,880 --> 00:27:07,620 We're going to test your knowledge. 645 00:27:17,720 --> 00:27:20,836 Channel 41, for those of you who have to adjust it. 646 00:27:20,836 --> 00:27:32,380 [CHATTER] 647 00:27:32,380 --> 00:27:32,880 Wow. 648 00:27:37,100 --> 00:27:38,100 That's kind of worrying. 649 00:27:42,980 --> 00:27:43,480 Aha. 650 00:27:58,400 --> 00:28:01,780 OK, ready? 651 00:28:01,780 --> 00:28:07,010 OK, channel 41, and here we go. 652 00:28:18,300 --> 00:28:19,789 So go ahead and start now. 653 00:28:19,789 --> 00:28:21,830 Sorry, there was a little technical glitch there. 654 00:28:21,830 --> 00:28:24,640 So psi 1 and psi 2 are eigenstates. 655 00:28:24,640 --> 00:28:27,590 They're non-degenerate, meaning the energies are different. 656 00:28:27,590 --> 00:28:30,020 Is a superposition psi 1 plus psi 2 also an eigenstate? 657 00:28:43,398 --> 00:28:45,580 All right, four more seconds. 658 00:28:48,610 --> 00:28:49,560 All right. 659 00:28:49,560 --> 00:28:52,910 I want everyone to turn to the person next to you 660 00:28:52,910 --> 00:28:54,050 and discuss this. 661 00:28:54,050 --> 00:28:56,538 You've got about 30 seconds to discuss, or a minute. 662 00:28:56,538 --> 00:29:39,780 [CHATTER] 663 00:29:39,780 --> 00:29:41,940 All right. 664 00:29:41,940 --> 00:29:45,230 I want everyone, now that you've got an answer, click again, 665 00:29:45,230 --> 00:29:47,300 put in your current best guess. 666 00:29:50,564 --> 00:29:51,230 Oh, wait, sorry. 667 00:29:51,230 --> 00:29:54,180 For some reason I have to start over again. 668 00:29:54,180 --> 00:29:56,080 OK, now click. 669 00:30:02,690 --> 00:30:05,090 This is the best. 670 00:30:05,090 --> 00:30:08,180 I'm such a convert to clickers, this is just fantastic. 671 00:30:08,180 --> 00:30:11,750 So you guys went from, so roughly you all 672 00:30:11,750 --> 00:30:20,480 went from about 30, 60, 10, to what are we now? 673 00:30:20,480 --> 00:30:26,990 8, 82, and 10. 674 00:30:26,990 --> 00:30:29,230 So it sounds like you guys are predicting answer b. 675 00:30:29,230 --> 00:30:30,200 And the answer is-- 676 00:30:33,587 --> 00:30:34,420 I like the suspense. 677 00:30:34,420 --> 00:30:37,170 There we go. 678 00:30:37,170 --> 00:30:38,150 B, good. 679 00:30:38,150 --> 00:30:40,160 So here's a quick question. 680 00:30:42,890 --> 00:30:44,020 So why? 681 00:30:44,020 --> 00:30:49,910 And the reason why is that if we have E on psi 1 plus psi 2, 682 00:30:49,910 --> 00:30:56,540 this is equal to E on psi 1 plus E on psi 2, operator, operator, 683 00:30:56,540 --> 00:30:59,200 operator, but this is equal to E 1 684 00:30:59,200 --> 00:31:05,380 psi 1 E 2 psi 2, which if E1 and E2 are not equal, 685 00:31:05,380 --> 00:31:09,070 which is not equal to E times psi 1 plus psi 2. 686 00:31:09,070 --> 00:31:10,350 Right? 687 00:31:10,350 --> 00:31:14,740 Not equal to E anything times psi 1 plus psi 2. 688 00:31:14,740 --> 00:31:16,280 And it needs to be, in order to be 689 00:31:16,280 --> 00:31:18,300 an eigenfunction, an eigenfunction of the energy 690 00:31:18,300 --> 00:31:20,960 operator. 691 00:31:20,960 --> 00:31:22,538 Yeah. 692 00:31:22,538 --> 00:31:24,162 AUDIENCE: So I was thinking about this, 693 00:31:24,162 --> 00:31:26,037 if this was kind of a silly random case where 694 00:31:26,037 --> 00:31:27,140 one of the energies is 0. 695 00:31:27,140 --> 00:31:29,640 Does this only happen if you have something that's infinite? 696 00:31:29,640 --> 00:31:31,360 PROFESSOR: Yeah, that's a really good question. 697 00:31:31,360 --> 00:31:33,340 So first off, how do you measure an energy? 698 00:31:36,240 --> 00:31:38,790 Do you ever measure an energy? 699 00:31:38,790 --> 00:31:41,365 Do you ever measure a voltage, the actual value 700 00:31:41,365 --> 00:31:43,490 of the scalar potential, the electromagnetic scalar 701 00:31:43,490 --> 00:31:43,670 potential? 702 00:31:43,670 --> 00:31:44,169 No. 703 00:31:44,169 --> 00:31:46,259 You measure a difference. 704 00:31:46,259 --> 00:31:47,550 Do you ever measure the energy? 705 00:31:47,550 --> 00:31:49,296 No, you measure a difference in energy. 706 00:31:49,296 --> 00:31:51,670 So the absolute value of energy is sort of a silly thing. 707 00:31:51,670 --> 00:31:54,440 But we always talk about it as if it's not. 708 00:31:54,440 --> 00:31:55,680 We say, that's got energy 14. 709 00:31:55,680 --> 00:31:58,200 It's a little bit suspicious. 710 00:31:58,200 --> 00:32:00,480 So to answer your question, there's 711 00:32:00,480 --> 00:32:02,290 nothing hallowed about the number 0, 712 00:32:02,290 --> 00:32:04,282 although we will often refer to zero energy 713 00:32:04,282 --> 00:32:05,490 with a very specific meaning. 714 00:32:05,490 --> 00:32:06,980 What we really mean in that case is 715 00:32:06,980 --> 00:32:09,420 the value of the potential energy at infinity. 716 00:32:09,420 --> 00:32:11,220 So when I say energy, usually what I mean 717 00:32:11,220 --> 00:32:12,790 is relative to the value at infinity. 718 00:32:12,790 --> 00:32:14,415 So then let me ask your question again. 719 00:32:14,415 --> 00:32:16,510 Your question is it possible to have energy 0? 720 00:32:16,510 --> 00:32:17,802 Absolutely, and we'll see that. 721 00:32:17,802 --> 00:32:20,093 And it's actually going to be really interesting what's 722 00:32:20,093 --> 00:32:21,910 true of states with energy 0 in that sense. 723 00:32:21,910 --> 00:32:23,451 Second part of your question, though, 724 00:32:23,451 --> 00:32:25,690 is how does energy being 0 fit into this? 725 00:32:25,690 --> 00:32:26,970 Well, does that save us? 726 00:32:26,970 --> 00:32:29,210 Suppose one of the energies is 0. 727 00:32:29,210 --> 00:32:31,690 Then that says E on psi 1 plus psi 2 is equal to, 728 00:32:31,690 --> 00:32:34,790 let's say E2 is 0. 729 00:32:34,790 --> 00:32:35,950 Well, that term is gone. 730 00:32:35,950 --> 00:32:37,074 So there's just the one E1. 731 00:32:37,074 --> 00:32:38,657 Are we in energy eigenstate? 732 00:32:38,657 --> 00:32:40,240 No, because it's still not of the form 733 00:32:40,240 --> 00:32:42,290 E times psi 1 plus psi 2. 734 00:32:42,290 --> 00:32:44,740 So it doesn't save us, but it's an interesting question 735 00:32:44,740 --> 00:32:46,200 for the future. 736 00:32:46,200 --> 00:32:48,290 All right. 737 00:32:48,290 --> 00:32:50,935 Next question, four parts. 738 00:33:07,100 --> 00:33:10,250 So the question says x and p commute to i h bar. 739 00:33:10,250 --> 00:33:11,470 We've shown this. 740 00:33:11,470 --> 00:33:14,580 Is p x equal to i h bar, and is ip plus cx 741 00:33:14,580 --> 00:33:15,910 the same as cx plus ip? 742 00:33:29,920 --> 00:33:33,740 If you're really unsure you can ask the person next to you, 743 00:33:33,740 --> 00:33:37,170 but you don't have to. 744 00:33:37,170 --> 00:33:41,290 OK, so this is looking good. 745 00:33:41,290 --> 00:33:44,060 Everyone have an answer in? 746 00:33:44,060 --> 00:33:44,560 No? 747 00:33:49,770 --> 00:33:55,310 Five, four, three, two, one, OK, good. 748 00:33:55,310 --> 00:34:03,242 So the answer is C, which most of you got, but not everyone. 749 00:34:03,242 --> 00:34:05,200 A bunch of you put D. So let's talk through it. 750 00:34:05,200 --> 00:34:10,639 So remember what the definition of the commutator is. 751 00:34:10,639 --> 00:34:15,219 x with p by definition is equal to xp minus px. 752 00:34:15,219 --> 00:34:18,500 If we change the order here, px is 753 00:34:18,500 --> 00:34:22,630 equal to minus this, px minus xp. 754 00:34:22,630 --> 00:34:24,620 It's just the definition of the commutator. 755 00:34:24,620 --> 00:34:27,800 So on the other hand, if you add things, does 7 plus 6 756 00:34:27,800 --> 00:34:28,940 equal 6 plus 7? 757 00:34:28,940 --> 00:34:29,440 Yeah. 758 00:34:29,440 --> 00:34:31,116 Well, of course 6 times 7 is 7 times 6. 759 00:34:31,116 --> 00:34:32,699 So that's not a terribly good analogy. 760 00:34:36,880 --> 00:34:39,880 Does the order of addition of operators matter? 761 00:34:39,880 --> 00:34:42,540 No. 762 00:34:42,540 --> 00:34:44,639 Yeah. 763 00:34:44,639 --> 00:34:45,630 Yeah, exactly. 764 00:34:45,630 --> 00:34:46,130 Exactly. 765 00:34:46,130 --> 00:34:47,130 So it's slightly sneaky. 766 00:34:47,130 --> 00:34:49,840 OK, next question. 767 00:34:49,840 --> 00:34:51,170 OK, this one has five. 768 00:34:54,860 --> 00:34:56,670 f and g are both wave functions. 769 00:34:56,670 --> 00:34:58,090 c is a constant. 770 00:34:58,090 --> 00:35:03,150 Then if we take the inner product c times f with g, 771 00:35:03,150 --> 00:35:05,160 this is equal to what? 772 00:35:21,000 --> 00:35:24,240 Three, two, one, OK. 773 00:35:24,240 --> 00:35:25,720 So the answer is-- 774 00:35:25,720 --> 00:35:27,610 so this one definitely discuss. 775 00:35:27,610 --> 00:35:29,778 Discuss with the person next to you. 776 00:35:29,778 --> 00:35:55,520 [CHATTER] 777 00:35:55,520 --> 00:35:56,280 All right. 778 00:36:01,440 --> 00:36:05,910 OK, go ahead and enter your guess again, 779 00:36:05,910 --> 00:36:08,490 or your answer again, let it not be a guess. 780 00:36:11,220 --> 00:36:12,855 OK, 10 seconds. 781 00:36:15,450 --> 00:36:15,950 Wow. 782 00:36:15,950 --> 00:36:16,630 OK, fantastic. 783 00:36:16,630 --> 00:36:17,630 That works like a champ. 784 00:36:17,630 --> 00:36:19,040 So what's the answer? 785 00:36:19,040 --> 00:36:20,970 Yes, complex conjugation. 786 00:36:20,970 --> 00:36:21,970 Don't screw that one up. 787 00:36:21,970 --> 00:36:24,110 It's very easy to forget, but it matters a lot. 788 00:36:28,297 --> 00:36:29,380 Cursor keeps disappearing. 789 00:36:29,380 --> 00:36:32,360 OK, next one. 790 00:36:32,360 --> 00:36:40,120 A wave function has been expressed 791 00:36:40,120 --> 00:36:41,570 as a sum of energy eigenfunctions. 792 00:36:41,570 --> 00:36:45,445 Here I'm calling them mu rather than phi, but same thing. 793 00:36:45,445 --> 00:36:47,070 Compared to the original wave function, 794 00:36:47,070 --> 00:36:49,920 the set of coefficients, given that we're using the energy 795 00:36:49,920 --> 00:36:52,730 basis, the set of coefficients contains more or less 796 00:36:52,730 --> 00:36:55,390 the same information, or it can't be determined. 797 00:37:00,650 --> 00:37:02,360 OK, five seconds. 798 00:37:06,990 --> 00:37:07,780 All right. 799 00:37:07,780 --> 00:37:11,660 And the answer is C, great. 800 00:37:11,660 --> 00:37:13,390 OK, next one. 801 00:37:16,429 --> 00:37:17,720 So right now we're normalizing. 802 00:37:22,540 --> 00:37:23,470 OK. 803 00:37:23,470 --> 00:37:25,820 All stationary states, or all energy eigenstates, 804 00:37:25,820 --> 00:37:28,570 have the form that spatial and time dependence 805 00:37:28,570 --> 00:37:30,760 is the spatial dependence, the energy eigenfunction, 806 00:37:30,760 --> 00:37:35,110 times a phase, so that the norm squared is time independent. 807 00:37:35,110 --> 00:37:37,290 Consider the sum of two non-degenerate energy 808 00:37:37,290 --> 00:37:40,280 eigenstates psi 1 and psi 2. 809 00:37:40,280 --> 00:37:42,990 Non-degenerate means they have different energy. 810 00:37:45,540 --> 00:37:47,140 Is the wave function stationary? 811 00:37:47,140 --> 00:37:48,840 Is the probability distribution time 812 00:37:48,840 --> 00:37:50,420 independent or is it time dependent? 813 00:38:00,600 --> 00:38:02,980 This one's not trivial. 814 00:38:02,980 --> 00:38:03,530 Oh, shoot. 815 00:38:03,530 --> 00:38:04,730 I forgot to get it started. 816 00:38:04,730 --> 00:38:06,399 Sorry. 817 00:38:06,399 --> 00:38:08,940 It's particularly non-trivial if you can't enter your answer. 818 00:38:08,940 --> 00:38:09,210 Right. 819 00:38:09,210 --> 00:38:10,626 So go ahead and enter your answer. 820 00:38:17,438 --> 00:38:19,420 Whoo, yeah. 821 00:38:19,420 --> 00:38:20,715 This one always kills people. 822 00:38:31,315 --> 00:38:32,190 No chatting just yet. 823 00:38:32,190 --> 00:38:33,565 Test yourself, not your neighbor. 824 00:38:38,427 --> 00:38:40,010 It's fine to look deep into your soul, 825 00:38:40,010 --> 00:38:41,980 but don't look deep into the soul of the person sitting next 826 00:38:41,980 --> 00:38:42,480 to you. 827 00:38:46,660 --> 00:38:47,590 All right. 828 00:38:47,590 --> 00:38:51,840 So at this point, chat with your neighbor. 829 00:38:51,840 --> 00:38:54,086 Let me just give you some presage. 830 00:38:54,086 --> 00:38:57,720 The parallel strategy's probably not so good, because about half 831 00:38:57,720 --> 00:39:00,273 of you got it right, and about half of you got it wrong. 832 00:39:00,273 --> 00:40:07,070 [CHATTER] 833 00:40:07,070 --> 00:40:07,950 All right. 834 00:40:07,950 --> 00:40:09,375 Let's vote again. 835 00:40:11,980 --> 00:40:15,720 And hold on, starting now. 836 00:40:15,720 --> 00:40:16,480 OK, vote again. 837 00:40:16,480 --> 00:40:18,063 You've got 10 seconds to enter a vote. 838 00:40:24,520 --> 00:40:25,710 Wow. 839 00:40:25,710 --> 00:40:28,110 OK, two seconds. 840 00:40:28,110 --> 00:40:29,020 Good. 841 00:40:29,020 --> 00:40:36,940 So the distribution on this one went from 30, 50, 842 00:40:36,940 --> 00:40:47,290 20 initially, to now it is 10, 80, and 10. 843 00:40:47,290 --> 00:40:51,030 Amazingly, you guys got worse. 844 00:40:51,030 --> 00:40:55,620 The answer is C. And I want you to discuss with each other 845 00:40:55,620 --> 00:40:57,174 why it's C. 846 00:40:57,174 --> 00:41:27,840 [CHATTER] 847 00:41:27,840 --> 00:41:29,420 All right. 848 00:41:29,420 --> 00:41:32,060 OK. 849 00:41:32,060 --> 00:41:33,830 So let me talk you through it. 850 00:41:33,830 --> 00:41:36,230 So the wave function, we've said psi of x and t 851 00:41:36,230 --> 00:41:45,865 is equal to phi 1 at x, e to the minus i omega 1 t plus phi 852 00:41:45,865 --> 00:41:50,940 2 of x e to the minus i omega 2 t. 853 00:41:50,940 --> 00:41:52,670 So great, we take the norm squared. 854 00:41:52,670 --> 00:41:55,342 What's the probability to find it at x at time t. 855 00:41:55,342 --> 00:41:56,800 The probability density is the norm 856 00:41:56,800 --> 00:41:59,260 squared of this guy, psi squared, which 857 00:41:59,260 --> 00:42:04,590 is equal to phi 1 complex conjugate e to the plus i omega 858 00:42:04,590 --> 00:42:12,200 1 t plus phi 2 complex conjugate e to the plus i omega 2t times 859 00:42:12,200 --> 00:42:16,210 the thing itself phi 1 of x e to the minus i 860 00:42:16,210 --> 00:42:23,320 omega 1 t plus phi 2 of x e to the minus i omega 2t, right? 861 00:42:23,320 --> 00:42:25,870 So this has four terms. 862 00:42:25,870 --> 00:42:28,470 The first term is psi 1 norm squared. 863 00:42:28,470 --> 00:42:30,650 The phases cancel, right? 864 00:42:30,650 --> 00:42:33,340 You're going to see this happen a billion times in 804. 865 00:42:33,340 --> 00:42:35,850 The first term is going to be phi 1 norm squared. 866 00:42:35,850 --> 00:42:37,970 There's another term, which is phi 2 norm squared. 867 00:42:37,970 --> 00:42:40,345 Again the phases exactly cancel, even the minus i omega 2 868 00:42:40,345 --> 00:42:42,080 t to the plus i omega 2 t. 869 00:42:42,080 --> 00:42:44,470 Plus phi 2 squared. 870 00:42:44,470 --> 00:42:47,280 But then there are two cross terms, the interference terms. 871 00:42:47,280 --> 00:42:55,180 Plus phi 1 complex conjugate phi 2 e to the i omega 1 t 872 00:42:55,180 --> 00:43:00,670 e to the plus i omega 1 t, i omega 1 t, and e to the minus 873 00:43:00,670 --> 00:43:01,869 i omega 2t, minus omega 2. 874 00:43:01,869 --> 00:43:04,160 So we have a cross-term which depends on the difference 875 00:43:04,160 --> 00:43:05,194 in frequencies. 876 00:43:05,194 --> 00:43:07,110 Frequencies are like energies modulo on h-bar, 877 00:43:07,110 --> 00:43:09,240 so it's a difference in energies. 878 00:43:09,240 --> 00:43:10,740 And then there's another term, which 879 00:43:10,740 --> 00:43:12,670 is the complex conjugate of this guy, 880 00:43:12,670 --> 00:43:15,950 phi 2 star times phi 1 phi 2 complex conjugate phi 1 881 00:43:15,950 --> 00:43:17,825 and the phases are also the complex conjugate 882 00:43:17,825 --> 00:43:28,350 e to the minus i omega 1 minus omega 2 t of x of x of x of x. 883 00:43:28,350 --> 00:43:31,880 So is there time dependence in this, in principle? 884 00:43:31,880 --> 00:43:33,890 Absolutely, from the interference terms. 885 00:43:33,890 --> 00:43:35,500 Were we not in the superposition, 886 00:43:35,500 --> 00:43:37,470 we would not have interference terms. 887 00:43:37,470 --> 00:43:40,330 Time dependence comes from interference, when we expand 888 00:43:40,330 --> 00:43:41,600 in energy eigenfunctions. 889 00:43:41,600 --> 00:43:43,187 Cool? 890 00:43:43,187 --> 00:43:44,270 However, can these vanish? 891 00:43:44,270 --> 00:43:44,769 When? 892 00:43:48,450 --> 00:43:50,460 Sorry, say again? 893 00:43:50,460 --> 00:43:54,577 Great, so when omega 1 equals omega 2, what happens? 894 00:43:54,577 --> 00:43:55,660 Time dependence goes away. 895 00:43:55,660 --> 00:44:00,750 But omega 1 is e 1 over h bar, omega 2 is e 2 over h bar, 896 00:44:00,750 --> 00:44:03,595 and we started out by saying these are non-degenerate. 897 00:44:03,595 --> 00:44:05,970 So if they're non-degenerate, the energies are different, 898 00:44:05,970 --> 00:44:08,361 the frequencies are different, so that doesn't help us. 899 00:44:08,361 --> 00:44:09,860 How do we kill this time dependence? 900 00:44:12,860 --> 00:44:13,399 Yes. 901 00:44:13,399 --> 00:44:15,190 If the two functions aren't just orthogonal 902 00:44:15,190 --> 00:44:17,150 in a functional sense, but if we have the following. 903 00:44:17,150 --> 00:44:18,410 Suppose phi 1 is like this. 904 00:44:18,410 --> 00:44:21,430 It's 0 everywhere except for in some lump that's phi 1, 905 00:44:21,430 --> 00:44:25,230 and phi 2 is 0 everywhere except here. 906 00:44:25,230 --> 00:44:27,670 Then anywhere that phi 1 is non-zero, phi 2 is zero. 907 00:44:27,670 --> 00:44:30,790 And anywhere where phi 2 is non-zero, phi 1 is zero. 908 00:44:30,790 --> 00:44:33,940 So this can point-wise vanish. 909 00:44:33,940 --> 00:44:37,250 Do you expect this to happen generically? 910 00:44:37,250 --> 00:44:39,850 Does it happen for the energy eigenfunctions 911 00:44:39,850 --> 00:44:41,270 in the infinite square well? 912 00:44:44,710 --> 00:44:45,360 Sine waves? 913 00:44:47,981 --> 00:44:48,480 No. 914 00:44:48,480 --> 00:44:50,100 They have zero at isolated points, 915 00:44:50,100 --> 00:44:53,019 but they're non-zero generically. 916 00:44:53,019 --> 00:44:54,310 Yeah, so it doesn't work there. 917 00:44:54,310 --> 00:44:55,875 What about for the free particle? 918 00:44:55,875 --> 00:44:57,250 Well, those are just plain waves. 919 00:44:57,250 --> 00:44:58,680 Does that ever happen? 920 00:44:58,680 --> 00:44:59,610 No. 921 00:44:59,610 --> 00:45:02,540 OK, so this is an incredibly special case. 922 00:45:02,540 --> 00:45:04,040 We'll actually see it in one problem 923 00:45:04,040 --> 00:45:05,485 on a problem set later on. 924 00:45:05,485 --> 00:45:07,130 It's a very special case. 925 00:45:07,130 --> 00:45:09,935 So technically, the answer is C. And I 926 00:45:09,935 --> 00:45:11,570 want you guys to keep your minds open 927 00:45:11,570 --> 00:45:15,310 on these sorts of questions, when does a spatial dependence 928 00:45:15,310 --> 00:45:17,370 matter and when are there interference terms. 929 00:45:17,370 --> 00:45:18,260 Those are two different questions, 930 00:45:18,260 --> 00:45:19,718 and I want you to tease them apart. 931 00:45:19,718 --> 00:45:21,490 OK? 932 00:45:21,490 --> 00:45:22,590 Cool? 933 00:45:22,590 --> 00:45:23,784 Yeah? 934 00:45:23,784 --> 00:45:26,462 AUDIENCE: Is a valid way to think about this 935 00:45:26,462 --> 00:45:34,594 to think that you're fixing the initial [INAUDIBLE] 936 00:45:34,594 --> 00:45:36,760 PROFESSOR: That's a very good way to think about it. 937 00:45:36,760 --> 00:45:39,150 That's exactly right. 938 00:45:39,150 --> 00:45:41,082 That's a very, very good question. 939 00:45:41,082 --> 00:45:43,040 Let me say that subtly differently, and tell me 940 00:45:43,040 --> 00:45:44,956 if this agrees with what you were just saying. 941 00:45:44,956 --> 00:45:47,806 So I can look at this wave function, 942 00:45:47,806 --> 00:45:49,930 and I already know that the overall phase of a wave 943 00:45:49,930 --> 00:45:51,150 function doesn't matter. 944 00:45:51,150 --> 00:45:52,960 That's what it is to say a stationary state is stationary. 945 00:45:52,960 --> 00:45:54,960 It's got an overall phase that's the only thing, 946 00:45:54,960 --> 00:45:56,540 norm squared it goes away. 947 00:45:56,540 --> 00:46:01,950 So I can write this as e to the minus i omega 1 t times phi 948 00:46:01,950 --> 00:46:08,750 1 of x plus phi 2 of x e to the minus i omega 949 00:46:08,750 --> 00:46:12,506 2 minus omega 1 t. 950 00:46:12,506 --> 00:46:13,920 Is that what you mean? 951 00:46:13,920 --> 00:46:15,170 So that's one way to do this. 952 00:46:15,170 --> 00:46:16,503 We could also do something else. 953 00:46:16,503 --> 00:46:22,752 We could do e to the minus i omega 1 plus omega 2 upon 2 t. 954 00:46:22,752 --> 00:46:24,210 And this is more, I think, what you 955 00:46:24,210 --> 00:46:26,085 were thinking of, a sort of average frequency 956 00:46:26,085 --> 00:46:28,070 and then a relative frequency, and then 957 00:46:28,070 --> 00:46:31,050 the change in the frequencies on these two terms. 958 00:46:31,050 --> 00:46:31,760 Absolutely. 959 00:46:31,760 --> 00:46:36,720 So you can organize this in many, many ways. 960 00:46:36,720 --> 00:46:39,670 But your question gets at a very important point, 961 00:46:39,670 --> 00:46:41,670 which is that the overall phase doesn't matter. 962 00:46:41,670 --> 00:46:46,400 But relative phases in a superposition do matter. 963 00:46:46,400 --> 00:46:48,727 So when does a phase matter in a wave function? 964 00:46:48,727 --> 00:46:50,560 It does not matter if it's an overall phase. 965 00:46:50,560 --> 00:46:53,170 But it does matter if it's a relative phase between terms 966 00:46:53,170 --> 00:46:54,790 in a superposition. 967 00:46:54,790 --> 00:46:55,870 Cool? 968 00:46:55,870 --> 00:46:57,510 Very good question. 969 00:46:57,510 --> 00:47:00,330 Other questions? 970 00:47:00,330 --> 00:47:03,550 If not, then I have some. 971 00:47:03,550 --> 00:47:07,930 So, consider a system which is in the state-- 972 00:47:07,930 --> 00:47:10,227 so I should give you five-- 973 00:47:10,227 --> 00:47:12,810 system is in a state which is a linear combination of n equals 974 00:47:12,810 --> 00:47:14,280 1 and n equals 2 eigenstates. 975 00:47:16,956 --> 00:47:18,580 What's the probability that measurement 976 00:47:18,580 --> 00:47:20,082 will give us energy E1? 977 00:47:20,082 --> 00:47:21,373 And it's in this superposition. 978 00:47:28,390 --> 00:47:29,215 OK, five seconds. 979 00:47:33,780 --> 00:47:34,660 OK, fantastic. 980 00:47:34,660 --> 00:47:36,450 What's the answer? 981 00:47:36,450 --> 00:47:37,497 Yes, C, great. 982 00:47:37,497 --> 00:47:38,580 OK, everyone got that one. 983 00:47:38,580 --> 00:47:42,100 So one's a slightly more interesting question. 984 00:47:45,580 --> 00:47:55,210 Suppose I have an infinite well with width L. 985 00:47:55,210 --> 00:47:57,150 How does the energy, the ground state energy, 986 00:47:57,150 --> 00:48:01,175 compare to that of a system with a wider well? 987 00:48:16,070 --> 00:48:21,335 So L versus a larger L. OK, four seconds. 988 00:48:25,140 --> 00:48:28,940 OK, quickly discuss amongst yourselves, like 10 seconds. 989 00:48:28,940 --> 00:48:45,160 [CHATTER] 990 00:48:45,160 --> 00:48:47,300 All right. 991 00:48:47,300 --> 00:48:48,185 Now click again. 992 00:48:51,493 --> 00:48:51,993 Yeah. 993 00:48:56,080 --> 00:48:57,520 All right. 994 00:48:57,520 --> 00:48:58,260 Five seconds. 995 00:48:58,260 --> 00:49:02,270 One, two, three, four, five, great. 996 00:49:02,270 --> 00:49:08,470 OK, the answer is A. OK, great, because the energy 997 00:49:08,470 --> 00:49:11,530 of the infinite well goes like K squared. 998 00:49:11,530 --> 00:49:18,974 K goes like 1 over L. So the energy is, if we make it wider, 999 00:49:18,974 --> 00:49:21,140 the energy if we make it wider is going to be lower. 1000 00:49:27,260 --> 00:49:28,725 And last couple of questions. 1001 00:49:31,510 --> 00:49:34,900 OK, so t equals 0. 1002 00:49:34,900 --> 00:49:38,350 Could the wave function for an electron in an infinite square 1003 00:49:38,350 --> 00:49:44,880 well of width a, rather than L, be A sine squared of pi x 1004 00:49:44,880 --> 00:49:48,680 upon a, where A is suitably chosen to be normalized? 1005 00:50:10,460 --> 00:50:13,870 All right, you've got about five seconds left. 1006 00:50:17,550 --> 00:50:21,420 And OK, we are at chance. 1007 00:50:21,420 --> 00:50:24,710 We are at even odds, and the answer 1008 00:50:24,710 --> 00:50:28,380 is not a superposition of A and B, 1009 00:50:28,380 --> 00:50:31,130 so I encourage you to discuss with the people around you. 1010 00:50:31,130 --> 00:50:34,567 [CHATTER] 1011 00:50:51,731 --> 00:50:52,230 Great. 1012 00:50:52,230 --> 00:50:54,400 What properties had it better satisfy in order 1013 00:50:54,400 --> 00:50:56,774 to be a viable wave function? 1014 00:50:56,774 --> 00:50:58,440 What properties should the wave function 1015 00:50:58,440 --> 00:51:00,391 have so that it's reasonable? 1016 00:51:00,391 --> 00:51:00,890 Yeah. 1017 00:51:00,890 --> 00:51:02,080 Is it zero at the ends? 1018 00:51:02,080 --> 00:51:03,001 Yeah. 1019 00:51:03,001 --> 00:51:03,500 Good. 1020 00:51:03,500 --> 00:51:04,841 Is it smooth? 1021 00:51:04,841 --> 00:51:07,100 Yeah. 1022 00:51:07,100 --> 00:51:07,830 Exactly. 1023 00:51:07,830 --> 00:51:09,691 And so you can write it as a superposition. 1024 00:51:09,691 --> 00:51:10,190 Excellent. 1025 00:51:10,190 --> 00:51:11,692 So the answer is? 1026 00:51:11,692 --> 00:51:12,192 Yeah. 1027 00:51:17,516 --> 00:51:18,015 All right. 1028 00:51:20,780 --> 00:51:21,750 Vote again. 1029 00:51:25,170 --> 00:51:26,794 OK, I might have missed a few people. 1030 00:51:26,794 --> 00:51:27,710 So go ahead and start. 1031 00:51:31,800 --> 00:51:33,185 OK, five more seconds. 1032 00:51:37,230 --> 00:51:37,950 All right. 1033 00:51:37,950 --> 00:51:43,820 So we went from 50-50 to 77-23. 1034 00:51:43,820 --> 00:51:44,820 That's pretty good. 1035 00:51:44,820 --> 00:51:47,260 What's the answer? 1036 00:51:47,260 --> 00:51:47,890 A. Why? 1037 00:51:51,710 --> 00:51:53,892 Is this an energy eigenstate? 1038 00:51:53,892 --> 00:51:55,160 No. 1039 00:51:55,160 --> 00:51:56,960 Does that matter? 1040 00:51:56,960 --> 00:51:57,620 No. 1041 00:51:57,620 --> 00:51:59,710 What properties had this wave function better 1042 00:51:59,710 --> 00:52:05,460 satisfy to be a reasonable wave function in this potential? 1043 00:52:05,460 --> 00:52:06,660 Say again? 1044 00:52:06,660 --> 00:52:08,020 It's got to vanish at the walls. 1045 00:52:08,020 --> 00:52:09,853 It's got to satisfy the boundary conditions. 1046 00:52:09,853 --> 00:52:12,290 What else must be true of this wave function? 1047 00:52:12,290 --> 00:52:13,010 Normalizable. 1048 00:52:13,010 --> 00:52:14,240 Is it normalizable? 1049 00:52:14,240 --> 00:52:14,880 Yeah. 1050 00:52:14,880 --> 00:52:16,400 What else? 1051 00:52:16,400 --> 00:52:16,900 Continuous. 1052 00:52:16,900 --> 00:52:18,524 It better not have any discontinuities. 1053 00:52:18,524 --> 00:52:19,471 Is it continuous? 1054 00:52:19,471 --> 00:52:19,970 Great. 1055 00:52:19,970 --> 00:52:20,470 OK. 1056 00:52:20,470 --> 00:52:23,691 Is there any reason that this is a stupid wave function? 1057 00:52:23,691 --> 00:52:24,190 No. 1058 00:52:24,190 --> 00:52:25,500 It's perfectly reasonable. 1059 00:52:25,500 --> 00:52:29,930 It's not an energy eigenfunction, but-- 1060 00:52:29,930 --> 00:52:31,330 Yeah, cool? 1061 00:52:31,330 --> 00:52:32,150 Yeah. 1062 00:52:32,150 --> 00:52:34,252 AUDIENCE: This is sort of like a math question. 1063 00:52:34,252 --> 00:52:35,768 So to write that at a superposition, 1064 00:52:35,768 --> 00:52:38,636 you have to write it like basically a Fourier sign 1065 00:52:38,636 --> 00:52:39,592 series? 1066 00:52:39,592 --> 00:52:42,940 Isn't the [INAUDIBLE] function even, though? 1067 00:52:42,940 --> 00:52:46,426 PROFESSOR: On this domain, that and the sines are even. 1068 00:52:46,426 --> 00:52:48,925 So this is actually odd, but we're only looking at it from 0 1069 00:52:48,925 --> 00:52:53,480 to L. So, I mean that half of it. 1070 00:52:53,480 --> 00:52:56,080 The sines are odd, but we're only looking at the first peak. 1071 00:52:56,080 --> 00:52:56,780 So you could just as well have written 1072 00:52:56,780 --> 00:52:59,420 that as cosine of the midpoint plus the distance 1073 00:52:59,420 --> 00:53:01,595 from the midpoint. 1074 00:53:01,595 --> 00:53:03,470 Actually, let me say that again, because it's 1075 00:53:03,470 --> 00:53:05,511 a much better question I just give it shrift for. 1076 00:53:10,050 --> 00:53:11,010 So here's the question. 1077 00:53:11,010 --> 00:53:15,359 The question is, look, so sine is an odd function, 1078 00:53:15,359 --> 00:53:16,900 but sine squared is an even function. 1079 00:53:16,900 --> 00:53:19,670 So how can you expand sine squared, an even function, 1080 00:53:19,670 --> 00:53:21,780 in terms of sines, an odd function? 1081 00:53:21,780 --> 00:53:23,860 But think about this physically. 1082 00:53:23,860 --> 00:53:30,570 Here's sine squared in our domain, and here's sine. 1083 00:53:30,570 --> 00:53:31,810 Now what do you mean by even? 1084 00:53:31,810 --> 00:53:33,905 Usually by even we mean reflection around zero. 1085 00:53:33,905 --> 00:53:35,780 But I could just as well have said reflection 1086 00:53:35,780 --> 00:53:36,770 around the origin. 1087 00:53:36,770 --> 00:53:39,030 This potential is symmetric. 1088 00:53:39,030 --> 00:53:41,745 And the energy eigenfunctions are symmetric about the origin. 1089 00:53:41,745 --> 00:53:44,120 They're not symmetric about reflection around this point. 1090 00:53:44,120 --> 00:53:46,380 But they are symmetric about reflection around this point. 1091 00:53:46,380 --> 00:53:48,210 That's a particularly natural place to call it 0. 1092 00:53:48,210 --> 00:53:49,930 So I was calling them sine because I was calling this 0, 1093 00:53:49,930 --> 00:53:51,304 but I could have called it cosine 1094 00:53:51,304 --> 00:53:55,470 if I called this 0, for the same Kx. 1095 00:53:55,470 --> 00:53:58,570 And indeed, can we expand this sine 1096 00:53:58,570 --> 00:54:01,570 squared function in terms of a basis of these sines 1097 00:54:01,570 --> 00:54:02,610 on the domain 0 to L? 1098 00:54:02,610 --> 00:54:04,030 Absolutely. 1099 00:54:04,030 --> 00:54:07,220 Very good question. 1100 00:54:07,220 --> 00:54:09,355 And lastly, last clicker question. 1101 00:54:12,700 --> 00:54:13,460 Oops. 1102 00:54:13,460 --> 00:54:16,030 Whatever. 1103 00:54:16,030 --> 00:54:17,210 OK. 1104 00:54:17,210 --> 00:54:20,600 At t equals 0, a particle is described by the wave function 1105 00:54:20,600 --> 00:54:21,130 we just saw. 1106 00:54:23,461 --> 00:54:25,710 Which of the following is true about the wave function 1107 00:54:25,710 --> 00:54:26,600 at subsequent times? 1108 00:54:37,650 --> 00:54:38,210 5 seconds. 1109 00:54:42,776 --> 00:54:43,730 Whew. 1110 00:54:43,730 --> 00:54:44,650 Oh, OK. 1111 00:54:44,650 --> 00:54:47,360 In the last few seconds we had an explosive burst for A, B, 1112 00:54:47,360 --> 00:54:49,870 and C. So our current distribution 1113 00:54:49,870 --> 00:54:54,960 is 8, 16, 10, and 67, sounds like 67 is popular. 1114 00:54:54,960 --> 00:54:58,630 Discuss quickly, very quickly, with the person next to you. 1115 00:54:58,630 --> 00:55:02,510 [CHATTER] 1116 00:55:13,670 --> 00:55:17,093 OK, and vote again. 1117 00:55:21,580 --> 00:55:22,980 OK, five seconds. 1118 00:55:22,980 --> 00:55:24,180 Get your last vote in. 1119 00:55:28,240 --> 00:55:29,300 All right. 1120 00:55:29,300 --> 00:55:33,472 And the answer is D. Yay. 1121 00:55:37,289 --> 00:55:38,580 So let's think about the logic. 1122 00:55:38,580 --> 00:55:39,950 Let's go through the logic here. 1123 00:55:39,950 --> 00:55:42,360 So as was pointed out by a student up here earlier, 1124 00:55:42,360 --> 00:55:46,070 the wave function sine squared of pi x 1125 00:55:46,070 --> 00:55:49,100 can be expanded in terms of the energy eigenfunction. 1126 00:55:49,100 --> 00:55:51,150 Any reasonable function can be expanded 1127 00:55:51,150 --> 00:55:53,800 in terms of a superposition of definite energy states 1128 00:55:53,800 --> 00:55:55,600 of energy eigenfunctions. 1129 00:55:55,600 --> 00:55:58,490 So that means we can write psi at some time 1130 00:55:58,490 --> 00:56:04,060 as a superposition Cn sine of n pi x upon a e to the minus i e 1131 00:56:04,060 --> 00:56:09,450 n t upon h bar, since those are, in fact, the eigenfunctions. 1132 00:56:09,450 --> 00:56:10,200 So we can do that. 1133 00:56:10,200 --> 00:56:11,877 Now, when we look at the time evolution, 1134 00:56:11,877 --> 00:56:13,710 we know that each term in that superposition 1135 00:56:13,710 --> 00:56:15,450 evolves with a phase. 1136 00:56:15,450 --> 00:56:19,220 The overall wave function does not evolve with a phase. 1137 00:56:19,220 --> 00:56:21,360 It is not an energy eigenstate. 1138 00:56:21,360 --> 00:56:23,460 There are going to be interference terms due 1139 00:56:23,460 --> 00:56:26,207 to the fact that it's a superposition. 1140 00:56:26,207 --> 00:56:28,540 So its probability distribution is not time-independent. 1141 00:56:28,540 --> 00:56:29,640 It is a superposition. 1142 00:56:29,640 --> 00:56:33,430 And so the wave function doesn't rotate by an overall phase. 1143 00:56:33,430 --> 00:56:37,030 However, we can solve the Schrodinger equation, 1144 00:56:37,030 --> 00:56:37,810 as we did before. 1145 00:56:37,810 --> 00:56:40,440 The wave function is expanded at time 0 1146 00:56:40,440 --> 00:56:43,544 as the energy eigenfunctions times some set of coefficients. 1147 00:56:43,544 --> 00:56:44,960 And the time evolution corresponds 1148 00:56:44,960 --> 00:56:48,230 to adding two each independent term in the superposition 1149 00:56:48,230 --> 00:56:52,120 the appropriate phase for that energy eigenstate. 1150 00:56:52,120 --> 00:56:54,040 Cool? 1151 00:56:54,040 --> 00:56:54,540 All right. 1152 00:56:54,540 --> 00:57:03,220 So the answer is D. And that's it for the clicker questions. 1153 00:57:03,220 --> 00:57:07,050 OK, so any questions on the clicker questions so far? 1154 00:57:07,050 --> 00:57:09,050 OK, those are going to be posted on the web site 1155 00:57:09,050 --> 00:57:12,190 so you can go over them. 1156 00:57:12,190 --> 00:57:14,250 And now back to energy eigenfunctions. 1157 00:57:24,770 --> 00:57:27,730 So what I want to talk about now is the qualitative behavior 1158 00:57:27,730 --> 00:57:29,140 of energy eigenfunctions. 1159 00:57:29,140 --> 00:57:31,460 Suppose I know I have an energy eigenfunction. 1160 00:57:31,460 --> 00:57:33,955 What can I say generally about its structure? 1161 00:57:38,870 --> 00:57:41,175 So let me ask the question, qualitative behavior. 1162 00:57:47,270 --> 00:57:49,670 So suppose someone hands you a potential U of x. 1163 00:57:49,670 --> 00:57:51,990 Someone hands you some potential, U of x, 1164 00:57:51,990 --> 00:57:54,900 and says, look, I've got this potential. 1165 00:57:54,900 --> 00:57:57,580 Maybe I'll draw it for you. 1166 00:57:57,580 --> 00:57:59,850 It's got some wiggles, and then a big wiggle, 1167 00:57:59,850 --> 00:58:01,880 and then it's got a big wiggle, and then-- 1168 00:58:01,880 --> 00:58:03,195 do I want to do that? 1169 00:58:03,195 --> 00:58:04,275 Yeah, let's do that. 1170 00:58:04,275 --> 00:58:08,089 Then a big wiggle, and something like this. 1171 00:58:08,089 --> 00:58:09,630 And someone shows you this potential. 1172 00:58:09,630 --> 00:58:11,921 And they say, look, what are the energy eigenfunctions? 1173 00:58:13,770 --> 00:58:17,510 Well, OK, free particle was easy. 1174 00:58:17,510 --> 00:58:19,070 The infinite square well was easy. 1175 00:58:19,070 --> 00:58:21,207 We could solve that analytically. 1176 00:58:21,207 --> 00:58:23,290 The next involved solving a differential equation. 1177 00:58:23,290 --> 00:58:25,360 So what differential equation is this going to lead us to? 1178 00:58:25,360 --> 00:58:27,570 Well, we know that the energy eigenvalue equation 1179 00:58:27,570 --> 00:58:32,070 is minus h bar squared upon 2 m phi 1180 00:58:32,070 --> 00:58:38,570 prime prime of x plus U of x phi x, so that's the energy 1181 00:58:38,570 --> 00:58:41,620 operator acting on phi, is equal to, 1182 00:58:41,620 --> 00:58:47,949 saying that it's an energy eigenfunction, phi sub E, 1183 00:58:47,949 --> 00:58:49,740 says that it's equal to the energy operator 1184 00:58:49,740 --> 00:58:52,670 acting on this eigenfunction is just a constant E phi sub 1185 00:58:52,670 --> 00:58:54,870 E of x. 1186 00:58:54,870 --> 00:58:56,680 And I'm going to work at moment in time, 1187 00:58:56,680 --> 00:59:00,420 so we're going to drop all the t dependence for the moment. 1188 00:59:00,420 --> 00:59:01,920 So this is the differential equation 1189 00:59:01,920 --> 00:59:07,072 we need to solve where U of x is this god-awful function. 1190 00:59:07,072 --> 00:59:08,780 Do you think it's very likely that you're 1191 00:59:08,780 --> 00:59:11,670 going to be able to solve this analytically? 1192 00:59:11,670 --> 00:59:13,700 Probably not. 1193 00:59:13,700 --> 00:59:18,761 However, some basic ideas will help 1194 00:59:18,761 --> 00:59:21,010 you get an intuition for what the wave function should 1195 00:59:21,010 --> 00:59:22,240 look like. 1196 00:59:22,240 --> 00:59:25,910 And I cannot overstate the importance of being able 1197 00:59:25,910 --> 00:59:29,560 to eyeball a system and guess the qualitative features of its 1198 00:59:29,560 --> 00:59:32,030 wave functions, because that intuition, 1199 00:59:32,030 --> 00:59:34,590 that ability to estimate, is going to contain an awful lot 1200 00:59:34,590 --> 00:59:35,340 of physics. 1201 00:59:35,340 --> 00:59:36,790 So let's try to extract it. 1202 00:59:36,790 --> 00:59:38,930 So in order to do so, I want to start 1203 00:59:38,930 --> 00:59:40,945 by massaging this equation into a form which 1204 00:59:40,945 --> 00:59:42,070 is particularly convenient. 1205 00:59:44,790 --> 00:59:47,910 So in particular, I'm going to write this equation as phi sub 1206 00:59:47,910 --> 00:59:49,449 E prime prime. 1207 00:59:49,449 --> 00:59:51,740 So what I'm going to do is I'm going to take this term, 1208 00:59:51,740 --> 00:59:53,615 I'm going to notice this has two derivatives, 1209 00:59:53,615 --> 00:59:56,026 this has no derivatives, this has no derivatives. 1210 00:59:56,026 --> 00:59:59,130 And I'm going to move this term over here and combine 1211 00:59:59,130 --> 01:00:01,600 these terms into E minus U of x, and I'm 1212 01:00:01,600 --> 01:00:05,320 going to divide each side by 2m upon h bar squared with a minus 1213 01:00:05,320 --> 01:00:08,260 sign, giving me that phi prime prime of E 1214 01:00:08,260 --> 01:00:14,420 of x upon phi E of x dividing through by this phi E 1215 01:00:14,420 --> 01:00:18,250 is equal to minus 2m over h bar squared. 1216 01:00:21,682 --> 01:00:23,140 And let's just get our signs right. 1217 01:00:23,140 --> 01:00:24,560 We've got the minus from here, so this 1218 01:00:24,560 --> 01:00:25,960 is going to be E minus U of x. 1219 01:00:33,931 --> 01:00:35,680 So you might look at that and think, well, 1220 01:00:35,680 --> 01:00:40,010 why is that any better than what I've just written down. 1221 01:00:40,010 --> 01:00:42,724 But what is the second derivative of function? 1222 01:00:42,724 --> 01:00:44,890 It's telling you not its slope, but it's telling you 1223 01:00:44,890 --> 01:00:46,550 how the slope changes. 1224 01:00:46,550 --> 01:00:49,588 It's telling about the curvature of the function. 1225 01:00:49,588 --> 01:00:52,046 And what this is telling me is something very, very useful. 1226 01:00:54,179 --> 01:00:55,970 So for example, let's look at the function. 1227 01:00:55,970 --> 01:00:57,360 Let's assume that the function is real, 1228 01:00:57,360 --> 01:00:58,901 although we know in general it's not. 1229 01:00:58,901 --> 01:01:02,100 Let's assume that the function is real for simplicity. 1230 01:01:02,100 --> 01:01:05,910 So we're going to plot the real part of phi 1231 01:01:05,910 --> 01:01:07,210 in the vertical axis. 1232 01:01:07,210 --> 01:01:08,630 And this is x. 1233 01:01:08,630 --> 01:01:15,130 Suppose the real part of phi is positive at some point. 1234 01:01:15,130 --> 01:01:17,070 Phi prime prime, if it's positive, 1235 01:01:17,070 --> 01:01:19,297 tells us that not only is the slope positive, 1236 01:01:19,297 --> 01:01:20,130 but it's increasing. 1237 01:01:20,130 --> 01:01:21,600 Or it doesn't tell us anything about the slope, 1238 01:01:21,600 --> 01:01:23,080 but it tells us that whatever the slope, it's increasing. 1239 01:01:23,080 --> 01:01:25,650 If it's negative, the slope is increasing as we increase x. 1240 01:01:25,650 --> 01:01:28,210 If it's positive, it's increasing as we increase x. 1241 01:01:28,210 --> 01:01:31,450 So it's telling us that the wave function looks like this, 1242 01:01:31,450 --> 01:01:34,880 locally, something like that. 1243 01:01:34,880 --> 01:01:39,040 If phi is negative, if phi is negative, 1244 01:01:39,040 --> 01:01:41,254 then if this quantity is positive, 1245 01:01:41,254 --> 01:01:42,920 then phi prime prime has to be negative. 1246 01:01:42,920 --> 01:01:45,150 But negative is curving down. 1247 01:01:50,210 --> 01:01:54,780 So if this quantity, which I will call the curvature, 1248 01:01:54,780 --> 01:02:01,260 if this quantity is positive, it curves away from the axis. 1249 01:02:01,260 --> 01:02:07,100 So this is phi prime prime over phi greater than 0. 1250 01:02:07,100 --> 01:02:09,700 If this quantity is positive, the function 1251 01:02:09,700 --> 01:02:11,050 curves away from the axis. 1252 01:02:11,050 --> 01:02:12,440 Cool? 1253 01:02:12,440 --> 01:02:16,270 If this quantity is negative, phi prime prime upon 1254 01:02:16,270 --> 01:02:19,490 phi less than 0, exactly the opposite. 1255 01:02:19,490 --> 01:02:20,920 This has to be negative. 1256 01:02:20,920 --> 01:02:23,480 If phi is positive, then phi prime prime has to be negative. 1257 01:02:23,480 --> 01:02:24,907 It has to be curving down. 1258 01:02:27,710 --> 01:02:30,290 And similarly, if phi is negative, 1259 01:02:30,290 --> 01:02:33,170 then phi prime prime has to be positive, 1260 01:02:33,170 --> 01:02:34,500 and it has to curve up. 1261 01:02:34,500 --> 01:02:37,840 So if this quantity is positive, if the curvature is positive, 1262 01:02:37,840 --> 01:02:39,440 it curves away from the axis. 1263 01:02:39,440 --> 01:02:41,950 If the curvature is negative, if this quantity is negative, 1264 01:02:41,950 --> 01:02:44,120 it curves towards the axis. 1265 01:02:44,120 --> 01:02:47,590 So what does that tell you about solutions when the curvature is 1266 01:02:47,590 --> 01:02:49,206 positive or negative? 1267 01:02:49,206 --> 01:02:50,330 It tells you the following. 1268 01:02:52,930 --> 01:02:54,980 It tells you that, imagine we have 1269 01:02:54,980 --> 01:02:59,450 a function where phi prime prime over phi is constant. 1270 01:02:59,450 --> 01:03:03,050 And in particular, let's let phi prime prime over phi 1271 01:03:03,050 --> 01:03:06,760 be a constant, which is positive. 1272 01:03:06,760 --> 01:03:10,671 And I'll call that positive constant kappa squared. 1273 01:03:10,671 --> 01:03:12,170 And to emphasize that it's positive, 1274 01:03:12,170 --> 01:03:14,174 I'm going to call it kappa squared. 1275 01:03:14,174 --> 01:03:15,090 It's a positive thing. 1276 01:03:15,090 --> 01:03:16,980 It's a real number squared. 1277 01:03:16,980 --> 01:03:18,635 What does the solution look like? 1278 01:03:22,510 --> 01:03:24,120 Well, this quantity is positive. 1279 01:03:24,120 --> 01:03:25,661 It's always going to be curving away. 1280 01:03:25,661 --> 01:03:28,287 So we have solutions that look like this or solutions 1281 01:03:28,287 --> 01:03:29,120 that look like this. 1282 01:03:29,120 --> 01:03:29,990 Can it ever be 0? 1283 01:03:32,907 --> 01:03:34,740 Yeah, sure, it could be an inflection point. 1284 01:03:34,740 --> 01:03:36,698 So for example, here the curvature is positive, 1285 01:03:36,698 --> 01:03:40,170 but at this point the curvature has to switch to be like this. 1286 01:03:40,170 --> 01:03:41,960 What functions are of this form? 1287 01:03:44,752 --> 01:03:45,960 Let me give you another hint. 1288 01:03:45,960 --> 01:03:46,550 Here's one. 1289 01:03:46,550 --> 01:03:49,400 Is this curvature positive? 1290 01:03:49,400 --> 01:03:50,057 Yes. 1291 01:03:50,057 --> 01:03:50,890 What about this one? 1292 01:03:53,470 --> 01:03:53,970 Yup. 1293 01:03:53,970 --> 01:03:56,220 Those are all positive curvature. 1294 01:03:56,220 --> 01:03:57,910 And these are exponentials. 1295 01:03:57,910 --> 01:03:59,930 And the solution to this differential equation 1296 01:03:59,930 --> 01:04:06,670 is e to the plus kappa x or e to the minus kappa x. 1297 01:04:06,670 --> 01:04:08,940 And an arbitrary solution of this equation 1298 01:04:08,940 --> 01:04:12,130 is a superposition A e to the kappa x plus B 1299 01:04:12,130 --> 01:04:14,346 e to the minus kappa x. 1300 01:04:14,346 --> 01:04:16,244 Everyone cool with that? 1301 01:04:16,244 --> 01:04:17,660 When this quantity is positive, we 1302 01:04:17,660 --> 01:04:19,390 get growing and collapsing exponentials. 1303 01:04:21,890 --> 01:04:22,390 Yeah? 1304 01:04:25,610 --> 01:04:29,990 On the other hand, if phi prime prime over phi 1305 01:04:29,990 --> 01:04:34,010 is a negative number, i.e. minus what I'll call k 1306 01:04:34,010 --> 01:04:44,010 squared, then the curvature has to be negative. 1307 01:04:44,010 --> 01:04:47,460 And what functions have everywhere negative curvature? 1308 01:04:47,460 --> 01:04:49,169 Sinusoidals. 1309 01:04:49,169 --> 01:04:49,669 Cool? 1310 01:04:53,540 --> 01:05:00,445 And the general solution is A e to the i K x plus B 1311 01:05:00,445 --> 01:05:03,226 e to the minus i K x. 1312 01:05:03,226 --> 01:05:06,710 So that differential equation, also known as sine and cosine. 1313 01:05:09,170 --> 01:05:09,670 Cool? 1314 01:05:12,540 --> 01:05:19,765 So putting that together with our original function, 1315 01:05:19,765 --> 01:05:21,140 let's bring this up. 1316 01:05:24,110 --> 01:05:26,680 So we want to think about the wave functions here. 1317 01:05:26,680 --> 01:05:29,014 But in order to think about the energy eigenstates, 1318 01:05:29,014 --> 01:05:30,305 we need to decide on an energy. 1319 01:05:32,850 --> 01:05:35,794 We need to pick an energy, because you 1320 01:05:35,794 --> 01:05:37,960 can't find the solution without figuring the energy. 1321 01:05:37,960 --> 01:05:38,950 But notice something nice here. 1322 01:05:38,950 --> 01:05:40,260 So suppose the energy is e. 1323 01:05:40,260 --> 01:05:42,360 And let me just draw E. This is a constant. 1324 01:05:42,360 --> 01:05:43,187 The energy is this. 1325 01:05:43,187 --> 01:05:45,520 So this is the value of E. Here we're drawing potential. 1326 01:05:45,520 --> 01:05:48,489 But this is the value of the energy, which is a constant. 1327 01:05:48,489 --> 01:05:49,280 It's just a number. 1328 01:05:53,680 --> 01:05:57,200 If you had a classical particle moving in this potential, 1329 01:05:57,200 --> 01:05:58,760 what would happen? 1330 01:05:58,760 --> 01:05:59,650 It would roll around. 1331 01:05:59,650 --> 01:06:01,360 So for example, let's say you gave it this energy 1332 01:06:01,360 --> 01:06:02,210 by putting it here. 1333 01:06:02,210 --> 01:06:04,168 And think of this as a gravitational potential. 1334 01:06:04,168 --> 01:06:06,157 You put it here, you let go, and it falls down. 1335 01:06:06,157 --> 01:06:07,990 And it'll keep rolling until it gets up here 1336 01:06:07,990 --> 01:06:10,230 to the classical turning point. 1337 01:06:10,230 --> 01:06:12,080 And at that point, its kinetic energy 1338 01:06:12,080 --> 01:06:13,710 must be 0, because its potential energy 1339 01:06:13,710 --> 01:06:15,410 is its total energy, at which point 1340 01:06:15,410 --> 01:06:17,670 it will turn around and fall back. 1341 01:06:17,670 --> 01:06:18,359 Yes? 1342 01:06:18,359 --> 01:06:20,150 If you take your ball, and you put it here, 1343 01:06:20,150 --> 01:06:21,525 and you let it roll, does it ever 1344 01:06:21,525 --> 01:06:23,960 get here, to this position? 1345 01:06:23,960 --> 01:06:25,990 No, because it doesn't have enough energy. 1346 01:06:25,990 --> 01:06:29,620 Classically, this is a forbidden position. 1347 01:06:29,620 --> 01:06:31,790 So given an energy and given a potential, 1348 01:06:31,790 --> 01:06:38,200 we can break the system up into classically allowed zones 1349 01:06:38,200 --> 01:06:39,760 and classically forbidden zones. 1350 01:06:45,170 --> 01:06:47,120 Cool? 1351 01:06:47,120 --> 01:06:49,210 Now, in a classically allowed zone, 1352 01:06:49,210 --> 01:06:53,670 the energy is greater than the potential. 1353 01:06:53,670 --> 01:06:55,170 And in a classically forbidden zone, 1354 01:06:55,170 --> 01:06:57,300 the energy is less than the potential. 1355 01:07:00,530 --> 01:07:03,230 Everyone cool with that? 1356 01:07:03,230 --> 01:07:06,080 But this tells us something really nice. 1357 01:07:06,080 --> 01:07:09,429 If the energy is greater than the potential, 1358 01:07:09,429 --> 01:07:10,970 what do you know about the curvature? 1359 01:07:15,050 --> 01:07:15,550 Yeah. 1360 01:07:15,550 --> 01:07:17,175 If we're in a classically allowed zone, 1361 01:07:17,175 --> 01:07:20,150 so the energy is greater than the potential, 1362 01:07:20,150 --> 01:07:22,860 then this quantity is positive, there's a minus sign here, 1363 01:07:22,860 --> 01:07:23,920 so this is negative. 1364 01:07:23,920 --> 01:07:25,760 So the curvature is negative. 1365 01:07:28,920 --> 01:07:31,680 Remember, curvature is negative means 1366 01:07:31,680 --> 01:07:35,000 that we curve towards the axis. 1367 01:07:35,000 --> 01:07:37,015 So in a classically allowed region, 1368 01:07:37,015 --> 01:07:38,640 the wave function should be sinusoidal. 1369 01:07:43,840 --> 01:07:46,050 What about in the classically forbidden regions? 1370 01:07:46,050 --> 01:07:47,280 In the classically forbidden regions, 1371 01:07:47,280 --> 01:07:48,890 the energy is less than the potential. 1372 01:07:48,890 --> 01:07:50,848 That means in magnitude this is less than this, 1373 01:07:50,848 --> 01:07:53,270 this is a negative number, minus sign, 1374 01:07:53,270 --> 01:07:55,365 the curvature is going to be minus 1375 01:07:55,365 --> 01:07:59,050 times a minus is a positive, so the curvature's positive. 1376 01:08:02,260 --> 01:08:05,190 So the solutions are either growing exponentials 1377 01:08:05,190 --> 01:08:10,000 or shrinking exponentials or superpositions of them. 1378 01:08:10,000 --> 01:08:11,810 Everyone cool with that? 1379 01:08:11,810 --> 01:08:14,340 So let's think about a simple example. 1380 01:08:14,340 --> 01:08:16,452 Let's work through this in a simple example. 1381 01:08:16,452 --> 01:08:18,575 And let me give you a little bit more board space. 1382 01:08:22,430 --> 01:08:26,122 Simple example would be a potential that looks like this. 1383 01:08:26,122 --> 01:08:27,580 And let's just suppose that we want 1384 01:08:27,580 --> 01:08:33,760 to find an energy eigenfunction with energy that's E. Well, 1385 01:08:33,760 --> 01:08:36,250 this is a classically allowed zone, 1386 01:08:36,250 --> 01:08:39,939 and these are the classically forbidden regions. 1387 01:08:39,939 --> 01:08:43,370 Now I want to ask, what does the wave function look like? 1388 01:08:43,370 --> 01:08:46,630 And I don't want to draw it on top of the energy diagram, 1389 01:08:46,630 --> 01:08:49,037 because wave function is not an energy. 1390 01:08:49,037 --> 01:08:50,620 Wave function is a different quantity, 1391 01:08:50,620 --> 01:08:52,370 because it's got different axes and I want 1392 01:08:52,370 --> 01:08:54,279 it drawn on a different plot. 1393 01:08:54,279 --> 01:08:57,960 So but as a function of x-- 1394 01:08:57,960 --> 01:09:00,935 so just to get the positions straight, 1395 01:09:00,935 --> 01:09:02,560 these are the bounds of the classically 1396 01:09:02,560 --> 01:09:04,899 allowed and forbidden regions. 1397 01:09:04,899 --> 01:09:07,140 What do we expect? 1398 01:09:07,140 --> 01:09:09,830 Well, we expect that it's going to be sinusoidal in here. 1399 01:09:13,160 --> 01:09:15,770 We expect that it's going to be exponential growing 1400 01:09:15,770 --> 01:09:20,399 or converging out here, exp. 1401 01:09:20,399 --> 01:09:25,100 But one last important thing is that not only is the curvature 1402 01:09:25,100 --> 01:09:28,270 negative in here in these classically allowed regions, 1403 01:09:28,270 --> 01:09:30,080 but the magnitude of the curvature, 1404 01:09:30,080 --> 01:09:32,870 how rapidly it's turning over, how big that second derivative 1405 01:09:32,870 --> 01:09:35,040 is, depends on the difference between the energy 1406 01:09:35,040 --> 01:09:35,865 and the potential. 1407 01:09:35,865 --> 01:09:38,240 The greater the difference, the more rapid the curvature, 1408 01:09:38,240 --> 01:09:40,500 the more rapid the turning over and fluctuation. 1409 01:09:40,500 --> 01:09:43,450 If the differences between the potential and the true energy, 1410 01:09:43,450 --> 01:09:46,720 the total energy, is small, then the curvature is very small. 1411 01:09:46,720 --> 01:09:49,990 So the derivative changes very gradually. 1412 01:09:49,990 --> 01:09:51,660 What does that tell us? 1413 01:09:51,660 --> 01:09:54,194 That tells us that in here the wave function is oscillating 1414 01:09:54,194 --> 01:09:56,110 rapidly, because the curvature, the difference 1415 01:09:56,110 --> 01:09:58,320 between the energy and the potential is large, 1416 01:09:58,320 --> 01:10:01,690 and so the wave function is oscillating rapidly. 1417 01:10:01,690 --> 01:10:04,200 As we get out towards the classical turning points, 1418 01:10:04,200 --> 01:10:07,640 the wave function will be oscillating less rapidly. 1419 01:10:07,640 --> 01:10:09,865 The slope will be changing more gradually. 1420 01:10:09,865 --> 01:10:11,810 And as a consequence, two things happen. 1421 01:10:11,810 --> 01:10:13,768 Let me actually draw this slightly differently. 1422 01:10:17,647 --> 01:10:19,230 So as a consequence two things happen. 1423 01:10:19,230 --> 01:10:21,130 One is the wavelength gets longer, 1424 01:10:21,130 --> 01:10:22,847 because the curvature is smaller. 1425 01:10:22,847 --> 01:10:24,680 And the second is the amplitude gets larger, 1426 01:10:24,680 --> 01:10:27,280 because it keeps on having a positive slope for longer 1427 01:10:27,280 --> 01:10:31,520 and longer, and it takes longer to curve back down. 1428 01:10:31,520 --> 01:10:34,260 So here we have rapid oscillations. 1429 01:10:34,260 --> 01:10:41,430 And then the oscillations get longer and longer wavelength, 1430 01:10:41,430 --> 01:10:43,892 until we get out to the classical turning point. 1431 01:10:43,892 --> 01:10:45,225 And at this point, what happens? 1432 01:10:47,704 --> 01:10:49,120 Yeah, it's got to be [INAUDIBLE].. 1433 01:10:49,120 --> 01:10:51,860 Now, here we have some sine, and some superpositions 1434 01:10:51,860 --> 01:10:53,647 of sine and cosines, exponentials. 1435 01:10:53,647 --> 01:10:55,730 And in particular, it arrives here with some slope 1436 01:10:55,730 --> 01:10:57,490 and with some value. 1437 01:10:57,490 --> 01:11:00,140 We know this side we've got to get exponentials. 1438 01:11:00,140 --> 01:11:02,330 And so this sum of sines and cosines at this point 1439 01:11:02,330 --> 01:11:06,400 must match the sum of exponentials. 1440 01:11:06,400 --> 01:11:08,282 How must it do so? 1441 01:11:08,282 --> 01:11:10,490 What must be true of the wave function at this point? 1442 01:11:13,880 --> 01:11:16,090 Can it be discontinuous? 1443 01:11:16,090 --> 01:11:17,850 Can its derivative be discontinuous? 1444 01:11:17,850 --> 01:11:18,350 No. 1445 01:11:18,350 --> 01:11:21,800 So the value and the derivative must be continuous. 1446 01:11:21,800 --> 01:11:24,690 So that tells us precisely which linear combination 1447 01:11:24,690 --> 01:11:29,089 of positive growing and shrinking exponentials we get. 1448 01:11:29,089 --> 01:11:30,630 So we'll get some linear combination, 1449 01:11:30,630 --> 01:11:32,360 which may do this for awhile. 1450 01:11:32,360 --> 01:11:33,860 But since it's got some contribution 1451 01:11:33,860 --> 01:11:36,490 of positive exponential, it'll just grow exponentially off 1452 01:11:36,490 --> 01:11:37,982 to infinity. 1453 01:11:37,982 --> 01:11:39,940 And as the energy gets further and further away 1454 01:11:39,940 --> 01:11:42,780 from the potential, now in their negative sine, what 1455 01:11:42,780 --> 01:11:45,444 happens to the rate of growth? 1456 01:11:45,444 --> 01:11:46,610 It gets more and more rapid. 1457 01:11:46,610 --> 01:11:48,540 So this just diverges more and more rapidly. 1458 01:11:48,540 --> 01:11:51,554 Similarly, out here we have to match the slope. 1459 01:11:51,554 --> 01:11:52,970 And we know that the curvature has 1460 01:11:52,970 --> 01:11:55,530 to be now positive, so it has to do this. 1461 01:11:59,120 --> 01:12:00,040 So two questions. 1462 01:12:00,040 --> 01:12:03,820 First off, is this sketch of the wave function 1463 01:12:03,820 --> 01:12:05,560 a reasonable sketch, given what we 1464 01:12:05,560 --> 01:12:08,570 know about curvature and this potential of a wave function 1465 01:12:08,570 --> 01:12:09,280 with that energy? 1466 01:12:13,007 --> 01:12:14,840 Are there ways in which it's a bad estimate? 1467 01:12:17,798 --> 01:12:19,590 AUDIENCE: [INAUDIBLE] 1468 01:12:19,590 --> 01:12:21,685 PROFESSOR: OK, excellent. 1469 01:12:21,685 --> 01:12:24,600 AUDIENCE: On the right side, could it have crossed zero? 1470 01:12:24,600 --> 01:12:25,720 PROFESSOR: Absolutely, it could have crossed zero. 1471 01:12:25,720 --> 01:12:26,920 So I may have drawn this badly. 1472 01:12:26,920 --> 01:12:28,060 It turned out it was a little subtle. 1473 01:12:28,060 --> 01:12:28,740 It's not obvious. 1474 01:12:28,740 --> 01:12:30,340 Maybe it actually punched all the way through zero, 1475 01:12:30,340 --> 01:12:31,660 and then it diverged down negative. 1476 01:12:31,660 --> 01:12:32,785 That's absolutely positive. 1477 01:12:32,785 --> 01:12:35,075 So that was one of the quibbles you could have. 1478 01:12:35,075 --> 01:12:36,450 Another quibble you could have is 1479 01:12:36,450 --> 01:12:39,200 that it looks like I have constant wavelength in here. 1480 01:12:39,200 --> 01:12:41,010 But the potential's actually changing. 1481 01:12:41,010 --> 01:12:43,070 And what you should chalk this up to, 1482 01:12:43,070 --> 01:12:46,299 if you'll pardon the pun, is my artistic skills are limited. 1483 01:12:46,299 --> 01:12:48,340 So this is always going to be sort of inescapable 1484 01:12:48,340 --> 01:12:50,505 when you qualitatively draw something. 1485 01:12:50,505 --> 01:12:53,130 On a test, I'm not going to bag you points on things like that. 1486 01:12:53,130 --> 01:12:54,584 That's what I want to emphasize. 1487 01:12:54,584 --> 01:12:56,250 But the second thing, is there something 1488 01:12:56,250 --> 01:12:57,622 bad about this wave function? 1489 01:12:57,622 --> 01:12:58,830 Yes, you've already named it. 1490 01:12:58,830 --> 01:13:01,090 What's bad about this wave function? 1491 01:13:01,090 --> 01:13:03,150 It's badly non-normalizable. 1492 01:13:03,150 --> 01:13:06,820 It diverges off to infinity out here and out here. 1493 01:13:06,820 --> 01:13:10,110 What does that tell you? 1494 01:13:10,110 --> 01:13:10,860 It's not physical. 1495 01:13:10,860 --> 01:13:11,170 Good. 1496 01:13:11,170 --> 01:13:13,045 What else does it tell you about this system? 1497 01:13:16,110 --> 01:13:18,220 Sorry? 1498 01:13:18,220 --> 01:13:18,870 Excellent. 1499 01:13:18,870 --> 01:13:21,670 Is this an allowable energy? 1500 01:13:21,670 --> 01:13:22,840 No. 1501 01:13:22,840 --> 01:13:25,130 If the wave function has this energy, 1502 01:13:25,130 --> 01:13:28,340 it is impossible to make it continuous, 1503 01:13:28,340 --> 01:13:30,010 assuming that I drew it correctly, 1504 01:13:30,010 --> 01:13:31,530 and have it converge. 1505 01:13:31,530 --> 01:13:33,640 Is this wave function allowable? 1506 01:13:33,640 --> 01:13:36,180 No, because it does not satisfy our boundary conditions. 1507 01:13:36,180 --> 01:13:38,763 Our boundary conditions are that the wave function must vanish 1508 01:13:38,763 --> 01:13:41,710 out here and it must vanish out here at infinity 1509 01:13:41,710 --> 01:13:43,785 in order to be normalizable. 1510 01:13:43,785 --> 01:13:44,410 Here we failed. 1511 01:13:47,590 --> 01:13:49,610 Now, you can imagine that-- so let's decrease 1512 01:13:49,610 --> 01:13:50,610 the energy a little bit. 1513 01:13:50,610 --> 01:13:53,760 If we decrease the energy, our trial energy just a little 1514 01:13:53,760 --> 01:13:55,200 tiny bit, what happens? 1515 01:13:55,200 --> 01:13:58,570 Well, that's going to decrease the curvature in here. 1516 01:13:58,570 --> 01:14:01,260 We decrease, we bring the energy in just a little tiny bit. 1517 01:14:01,260 --> 01:14:03,550 That means this is a little bit smaller. 1518 01:14:03,550 --> 01:14:05,130 The potential stays the same. 1519 01:14:05,130 --> 01:14:06,760 So the curvature in the allowed region 1520 01:14:06,760 --> 01:14:09,010 is just a little tiny bit smaller. 1521 01:14:09,010 --> 01:14:10,600 And meanwhile, the allowed region 1522 01:14:10,600 --> 01:14:12,649 has got just a little bit thinner. 1523 01:14:12,649 --> 01:14:14,940 And what that will do is the curvature's a little less, 1524 01:14:14,940 --> 01:14:17,500 the region's a little less, so now we have-- 1525 01:14:28,130 --> 01:14:30,885 Sorry, I get excited. 1526 01:14:30,885 --> 01:14:33,010 And if we tweak the energy, what's going to happen? 1527 01:14:33,010 --> 01:14:35,176 Well, it's going to arrive here a little bit sooner. 1528 01:14:38,220 --> 01:14:40,490 And let's imagine something like this. 1529 01:14:40,490 --> 01:14:45,250 And if we chose the energy just right, 1530 01:14:45,250 --> 01:14:48,120 we would get it to match to a linear combination 1531 01:14:48,120 --> 01:14:50,560 of collapsing and growing exponentials, 1532 01:14:50,560 --> 01:14:53,085 where the contribution from the growing exponential 1533 01:14:53,085 --> 01:14:54,210 in this direction vanishes. 1534 01:14:57,180 --> 01:14:59,320 There's precisely one value of the energy 1535 01:14:59,320 --> 01:15:01,992 that lets me do that with this number of wiggles. 1536 01:15:01,992 --> 01:15:03,950 And so then it goes through and does its thing. 1537 01:15:07,860 --> 01:15:11,570 And we need it to happen on both sides. 1538 01:15:11,570 --> 01:15:14,510 Now if I take that solution, so that it achieves convergence 1539 01:15:14,510 --> 01:15:18,740 out here, and it achieves convergence out here, 1540 01:15:18,740 --> 01:15:21,630 and I take that energy and I increase it by epsilon, 1541 01:15:21,630 --> 01:15:24,450 by just the tiniest little bit, what will happen to this wave 1542 01:15:24,450 --> 01:15:26,057 function? 1543 01:15:26,057 --> 01:15:26,640 It'll diverge. 1544 01:15:26,640 --> 01:15:28,670 It will no longer be normalizable. 1545 01:15:28,670 --> 01:15:31,070 When you have classically forbidden regions, 1546 01:15:31,070 --> 01:15:36,010 are the allowed energies continuous or discrete? 1547 01:15:36,010 --> 01:15:38,820 And that answers a question from earlier in the class. 1548 01:15:38,820 --> 01:15:40,570 And it also is going to be the beginning 1549 01:15:40,570 --> 01:15:43,153 of the answer to the question, why is the spectrum of hydrogen 1550 01:15:43,153 --> 01:15:44,000 discrete. 1551 01:15:44,000 --> 01:15:45,960 See you next time.