1 00:00:00,060 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,800 under a Creative Commons license. 3 00:00:03,800 --> 00:00:06,530 Your support will help MIT OpenCourseWare continue 4 00:00:06,530 --> 00:00:10,130 to offer high quality educational resources for free. 5 00:00:10,130 --> 00:00:12,700 To make a donation or to view additional materials 6 00:00:12,700 --> 00:00:16,605 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,605 --> 00:00:17,575 at ocw.mit.edu. 8 00:00:23,760 --> 00:00:25,750 PROFESSOR: So today is going to be 9 00:00:25,750 --> 00:00:29,600 our last pass at bound states. 10 00:00:29,600 --> 00:00:32,750 So starting next week or actually starting next lecture, 11 00:00:32,750 --> 00:00:34,389 we're going to look at scattering. 12 00:00:34,389 --> 00:00:35,680 Scattering's going to be great. 13 00:00:35,680 --> 00:00:38,970 But we need to close out bound states. 14 00:00:38,970 --> 00:00:45,600 So today's topic is the finite well, 15 00:00:45,600 --> 00:00:46,695 finite the potential well. 16 00:00:52,892 --> 00:00:54,350 We've sort of sketched this when we 17 00:00:54,350 --> 00:00:56,560 looked at qualitative structure of wave functions of energy 18 00:00:56,560 --> 00:00:57,060 eigenstates. 19 00:01:00,280 --> 00:01:02,090 But we're going to solve the system today. 20 00:01:06,830 --> 00:01:10,220 So good. 21 00:01:10,220 --> 00:01:14,185 So the system we're interested in is going to be-- 22 00:01:14,185 --> 00:01:15,560 The system we're interested in is 23 00:01:15,560 --> 00:01:18,009 going to be a system with a finite depth 24 00:01:18,009 --> 00:01:18,800 and a finite width. 25 00:01:18,800 --> 00:01:21,531 And I'll go into detail and give you parameters in a bit. 26 00:01:21,531 --> 00:01:23,030 But first I want to just think about 27 00:01:23,030 --> 00:01:26,020 how do we find energy eigenfunctions of a potential 28 00:01:26,020 --> 00:01:31,470 of this form, v of x, which is piecewise constant. 29 00:01:31,470 --> 00:01:34,370 So first off, is this is a terribly realistic potential? 30 00:01:34,370 --> 00:01:36,360 Will you ever in the real world find a system 31 00:01:36,360 --> 00:01:39,560 that has a potential which is piecewise constant? 32 00:01:39,560 --> 00:01:40,210 Probably not. 33 00:01:40,210 --> 00:01:41,310 It's discontinuous. 34 00:01:41,310 --> 00:01:41,840 Right? 35 00:01:41,840 --> 00:01:43,210 So it's rather unphysical. 36 00:01:43,210 --> 00:01:45,130 But it's a very useful toy model. 37 00:01:45,130 --> 00:01:47,730 So for example, if you take a couple of capacitor plates, 38 00:01:47,730 --> 00:01:52,470 then you can induce a situation where the electric field 39 00:01:52,470 --> 00:01:56,050 is nonzero in between the capacitor plates 40 00:01:56,050 --> 00:01:59,621 and zero outside of the capacitor plates. 41 00:01:59,621 --> 00:02:00,120 Right. 42 00:02:00,120 --> 00:02:04,209 So at a superficial level, this looks discontinuous. 43 00:02:04,209 --> 00:02:05,750 It looks like the electric field is-- 44 00:02:05,750 --> 00:02:07,276 But actually, you know that microscopically there 45 00:02:07,276 --> 00:02:08,859 are a bunch of charges, and everything 46 00:02:08,859 --> 00:02:11,250 is nice and continuous except for the behavior 47 00:02:11,250 --> 00:02:12,820 right at the charges. 48 00:02:12,820 --> 00:02:16,150 So but it's reasonable to model this 49 00:02:16,150 --> 00:02:18,680 as a step function for an electric field. 50 00:02:18,680 --> 00:02:20,490 So this is going to be an idealization, 51 00:02:20,490 --> 00:02:22,660 but it's going to be a very useful idealization, 52 00:02:22,660 --> 00:02:25,855 the constant potential. 53 00:02:25,855 --> 00:02:26,355 OK. 54 00:02:30,822 --> 00:02:31,780 So what's the equation? 55 00:02:31,780 --> 00:02:33,110 What are we trying to do? 56 00:02:33,110 --> 00:02:34,630 We want to find the energy eigenstates for this 57 00:02:34,630 --> 00:02:36,190 because we want to study the time evolution. 58 00:02:36,190 --> 00:02:38,220 And the easiest way to solve the Schrodinger equation, 59 00:02:38,220 --> 00:02:40,345 the time evolution equation, is to expand an energy 60 00:02:40,345 --> 00:02:41,210 eigenstates. 61 00:02:41,210 --> 00:02:46,220 So the equation we want to solve is energy eigenvalue times phi 62 00:02:46,220 --> 00:02:51,790 e of x is equal to minus h bar squared on 2m phi 63 00:02:51,790 --> 00:02:55,610 e phi prime plus v of x. 64 00:03:00,820 --> 00:03:03,900 And I'm going to put it in the form we've been using. 65 00:03:03,900 --> 00:03:08,910 Phi prime prime e of x is equal to 2m upon h bar 66 00:03:08,910 --> 00:03:13,980 squared v of x minus e. 67 00:03:13,980 --> 00:03:14,480 OK. 68 00:03:14,480 --> 00:03:15,862 So this is the form that I'm going 69 00:03:15,862 --> 00:03:18,278 to use today to solve for the energy eigenvalue equations. 70 00:03:18,278 --> 00:03:19,890 e is some constant. 71 00:03:19,890 --> 00:03:25,130 Do you expect the allowed energies to be arbitrary? 72 00:03:25,130 --> 00:03:25,630 No. 73 00:03:25,630 --> 00:03:28,567 They should be discrete. 74 00:03:28,567 --> 00:03:29,150 Yeah, exactly. 75 00:03:29,150 --> 00:03:32,040 So we expect that there should be discrete lowest energy 76 00:03:32,040 --> 00:03:35,730 state, some number of bound states. 77 00:03:35,730 --> 00:03:37,850 And then, eventually, if the energy 78 00:03:37,850 --> 00:03:41,220 is greater than the potential everywhere, 79 00:03:41,220 --> 00:03:45,030 the energies will be continuous. 80 00:03:45,030 --> 00:03:47,035 Any energy will be allowed above the potential. 81 00:03:47,035 --> 00:03:52,240 So we'll have a continuum of states above the potential. 82 00:03:52,240 --> 00:03:54,650 And we'll have a discrete set of bound states-- 83 00:03:54,650 --> 00:03:57,000 Probably, it's reasonable to expect some finite number 84 00:03:57,000 --> 00:03:58,416 of bound states just by intuition. 85 00:04:01,240 --> 00:04:03,860 --from the infinite well. 86 00:04:03,860 --> 00:04:06,460 So we expect to have a finite number of discrete energies 87 00:04:06,460 --> 00:04:09,890 and then a continuous set of energies above zero. 88 00:04:09,890 --> 00:04:14,250 So if this is the asymptotic value potential of zero. 89 00:04:14,250 --> 00:04:14,750 OK. 90 00:04:14,750 --> 00:04:16,566 And this is intuition gained from our study 91 00:04:16,566 --> 00:04:18,649 of qualitative structure of energy eigenfunctions. 92 00:04:18,649 --> 00:04:21,390 So we are going to talk today about the bound states. 93 00:04:21,390 --> 00:04:24,980 And in recitation, leaders should 94 00:04:24,980 --> 00:04:29,720 discuss the continuum above zero energy. 95 00:04:29,720 --> 00:04:32,440 OK. 96 00:04:32,440 --> 00:04:35,340 So to solve for the actual energy eigenfunctions 97 00:04:35,340 --> 00:04:37,750 and the energy eigenvalues, what we need to do 98 00:04:37,750 --> 00:04:44,830 is we need to solve this equation 99 00:04:44,830 --> 00:04:46,900 subject to some boundary conditions. 100 00:04:46,900 --> 00:04:49,233 And the boundary conditions we're going to want to solve 101 00:04:49,233 --> 00:04:51,240 are going to be finite. 102 00:04:51,240 --> 00:04:52,490 So it's normalizable infinity. 103 00:04:52,490 --> 00:04:55,640 The solution should be vanishing far away. 104 00:04:55,640 --> 00:04:58,390 And the wave function should be everywhere smooth. 105 00:04:58,390 --> 00:05:00,059 Well, at least it should be continuous. 106 00:05:00,059 --> 00:05:02,350 So let's talk about what exactly boundary conditions we 107 00:05:02,350 --> 00:05:02,974 want to impose. 108 00:05:05,770 --> 00:05:08,290 And so in particular, we're going 109 00:05:08,290 --> 00:05:12,870 to want to solve for the energy eigenfunctions in the regions 110 00:05:12,870 --> 00:05:16,150 where the potential is constant and then patch together 111 00:05:16,150 --> 00:05:18,460 solutions at these boundaries. 112 00:05:18,460 --> 00:05:20,749 We know how to solve for the energy eigenfunctions 113 00:05:20,749 --> 00:05:22,040 when the potential is constant. 114 00:05:22,040 --> 00:05:23,498 What are the energy eigenfunctions? 115 00:05:26,501 --> 00:05:27,000 Yeah. 116 00:05:27,000 --> 00:05:29,420 Suppose I have a potential, which is constant. 117 00:05:29,420 --> 00:05:30,564 v is equal to 0. 118 00:05:30,564 --> 00:05:31,980 What are the energy eigenfunctions 119 00:05:31,980 --> 00:05:33,065 of this potential? 120 00:05:33,065 --> 00:05:34,220 AUDIENCE: [INAUDIBLE] 121 00:05:34,220 --> 00:05:36,031 PROFESSOR: Yeah, e to the ikx. 122 00:05:36,031 --> 00:05:36,530 Yeah. 123 00:05:36,530 --> 00:05:37,988 Now, what if I happen to tell you-- 124 00:05:37,988 --> 00:05:41,315 So we're at h bar squared k squared upon 2m 125 00:05:41,315 --> 00:05:42,829 is equal to the energy. 126 00:05:42,829 --> 00:05:45,120 Suppose I happen to tell you that here's the potential. 127 00:05:45,120 --> 00:05:47,730 And I want to find the solution in this region 128 00:05:47,730 --> 00:05:50,397 where the energy is here less than the potential. 129 00:05:50,397 --> 00:05:51,355 What are the solutions? 130 00:05:54,017 --> 00:05:55,850 That's a second order differential equation. 131 00:05:55,850 --> 00:05:56,670 There should be two solutions. 132 00:05:56,670 --> 00:05:58,500 What are the solutions to that differential equation when 133 00:05:58,500 --> 00:06:00,083 the energy is less than the potential? 134 00:06:03,689 --> 00:06:04,980 AUDIENCE: Decaying and growing. 135 00:06:04,980 --> 00:06:06,855 PROFESSOR: Decaying and growing exponentials. 136 00:06:06,855 --> 00:06:11,930 Exactly. e to the plus alpha x and e to the minus alpha x. 137 00:06:11,930 --> 00:06:15,720 And the reason is these are sinusoidal, 138 00:06:15,720 --> 00:06:18,320 and these have the opposite concavity. 139 00:06:18,320 --> 00:06:21,551 They are growing and dying exponentials. 140 00:06:21,551 --> 00:06:22,185 Cool? 141 00:06:22,185 --> 00:06:22,685 OK. 142 00:06:22,685 --> 00:06:24,163 So we've studied that. 143 00:06:24,163 --> 00:06:25,030 Yeah? 144 00:06:25,030 --> 00:06:27,412 AUDIENCE: Shouldn't that be a phi e of x? 145 00:06:27,412 --> 00:06:28,120 PROFESSOR: Sorry. 146 00:06:28,120 --> 00:06:29,510 Oh, oh, yes, indeed. 147 00:06:29,510 --> 00:06:30,150 Sorry. 148 00:06:30,150 --> 00:06:31,470 Thank you. 149 00:06:31,470 --> 00:06:34,600 Phi e of x. 150 00:06:34,600 --> 00:06:36,600 It's early, and I'm still working on the coffee. 151 00:06:40,310 --> 00:06:41,390 It won't take long. 152 00:06:46,120 --> 00:06:46,620 Good. 153 00:06:46,620 --> 00:06:49,520 So we know how to solve the energy eigenvalue equation 154 00:06:49,520 --> 00:06:51,990 in all these regions where the potential is constant. 155 00:06:51,990 --> 00:06:54,690 So our job is going to be to find a solution where we patch 156 00:06:54,690 --> 00:06:56,482 them together at these interfaces. 157 00:06:56,482 --> 00:06:57,440 We patch them together. 158 00:06:57,440 --> 00:07:00,520 And what condition should we impose? 159 00:07:00,520 --> 00:07:02,250 So the basic condition is going to be 160 00:07:02,250 --> 00:07:13,580 continuity of the wave functions phi e of x. 161 00:07:13,580 --> 00:07:15,980 And so what are the conditions that we need? 162 00:07:15,980 --> 00:07:17,380 Well, if v of x-- 163 00:07:17,380 --> 00:07:19,210 Here's the way I like to think about this. 164 00:07:19,210 --> 00:07:20,550 Suppose v of x is continuous. 165 00:07:26,040 --> 00:07:28,512 So if the potential is continuous, then 166 00:07:28,512 --> 00:07:29,220 what can you say? 167 00:07:29,220 --> 00:07:33,450 You can say that phi prime prime is a continuous function times 168 00:07:33,450 --> 00:07:35,635 phi of x. 169 00:07:35,635 --> 00:07:37,260 So very roughly, if we look at a region 170 00:07:37,260 --> 00:07:39,860 where phi isn't varying very much, so if we 171 00:07:39,860 --> 00:07:42,540 have a potential that's varying in some way, then 172 00:07:42,540 --> 00:07:45,890 phi prime prime, in a region where it's not 173 00:07:45,890 --> 00:07:50,310 varying much compared to its value as a function of x, 174 00:07:50,310 --> 00:07:51,960 does something smooth because it's 175 00:07:51,960 --> 00:07:55,050 varying with the potential. 176 00:07:55,050 --> 00:07:57,510 And so phi prime, which is just going 177 00:07:57,510 --> 00:07:58,770 to be the integral of this-- 178 00:07:58,770 --> 00:08:00,145 The integral of a smooth function 179 00:08:00,145 --> 00:08:02,810 is again a smooth function. 180 00:08:02,810 --> 00:08:04,980 --and phi, the integral of that function 181 00:08:04,980 --> 00:08:08,150 is also going to be a nice, smooth function. 182 00:08:08,150 --> 00:08:08,710 OK? 183 00:08:08,710 --> 00:08:11,320 I've drawn it badly, but-- 184 00:08:11,320 --> 00:08:14,430 So the key thing here is that if the potential is continuous, 185 00:08:14,430 --> 00:08:17,360 then the energy eigenfunction has a second derivative, which 186 00:08:17,360 --> 00:08:19,914 is also continuous. 187 00:08:19,914 --> 00:08:21,830 That means its first derivative is continuous. 188 00:08:21,830 --> 00:08:23,830 So that means the function itself is continuous. 189 00:08:27,720 --> 00:08:29,240 Everyone agree with this? 190 00:08:29,240 --> 00:08:31,262 Questions? 191 00:08:31,262 --> 00:08:33,220 So in regions where the potential's continuous, 192 00:08:33,220 --> 00:08:35,344 the wave function and its first two derivatives all 193 00:08:35,344 --> 00:08:36,816 have to be continuous. 194 00:08:36,816 --> 00:08:39,980 On the other hand, suppose the potential has a step. 195 00:08:39,980 --> 00:08:42,209 v of x has a step discontinuity. 196 00:08:45,480 --> 00:08:46,030 OK. 197 00:08:46,030 --> 00:08:48,190 So the potential does one of those. 198 00:08:50,399 --> 00:08:52,440 So what does that tell you about phi prime prime? 199 00:08:56,416 --> 00:08:57,290 It's a function of x. 200 00:09:03,444 --> 00:09:05,360 So for example, let's look at that first step. 201 00:09:05,360 --> 00:09:08,420 Suppose the potential is that first step down by some amount. 202 00:09:08,420 --> 00:09:12,390 Then phi prime prime is going to decrease precipitously 203 00:09:12,390 --> 00:09:13,745 at some point. 204 00:09:13,745 --> 00:09:15,370 And the actual amount that it decreases 205 00:09:15,370 --> 00:09:17,300 depends on the value of phi because the change 206 00:09:17,300 --> 00:09:21,770 in the potential time is the actual value of phi. 207 00:09:21,770 --> 00:09:23,730 And if that's true of phi prime prime what can 208 00:09:23,730 --> 00:09:24,629 you say of phi prime? 209 00:09:24,629 --> 00:09:25,670 So this is discontinuous. 210 00:09:30,900 --> 00:09:32,960 Phi prime of x, however, is the integral 211 00:09:32,960 --> 00:09:35,627 of this discontinuous function. 212 00:09:35,627 --> 00:09:36,460 And what does it do? 213 00:09:36,460 --> 00:09:38,335 Well, it's linearly increasing in this region 214 00:09:38,335 --> 00:09:40,340 because its derivative is constant. 215 00:09:40,340 --> 00:09:44,410 And here, it's linearly increasing less. 216 00:09:44,410 --> 00:09:49,590 So it's not differentiably smooth, but it's continuous. 217 00:09:49,590 --> 00:09:52,060 And then let's look at the actual function phi of x. 218 00:09:56,770 --> 00:09:57,270 OK. 219 00:09:57,270 --> 00:09:57,978 What is it doing? 220 00:09:57,978 --> 00:09:59,430 Well, it's quadratic. 221 00:09:59,430 --> 00:10:02,907 And then here it's quadratic a little less afterwards. 222 00:10:02,907 --> 00:10:04,490 But that still continuous because it's 223 00:10:04,490 --> 00:10:07,642 the integral of a continuous function. 224 00:10:07,642 --> 00:10:09,150 Everyone we cool with that? 225 00:10:09,150 --> 00:10:12,150 So even when we have a step discontinuity in our potential, 226 00:10:12,150 --> 00:10:14,470 we still have that our derivative 227 00:10:14,470 --> 00:10:16,685 and the value of the function are continuous. 228 00:10:19,240 --> 00:10:20,210 Yeah? 229 00:10:20,210 --> 00:10:26,340 However, imagine the potential has this delta function. 230 00:10:26,340 --> 00:10:28,200 Let's just really push it. 231 00:10:28,200 --> 00:10:30,330 What happens if our potential has a delta function 232 00:10:30,330 --> 00:10:31,616 singularity? 233 00:10:31,616 --> 00:10:32,740 Really badly discontinuous. 234 00:10:32,740 --> 00:10:39,424 Then what can you say about phi prime prime as a function of x? 235 00:10:39,424 --> 00:10:40,840 So it has a delta function, right? 236 00:10:40,840 --> 00:10:43,650 So the phi prime prime has to look like something relatively 237 00:10:43,650 --> 00:10:48,580 slowly varying, and then a step delta function. 238 00:10:48,580 --> 00:10:51,070 So what does that tell you about the derivative of the wave 239 00:10:51,070 --> 00:10:52,799 function? 240 00:10:52,799 --> 00:10:53,840 It's got a step function. 241 00:10:53,840 --> 00:10:54,190 Exactly. 242 00:10:54,190 --> 00:10:56,689 Because it's the integral of this, and the integral is 0, 1. 243 00:10:59,810 --> 00:11:00,310 Whoops. 244 00:11:00,310 --> 00:11:00,810 I missed. 245 00:11:06,510 --> 00:11:07,660 So this is a delta. 246 00:11:07,660 --> 00:11:09,796 This is a step. 247 00:11:09,796 --> 00:11:11,170 And then the wave function itself 248 00:11:11,170 --> 00:11:14,615 is, well, it's the integral of a step, so it's continuous. 249 00:11:18,840 --> 00:11:20,960 Sorry. 250 00:11:20,960 --> 00:11:22,430 It's certainly not differentiable. 251 00:11:22,430 --> 00:11:23,763 Its derivative is discontinuous. 252 00:11:26,964 --> 00:11:28,880 Well, it's differentiable but its derivative's 253 00:11:28,880 --> 00:11:29,671 not supposed to be. 254 00:11:29,671 --> 00:11:31,110 It is not continuous, indeed. 255 00:11:31,110 --> 00:11:33,750 So this is continuous. 256 00:11:33,750 --> 00:11:36,670 So we've learned something nice, that unless our potential 257 00:11:36,670 --> 00:11:39,100 is so stupid as to have delta functions, which 258 00:11:39,100 --> 00:11:41,690 sounds fairly unphysical-- 259 00:11:41,690 --> 00:11:44,650 We'll come back to that later in today's lecture. 260 00:11:44,650 --> 00:11:46,900 --unless our potential has delta functions, 261 00:11:46,900 --> 00:11:50,473 the wave function and its first derivative must be smooth. 262 00:11:50,473 --> 00:11:51,160 Yeah. 263 00:11:51,160 --> 00:11:53,590 This is just from the energy eigenvalue equation. 264 00:11:53,590 --> 00:11:56,589 Now, we actually argued this from the well definedness 265 00:11:56,589 --> 00:11:58,130 and the finiteness of the expectation 266 00:11:58,130 --> 00:12:00,800 value of the momentum earlier in the semester. 267 00:12:00,800 --> 00:12:02,992 But I wanted to give you this argument for it 268 00:12:02,992 --> 00:12:04,700 because it's going to play a useful role. 269 00:12:04,700 --> 00:12:08,040 And it also tells us that if we do have a delta function 270 00:12:08,040 --> 00:12:10,510 singularity in the potential, then the upshot 271 00:12:10,510 --> 00:12:13,234 is that the wave function is going to be continuous, 272 00:12:13,234 --> 00:12:14,650 but its first derivative will not. 273 00:12:14,650 --> 00:12:17,935 Its first derivative will jump at the wave function. 274 00:12:17,935 --> 00:12:19,560 And that means that the wave function-- 275 00:12:19,560 --> 00:12:21,250 Let me draw this slightly differently. 276 00:12:21,250 --> 00:12:23,790 --the wave function will have a kink. 277 00:12:23,790 --> 00:12:26,080 Its derivative will not be continuous. 278 00:12:26,080 --> 00:12:26,580 OK. 279 00:12:26,580 --> 00:12:28,330 So anywhere where we have a delta function 280 00:12:28,330 --> 00:12:31,760 in the potential, we will have a kink in the wave function 281 00:12:31,760 --> 00:12:35,630 where the first derivative is discontinuous. 282 00:12:35,630 --> 00:12:36,470 Cool? 283 00:12:36,470 --> 00:12:37,410 Yeah. 284 00:12:37,410 --> 00:12:39,022 AUDIENCE: What does that mean as far 285 00:12:39,022 --> 00:12:41,017 as the expectation value of the momentum? 286 00:12:41,017 --> 00:12:42,850 PROFESSOR: Ah, that's an excellent question. 287 00:12:42,850 --> 00:12:46,582 So what do you expect to happen at such a point? 288 00:12:46,582 --> 00:12:48,470 AUDIENCE: Well, your momentum blows up. 289 00:12:48,470 --> 00:12:49,511 PROFESSOR: Yeah, exactly. 290 00:12:49,511 --> 00:12:51,520 So we're going to have some pathologies 291 00:12:51,520 --> 00:12:53,569 with expectation values in the momentum. 292 00:12:53,569 --> 00:12:56,110 Let's come back to that when we talk about the delta function 293 00:12:56,110 --> 00:12:57,360 potential, which should be at the end of today. 294 00:12:57,360 --> 00:12:58,480 Hold that question in the back of your head. 295 00:12:58,480 --> 00:12:59,580 It's a good question. 296 00:12:59,580 --> 00:13:02,430 Other questions? 297 00:13:02,430 --> 00:13:03,020 OK. 298 00:13:03,020 --> 00:13:07,426 So what we're going to do now is we're going to use, you know, 299 00:13:07,426 --> 00:13:10,960 math to find the energy eigenfunctions 300 00:13:10,960 --> 00:13:13,766 and eigenvalues for the finite well potential, i.e. 301 00:13:13,766 --> 00:13:15,640 we're going to solve this equation contingent 302 00:13:15,640 --> 00:13:17,330 on the boundary conditions that the wave 303 00:13:17,330 --> 00:13:19,480 function and its derivative are smooth everywhere. 304 00:13:19,480 --> 00:13:22,389 And in particular, they must be smooth here and here. 305 00:13:22,389 --> 00:13:24,180 I mean they've got to be smooth everywhere. 306 00:13:24,180 --> 00:13:25,790 We know what the solution is inside here. 307 00:13:25,790 --> 00:13:27,331 We know what the solution is in here. 308 00:13:27,331 --> 00:13:30,180 All we have to worry about is what happens at the interface. 309 00:13:30,180 --> 00:13:32,830 And we're going to use smoothness of the wave function 310 00:13:32,830 --> 00:13:35,510 and its derivative to impose conditions that 311 00:13:35,510 --> 00:13:37,950 allow us to match across that step. 312 00:13:37,950 --> 00:13:40,710 Everyone cool with that? 313 00:13:40,710 --> 00:13:42,540 OK. 314 00:13:42,540 --> 00:13:45,320 So let's do it. 315 00:13:45,320 --> 00:13:46,812 By the way, just a quick side note. 316 00:13:46,812 --> 00:13:48,020 Let me give you a definition. 317 00:13:48,020 --> 00:13:49,090 I've used this phrase many times, 318 00:13:49,090 --> 00:13:50,881 but I haven't given you a definition of it. 319 00:13:50,881 --> 00:13:53,260 So I've used the phrase, bound state. 320 00:13:53,260 --> 00:13:56,560 And its opposite is called a scattering state. 321 00:13:56,560 --> 00:13:58,204 So here's what I mean by a bound state. 322 00:13:58,204 --> 00:13:59,620 Intuitively, a bound state, if you 323 00:13:59,620 --> 00:14:01,161 think about this classically, imagine 324 00:14:01,161 --> 00:14:02,220 I have a potential well. 325 00:14:02,220 --> 00:14:04,910 And I have a marble, and I let go from here. 326 00:14:04,910 --> 00:14:06,220 This marble is bound. 327 00:14:06,220 --> 00:14:06,720 Right? 328 00:14:06,720 --> 00:14:08,178 It's never getting out of the well. 329 00:14:08,178 --> 00:14:09,073 It's stuck. 330 00:14:09,073 --> 00:14:10,040 Yeah? 331 00:14:10,040 --> 00:14:13,130 And so I would call that a bound marble. 332 00:14:13,130 --> 00:14:17,180 On the other hand, a marble that I give a huge kick to, 333 00:14:17,180 --> 00:14:19,240 big velocity so that it can get out, 334 00:14:19,240 --> 00:14:21,400 well then it's not bound to this potential well. 335 00:14:21,400 --> 00:14:23,910 So I'll call that a scattering state just to give it a name. 336 00:14:23,910 --> 00:14:26,760 And next lecture we'll see why we call it a scattering state. 337 00:14:26,760 --> 00:14:30,032 The important thing is that bound configurations 338 00:14:30,032 --> 00:14:31,490 of a classical potential are things 339 00:14:31,490 --> 00:14:33,870 basically in a well that are stuck. 340 00:14:33,870 --> 00:14:34,370 OK. 341 00:14:34,370 --> 00:14:36,786 So the quantum version of that statement is the following. 342 00:14:36,786 --> 00:14:39,142 Suppose I take this potential, and I 343 00:14:39,142 --> 00:14:41,350 treat it quantum mechanically, and I consider a state 344 00:14:41,350 --> 00:14:43,780 with total energy like this. 345 00:14:43,780 --> 00:14:45,760 Well, among other things, the total energy 346 00:14:45,760 --> 00:14:49,290 is less than the value of the potential asymptotically 347 00:14:49,290 --> 00:14:49,847 far away. 348 00:14:49,847 --> 00:14:51,430 So what is the form of a wave function 349 00:14:51,430 --> 00:14:54,529 in this region with this energy? 350 00:14:54,529 --> 00:14:55,320 Exponential, right? 351 00:14:55,320 --> 00:14:57,770 It's e to the minus some alpha x, 352 00:14:57,770 --> 00:15:01,440 where alpha squared is roughly the difference, alpha x. 353 00:15:01,440 --> 00:15:05,009 And out here it's going to be e to the plus alpha x. 354 00:15:05,009 --> 00:15:06,300 And that's for normalizability. 355 00:15:06,300 --> 00:15:09,540 We want to have a single particle in this state. 356 00:15:09,540 --> 00:15:14,220 So what that tells us is that the wave function falls off 357 00:15:14,220 --> 00:15:19,550 in these classically disallowed regions exponentially. 358 00:15:19,550 --> 00:15:22,510 And so the probability of measuring the particle 359 00:15:22,510 --> 00:15:27,360 at an arbitrarily far position goes to zero. 360 00:15:27,360 --> 00:15:29,640 And it goes to zero exponentially rapidly. 361 00:15:29,640 --> 00:15:30,140 Cool? 362 00:15:30,140 --> 00:15:34,030 So I will call a bound state, a quantum bound state, an energy 363 00:15:34,030 --> 00:15:38,300 eigenstate, such that the probability falls off 364 00:15:38,300 --> 00:15:42,252 exponentially as we go far away from wherever we think 365 00:15:42,252 --> 00:15:44,460 is an interesting point, like the bottom of the well. 366 00:15:44,460 --> 00:15:45,340 Cool? 367 00:15:45,340 --> 00:15:46,960 A bound state is just a state which 368 00:15:46,960 --> 00:15:48,720 is exponentially localized. 369 00:15:48,720 --> 00:15:51,760 If you put it there, it will stay there. 370 00:15:51,760 --> 00:15:52,680 Yeah. 371 00:15:52,680 --> 00:15:54,680 And it's important that when I say a bound state 372 00:15:54,680 --> 00:15:57,810 I'm talking about energy eigenstates. 373 00:15:57,810 --> 00:15:59,455 And the reason is this. 374 00:15:59,455 --> 00:16:01,320 Bound state equals energy eigenstate. 375 00:16:04,890 --> 00:16:07,630 The reason is that, consider by contrast 376 00:16:07,630 --> 00:16:10,610 a free particle, so a free particle 377 00:16:10,610 --> 00:16:12,230 with constant potential. 378 00:16:12,230 --> 00:16:13,790 What are the wave functions? 379 00:16:13,790 --> 00:16:15,340 What are the energy eigenfunctions? 380 00:16:15,340 --> 00:16:16,670 Well, they're plane waves. 381 00:16:16,670 --> 00:16:17,170 Right. 382 00:16:17,170 --> 00:16:19,860 So are these bound states? 383 00:16:19,860 --> 00:16:20,360 No. 384 00:16:20,360 --> 00:16:20,700 Good. 385 00:16:20,700 --> 00:16:21,000 OK. 386 00:16:21,000 --> 00:16:22,458 On the other hand, I claimed that I 387 00:16:22,458 --> 00:16:27,180 can build a wave packet, a perfectly reasonable wave 388 00:16:27,180 --> 00:16:30,600 packet, which is a Gaussian. 389 00:16:30,600 --> 00:16:31,780 OK. 390 00:16:31,780 --> 00:16:35,940 This is some wave function, psi of x times 0. 391 00:16:35,940 --> 00:16:36,630 It's a Gaussian. 392 00:16:36,630 --> 00:16:37,630 It's nice and narrow. 393 00:16:37,630 --> 00:16:41,350 Is that a bound state? 394 00:16:41,350 --> 00:16:45,550 Well, it's localized at this moment in time. 395 00:16:45,550 --> 00:16:47,880 But will it remain, and in particular its probability 396 00:16:47,880 --> 00:16:49,470 distribution, which is this thing norm 397 00:16:49,470 --> 00:16:51,980 squared is localized in space-- 398 00:16:51,980 --> 00:16:52,480 Sorry. 399 00:16:52,480 --> 00:16:55,237 This was zero. 400 00:16:55,237 --> 00:16:56,320 --it's localized in space. 401 00:16:56,320 --> 00:16:57,270 The probability of it being out here 402 00:16:57,270 --> 00:16:58,527 is not just exponentially small, it's 403 00:16:58,527 --> 00:17:00,277 Gaussian so it's e to the minus x squared. 404 00:17:00,277 --> 00:17:02,560 It's really not out here. 405 00:17:02,560 --> 00:17:04,329 But what happens if I let go? 406 00:17:04,329 --> 00:17:08,019 What happens if I look at the system at time t? 407 00:17:08,019 --> 00:17:09,060 It's going to spread out. 408 00:17:09,060 --> 00:17:09,650 It's going to disperse. 409 00:17:09,650 --> 00:17:11,816 We're going to talk about that in more detail later. 410 00:17:11,816 --> 00:17:13,339 So it's going to spread out. 411 00:17:13,339 --> 00:17:18,030 And eventually, it will get out arbitrarily far away 412 00:17:18,030 --> 00:17:19,670 with whatever probably you like. 413 00:17:19,670 --> 00:17:23,538 So the probability distribution is not time invariant. 414 00:17:23,538 --> 00:17:25,329 That is to say it's not a stationary state. 415 00:17:25,329 --> 00:17:26,780 It's not an energy eigenstate. 416 00:17:26,780 --> 00:17:31,140 Saying something is bound means that it never gets away. 417 00:17:31,140 --> 00:17:34,071 So bound states are specifically energy eigenstates 418 00:17:34,071 --> 00:17:35,820 that are strictly localized, that fall off 419 00:17:35,820 --> 00:17:38,870 at least exponentially as we go away from the origin. 420 00:17:38,870 --> 00:17:39,780 Cool? 421 00:17:39,780 --> 00:17:43,110 It's just terminology, what I mean by a bound state. 422 00:17:43,110 --> 00:17:44,760 Questions? 423 00:17:44,760 --> 00:17:45,280 OK. 424 00:17:45,280 --> 00:17:46,840 So let's talk about the finite well. 425 00:17:50,680 --> 00:17:56,690 So I need to give you definitions of the parameters. 426 00:17:56,690 --> 00:17:58,520 Let's draw this more precisely. 427 00:17:58,520 --> 00:17:59,850 Here's my well. 428 00:17:59,850 --> 00:18:02,920 Asymptotically, the potential is zero. 429 00:18:02,920 --> 00:18:07,010 The potential depth, I'm going to call minus v naught. 430 00:18:07,010 --> 00:18:07,630 OK. 431 00:18:07,630 --> 00:18:10,060 And I'm going to center the well around zero. 432 00:18:10,060 --> 00:18:12,330 And I'll call the sides minus l and l. 433 00:18:18,590 --> 00:18:21,020 And I want to find bound states of this potential, 434 00:18:21,020 --> 00:18:22,880 just like we found bound states of the harmonic oscillator, 435 00:18:22,880 --> 00:18:23,400 i.e. 436 00:18:23,400 --> 00:18:27,780 states with energy e, which is less than zero. 437 00:18:27,780 --> 00:18:30,640 So these are going to give us bound states because we're 438 00:18:30,640 --> 00:18:32,910 going to have exponential fall offs far away. 439 00:18:37,830 --> 00:18:39,120 So a couple of things to note. 440 00:18:39,120 --> 00:18:40,494 The first is on, I think, problem 441 00:18:40,494 --> 00:18:43,080 set three or four you showed that if you 442 00:18:43,080 --> 00:18:45,540 have a potential, which is symmetric, 443 00:18:45,540 --> 00:18:48,935 which is even, under x goes to minus x, then every energy 444 00:18:48,935 --> 00:18:51,060 eigenfunction, or at least every bound state energy 445 00:18:51,060 --> 00:18:53,890 eigenfunction, every energy eigenfunction can be written 446 00:18:53,890 --> 00:19:02,450 as phi e symmetric or phi e anti-symmetric. 447 00:19:02,450 --> 00:19:03,940 So it's either even or odd. 448 00:19:06,880 --> 00:19:10,122 It's either even or odd under the exchange of x to minus x. 449 00:19:10,122 --> 00:19:11,580 So when our potential is symmetric, 450 00:19:11,580 --> 00:19:13,590 the wave function or the energy eigenfunctions 451 00:19:13,590 --> 00:19:16,410 are either symmetric or anti-symmetric. 452 00:19:16,410 --> 00:19:17,580 OK. 453 00:19:17,580 --> 00:19:19,880 So we want to solve for the actual eigenfunctions. 454 00:19:19,880 --> 00:19:21,745 So we want to solve that equation. 455 00:19:25,930 --> 00:19:28,300 And we have this nice simple fact 456 00:19:28,300 --> 00:19:30,440 that we know the solutions in this region. 457 00:19:30,440 --> 00:19:31,820 We know the general solution in this region. 458 00:19:31,820 --> 00:19:33,653 We know the general solution in this region. 459 00:19:33,653 --> 00:19:38,189 So I'm going to call these regions one, two, and three, 460 00:19:38,189 --> 00:19:39,230 just to give them a name. 461 00:19:39,230 --> 00:19:40,290 So in region one-- 462 00:19:43,430 --> 00:19:45,290 That's actually sort of stupid. 463 00:19:45,290 --> 00:19:49,304 Let's call this inside, left, and right. 464 00:19:49,304 --> 00:19:49,804 Good. 465 00:19:54,730 --> 00:19:55,230 OK. 466 00:19:55,230 --> 00:20:02,730 So let's look at this equation. 467 00:20:06,610 --> 00:20:08,170 We have two cases. 468 00:20:08,170 --> 00:20:13,220 If the energy is greater than the potential in some region, 469 00:20:13,220 --> 00:20:17,020 then this is of the form phi prime prime 470 00:20:17,020 --> 00:20:19,050 is equal to energy greater than potential. 471 00:20:19,050 --> 00:20:20,635 This is a negative number. 472 00:20:24,050 --> 00:20:29,136 And so this is a minus k squared phi. 473 00:20:29,136 --> 00:20:31,750 And we get exponentials. 474 00:20:31,750 --> 00:20:38,450 And if, on the other hand, e is less than v of x, then 475 00:20:38,450 --> 00:20:45,530 phi prime prime is equal to plus alpha squared phi. 476 00:20:45,530 --> 00:20:46,570 I should say oscillator. 477 00:20:54,240 --> 00:21:00,120 And in particular here, I want the k squared is equal to 2m 478 00:21:00,120 --> 00:21:03,130 upon h bar squared. 479 00:21:03,130 --> 00:21:07,860 It's just the coefficient v minus e. 480 00:21:07,860 --> 00:21:17,330 And alpha squared is equal to 2m over h bar squared e minus v. 481 00:21:17,330 --> 00:21:17,830 OK. 482 00:21:17,830 --> 00:21:19,510 So let's apply that here. 483 00:21:19,510 --> 00:21:22,294 So in this region, we're going to get oscillations 484 00:21:22,294 --> 00:21:24,460 because we're in a classically allowed region, where 485 00:21:24,460 --> 00:21:26,168 the energy is greater than the potential. 486 00:21:26,168 --> 00:21:27,870 So we'll get oscillatory solutions. 487 00:21:27,870 --> 00:21:31,390 And the salient value of k inside, is k 488 00:21:31,390 --> 00:21:39,120 is equal to the square root of 2m over h bar squared times 489 00:21:39,120 --> 00:21:40,940 v minus e. 490 00:21:40,940 --> 00:21:42,255 So that's minus v naught. 491 00:21:48,920 --> 00:21:51,560 What did I do? 492 00:21:51,560 --> 00:21:53,452 I did. 493 00:21:53,452 --> 00:21:54,670 It's e minus v. I thank you. 494 00:21:54,670 --> 00:21:55,170 Yes. 495 00:21:59,260 --> 00:22:03,170 And I want the other one to be l so it'd be minus v. Good. 496 00:22:03,170 --> 00:22:03,950 Good. 497 00:22:03,950 --> 00:22:04,690 Excellent. 498 00:22:04,690 --> 00:22:06,326 So root 2 over h bar squared. 499 00:22:06,326 --> 00:22:07,950 And now we have e minus v naught, which 500 00:22:07,950 --> 00:22:10,520 is the actual value of e, which is negative. 501 00:22:10,520 --> 00:22:11,310 Right? 502 00:22:11,310 --> 00:22:15,960 But minus v naught or plus v naught 503 00:22:15,960 --> 00:22:21,640 is positive and greater in magnitude. 504 00:22:21,640 --> 00:22:23,350 So this is a nice positive number, 505 00:22:23,350 --> 00:22:25,210 and k is the square root of it. 506 00:22:25,210 --> 00:22:27,520 This is controlling how rapidly the wave 507 00:22:27,520 --> 00:22:30,720 function oscillates in this region. 508 00:22:30,720 --> 00:22:33,370 Similarly, out here we have alpha is equal to-- 509 00:22:33,370 --> 00:22:35,020 Well, here the potential is zero. 510 00:22:35,020 --> 00:22:36,310 So it's particularly easy. 511 00:22:36,310 --> 00:22:38,896 Alpha is equal to the square root of 2m 512 00:22:38,896 --> 00:22:41,530 upon h bar squared of-- 513 00:22:41,530 --> 00:22:45,850 Now z minus e is zero minus e, which is a negative number. 514 00:22:45,850 --> 00:22:49,950 So we can just write e absolute value. 515 00:22:55,770 --> 00:23:03,450 So we can write the general solution of this eigenfunction, 516 00:23:03,450 --> 00:23:11,941 of this eigenvalue equation, as phi e of x is equal to-- 517 00:23:11,941 --> 00:23:13,690 Let's break it up into inside and outside. 518 00:23:13,690 --> 00:23:19,340 Well, inside we know, since it's constant with this value of k, 519 00:23:19,340 --> 00:23:23,055 we get superpositions of oscillatory solutions. 520 00:23:23,055 --> 00:23:24,680 It' a second-order difference equation. 521 00:23:24,680 --> 00:23:26,929 There are two solutions and two integration constants. 522 00:23:26,929 --> 00:23:34,065 So first we have a cosine of kx plus b sine kx. 523 00:23:37,172 --> 00:23:38,610 This is inside. 524 00:23:44,340 --> 00:23:52,520 And then on the left we have a combination 525 00:23:52,520 --> 00:23:54,770 of exponentially growing and exponentially decreasing. 526 00:23:54,770 --> 00:23:57,630 So the exponentially growing is e to the alpha x 527 00:23:57,630 --> 00:24:02,180 plus de the minus alpha x. 528 00:24:02,180 --> 00:24:05,490 And on the right, similarly by symmetry, 529 00:24:05,490 --> 00:24:07,088 we have some combination of e-- 530 00:24:07,088 --> 00:24:09,046 But I don't want to call it the energy, so I'll 531 00:24:09,046 --> 00:24:10,080 call it the curly e. 532 00:24:15,005 --> 00:24:22,240 --to the alpha x plus fe to the minus alpha x. 533 00:24:22,240 --> 00:24:23,240 OK. 534 00:24:23,240 --> 00:24:24,630 So that's the general solution. 535 00:24:24,630 --> 00:24:27,770 We solve the problem as a superposition of the two 536 00:24:27,770 --> 00:24:29,820 oscillatory solutions or a superposition 537 00:24:29,820 --> 00:24:32,590 of the two exponentially growing and damped solutions 538 00:24:32,590 --> 00:24:35,370 or exponentially growing and collapsing 539 00:24:35,370 --> 00:24:37,090 functions on the left and right. 540 00:24:42,620 --> 00:24:44,400 Questions? 541 00:24:44,400 --> 00:24:46,530 So a couple of things to note at this point. 542 00:24:49,500 --> 00:24:53,470 So the first is we have boundary conditions to impose. 543 00:24:53,470 --> 00:24:56,730 We have boundary conditions at these two interfaces. 544 00:24:56,730 --> 00:24:59,610 But we also have boundary conditions off in infinity. 545 00:24:59,610 --> 00:25:02,470 What are the boundary conditions at infinity? 546 00:25:02,470 --> 00:25:03,292 Yeah, exactly. 547 00:25:03,292 --> 00:25:04,000 It should vanish. 548 00:25:04,000 --> 00:25:05,708 So we want the system to be normalizable. 549 00:25:11,610 --> 00:25:13,740 So normalizable is going to say that phi 550 00:25:13,740 --> 00:25:18,320 goes to zero at minus infinity. 551 00:25:18,320 --> 00:25:25,930 Phi of x goes to minus infinity goes to zero, which it equals. 552 00:25:25,930 --> 00:25:31,260 And phi of x goes to plus infinity should also be zero. 553 00:25:31,260 --> 00:25:32,110 OK. 554 00:25:32,110 --> 00:25:33,484 And then we're also going to have 555 00:25:33,484 --> 00:25:37,454 the conditions at the left boundary, 556 00:25:37,454 --> 00:25:38,870 and we're going to have conditions 557 00:25:38,870 --> 00:25:39,786 at the right boundary. 558 00:25:45,963 --> 00:25:50,382 [LAUGHTER] 559 00:25:50,382 --> 00:25:52,619 All right. 560 00:25:52,619 --> 00:25:54,910 So what are the boundary condition at the left boundary 561 00:25:54,910 --> 00:25:56,840 condition? 562 00:25:56,840 --> 00:25:58,830 So first off, what are the boundary conditions 563 00:25:58,830 --> 00:26:00,955 we want to impose at the left and right boundaries? 564 00:26:00,955 --> 00:26:02,200 AUDIENCE: Continuous. 565 00:26:02,200 --> 00:26:03,360 PROFESSOR: Continuous, and the derivative 566 00:26:03,360 --> 00:26:04,000 should be continuous. 567 00:26:04,000 --> 00:26:04,499 Exactly. 568 00:26:04,499 --> 00:26:08,060 So we have that phi is continuous, and phi prime-- 569 00:26:08,060 --> 00:26:09,110 Good god. 570 00:26:09,110 --> 00:26:10,960 --phi prime is continuous. 571 00:26:10,960 --> 00:26:14,775 Similarly, phi continuous, phi prime continuous. 572 00:26:18,040 --> 00:26:20,690 OK. 573 00:26:20,690 --> 00:26:23,880 Do we have enough boundary conditions 574 00:26:23,880 --> 00:26:27,360 to specify our function? 575 00:26:27,360 --> 00:26:28,910 So we have now for our solution, we 576 00:26:28,910 --> 00:26:31,700 have six undetermined coefficients. 577 00:26:31,700 --> 00:26:33,390 And we have six boundary conditions. 578 00:26:33,390 --> 00:26:34,549 So that looks good. 579 00:26:34,549 --> 00:26:35,590 Are they all independent? 580 00:26:38,204 --> 00:26:38,870 Ponder that one. 581 00:26:43,300 --> 00:26:45,960 So in particular, let's start with the normalizable. 582 00:26:45,960 --> 00:26:49,400 So in order for phi to go to zero at minus infinity deep 583 00:26:49,400 --> 00:26:53,631 out on the left, what should be true? 584 00:26:53,631 --> 00:26:54,715 Yeah, d goes to zero. 585 00:26:57,390 --> 00:27:00,200 Oops, equals zero. 586 00:27:00,200 --> 00:27:03,040 And on the right? 587 00:27:03,040 --> 00:27:05,800 Yeah, that curly e equals 0, which 588 00:27:05,800 --> 00:27:08,860 is nice so I don't ever have to write it again. 589 00:27:08,860 --> 00:27:09,690 So that's zero. 590 00:27:09,690 --> 00:27:10,440 And that's zero. 591 00:27:10,440 --> 00:27:10,940 OK. 592 00:27:10,940 --> 00:27:11,470 That's good. 593 00:27:16,280 --> 00:27:18,754 We can take advantage though of something nice. 594 00:27:18,754 --> 00:27:20,170 We know that the wave function has 595 00:27:20,170 --> 00:27:22,240 to be either symmetric or anti-symmetric. 596 00:27:22,240 --> 00:27:22,920 Right? 597 00:27:22,920 --> 00:27:24,840 So we can exploit that and say, look, 598 00:27:24,840 --> 00:27:28,420 the wave function is going to be different from our boundary 599 00:27:28,420 --> 00:27:30,827 conditions, but it's a true fact, 600 00:27:30,827 --> 00:27:32,160 and we can take advantage of it. 601 00:27:32,160 --> 00:27:33,590 We can use the parody of the well. 602 00:27:36,610 --> 00:27:38,090 I can never-- 603 00:27:38,090 --> 00:27:40,450 So we can use the parody of the potential 604 00:27:40,450 --> 00:27:47,380 to say that the system is either symmetric or anti-symmetric. 605 00:27:47,380 --> 00:27:51,669 And these are often said as even or odd 606 00:27:51,669 --> 00:27:53,460 because the function will be, as a function 607 00:27:53,460 --> 00:27:54,501 of x, either even or odd. 608 00:27:54,501 --> 00:27:57,150 You either pick up a minus sign or a plus sign 609 00:27:57,150 --> 00:28:00,109 under taking x to minus x. 610 00:28:00,109 --> 00:28:01,650 So if the system is even, what can we 611 00:28:01,650 --> 00:28:02,858 say about these coefficients? 612 00:28:08,110 --> 00:28:10,157 What must be true of b, for example? 613 00:28:10,157 --> 00:28:10,990 AUDIENCE: It's zero. 614 00:28:10,990 --> 00:28:11,656 PROFESSOR: Yeah. 615 00:28:11,656 --> 00:28:13,130 So b equals 0. 616 00:28:13,130 --> 00:28:14,534 And what else? 617 00:28:14,534 --> 00:28:15,450 AUDIENCE: It equals f. 618 00:28:15,450 --> 00:28:16,220 PROFESSOR: This equals f. 619 00:28:16,220 --> 00:28:16,540 Yeah, good. 620 00:28:16,540 --> 00:28:17,040 OK. 621 00:28:17,040 --> 00:28:17,750 This equals f. 622 00:28:17,750 --> 00:28:19,310 And if the system is anti-symmetric, 623 00:28:19,310 --> 00:28:21,440 then a equal to 0. 624 00:28:21,440 --> 00:28:24,845 And we see that c is equal to minus f. 625 00:28:24,845 --> 00:28:25,345 Yeah? 626 00:28:28,269 --> 00:28:29,685 So that's a useful simplification. 627 00:28:32,580 --> 00:28:35,852 So it's easy to see that we could do this either way. 628 00:28:35,852 --> 00:28:37,810 We could do either symmetric or anti-symmetric. 629 00:28:37,810 --> 00:28:39,476 I'm going to, for simplicity in lecture, 630 00:28:39,476 --> 00:28:41,050 focus on the even case. 631 00:28:41,050 --> 00:28:45,010 b is equal to 0, and c is equal to minus f. 632 00:28:45,010 --> 00:28:46,650 Sorry, c is equal to f. 633 00:28:46,650 --> 00:28:47,757 So plus c. 634 00:28:50,310 --> 00:28:55,670 So now we're specifically working 635 00:28:55,670 --> 00:28:58,800 with the even solutions. 636 00:28:58,800 --> 00:29:01,179 And on your problem set, you'll repeat this calculation 637 00:29:01,179 --> 00:29:02,095 for the odd functions. 638 00:29:05,420 --> 00:29:08,379 So we're going to focus on the even solutions. 639 00:29:08,379 --> 00:29:09,920 And now what we have to do is we have 640 00:29:09,920 --> 00:29:14,560 to impose the boundary conditions for phi and phi 641 00:29:14,560 --> 00:29:16,680 prime. 642 00:29:16,680 --> 00:29:17,750 So that's easy enough. 643 00:29:23,010 --> 00:29:24,525 We have the function. 644 00:29:24,525 --> 00:29:28,995 All we have to do is impose that the values are the same. 645 00:29:28,995 --> 00:29:31,495 So for example, let's focus on the left boundary conditions. 646 00:29:36,250 --> 00:29:36,750 Sorry. 647 00:29:36,750 --> 00:29:38,166 Let's focus on the right because I 648 00:29:38,166 --> 00:29:40,280 don't want to deal with that minus sign. 649 00:29:40,280 --> 00:29:42,280 So let's focus on the right boundary conditions. 650 00:29:42,280 --> 00:29:46,240 So this is x is equal to plus l. 651 00:29:46,240 --> 00:29:48,900 So when x is equal to plus l, what must be true? 652 00:29:48,900 --> 00:29:51,150 Phi and phi prime must be continuous. 653 00:29:51,150 --> 00:29:51,820 So what's phi? 654 00:29:51,820 --> 00:29:53,520 So phi is equal to-- 655 00:29:56,540 --> 00:30:00,660 Well, from inside, it's a cosine of kl. 656 00:30:05,310 --> 00:30:06,710 Yeah. 657 00:30:06,710 --> 00:30:09,690 And b is gone because we're only looking at the even functions. 658 00:30:09,690 --> 00:30:15,060 On the right, however, it's equal to c e to the minus alpha 659 00:30:15,060 --> 00:30:18,170 l because we're evaluating at the right boundary. 660 00:30:18,170 --> 00:30:19,320 Yeah? 661 00:30:19,320 --> 00:30:19,820 OK. 662 00:30:19,820 --> 00:30:20,486 So this is cool. 663 00:30:20,486 --> 00:30:22,720 It allows us to determine c in terms of a. 664 00:30:22,720 --> 00:30:25,100 And if we solve that equation for c in terms of a, 665 00:30:25,100 --> 00:30:29,150 we'll get an eigenfunction, phi even, 666 00:30:29,150 --> 00:30:32,760 with one overall normalization coefficient, a. 667 00:30:32,760 --> 00:30:34,260 And then we can fix that to whatever 668 00:30:34,260 --> 00:30:36,550 it has to be so that everything integrates to one. 669 00:30:36,550 --> 00:30:37,630 Yeah? 670 00:30:37,630 --> 00:30:38,980 So that seems fine. 671 00:30:38,980 --> 00:30:42,076 It seems like we can solve for c in terms of a. 672 00:30:42,076 --> 00:30:43,030 c is equal to-- 673 00:30:43,030 --> 00:30:46,355 This is weight. 674 00:30:46,355 --> 00:30:47,030 Well, OK. 675 00:30:47,030 --> 00:30:53,890 So c is equal to a cosine e to the plus alpha l. 676 00:30:53,890 --> 00:30:57,002 On the other hand, we also have a condition on the derivative. 677 00:30:57,002 --> 00:30:58,460 And the condition on the derivative 678 00:30:58,460 --> 00:31:02,246 is that phi prime is continuous. 679 00:31:02,246 --> 00:31:04,120 And the derivative of this, well that's easy. 680 00:31:04,120 --> 00:31:05,790 It's the derivative of cosine. 681 00:31:05,790 --> 00:31:08,880 So this is going to be minus sine. 682 00:31:08,880 --> 00:31:10,920 But we pull out a factor of k, because we're 683 00:31:10,920 --> 00:31:12,420 taking derivative with respect to x. 684 00:31:12,420 --> 00:31:14,470 Minus k is sine of kx. 685 00:31:14,470 --> 00:31:18,040 Evaluate it out, sine of kl, l. 686 00:31:18,040 --> 00:31:20,930 And this is going to be equal to minus 687 00:31:20,930 --> 00:31:24,706 alpha c e to the minus alpha l. 688 00:31:24,706 --> 00:31:26,159 AUDIENCE: You forgot an a. 689 00:31:26,159 --> 00:31:26,950 PROFESSOR: Oh, yes. 690 00:31:26,950 --> 00:31:30,060 There should be an a. 691 00:31:30,060 --> 00:31:32,150 Thank you. 692 00:31:32,150 --> 00:31:32,789 OK. 693 00:31:32,789 --> 00:31:34,580 So but now we've got a problem because this 694 00:31:34,580 --> 00:31:43,190 says that c is equal to minus a times k over alpha sine 695 00:31:43,190 --> 00:31:47,918 of kl times e to the alpha l. 696 00:31:47,918 --> 00:31:51,670 And that's bad because c can't be 697 00:31:51,670 --> 00:31:55,350 equal to two different numbers at the same time. 698 00:31:55,350 --> 00:31:57,950 There's a certain monogamy of mathematical equations. 699 00:31:57,950 --> 00:31:59,400 It just doesn't work. 700 00:31:59,400 --> 00:32:00,972 So how do we deal with this? 701 00:32:00,972 --> 00:32:03,055 Well, let's think about what these equations would 702 00:32:03,055 --> 00:32:03,370 have meant. 703 00:32:03,370 --> 00:32:05,110 Forget this one for the moment, and just 704 00:32:05,110 --> 00:32:06,610 focus on that first expression. 705 00:32:06,610 --> 00:32:08,240 I'm going to rewrite this slightly. 706 00:32:08,240 --> 00:32:11,812 a cosine of kl. 707 00:32:11,812 --> 00:32:13,120 OK. 708 00:32:13,120 --> 00:32:14,610 So what does this expression say? 709 00:32:14,610 --> 00:32:16,320 Well, it seems like it's just saying 710 00:32:16,320 --> 00:32:20,100 if we fix c to be equal to this value, for fixed value of kappa 711 00:32:20,100 --> 00:32:22,810 l and alpha, then there's a solution. 712 00:32:22,810 --> 00:32:25,250 However, what is k? 713 00:32:25,250 --> 00:32:26,715 What are k and alpha. 714 00:32:26,715 --> 00:32:30,420 k and alpha are functions of the energy. 715 00:32:30,420 --> 00:32:32,150 So it would seem from this point of view, 716 00:32:32,150 --> 00:32:34,830 like for any value of the energy, 717 00:32:34,830 --> 00:32:38,686 we get a solution to this equation. 718 00:32:38,686 --> 00:32:40,150 Everyone see that? 719 00:32:40,150 --> 00:32:42,010 But we know that can't possibly be right 720 00:32:42,010 --> 00:32:45,110 because we expect the solutions to be discrete. 721 00:32:45,110 --> 00:32:47,410 We don't expect any value of energy 722 00:32:47,410 --> 00:32:50,610 to lead to a solution of the energy eigenvalue equation. 723 00:32:50,610 --> 00:32:52,610 There should be only discrete set of energies. 724 00:32:52,610 --> 00:32:53,003 Yeah? 725 00:32:53,003 --> 00:32:54,669 AUDIENCE: Did you pick up an extra minus 726 00:32:54,669 --> 00:32:56,951 sign in the expression for c? 727 00:32:56,951 --> 00:32:57,700 PROFESSOR: You do. 728 00:32:57,700 --> 00:32:58,200 Thank you. 729 00:32:58,200 --> 00:32:59,050 The sign's cancel. 730 00:32:59,050 --> 00:32:59,737 Yes, excellent. 731 00:32:59,737 --> 00:33:01,570 I've never written this equation in my life. 732 00:33:01,570 --> 00:33:03,530 So thank you. 733 00:33:03,530 --> 00:33:04,590 Yes, extra minus sign. 734 00:33:04,590 --> 00:33:05,740 So what's going on here? 735 00:33:05,740 --> 00:33:08,510 Well, what we see is that we've written down 736 00:33:08,510 --> 00:33:09,970 the general form of the solution. 737 00:33:09,970 --> 00:33:12,428 Here were imposing that we've already imposed the condition 738 00:33:12,428 --> 00:33:14,210 that we're normalizable at infinity. 739 00:33:14,210 --> 00:33:16,250 Here, we're imposing the continuity condition 740 00:33:16,250 --> 00:33:19,400 on the right. 741 00:33:19,400 --> 00:33:21,690 And if we impose just the continuity condition 742 00:33:21,690 --> 00:33:25,800 for the wave function, we can find a solution. 743 00:33:25,800 --> 00:33:28,420 Similarly, if we impose only the continuity condition 744 00:33:28,420 --> 00:33:30,170 for the derivative, we can find a solution 745 00:33:30,170 --> 00:33:31,860 for arbitrary values of the energy. 746 00:33:31,860 --> 00:33:33,900 But in order to find a solution where 747 00:33:33,900 --> 00:33:39,390 the wave function and its derivative are both continuous, 748 00:33:39,390 --> 00:33:43,059 it can't be true that the energy takes just any value because it 749 00:33:43,059 --> 00:33:45,100 would tell you that c takes two different values. 750 00:33:45,100 --> 00:33:45,720 Right? 751 00:33:45,720 --> 00:33:47,260 So there's a consistency condition. 752 00:33:47,260 --> 00:33:50,410 For what values of energy or equivalently, for what values 753 00:33:50,410 --> 00:33:54,090 of k and alpha are these two expressions equal 754 00:33:54,090 --> 00:33:56,850 to the same thing? 755 00:33:56,850 --> 00:33:58,214 Cool? 756 00:33:58,214 --> 00:33:59,630 So we can get that by saying, look 757 00:33:59,630 --> 00:34:02,050 we want both of these equations to be true. 758 00:34:02,050 --> 00:34:03,720 And this is easy. 759 00:34:03,720 --> 00:34:07,510 I can take this equation and divide it by this equation. 760 00:34:07,510 --> 00:34:09,030 And I will lose my coefficients c. 761 00:34:09,030 --> 00:34:10,280 I will lose my coefficients a. 762 00:34:10,280 --> 00:34:10,960 What do we get? 763 00:34:10,960 --> 00:34:12,739 On the right hand side, we get-- 764 00:34:12,739 --> 00:34:14,929 And I'm going to put a minus sign on everything. 765 00:34:14,929 --> 00:34:18,560 So minus, minus, minus. 766 00:34:18,560 --> 00:34:21,310 So if we take this equation and we divide it by this equation, 767 00:34:21,310 --> 00:34:24,040 on the right hand side, we get alpha, 768 00:34:24,040 --> 00:34:26,170 because the c exponential drops off. 769 00:34:26,170 --> 00:34:27,929 And on this side, we lose the a. 770 00:34:27,929 --> 00:34:29,830 We get a k. 771 00:34:29,830 --> 00:34:32,170 And then we get sine over cosine of kl, also known 772 00:34:32,170 --> 00:34:34,210 as tangent of kl. 773 00:34:37,280 --> 00:34:38,080 Here we have a kl. 774 00:34:38,080 --> 00:34:39,564 Here we have a k. 775 00:34:39,564 --> 00:34:40,980 These are all dimensionful things. 776 00:34:40,980 --> 00:34:43,752 Let's multiply everything by an l. 777 00:34:43,752 --> 00:34:45,210 And this is nice and dimensionless. 778 00:34:45,210 --> 00:34:46,650 Both sides are dimensionless. 779 00:34:46,650 --> 00:34:47,960 So we get this condition. 780 00:34:47,960 --> 00:34:49,989 This is the consistency condition, 781 00:34:49,989 --> 00:34:52,530 such that both the wave function and its derivative 782 00:34:52,530 --> 00:34:56,230 can be continuous at the right boundary. 783 00:34:56,230 --> 00:34:58,650 OK? 784 00:34:58,650 --> 00:35:01,390 And this is a pretty nontrivial condition. 785 00:35:01,390 --> 00:35:03,790 It says, given a value of k, you can always 786 00:35:03,790 --> 00:35:06,290 determine the value of alpha, such as this equation as true. 787 00:35:06,290 --> 00:35:08,300 But remember that k and alpha are both known 788 00:35:08,300 --> 00:35:09,450 functions of the energy. 789 00:35:09,450 --> 00:35:12,440 So this is really an equation, a complicated, nonlinear equation 790 00:35:12,440 --> 00:35:13,220 for the energy. 791 00:35:17,020 --> 00:35:23,180 So this is equal to a horrible expression, a condition, badly 792 00:35:23,180 --> 00:35:25,450 nonlinear, in fact, transcendental condition 793 00:35:25,450 --> 00:35:26,130 on the energy. 794 00:35:29,846 --> 00:35:30,970 And where's it coming from? 795 00:35:30,970 --> 00:35:34,860 It's coming from normalizability and continuity everywhere. 796 00:35:39,280 --> 00:35:41,020 And a useful thing to check, and I 797 00:35:41,020 --> 00:35:43,105 invite you to do this on your own, 798 00:35:43,105 --> 00:35:51,020 is to check that the boundary conditions at the left wall 799 00:35:51,020 --> 00:35:52,305 give the same expression. 800 00:35:55,870 --> 00:35:57,581 Yeah. 801 00:35:57,581 --> 00:36:00,443 AUDIENCE: For our final form of that equation, 802 00:36:00,443 --> 00:36:04,259 is there a reason that we prefer to multiply both sides by l 803 00:36:04,259 --> 00:36:05,690 than divide both sides by k? 804 00:36:05,690 --> 00:36:06,400 PROFESSOR: Yeah. 805 00:36:06,400 --> 00:36:08,275 And it'll be little more obvious in a second. 806 00:36:08,275 --> 00:36:10,920 But here's the reason. 807 00:36:10,920 --> 00:36:12,760 So let's divide through by l. 808 00:36:12,760 --> 00:36:15,439 This is the form that we got. 809 00:36:15,439 --> 00:36:16,230 What are the units? 810 00:36:16,230 --> 00:36:18,585 What are the dimensions of this expression? 811 00:36:18,585 --> 00:36:23,140 k is a wave number, so it has units of 1 upon length. 812 00:36:23,140 --> 00:36:23,640 Right? 813 00:36:23,640 --> 00:36:26,223 And that's good because that's 1 upon length times the length, 814 00:36:26,223 --> 00:36:29,370 and you'd better have something dimensionless inside a tangent. 815 00:36:29,370 --> 00:36:31,632 But it seems there are two things to say about this. 816 00:36:31,632 --> 00:36:34,090 The first is it seems like l is playing an independent role 817 00:36:34,090 --> 00:36:36,200 from k in this equation. 818 00:36:36,200 --> 00:36:38,080 But this is dimensionless. 819 00:36:38,080 --> 00:36:40,407 These are both dimensionful units of 1 over length. 820 00:36:40,407 --> 00:36:42,490 So we can make the entire expression dimensionless 821 00:36:42,490 --> 00:36:46,680 and make it clear that k and l don't have an independent life. 822 00:36:46,680 --> 00:36:49,020 The dimensionless quantity, kl, times the tangent 823 00:36:49,020 --> 00:36:50,410 of that dimensionless quantity is 824 00:36:50,410 --> 00:36:52,129 equal to this dimensionless quantity. 825 00:36:52,129 --> 00:36:54,170 So the reason that this is preferable is twofold. 826 00:36:54,170 --> 00:36:56,430 First off, it makes it sort of obvious that k 827 00:36:56,430 --> 00:36:59,740 and l, you can't vary them independently in this sense. 828 00:36:59,740 --> 00:37:04,190 But the second is that it makes it nice and dimensionless. 829 00:37:04,190 --> 00:37:05,840 And you'll always, whenever possible, 830 00:37:05,840 --> 00:37:07,548 want to put things in dimensionless form. 831 00:37:10,434 --> 00:37:11,850 I mean it's just multiplying by l. 832 00:37:11,850 --> 00:37:13,349 So it's obviously not all that deep. 833 00:37:13,349 --> 00:37:16,960 But it's a convenient bit of multiplication by l. 834 00:37:16,960 --> 00:37:19,360 Other questions? 835 00:37:19,360 --> 00:37:20,960 OK. 836 00:37:20,960 --> 00:37:23,395 So where are we? 837 00:37:23,395 --> 00:37:25,520 So I'd like to find the solutions of this equation. 838 00:37:25,520 --> 00:37:26,680 So again, just to-- 839 00:37:29,440 --> 00:37:37,920 Let me write this slightly differently where k squared is 840 00:37:37,920 --> 00:37:44,320 equal to 2m upon h bar squared v0 plus e. 841 00:37:44,320 --> 00:37:52,420 And alpha squared is equal to 2m upon h bar squared e, 842 00:37:52,420 --> 00:37:55,140 the positive value of e. 843 00:37:55,140 --> 00:37:58,670 So this is a really complicated expression as a function of e. 844 00:37:58,670 --> 00:38:00,980 So I'd like to solve for the actual energy eigenvalues. 845 00:38:00,980 --> 00:38:05,420 I want to know what are the energy eigenvalues of the bound 846 00:38:05,420 --> 00:38:08,654 states of the finite potential well, as a function of l, 847 00:38:08,654 --> 00:38:09,154 for example. 848 00:38:11,820 --> 00:38:14,389 Sadly, I can't solve this equation. 849 00:38:14,389 --> 00:38:15,680 It's a transcendental equation. 850 00:38:15,680 --> 00:38:18,570 It's a sort of canonically hard problem to solve. 851 00:38:18,570 --> 00:38:21,180 You can't write down a closed from expression for it. 852 00:38:21,180 --> 00:38:24,080 However, there are a bunch of ways to easily solve it. 853 00:38:24,080 --> 00:38:27,760 One is take your convenient nearby laptop. 854 00:38:27,760 --> 00:38:30,670 Open up Mathematica, and ask it to numerically find solutions 855 00:38:30,670 --> 00:38:31,170 to this. 856 00:38:31,170 --> 00:38:32,075 And you can do this. 857 00:38:32,075 --> 00:38:32,950 It's a good exercise. 858 00:38:35,407 --> 00:38:37,490 I will encourage you to do so on your problem set. 859 00:38:37,490 --> 00:38:39,450 And in fact, on the problem set, it 860 00:38:39,450 --> 00:38:40,960 asks you to do a calculation. 861 00:38:40,960 --> 00:38:43,100 And it encourages you do it using Mathematica. 862 00:38:43,100 --> 00:38:45,200 Let me rephrase the statement in the problem set. 863 00:38:45,200 --> 00:38:48,900 It would be crazy for you to try to do it only by hand. 864 00:38:48,900 --> 00:38:50,707 You should do it by hand and on computer 865 00:38:50,707 --> 00:38:51,790 because they're both easy. 866 00:38:51,790 --> 00:38:53,560 And you can check against each other. 867 00:38:53,560 --> 00:38:56,680 They make different things obvious. 868 00:38:56,680 --> 00:38:59,455 This should be your default is to also check on Mathematica. 869 00:38:59,455 --> 00:39:02,080 The second thing we can do is we can get a qualitative solution 870 00:39:02,080 --> 00:39:03,890 of this equation just graphically. 871 00:39:03,890 --> 00:39:06,280 And since this is such a useful technique, not just here, 872 00:39:06,280 --> 00:39:08,122 but throughout physics to graphically solve 873 00:39:08,122 --> 00:39:09,580 transcendental equations, I'm going 874 00:39:09,580 --> 00:39:11,114 to walk through it a little bit. 875 00:39:11,114 --> 00:39:13,030 So this is going to be the graphical solution. 876 00:39:17,336 --> 00:39:18,710 And we can extract, it turns out, 877 00:39:18,710 --> 00:39:22,420 an awful lot of the physics of these energy 878 00:39:22,420 --> 00:39:24,700 eigenstates and their energies through 879 00:39:24,700 --> 00:39:27,464 this graphical technique. 880 00:39:27,464 --> 00:39:28,880 So the first thing is I write this 881 00:39:28,880 --> 00:39:31,232 in nice, dimensionless form. 882 00:39:31,232 --> 00:39:33,440 And let me give those dimensionless variables a name. 883 00:39:33,440 --> 00:39:37,720 Let me call kl is equal to z, just define a parameter z. 884 00:39:37,720 --> 00:39:39,660 And alpha l is a parameter y. 885 00:39:42,530 --> 00:39:48,330 And I want to note that z squared plus y squared is 886 00:39:48,330 --> 00:39:52,170 equal to a constant, which is, I will call if you just 887 00:39:52,170 --> 00:39:54,620 plug these guys out, kl squared plus al squared. 888 00:39:54,620 --> 00:39:55,320 That's easy. 889 00:39:55,320 --> 00:39:58,460 Kl squared is this guy times l squared. 890 00:39:58,460 --> 00:40:01,070 Al squared is this guy times l squared. 891 00:40:01,070 --> 00:40:03,680 And so the e and the minus e cancel 892 00:40:03,680 --> 00:40:05,210 when we add them together. 893 00:40:05,210 --> 00:40:09,690 So we just get 2mv0 over h bar squared times l squared. 894 00:40:09,690 --> 00:40:16,830 So 2m upon h bar squared l squared v0. 895 00:40:16,830 --> 00:40:21,420 And I'm going to call this r naught in something 896 00:40:21,420 --> 00:40:24,670 of a pathological abusive notation. 897 00:40:24,670 --> 00:40:25,170 OK. 898 00:40:25,170 --> 00:40:26,253 So this is our expression. 899 00:40:27,827 --> 00:40:29,660 And I actually want to call this r0 squared. 900 00:40:32,260 --> 00:40:32,760 I know. 901 00:40:32,760 --> 00:40:33,540 I know. 902 00:40:33,540 --> 00:40:34,120 It's awful. 903 00:40:34,120 --> 00:40:35,578 But the reason I want to do this is 904 00:40:35,578 --> 00:40:37,670 that this is the equation for a circle. 905 00:40:37,670 --> 00:40:38,620 Yeah? 906 00:40:38,620 --> 00:40:40,220 And a circle has a radius. 907 00:40:40,220 --> 00:40:42,680 The thing that goes over here is r squared. 908 00:40:42,680 --> 00:40:43,180 OK. 909 00:40:43,180 --> 00:40:45,513 So at this point, you're thinking like, come on, circle. 910 00:40:45,513 --> 00:40:46,229 So let's plot it. 911 00:40:46,229 --> 00:40:48,020 So how are we going to solve this equation? 912 00:40:48,020 --> 00:40:49,260 Here's what I want to solve. 913 00:40:49,260 --> 00:40:55,340 I have now two equations relating z and y. 914 00:40:55,340 --> 00:41:02,020 We have that from this equation z tangent z is equal to y. 915 00:41:02,020 --> 00:41:04,890 And from this equation we have that z squared plus y 916 00:41:04,890 --> 00:41:08,980 squared is a constant r0 squared. 917 00:41:08,980 --> 00:41:11,892 Where r0 squared depends on the potential and the width 918 00:41:11,892 --> 00:41:14,100 in a very specific way, on the depth of the potential 919 00:41:14,100 --> 00:41:15,740 and the width in a very specific way. 920 00:41:15,740 --> 00:41:16,580 So we want to find-- 921 00:41:16,580 --> 00:41:17,079 Bless you. 922 00:41:17,079 --> 00:41:20,640 --simultaneously, we want to find simultaneous solutions 923 00:41:20,640 --> 00:41:22,910 of these two equations. 924 00:41:22,910 --> 00:41:25,720 Yeah? 925 00:41:25,720 --> 00:41:27,770 So that's relatively easy. 926 00:41:27,770 --> 00:41:30,300 So here's y, and here's z. 927 00:41:34,790 --> 00:41:36,940 So this equation has solutions. 928 00:41:36,940 --> 00:41:38,424 Any time that y plus z squared is 929 00:41:38,424 --> 00:41:40,840 equal to r0 squared, that means any time we have a circle. 930 00:41:40,840 --> 00:41:45,350 So solutions for fixed values of r0 lie on circles. 931 00:41:49,410 --> 00:41:52,853 Oh, I really should have drawn this under here. 932 00:41:52,853 --> 00:41:53,353 Sorry. 933 00:41:57,650 --> 00:41:58,200 y and z. 934 00:42:05,760 --> 00:42:06,860 So those are the circles. 935 00:42:06,860 --> 00:42:10,300 Notice that I'm only focusing on y and z, both positive. 936 00:42:10,300 --> 00:42:12,775 Why? 937 00:42:12,775 --> 00:42:14,930 Not yz, but W-H-Y . 938 00:42:14,930 --> 00:42:17,101 Why am I focusing on the variables y 939 00:42:17,101 --> 00:42:17,975 and z being positive? 940 00:42:21,335 --> 00:42:23,710 Because we started out defining them in terms of k and l, 941 00:42:23,710 --> 00:42:25,460 which were both positive, and alpha and l, 942 00:42:25,460 --> 00:42:26,600 which were both positive. 943 00:42:26,600 --> 00:42:28,183 Can we find solutions to this equation 944 00:42:28,183 --> 00:42:30,100 that have x and y negative? 945 00:42:30,100 --> 00:42:30,850 Sure. 946 00:42:30,850 --> 00:42:33,040 But they don't mean anything in terms of our original problem. 947 00:42:33,040 --> 00:42:35,020 So to map onto solutions of our original problem, 948 00:42:35,020 --> 00:42:37,200 we want to focus on the positive values of y and z. 949 00:42:37,200 --> 00:42:38,260 Cool? 950 00:42:38,260 --> 00:42:39,240 OK. 951 00:42:39,240 --> 00:42:40,260 So that's this one. 952 00:42:40,260 --> 00:42:41,570 The solutions lie on circles. 953 00:42:41,570 --> 00:42:44,130 So given a value of y, you can find a solution of z. 954 00:42:44,130 --> 00:42:46,877 But we want to also find a solution of this equation. 955 00:42:46,877 --> 00:42:49,210 And this equation is a little more entertaining to plot. 956 00:42:49,210 --> 00:42:50,130 Here's y. 957 00:42:50,130 --> 00:42:51,370 Here's z. 958 00:42:51,370 --> 00:42:53,370 So what does z tangent z do? 959 00:42:56,220 --> 00:42:57,257 Oh, shoot. 960 00:42:57,257 --> 00:42:58,340 I want to plot y vertical. 961 00:42:58,340 --> 00:43:00,930 Otherwise, it's going to a giant pain. 962 00:43:00,930 --> 00:43:04,656 Happily, this plot can be left identical. 963 00:43:04,656 --> 00:43:05,942 Let's plot y vertically. 964 00:43:05,942 --> 00:43:07,650 So the reason I want to plot y vertically 965 00:43:07,650 --> 00:43:09,380 is that this is z tangent z. 966 00:43:09,380 --> 00:43:13,205 So first off, what does tangent z look like? 967 00:43:13,205 --> 00:43:13,864 Yeah. 968 00:43:13,864 --> 00:43:14,530 This is awesome. 969 00:43:14,530 --> 00:43:16,020 Yeah, it looks like this. 970 00:43:16,020 --> 00:43:16,890 Yes, exactly. 971 00:43:16,890 --> 00:43:19,377 So tangent is sine over cosine. 972 00:43:19,377 --> 00:43:20,710 Sine is zero, and cosine is one. 973 00:43:20,710 --> 00:43:23,870 So it does this, as you go to a value 974 00:43:23,870 --> 00:43:26,170 where the argument, let's call the argument z. 975 00:43:26,170 --> 00:43:28,450 So if we just plot tangent-- 976 00:43:28,450 --> 00:43:29,080 OK. 977 00:43:29,080 --> 00:43:33,510 So when z is equal to pi over 2, then the denominator cosine 978 00:43:33,510 --> 00:43:35,205 vanishes, and that diverges. 979 00:43:39,700 --> 00:43:41,750 Oops. 980 00:43:41,750 --> 00:43:42,250 OK. 981 00:43:42,250 --> 00:43:43,960 So here's pi over 2. 982 00:43:43,960 --> 00:43:45,620 Here's pi. 983 00:43:45,620 --> 00:43:46,230 Whoops. 984 00:43:46,230 --> 00:43:54,670 Pi, pi over 2, and here's 3pi over 2, and so on. 985 00:43:54,670 --> 00:43:56,840 Now, we're only interested in the first quadrant. 986 00:43:56,840 --> 00:44:00,540 So I'm just ignore down here. 987 00:44:00,540 --> 00:44:01,300 OK. 988 00:44:01,300 --> 00:44:03,760 So this is pi over 2. 989 00:44:03,760 --> 00:44:07,760 This is pi, 3pi over 2. 990 00:44:07,760 --> 00:44:08,832 OK. 991 00:44:08,832 --> 00:44:10,540 But this is not what we're interested in. 992 00:44:10,540 --> 00:44:13,100 We're not interested in tangent of z. 993 00:44:13,100 --> 00:44:15,120 We're interested in z tangent z. 994 00:44:15,120 --> 00:44:16,760 And what does z tangent z look like? 995 00:44:16,760 --> 00:44:18,250 Well, it's basically the same. 996 00:44:18,250 --> 00:44:18,750 Right? 997 00:44:18,750 --> 00:44:21,450 z tangent z, it has an extra factor of zero here 998 00:44:21,450 --> 00:44:23,270 and remains extra small at the beginning. 999 00:44:23,270 --> 00:44:25,020 But it still curves off roughly like this. 1000 00:44:25,020 --> 00:44:28,760 And z is just nice and linear, nice 1001 00:44:28,760 --> 00:44:29,940 and regular throughout this. 1002 00:44:29,940 --> 00:44:31,481 So it doesn't change the fact that we 1003 00:44:31,481 --> 00:44:33,350 have a divergence at pi over 2. 1004 00:44:33,350 --> 00:44:36,330 And it doesn't change the fact that it vanishes again at pi 1005 00:44:36,330 --> 00:44:37,500 and becomes positive again. 1006 00:44:37,500 --> 00:44:39,480 It just changes the shape of the curve. 1007 00:44:39,480 --> 00:44:41,030 And in fact, the way it changes the shape of the curve 1008 00:44:41,030 --> 00:44:43,180 is this becomes a little fatter around the bottom. 1009 00:44:43,180 --> 00:44:44,890 It's just a little more round. 1010 00:44:44,890 --> 00:44:46,829 And when we get out to large values of z, 1011 00:44:46,829 --> 00:44:48,620 it's going to have a more pronounced effect 1012 00:44:48,620 --> 00:44:50,920 because that slope is, at every example 1013 00:44:50,920 --> 00:44:52,920 where it crosses z, that slope is getting larger 1014 00:44:52,920 --> 00:44:56,542 because the coefficient of z is getting larger. 1015 00:44:56,542 --> 00:44:57,244 OK. 1016 00:44:57,244 --> 00:44:59,160 So it's just going to get more and more sharp. 1017 00:44:59,160 --> 00:45:02,090 But anyway, with all that said, here's 0. 1018 00:45:02,090 --> 00:45:03,510 Here's pi. 1019 00:45:03,510 --> 00:45:05,470 Here's pi over 2. 1020 00:45:05,470 --> 00:45:06,910 Here's pi. 1021 00:45:06,910 --> 00:45:09,590 Here's 3pi over 2. 1022 00:45:09,590 --> 00:45:11,799 The second plot we want to plot, y is z tangent z. 1023 00:45:11,799 --> 00:45:12,840 We know how to plot this. 1024 00:45:27,390 --> 00:45:28,720 Cool? 1025 00:45:28,720 --> 00:45:31,140 And what we want to find are simultaneous solutions 1026 00:45:31,140 --> 00:45:34,570 of this, values of y and z, for which this equation is solved 1027 00:45:34,570 --> 00:45:38,410 and this equation is solved for the same value of y and z. 1028 00:45:38,410 --> 00:45:40,640 This is a graphical solution. 1029 00:45:40,640 --> 00:45:43,080 So let's combine them together. 1030 00:45:43,080 --> 00:45:45,610 And the combined plots look like this. 1031 00:45:45,610 --> 00:45:48,900 First we have pi over 2. 1032 00:45:51,600 --> 00:45:52,930 So let's plot the tangents. 1033 00:45:57,780 --> 00:46:00,557 And then we have these circles for various values of r. 1034 00:46:00,557 --> 00:46:02,390 So for a particular value of r, for example, 1035 00:46:02,390 --> 00:46:05,920 suppose this is the value of r. 1036 00:46:05,920 --> 00:46:06,610 This is r0. 1037 00:46:12,100 --> 00:46:15,250 So how many solutions do we have? 1038 00:46:15,250 --> 00:46:16,082 One. 1039 00:46:16,082 --> 00:46:18,310 One set of common points where at y and z 1040 00:46:18,310 --> 00:46:19,185 solve both equations. 1041 00:46:22,620 --> 00:46:25,962 So we immediately learn something really lovely. 1042 00:46:25,962 --> 00:46:27,670 What happens to the radius of that circle 1043 00:46:27,670 --> 00:46:28,753 as I make the well deeper? 1044 00:46:32,294 --> 00:46:35,124 Yeah, as I make the well deeper, that means v0 gets 1045 00:46:35,124 --> 00:46:37,290 larger and larger magnitude, the radius gets larger. 1046 00:46:37,290 --> 00:46:38,081 So does the circle. 1047 00:46:38,081 --> 00:46:40,730 So if I make the well deeper, I make this the circle larger. 1048 00:46:40,730 --> 00:46:42,345 Will I still have a solution? 1049 00:46:42,345 --> 00:46:43,720 Yeah, I'll still have a solution. 1050 00:46:43,720 --> 00:46:45,235 But check this out. 1051 00:46:45,235 --> 00:46:46,485 Now, I'll have a new solution. 1052 00:46:50,570 --> 00:46:55,200 And you can even see the critical value of the depth 1053 00:46:55,200 --> 00:46:57,290 and the width of the well. 1054 00:46:57,290 --> 00:47:01,030 In order to have exactly a new bound state appearing, 1055 00:47:01,030 --> 00:47:02,890 what must the value of r0 be? 1056 00:47:02,890 --> 00:47:07,951 Well, it's got to be that value, such that r0 squared is pi. 1057 00:47:07,951 --> 00:47:08,450 Yeah? 1058 00:47:10,792 --> 00:47:13,000 And similarly, let me ask you the following question. 1059 00:47:13,000 --> 00:47:15,000 As I make the well deeper and deeper and deeper, 1060 00:47:15,000 --> 00:47:17,780 holding the width, and make it deeper and deeper and deeper, 1061 00:47:17,780 --> 00:47:20,480 does the number of states increase or decrease? 1062 00:47:20,480 --> 00:47:21,025 It increases. 1063 00:47:21,025 --> 00:47:22,400 If you make it deeper and deeper, 1064 00:47:22,400 --> 00:47:24,691 the radius of that circle is getting bigger and bigger. 1065 00:47:24,691 --> 00:47:30,010 There are more points where this circle intersects this point. 1066 00:47:30,010 --> 00:47:33,730 So here's another one. 1067 00:47:33,730 --> 00:47:38,670 We've got one here, one here, one here, three solutions. 1068 00:47:38,670 --> 00:47:40,850 And the number of solutions just goes. 1069 00:47:40,850 --> 00:47:42,800 Every time we click over a new point 1070 00:47:42,800 --> 00:47:45,383 by increasing the radius of the circle, we get a new solution. 1071 00:47:45,383 --> 00:47:46,990 We get another bound state. 1072 00:47:46,990 --> 00:47:49,730 But here's the thing that I really want to focus on. 1073 00:47:49,730 --> 00:47:51,390 Let's make the well less and less deep. 1074 00:47:51,390 --> 00:47:54,060 Let's make it shallower and shallower. 1075 00:47:54,060 --> 00:48:00,220 At what depth do we lose that first bound state? 1076 00:48:00,220 --> 00:48:01,131 We never do. 1077 00:48:01,131 --> 00:48:01,630 Right? 1078 00:48:01,630 --> 00:48:03,190 There is no circle so small that it 1079 00:48:03,190 --> 00:48:06,030 doesn't intersect this curve. 1080 00:48:06,030 --> 00:48:09,590 In a 1D, finite well potential, there is always at least one 1081 00:48:09,590 --> 00:48:10,890 bound state. 1082 00:48:10,890 --> 00:48:12,931 There are never zero bound states. 1083 00:48:12,931 --> 00:48:15,430 This will turn out not to be true in three dimensions, which 1084 00:48:15,430 --> 00:48:16,960 is kind of interesting. 1085 00:48:16,960 --> 00:48:19,840 But it's true in one dimension that we always 1086 00:48:19,840 --> 00:48:21,970 have at least one bound state. 1087 00:48:21,970 --> 00:48:23,800 And in fact, you can decorate this. 1088 00:48:23,800 --> 00:48:25,790 You can use this and fancy it up a bit 1089 00:48:25,790 --> 00:48:28,770 to argue that in any potential in 1D, 1090 00:48:28,770 --> 00:48:31,445 there's always at least one bound state 1091 00:48:31,445 --> 00:48:32,820 unless the potential is constant, 1092 00:48:32,820 --> 00:48:36,025 I mean any potential that varies and goes to zero infinity. 1093 00:48:38,640 --> 00:48:39,405 Yeah? 1094 00:48:39,405 --> 00:48:41,030 And so we still don't have any numbers. 1095 00:48:41,030 --> 00:48:44,390 But we know an awful lot about the qualitative structure 1096 00:48:44,390 --> 00:48:49,476 of the set of energy eigenvalues of the spectrum of the energy. 1097 00:48:49,476 --> 00:48:49,975 Questions? 1098 00:48:52,580 --> 00:48:53,710 Yeah? 1099 00:48:53,710 --> 00:48:57,081 AUDIENCE: So what happens if r is bigger 1100 00:48:57,081 --> 00:49:03,622 than pi or y is bigger than pi and you get two solutions? 1101 00:49:03,622 --> 00:49:04,330 PROFESSOR: Great. 1102 00:49:04,330 --> 00:49:06,720 So when we have two solutions, what does that mean? 1103 00:49:06,720 --> 00:49:11,270 Well, you've got to bound states, two different energies. 1104 00:49:11,270 --> 00:49:13,175 Right? 1105 00:49:13,175 --> 00:49:14,050 It's a good question. 1106 00:49:14,050 --> 00:49:17,890 Every solution here corresponds to some particular value of y 1107 00:49:17,890 --> 00:49:19,430 and some particular value of z. 1108 00:49:19,430 --> 00:49:21,640 But those values of y and z are just 1109 00:49:21,640 --> 00:49:23,375 telling you what k and alpha are. 1110 00:49:26,050 --> 00:49:27,674 And so that's determining the energy. 1111 00:49:27,674 --> 00:49:29,090 So a different value of k is going 1112 00:49:29,090 --> 00:49:30,941 to give you a different value of the energy. 1113 00:49:30,941 --> 00:49:32,690 So we can just eyeball this in particular. 1114 00:49:32,690 --> 00:49:33,780 Let's look at alpha. 1115 00:49:33,780 --> 00:49:35,020 Alpha is just e. 1116 00:49:35,020 --> 00:49:36,090 Alpha squared is just e. 1117 00:49:36,090 --> 00:49:37,716 Yes? 1118 00:49:37,716 --> 00:49:38,840 So here's a quick question. 1119 00:49:38,840 --> 00:49:40,920 If alpha is just e, and alpha l is y-- 1120 00:49:40,920 --> 00:49:43,060 So this is our y value. 1121 00:49:43,060 --> 00:49:46,350 y is roughly alpha, the width, which 1122 00:49:46,350 --> 00:49:49,450 means it's roughly the energy square root. 1123 00:49:49,450 --> 00:49:51,720 So this value, the vertical value 1124 00:49:51,720 --> 00:49:54,760 of each of these intersection points on a given circle 1125 00:49:54,760 --> 00:49:57,030 corresponds to the square root of the energy 1126 00:49:57,030 --> 00:49:59,700 times some coefficients. 1127 00:49:59,700 --> 00:50:04,671 So which state has the largest value of the energy? 1128 00:50:04,671 --> 00:50:06,420 Absolute value, which state is most deeply 1129 00:50:06,420 --> 00:50:09,577 bound on this circle? 1130 00:50:09,577 --> 00:50:10,410 Yeah, the first one. 1131 00:50:10,410 --> 00:50:10,540 Right? 1132 00:50:10,540 --> 00:50:11,880 Because it's got the largest value of alpha. 1133 00:50:11,880 --> 00:50:12,630 So this is nice. 1134 00:50:12,630 --> 00:50:14,127 We see that the first state always 1135 00:50:14,127 --> 00:50:16,210 has a higher value of alpha than the second state, 1136 00:50:16,210 --> 00:50:18,420 which always has a higher value of alpha than the third state. 1137 00:50:18,420 --> 00:50:19,870 And every time we add a new state, 1138 00:50:19,870 --> 00:50:23,640 we make the depth of these guys the binding energy 1139 00:50:23,640 --> 00:50:25,820 of the already existing states. 1140 00:50:25,820 --> 00:50:27,952 We make it just a little bit deeper. 1141 00:50:27,952 --> 00:50:29,700 We make them a little more tightly bound. 1142 00:50:29,700 --> 00:50:31,420 And only eventually then do we get 1143 00:50:31,420 --> 00:50:32,890 a new bound state appearing. 1144 00:50:32,890 --> 00:50:34,764 And what's the energy of that new bound state 1145 00:50:34,764 --> 00:50:37,340 when it appears? 1146 00:50:37,340 --> 00:50:38,300 Zero energy. 1147 00:50:38,300 --> 00:50:41,200 It's appearing just at threshold. 1148 00:50:41,200 --> 00:50:42,010 OK. 1149 00:50:42,010 --> 00:50:44,343 So we'll explore that in more detail in the problem set. 1150 00:50:44,343 --> 00:50:45,930 But for now, let me leave it at that. 1151 00:50:45,930 --> 00:50:46,430 Questions? 1152 00:50:46,430 --> 00:50:47,096 Other questions? 1153 00:50:47,096 --> 00:50:47,750 Yeah. 1154 00:50:47,750 --> 00:50:49,708 AUDIENCE: You said that this can be generalized 1155 00:50:49,708 --> 00:50:53,250 to any nonconstant function that you'd like. 1156 00:50:53,250 --> 00:50:56,250 So there's always going to be at least one bound state. 1157 00:50:56,250 --> 00:50:58,250 What about, like with delta function? 1158 00:50:58,250 --> 00:50:58,790 PROFESSOR: Excellent question. 1159 00:50:58,790 --> 00:50:59,610 What about the delta function? 1160 00:50:59,610 --> 00:51:00,420 We're going to come back to that in just a few minutes. 1161 00:51:00,420 --> 00:51:01,503 It's a very good question. 1162 00:51:01,503 --> 00:51:04,380 So the question is, look, if any potential that 1163 00:51:04,380 --> 00:51:07,077 goes to zero infinity and wiggles inside, 1164 00:51:07,077 --> 00:51:09,410 if any potential like that in 1D has a bound state, what 1165 00:51:09,410 --> 00:51:10,451 about the delta function? 1166 00:51:10,451 --> 00:51:11,730 We briefly talked about that. 1167 00:51:11,730 --> 00:51:14,146 So we're going to come back to that in just a few minutes. 1168 00:51:14,146 --> 00:51:15,361 But it's a pressing question. 1169 00:51:15,361 --> 00:51:15,860 OK. 1170 00:51:15,860 --> 00:51:16,526 Other questions? 1171 00:51:22,550 --> 00:51:23,572 Yes. 1172 00:51:23,572 --> 00:51:25,947 AUDIENCE: So the energy is zero, but that's not possible. 1173 00:51:28,015 --> 00:51:28,890 PROFESSOR: Thank you. 1174 00:51:28,890 --> 00:51:29,070 OK. 1175 00:51:29,070 --> 00:51:29,240 Good. 1176 00:51:29,240 --> 00:51:31,040 So let me talk about that in a little more detail. 1177 00:51:31,040 --> 00:51:32,970 So I wasn't going to go into this, but-- 1178 00:51:32,970 --> 00:51:36,130 So when new bound states appear, so let's consider a point where 1179 00:51:36,130 --> 00:51:39,950 our r0 is, let's say, it's just the right value 1180 00:51:39,950 --> 00:51:43,291 so that r0 is equal to pi. 1181 00:51:43,291 --> 00:51:43,790 OK. 1182 00:51:43,790 --> 00:51:48,400 And we see that we're just about to develop a new bound state. 1183 00:51:48,400 --> 00:51:50,870 So let's think about what that bound state looks like. 1184 00:51:50,870 --> 00:51:52,440 So this is the new bound state. 1185 00:52:00,504 --> 00:52:02,170 And I'm going to put this in parentheses 1186 00:52:02,170 --> 00:52:04,545 because it's got bound state. 1187 00:52:04,545 --> 00:52:05,545 And we say at threshold. 1188 00:52:09,660 --> 00:52:11,100 OK. 1189 00:52:11,100 --> 00:52:13,124 At threshold, i.e. 1190 00:52:13,124 --> 00:52:21,199 at the energy is roughly zero, and r0 is equal to pi. 1191 00:52:21,199 --> 00:52:22,490 So this is really what we mean. 1192 00:52:22,490 --> 00:52:24,030 This new state, when r0 is pi and we 1193 00:52:24,030 --> 00:52:26,600 have a solution on that second branch. 1194 00:52:26,600 --> 00:52:27,101 Cool? 1195 00:52:27,101 --> 00:52:28,850 So what does this wave function look like? 1196 00:52:28,850 --> 00:52:30,350 What does it look like when you have 1197 00:52:30,350 --> 00:52:32,050 a wave function that just appeared? 1198 00:52:32,050 --> 00:52:33,252 It's just barely bound. 1199 00:52:33,252 --> 00:52:35,210 Well, first off, what does it mean to be bound? 1200 00:52:35,210 --> 00:52:36,370 Let's just step back and remember for now. 1201 00:52:36,370 --> 00:52:37,870 What does it mean to be a bound state? 1202 00:52:37,870 --> 00:52:39,495 It means you're an energy eigenfunction 1203 00:52:39,495 --> 00:52:40,930 and you're localized. 1204 00:52:40,930 --> 00:52:43,600 Your wave function falls off at infinity. 1205 00:52:43,600 --> 00:52:45,070 Now, if it falls off at infinity, 1206 00:52:45,070 --> 00:52:46,554 do these guys fall off at infinity? 1207 00:52:46,554 --> 00:52:48,095 These wave functions, sure, they fall 1208 00:52:48,095 --> 00:52:49,940 of with an exponential damping. 1209 00:52:49,940 --> 00:52:51,898 And in particular, let's look at the right hand 1210 00:52:51,898 --> 00:52:52,690 side of the well. 1211 00:52:52,690 --> 00:52:56,110 This new bound state is appearing just at zero energy. 1212 00:52:56,110 --> 00:52:58,130 So out here, what is the wave function? 1213 00:52:58,130 --> 00:52:59,860 It's e to the minus alpha x. 1214 00:52:59,860 --> 00:53:02,560 But what's alpha? 1215 00:53:02,560 --> 00:53:04,560 Zero, right? 1216 00:53:04,560 --> 00:53:07,020 There it is, zero. 1217 00:53:07,020 --> 00:53:09,572 So this is e to the minus alpha x where alpha is equal to 0. 1218 00:53:09,572 --> 00:53:10,280 This is constant. 1219 00:53:13,040 --> 00:53:15,190 So what does the wave function look like? 1220 00:53:15,190 --> 00:53:17,540 Well, the wave function, again over the same domain-- 1221 00:53:17,540 --> 00:53:19,360 Here's 0, l. 1222 00:53:23,607 --> 00:53:24,690 And here's the value zero. 1223 00:53:24,690 --> 00:53:28,184 We know that in this domain it's oscillatory, 1224 00:53:28,184 --> 00:53:29,600 and in this domain, it's constant. 1225 00:53:36,610 --> 00:53:40,515 And actually, since we know that it's the first excited state, 1226 00:53:40,515 --> 00:53:42,130 we know that it does this. 1227 00:53:46,340 --> 00:53:51,990 So if we make the well ever so slightly deeper, 1228 00:53:51,990 --> 00:53:54,070 ever so slightly deeper, which means 1229 00:53:54,070 --> 00:53:57,320 making the radius of the circle ever so slightly larger, 1230 00:53:57,320 --> 00:54:00,450 we will get a nonzero value for the alpha of this solution. 1231 00:54:00,450 --> 00:54:00,950 Right? 1232 00:54:00,950 --> 00:54:02,270 It'll be just tiny. 1233 00:54:02,270 --> 00:54:03,637 But it'll be nonzero. 1234 00:54:03,637 --> 00:54:05,720 So we make the well just a little tiny bit deeper. 1235 00:54:05,720 --> 00:54:06,428 We get something. 1236 00:54:06,428 --> 00:54:08,130 OK, good. 1237 00:54:08,130 --> 00:54:10,310 So what's going to happen to the wave function? 1238 00:54:10,310 --> 00:54:12,320 Well, instead of going flat. 1239 00:54:12,320 --> 00:54:13,400 This is going to curve. 1240 00:54:13,400 --> 00:54:14,649 It's got a little more energy. 1241 00:54:14,649 --> 00:54:16,050 There's a little more curvature. 1242 00:54:16,050 --> 00:54:18,950 So it curves just a little tiny bit more. 1243 00:54:18,950 --> 00:54:21,575 And then it matches onto a very gradually decaying exponential. 1244 00:54:24,710 --> 00:54:25,210 OK. 1245 00:54:25,210 --> 00:54:27,920 So what's happening as we take this bound state, the second 1246 00:54:27,920 --> 00:54:30,510 bound state, and we make the well a little more shallow? 1247 00:54:30,510 --> 00:54:32,450 We make the well a little more shallow. 1248 00:54:32,450 --> 00:54:34,240 It's a little less curved inside. 1249 00:54:34,240 --> 00:54:37,490 And the evanescent tails, the exponential tails 1250 00:54:37,490 --> 00:54:40,210 become longer and longer and longer and broader 1251 00:54:40,210 --> 00:54:43,530 until they go off to infinity, until they're infinitely wide. 1252 00:54:43,530 --> 00:54:45,404 And is that normalizable anymore? 1253 00:54:45,404 --> 00:54:46,570 No, that's not normalizable. 1254 00:54:46,570 --> 00:54:48,720 So the state really isn't strictly localized 1255 00:54:48,720 --> 00:54:49,350 at that point. 1256 00:54:49,350 --> 00:54:50,990 It's not really a normalizable state. 1257 00:54:50,990 --> 00:54:54,490 And just when the state ceases to be normalizable, 1258 00:54:54,490 --> 00:54:56,627 it disappears. 1259 00:54:56,627 --> 00:54:58,210 We make the well just a little deeper, 1260 00:54:58,210 --> 00:55:00,180 and there's no state there at all. 1261 00:55:00,180 --> 00:55:00,680 OK. 1262 00:55:00,680 --> 00:55:02,179 So this tells you a very nice thing. 1263 00:55:02,179 --> 00:55:04,730 It's a good bit of intuition that when states are appearing 1264 00:55:04,730 --> 00:55:08,620 or disappearing, when states are at threshold as you 1265 00:55:08,620 --> 00:55:13,020 vary the depth, those threshold bound states have exceedingly 1266 00:55:13,020 --> 00:55:18,544 long evanescent tails, and they're just barely bound. 1267 00:55:18,544 --> 00:55:19,087 OK? 1268 00:55:19,087 --> 00:55:21,420 This turns out to have all sorts of useful consequences, 1269 00:55:21,420 --> 00:55:22,290 but let me move on. 1270 00:55:22,290 --> 00:55:23,800 Did that answer your question? 1271 00:55:23,800 --> 00:55:24,300 Good. 1272 00:55:24,300 --> 00:55:25,023 Yeah? 1273 00:55:25,023 --> 00:55:27,861 AUDIENCE: So in this case, the radius, as you say, 1274 00:55:27,861 --> 00:55:29,753 is proportional to the length. 1275 00:55:29,753 --> 00:55:32,118 Right? 1276 00:55:32,118 --> 00:55:36,385 But also we have intuition that as we increase 1277 00:55:36,385 --> 00:55:37,884 the length of the well, the energy's 1278 00:55:37,884 --> 00:55:39,351 going to keep increasing-- 1279 00:55:39,351 --> 00:55:41,141 PROFESSOR: Fantastic. 1280 00:55:41,141 --> 00:55:41,640 OK. 1281 00:55:41,640 --> 00:55:43,160 So what's up with that? 1282 00:55:43,160 --> 00:55:44,020 Right. 1283 00:55:44,020 --> 00:55:45,020 So the question is this. 1284 00:55:45,020 --> 00:55:47,210 We already have intuition that is if we take a finite well 1285 00:55:47,210 --> 00:55:48,760 and we make it a little bit wider, 1286 00:55:48,760 --> 00:55:51,050 the ground state energy should decrease. 1287 00:55:51,050 --> 00:55:53,550 The energy of the ground state should get deeper and deeper, 1288 00:55:53,550 --> 00:55:54,925 or the magnitude should increase, 1289 00:55:54,925 --> 00:55:55,950 another way to say it. 1290 00:55:55,950 --> 00:55:56,450 Right? 1291 00:55:56,450 --> 00:55:57,470 That was our intuition. 1292 00:55:57,470 --> 00:55:58,790 So let's check if that's true. 1293 00:55:58,790 --> 00:56:01,790 What happens if I take the ground state 1294 00:56:01,790 --> 00:56:03,350 some value of the radius, and then I 1295 00:56:03,350 --> 00:56:05,077 make the well a little bit wider. 1296 00:56:05,077 --> 00:56:06,910 Well, if I make the well a little bit wider, 1297 00:56:06,910 --> 00:56:09,080 what happens to r0, the radius of the circle? 1298 00:56:09,080 --> 00:56:12,020 Well, if I double the length of the well, the width 1299 00:56:12,020 --> 00:56:13,520 of the well, then it will double r0, 1300 00:56:13,520 --> 00:56:15,780 and it will double the radius of the circle. 1301 00:56:15,780 --> 00:56:18,142 So if I make it wider, r0 gets bigger, 1302 00:56:18,142 --> 00:56:19,350 and we go to a bigger circle. 1303 00:56:19,350 --> 00:56:23,100 And what happened to the energy of this state? 1304 00:56:23,100 --> 00:56:26,330 Yeah, it got deeper and deeper and deeper. 1305 00:56:26,330 --> 00:56:28,210 And meanwhile, as we make it wider, 1306 00:56:28,210 --> 00:56:30,987 as we make the well wider, the circle is getting bigger again. 1307 00:56:30,987 --> 00:56:32,820 And we're going to get more and more states. 1308 00:56:32,820 --> 00:56:36,200 So as we make the well wider, holding the depth fixed, 1309 00:56:36,200 --> 00:56:37,390 we get more and more states. 1310 00:56:37,390 --> 00:56:40,210 As we make the well deeper, holding the width fixed, 1311 00:56:40,210 --> 00:56:42,110 we get more and more states. 1312 00:56:42,110 --> 00:56:43,650 And so how do you trade off? 1313 00:56:43,650 --> 00:56:45,537 If I make it twice as wide, how much-- 1314 00:56:45,537 --> 00:56:46,620 So here's a good question. 1315 00:56:46,620 --> 00:56:51,140 Suppose I take a well, and it has n states. 1316 00:56:51,140 --> 00:56:53,720 Suppose I then make it twice as wide. 1317 00:56:53,720 --> 00:56:55,690 What must I do to the energy so that it still 1318 00:56:55,690 --> 00:56:59,181 has the same states with the same energies? 1319 00:56:59,181 --> 00:57:01,730 AUDIENCE: Divide it by 4. 1320 00:57:01,730 --> 00:57:02,730 PROFESSOR: Yep, exactly. 1321 00:57:02,730 --> 00:57:04,289 I've got to divide it by 4. 1322 00:57:04,289 --> 00:57:05,830 Because I've doubled the length, that 1323 00:57:05,830 --> 00:57:07,600 means the radius has gone up by four. 1324 00:57:07,600 --> 00:57:09,200 But I get exactly the same solutions 1325 00:57:09,200 --> 00:57:10,700 if I just bring this out. 1326 00:57:10,700 --> 00:57:12,570 Well, that's almost true. 1327 00:57:12,570 --> 00:57:14,200 So if I put a factor of 1/4 here, 1328 00:57:14,200 --> 00:57:19,160 that's almost true, except for the value of y is unchanged, 1329 00:57:19,160 --> 00:57:22,391 but the alpha hasn't changed. 1330 00:57:22,391 --> 00:57:22,890 Sorry. 1331 00:57:22,890 --> 00:57:24,875 The alpha has changed because there's an l. 1332 00:57:24,875 --> 00:57:25,770 So y is fixed. 1333 00:57:25,770 --> 00:57:29,274 But alpha's changed because of the l. 1334 00:57:29,274 --> 00:57:31,190 So the reason that it's useful to write things 1335 00:57:31,190 --> 00:57:33,740 in these dimensionless forms is that you can see the play off 1336 00:57:33,740 --> 00:57:36,550 of the various different dimensionful parameters 1337 00:57:36,550 --> 00:57:39,270 in changing the answer. 1338 00:57:39,270 --> 00:57:41,210 Other questions. 1339 00:57:41,210 --> 00:57:42,550 OK. 1340 00:57:42,550 --> 00:57:43,745 So a couple of comments. 1341 00:57:43,745 --> 00:57:45,620 So the first is let's just check to make sure 1342 00:57:45,620 --> 00:57:46,630 that this makes sense. 1343 00:57:46,630 --> 00:57:49,262 We have already solved this problem. 1344 00:57:49,262 --> 00:57:50,720 We solved this problem a while ago, 1345 00:57:50,720 --> 00:57:52,950 but we solved it in a particular limit. 1346 00:57:52,950 --> 00:57:56,750 We solved it in the limit that the potential one's arbitrarily 1347 00:57:56,750 --> 00:57:58,804 deep. 1348 00:57:58,804 --> 00:58:01,039 Right? 1349 00:58:01,039 --> 00:58:03,080 So when the potential one's are arbitrarily deep, 1350 00:58:03,080 --> 00:58:05,390 holding the width fixed, that was the infinite well. 1351 00:58:05,390 --> 00:58:07,480 That was the first problem we solved. 1352 00:58:07,480 --> 00:58:11,100 So let's make sure that we recover this in that limit. 1353 00:58:11,100 --> 00:58:14,470 So what happens as we make the well arbitrarily deep, 1354 00:58:14,470 --> 00:58:18,030 holding the length fixed or the width fixed. 1355 00:58:18,030 --> 00:58:19,680 So if we make this arbitrarily deep, 1356 00:58:19,680 --> 00:58:21,270 v0 is getting arbitrarily large. 1357 00:58:21,270 --> 00:58:22,460 That means r0 is getting-- 1358 00:58:22,460 --> 00:58:25,410 We've got a huge circle. 1359 00:58:25,410 --> 00:58:27,520 So what do these solutions look like when 1360 00:58:27,520 --> 00:58:29,750 we have a huge circle. 1361 00:58:29,750 --> 00:58:31,110 Let me not do that here. 1362 00:58:33,696 --> 00:58:34,985 Let me do that here. 1363 00:58:38,290 --> 00:58:43,072 So if we make the potential nice and deep, 1364 00:58:43,072 --> 00:58:44,905 Let's think about what that plot looks like. 1365 00:58:47,640 --> 00:58:51,100 So again, that first plot looks identical with the tangents. 1366 00:59:09,160 --> 00:59:11,370 So on and so forth. 1367 00:59:11,370 --> 00:59:15,690 And what I want to do is I want to dot, dot, dot. 1368 00:59:15,690 --> 00:59:16,190 OK. 1369 00:59:16,190 --> 00:59:17,280 So this is way up there. 1370 00:59:22,580 --> 00:59:25,390 So these guys are basically vertical lines at this point. 1371 00:59:25,390 --> 00:59:28,770 So for very, very large values of y, and in particular, 1372 00:59:28,770 --> 00:59:31,470 for very large values that are of order the gigantically deep 1373 00:59:31,470 --> 00:59:36,819 radius r0, what does the circle look like? 1374 00:59:36,819 --> 00:59:39,360 So what does the second equation look like, the second curve? 1375 00:59:39,360 --> 00:59:40,970 Well, again it's circles. 1376 00:59:40,970 --> 00:59:43,490 But now it's a gigantic circle. 1377 00:59:43,490 --> 00:59:44,250 Yeah, exactly. 1378 00:59:44,250 --> 00:59:46,370 If it's a gigantic circle, it's basically flat. 1379 00:59:51,907 --> 00:59:53,740 It's not exactly flat, but it's almost flat. 1380 00:59:53,740 --> 00:59:54,390 It's a circle. 1381 01:00:02,620 --> 01:00:04,900 So what are the values? 1382 01:00:04,900 --> 01:00:06,090 And here's the key question. 1383 01:00:06,090 --> 01:00:07,400 What are the values of-- 1384 01:00:10,720 --> 01:00:11,750 Did do that right? 1385 01:00:11,750 --> 01:00:12,250 Yeah, OK. 1386 01:00:12,250 --> 01:00:12,750 Good. 1387 01:00:12,750 --> 01:00:20,010 So what are the values of the curve, where we get a solution? 1388 01:00:20,010 --> 01:00:26,436 The values now of z, are just exactly 1389 01:00:26,436 --> 01:00:28,310 on these vertical lines, on the separatrices. 1390 01:00:28,310 --> 01:00:30,955 The value of z is at pi over 2. 1391 01:00:34,410 --> 01:00:35,480 and 3pi over 2. 1392 01:00:38,220 --> 01:00:43,281 And then another one at 5pi over 2, and so on and so forth. 1393 01:00:43,281 --> 01:00:43,780 Right? 1394 01:00:43,780 --> 01:00:48,694 So what we find is that the allowed values of z-- 1395 01:00:48,694 --> 01:00:49,690 Sorry. 1396 01:00:49,690 --> 01:00:51,920 kl, yes, good. 1397 01:00:51,920 --> 01:00:58,140 --allowed values of z are equal to 2n plus 1 over pi. 1398 01:00:58,140 --> 01:01:00,159 Whoops 2n plus 1 over 2 times pi. 1399 01:01:00,159 --> 01:01:01,200 So let's just check that. 1400 01:01:01,200 --> 01:01:02,505 So n is 0. 1401 01:01:02,505 --> 01:01:04,630 That's 0 1/2 pi. 1402 01:01:04,630 --> 01:01:05,710 n is 1. 1403 01:01:05,710 --> 01:01:06,831 That's 3/2 pi. 1404 01:01:06,831 --> 01:01:07,330 Good. 1405 01:01:07,330 --> 01:01:09,420 So these are the values of z, which 1406 01:01:09,420 --> 01:01:18,580 says that kl is equal to 2n plus 1 upon 2 pi or k is equal to 2n 1407 01:01:18,580 --> 01:01:22,520 plus 1 over 2l, which is the width of this well 1408 01:01:22,520 --> 01:01:26,820 because it's from minus l to l pi. 1409 01:01:26,820 --> 01:01:29,355 So is this the correct answer for the infinite square well? 1410 01:01:33,330 --> 01:01:35,860 Are these the allowed values of k 1411 01:01:35,860 --> 01:01:41,330 inside the well for the infinite square well? 1412 01:01:41,330 --> 01:01:42,850 Almost. 1413 01:01:42,850 --> 01:01:45,720 Instead of 2n plus 1, it should just be n plus 1. 1414 01:01:45,720 --> 01:01:49,596 We seem to be missing about half of the energy eigenvalues. 1415 01:01:49,596 --> 01:01:51,096 AUDIENCE: That's only the even ones. 1416 01:01:51,096 --> 01:01:51,870 PROFESSOR: Yeah, thank you. 1417 01:01:51,870 --> 01:01:52,820 This is only the even ones. 1418 01:01:52,820 --> 01:01:54,030 We started out saying, oh, look. 1419 01:01:54,030 --> 01:01:55,510 Let's look only at the even ones. 1420 01:01:55,510 --> 01:01:58,710 Where do you think the odd ones are going to be? 1421 01:01:58,710 --> 01:02:02,450 Ah, well the odd ones, so this should be k even. 1422 01:02:02,450 --> 01:02:03,380 So what about k odd? 1423 01:02:03,380 --> 01:02:04,830 Well, we know the answer already-- 1424 01:02:04,830 --> 01:02:05,440 Whoops. 1425 01:02:05,440 --> 01:02:06,820 Odd, that's an odd spelling. 1426 01:02:06,820 --> 01:02:12,000 --should be equal to 2n over 2l plus 2 over 2l. 1427 01:02:12,000 --> 01:02:12,500 Whoops. 1428 01:02:12,500 --> 01:02:14,561 2 capital L, pi. 1429 01:02:14,561 --> 01:02:15,060 OK. 1430 01:02:15,060 --> 01:02:17,710 This is our guess just from matching on 1431 01:02:17,710 --> 01:02:18,902 to the infinite square well. 1432 01:02:18,902 --> 01:02:19,860 So what does that mean? 1433 01:02:19,860 --> 01:02:22,994 Well, that means it should be this one and this one. 1434 01:02:22,994 --> 01:02:25,160 So when you go through this exercise on your problem 1435 01:02:25,160 --> 01:02:28,215 set and you find the solutions for the odd, 1436 01:02:28,215 --> 01:02:30,740 and you repeat this analysis for the odd ground states. 1437 01:02:30,740 --> 01:02:31,710 What should you expect? 1438 01:02:31,710 --> 01:02:33,400 Well, you should expect to find this. 1439 01:02:33,400 --> 01:02:37,327 And what do you think the curves are 1440 01:02:37,327 --> 01:02:39,410 that you're going to use int he graphical solution 1441 01:02:39,410 --> 01:02:42,804 to do your transcendental equation? 1442 01:02:42,804 --> 01:02:44,470 Yeah, it's really tempting to say, look. 1443 01:02:44,470 --> 01:02:46,700 It's just going to be something shifted, like this. 1444 01:02:46,700 --> 01:02:47,200 OK. 1445 01:02:47,200 --> 01:02:53,850 So these are going to be the odd question mark, question mark. 1446 01:02:53,850 --> 01:02:55,330 OK. 1447 01:02:55,330 --> 01:02:58,245 So you'll check whether that's correct intuition or not 1448 01:02:58,245 --> 01:03:00,274 on your problem set. 1449 01:03:00,274 --> 01:03:01,230 OK. 1450 01:03:01,230 --> 01:03:03,514 Questions? 1451 01:03:03,514 --> 01:03:04,014 Yeah? 1452 01:03:04,014 --> 01:03:07,316 AUDIENCE: What about the lower end of these states, where 1453 01:03:07,316 --> 01:03:08,774 it won't have gone off high enough? 1454 01:03:08,774 --> 01:03:09,730 Is that [INAUDIBLE]? 1455 01:03:09,730 --> 01:03:10,605 PROFESSOR: Excellent. 1456 01:03:10,605 --> 01:03:12,930 So that's a really good question. 1457 01:03:12,930 --> 01:03:15,600 How to say it? 1458 01:03:15,600 --> 01:03:18,649 What we've done here to make this an infinite well-- 1459 01:03:18,649 --> 01:03:20,690 So the question, let me just repeat the question. 1460 01:03:20,690 --> 01:03:24,400 The question is, look, what about all the other states? 1461 01:03:24,400 --> 01:03:26,110 OK, it's true that r0 as gigantic. 1462 01:03:26,110 --> 01:03:28,650 But eventually, we'll go to a large enough z, where 1463 01:03:28,650 --> 01:03:30,570 it's the circle coming down over here too. 1464 01:03:30,570 --> 01:03:31,600 So what's up with that? 1465 01:03:31,600 --> 01:03:32,625 What are those states? 1466 01:03:32,625 --> 01:03:33,250 Where are they? 1467 01:03:33,250 --> 01:03:35,541 What do they mean in terms if the infinite square well? 1468 01:03:35,541 --> 01:03:39,798 Well, first off, what are the energies of those states? 1469 01:03:39,798 --> 01:03:40,652 AUDIENCE: Very low. 1470 01:03:40,652 --> 01:03:42,360 PROFESSOR: They're very low in magnitude, 1471 01:03:42,360 --> 01:03:44,555 which means they're close to what in absolute value? 1472 01:03:44,555 --> 01:03:45,180 AUDIENCE: Zero. 1473 01:03:45,180 --> 01:03:45,550 PROFESSOR: Zero. 1474 01:03:45,550 --> 01:03:46,110 They're close to zero. 1475 01:03:46,110 --> 01:03:47,819 So they're at the top of the finite well. 1476 01:03:47,819 --> 01:03:50,193 These are the states bound at the top of the finite well. 1477 01:03:50,193 --> 01:03:52,670 These are the states bound at the bottom the finite well. 1478 01:03:52,670 --> 01:03:55,824 But how many states are bound at the top of the finite well 1479 01:03:55,824 --> 01:03:58,240 when we take the limit that the well goes infinitely deep? 1480 01:04:01,630 --> 01:04:02,510 Yeah, none of them. 1481 01:04:02,510 --> 01:04:02,760 Right? 1482 01:04:02,760 --> 01:04:03,860 So when we make the well infinitely deep, 1483 01:04:03,860 --> 01:04:05,570 what we're saying is, pay no attention 1484 01:04:05,570 --> 01:04:07,050 to the top of the well. 1485 01:04:07,050 --> 01:04:08,550 Look only at the bottom of the well. 1486 01:04:08,550 --> 01:04:11,607 And if it's really deep, it's a pretty good approximation. 1487 01:04:11,607 --> 01:04:12,940 So that's what we're doing here. 1488 01:04:12,940 --> 01:04:13,731 We're saying, look. 1489 01:04:13,731 --> 01:04:15,190 Pay no attention. 1490 01:04:15,190 --> 01:04:16,960 There is no top of the well. 1491 01:04:16,960 --> 01:04:17,960 There's just the bottom. 1492 01:04:17,960 --> 01:04:19,030 And look at the energy eigenvalues. 1493 01:04:19,030 --> 01:04:19,880 Does that make sense? 1494 01:04:19,880 --> 01:04:21,770 So what we're saying is if you have a preposterously 1495 01:04:21,770 --> 01:04:23,750 deep well, the energies of a preposterously 1496 01:04:23,750 --> 01:04:25,870 deep well should be a good approximation 1497 01:04:25,870 --> 01:04:29,350 to the low-lying energies of an infinitely deep well. 1498 01:04:29,350 --> 01:04:33,180 Because it's way up there, what difference can it make? 1499 01:04:33,180 --> 01:04:35,180 And that's what we're seeing work out. 1500 01:04:35,180 --> 01:04:37,420 Did that answer your question? 1501 01:04:37,420 --> 01:04:38,350 AUDIENCE: Wait. 1502 01:04:38,350 --> 01:04:39,696 I might have this backward. 1503 01:04:39,696 --> 01:04:42,070 But when it says the high energy instead of looking right 1504 01:04:42,070 --> 01:04:43,275 at the top of the well-- 1505 01:04:43,275 --> 01:04:43,550 PROFESSOR: Yeah. 1506 01:04:43,550 --> 01:04:43,950 OK, good. 1507 01:04:43,950 --> 01:04:45,730 So this is an important bit of intuition. 1508 01:04:45,730 --> 01:04:49,590 So when we say this is the energy zero, 1509 01:04:49,590 --> 01:04:55,277 and the potential has a minimum at minus v0, 1510 01:04:55,277 --> 01:04:57,360 and we're measuring the energies relative to zero, 1511 01:04:57,360 --> 01:05:00,080 then the states at the top of the potential well 1512 01:05:00,080 --> 01:05:02,100 are the states with energy close to zero. 1513 01:05:02,100 --> 01:05:03,975 And the states at the bottom of the potential 1514 01:05:03,975 --> 01:05:07,550 well are those states with the energy of order v0. 1515 01:05:07,550 --> 01:05:08,050 cool? 1516 01:05:08,050 --> 01:05:12,010 And what are the energies of all these states? 1517 01:05:12,010 --> 01:05:12,920 They're order of v0. 1518 01:05:12,920 --> 01:05:13,672 Right? 1519 01:05:13,672 --> 01:05:14,380 Because this is-- 1520 01:05:14,380 --> 01:05:15,480 Are they exactly v0? 1521 01:05:18,080 --> 01:05:19,392 No, because this isn't linear. 1522 01:05:19,392 --> 01:05:20,350 It's actually a circle. 1523 01:05:20,350 --> 01:05:21,849 So there's going to be a correction, 1524 01:05:21,849 --> 01:05:23,720 and the correction is going to be quadratic. 1525 01:05:23,720 --> 01:05:25,636 If you work out that correction, it's correct. 1526 01:05:25,636 --> 01:05:28,410 The depth above the bottom of the potential 1527 01:05:28,410 --> 01:05:31,720 is correct for the energy of the corresponding infinite well 1528 01:05:31,720 --> 01:05:32,966 problem. 1529 01:05:32,966 --> 01:05:35,920 I'll leave that to you as an exercise. 1530 01:05:35,920 --> 01:05:37,800 Other questions? 1531 01:05:37,800 --> 01:05:39,210 OK. 1532 01:05:39,210 --> 01:05:41,050 So there's another limit of this system 1533 01:05:41,050 --> 01:05:43,010 that's fun to think about. 1534 01:05:43,010 --> 01:05:44,510 So this was the infinite well limit. 1535 01:05:47,110 --> 01:05:49,999 What I want to do is I want to take advantage 1536 01:05:49,999 --> 01:05:52,290 of the observation we made a second ago that as we make 1537 01:05:52,290 --> 01:05:54,680 the well deeper, we get more states. 1538 01:05:54,680 --> 01:05:58,720 As we make the well more narrow, we get fewer states. 1539 01:05:58,720 --> 01:06:05,300 To trade that off, consider the following limit. 1540 01:06:05,300 --> 01:06:14,829 I want to take a potential well, which has a ground state. 1541 01:06:14,829 --> 01:06:16,370 What does the ground state look like? 1542 01:06:16,370 --> 01:06:18,453 So the ground state wave function is going to be-- 1543 01:06:18,453 --> 01:06:19,940 So here's zero. 1544 01:06:19,940 --> 01:06:22,385 And here's exponentially growing. 1545 01:06:22,385 --> 01:06:23,800 Here's exponentially decreasing. 1546 01:06:23,800 --> 01:06:26,425 And if it's a ground state, how many nodes will it have inside? 1547 01:06:28,010 --> 01:06:29,760 How many nodes will the ground state have? 1548 01:06:29,760 --> 01:06:30,634 AUDIENCE: Zero. 1549 01:06:30,634 --> 01:06:31,300 PROFESSOR: Zero. 1550 01:06:31,300 --> 01:06:31,820 Good. 1551 01:06:31,820 --> 01:06:34,017 OK. 1552 01:06:34,017 --> 01:06:36,100 So the ground state will look something like this. 1553 01:06:36,100 --> 01:06:39,340 We more conventionally draw it like this. 1554 01:06:39,340 --> 01:06:42,870 But just for fun, I'm going to draw it in this fashion. 1555 01:06:42,870 --> 01:06:45,330 In particular, it has some slope here. 1556 01:06:45,330 --> 01:06:46,700 And it has some slope here. 1557 01:06:46,700 --> 01:06:47,280 Oh, shoot. 1558 01:06:47,280 --> 01:06:48,800 Did I? 1559 01:06:48,800 --> 01:06:50,028 Yes, I did. 1560 01:06:50,028 --> 01:06:51,780 Dammit. 1561 01:06:51,780 --> 01:06:55,430 I just erased the one thing that I wanted to hold onto. 1562 01:06:55,430 --> 01:06:55,930 OK. 1563 01:06:55,930 --> 01:06:57,164 So there's my wave function. 1564 01:06:57,164 --> 01:06:58,580 It has some particular slope here. 1565 01:06:58,580 --> 01:07:00,029 It has some particular slope here. 1566 01:07:00,029 --> 01:07:01,820 And this is the ground state wave function, 1567 01:07:01,820 --> 01:07:02,609 with some energy. 1568 01:07:02,609 --> 01:07:03,150 I don't know. 1569 01:07:03,150 --> 01:07:03,920 I'll call it this. 1570 01:07:08,565 --> 01:07:10,190 Now, what I want to do is we've already 1571 01:07:10,190 --> 01:07:14,130 shown that as we make the well more and more shallow and more 1572 01:07:14,130 --> 01:07:17,849 and more narrow, the energy of the ground state 1573 01:07:17,849 --> 01:07:19,140 gets closer and closer to zero. 1574 01:07:19,140 --> 01:07:20,870 But there remains always a bound state. 1575 01:07:20,870 --> 01:07:22,580 There is always at least one bound state. 1576 01:07:22,580 --> 01:07:24,230 We proved that. 1577 01:07:24,230 --> 01:07:27,776 Proved, as a physicist would. 1578 01:07:27,776 --> 01:07:28,650 So I want to do that. 1579 01:07:28,650 --> 01:07:29,500 I want to take this seriously. 1580 01:07:29,500 --> 01:07:31,166 But here's the limit I want to consider. 1581 01:07:31,166 --> 01:07:34,070 Consider the limit that we make the potential v goes 1582 01:07:34,070 --> 01:07:38,920 to infinity, v0, while making l go to zero. 1583 01:07:38,920 --> 01:07:41,010 So what I want to do is I want to take this thing, 1584 01:07:41,010 --> 01:07:44,464 and I want to make it deeper and deeper, but more 1585 01:07:44,464 --> 01:07:45,130 and more narrow. 1586 01:07:52,320 --> 01:07:56,090 If I do this repeatedly, eventually I 1587 01:07:56,090 --> 01:07:58,860 will get a delta function. 1588 01:07:58,860 --> 01:08:00,360 And I will get a delta function if I 1589 01:08:00,360 --> 01:08:02,820 hold the area of this guy fixed. 1590 01:08:02,820 --> 01:08:03,320 Yeah? 1591 01:08:03,320 --> 01:08:11,050 So if I do so, holding the area under this plot fixed, 1592 01:08:11,050 --> 01:08:13,876 I will get a delta function. 1593 01:08:13,876 --> 01:08:14,900 Everyone cool with that? 1594 01:08:18,347 --> 01:08:20,180 So let's think, though, quickly about what's 1595 01:08:20,180 --> 01:08:22,263 going to happen to the ground state wave function. 1596 01:08:27,240 --> 01:08:31,319 So as I make the potential, let's take this wave function, 1597 01:08:31,319 --> 01:08:34,979 and let's look at this version of the potential. 1598 01:08:34,979 --> 01:08:36,520 So as I make the potential deeper, 1599 01:08:36,520 --> 01:08:38,850 what happens to the rate of the oscillation inside 1600 01:08:38,850 --> 01:08:39,974 or to the curvature inside? 1601 01:08:43,290 --> 01:08:44,000 It increases. 1602 01:08:44,000 --> 01:08:44,665 Right? 1603 01:08:44,665 --> 01:08:48,450 So the system is oscillating more, it changes more rapidly, 1604 01:08:48,450 --> 01:08:51,060 because phi double dot or phi double 1605 01:08:51,060 --> 01:08:53,655 prime is equal to v minus e phi. 1606 01:08:53,655 --> 01:08:54,779 It oscillates more rapidly. 1607 01:08:54,779 --> 01:08:56,749 So to make it deeper, the system tends 1608 01:08:56,749 --> 01:08:57,939 to oscillate more rapidly. 1609 01:09:00,720 --> 01:09:04,290 However, as we make it more narrow, 1610 01:09:04,290 --> 01:09:08,389 the system doesn't have as far to oscillate. 1611 01:09:08,389 --> 01:09:09,930 So it oscillates more rapidly, but it 1612 01:09:09,930 --> 01:09:11,939 doesn't oscillate as far. 1613 01:09:11,939 --> 01:09:13,284 So what's going to happen? 1614 01:09:13,284 --> 01:09:16,410 Well, as we make it more and more narrow and deeper 1615 01:09:16,410 --> 01:09:21,450 and deeper, we again have the wave function coming in. 1616 01:09:21,450 --> 01:09:24,960 And now it oscillates very rapidly. 1617 01:09:24,960 --> 01:09:26,696 Let's do it again. 1618 01:09:26,696 --> 01:09:31,630 The wave function comes in, and it oscillates very rapidly. 1619 01:09:31,630 --> 01:09:33,609 And the it evanescent tail out. 1620 01:09:33,609 --> 01:09:39,450 And now as we have a delta function, exponential damping, 1621 01:09:39,450 --> 01:09:42,750 it oscillates extremely rapidly over an arbitrarily short 1622 01:09:42,750 --> 01:09:46,500 distance and gives us the kink that we 1623 01:09:46,500 --> 01:09:48,500 knew at the very beginning we should expect when 1624 01:09:48,500 --> 01:09:50,432 the potential is a delta function. 1625 01:09:50,432 --> 01:09:51,770 Right? 1626 01:09:51,770 --> 01:09:54,069 From our qualitative structure of the wave function 1627 01:09:54,069 --> 01:09:55,330 at the very beginning we saw that when 1628 01:09:55,330 --> 01:09:56,788 we have a delta function potential, 1629 01:09:56,788 --> 01:09:59,620 we should see a kink in the wave function. 1630 01:09:59,620 --> 01:10:03,780 Because again, if we have phi prime prime is delta function 1631 01:10:03,780 --> 01:10:06,220 discontinuous, phi prime is the integral of this. 1632 01:10:06,220 --> 01:10:08,805 This is a step, and phi is continuous. 1633 01:10:13,330 --> 01:10:14,970 And so here we have a step function. 1634 01:10:14,970 --> 01:10:18,360 We get a discontinuity in the second derivative. 1635 01:10:18,360 --> 01:10:20,340 Here we have a delta function in the potential, 1636 01:10:20,340 --> 01:10:22,440 and we get a discontinuity in the first derivative 1637 01:10:22,440 --> 01:10:26,160 if we get a kink in the wave function. 1638 01:10:26,160 --> 01:10:27,914 Yeah? 1639 01:10:27,914 --> 01:10:30,192 AUDIENCE: Would we get a jump there? 1640 01:10:30,192 --> 01:10:30,900 PROFESSOR: Sorry. 1641 01:10:30,900 --> 01:10:31,910 Say again. 1642 01:10:31,910 --> 01:10:32,631 For e1? 1643 01:10:32,631 --> 01:10:33,256 AUDIENCE: Yeah. 1644 01:10:33,256 --> 01:10:34,460 Would we get a jump? 1645 01:10:34,460 --> 01:10:36,036 PROFESSOR: Very good question. 1646 01:10:36,036 --> 01:10:37,410 So let me do this more seriously. 1647 01:10:37,410 --> 01:10:39,990 Let's do this more carefully. 1648 01:10:39,990 --> 01:10:42,130 So the question is, for the first excited state, 1649 01:10:42,130 --> 01:10:43,727 do we get a jump? 1650 01:10:43,727 --> 01:10:44,810 Do we get a discontinuity? 1651 01:10:44,810 --> 01:10:46,680 What do we get for the first excited state? 1652 01:10:46,680 --> 01:10:47,225 Right? 1653 01:10:47,225 --> 01:10:48,750 So let's talk about that in detail. 1654 01:10:48,750 --> 01:10:50,040 It's a very good question. 1655 01:10:50,040 --> 01:10:56,132 Example v is equal to minus v0 delta of x. 1656 01:10:56,132 --> 01:10:58,090 Now, here I want to just warn you of something. 1657 01:10:58,090 --> 01:11:00,670 This is totally standard notation for these problems, 1658 01:11:00,670 --> 01:11:02,950 but you should be careful about dimensions. 1659 01:11:02,950 --> 01:11:06,020 What are the dimensions of v0, the parameter of v0? 1660 01:11:09,070 --> 01:11:10,650 It's tempting to say energy. 1661 01:11:10,650 --> 01:11:11,377 That's an energy. 1662 01:11:11,377 --> 01:11:12,085 That's an energy. 1663 01:11:12,085 --> 01:11:12,890 But wait. 1664 01:11:12,890 --> 01:11:14,823 What are the dimensions of the delta function? 1665 01:11:14,823 --> 01:11:15,610 AUDIENCE: Length. 1666 01:11:15,610 --> 01:11:16,985 PROFESSOR: Whatever length right? 1667 01:11:16,985 --> 01:11:20,190 Because we know that delta of alpha x 1668 01:11:20,190 --> 01:11:23,500 is equal to 1 over norm alpha delta of x. 1669 01:11:27,290 --> 01:11:29,047 So if I write delta of x, which is always 1670 01:11:29,047 --> 01:11:31,380 a slightly ballsy thing to do because this should really 1671 01:11:31,380 --> 01:11:33,421 be dimensionless, but if I write delta of x, then 1672 01:11:33,421 --> 01:11:35,470 this has units of 1 over length, which 1673 01:11:35,470 --> 01:11:39,270 means this must have units of length times energy. 1674 01:11:42,081 --> 01:11:42,580 OK. 1675 01:11:42,580 --> 01:11:44,700 Just a little warning, when you check your answers 1676 01:11:44,700 --> 01:11:46,030 on a problem, you always want to make sure 1677 01:11:46,030 --> 01:11:47,820 that they're dimensionally consistent. 1678 01:11:47,820 --> 01:11:49,486 And so it will be important to make sure 1679 01:11:49,486 --> 01:11:52,810 that you use the energy times the length 1680 01:11:52,810 --> 01:11:55,330 for the dimensions of that beast. 1681 01:11:55,330 --> 01:11:57,910 So my question here is, is there a bound state? 1682 01:11:57,910 --> 01:11:59,580 So for this example, for this potential, 1683 01:11:59,580 --> 01:12:02,610 the delta function potential bound state, which 1684 01:12:02,610 --> 01:12:09,890 again is this guy, is there a bound state? 1685 01:12:12,460 --> 01:12:16,310 So again, we just ran through the intuition 1686 01:12:16,310 --> 01:12:19,410 where we made the potential deep and deeper and deeper, 1687 01:12:19,410 --> 01:12:26,860 v0 divided by epsilon over width epsilon. 1688 01:12:26,860 --> 01:12:33,710 So v0, in order for this to be an energy 1689 01:12:33,710 --> 01:12:35,689 has to be an energy times a length 1690 01:12:35,689 --> 01:12:37,480 because we're going to divide it by length. 1691 01:12:37,480 --> 01:12:39,313 So this is going to give us a delta function 1692 01:12:39,313 --> 01:12:40,290 potential in the limit. 1693 01:12:40,290 --> 01:12:41,748 We have an intuition that we should 1694 01:12:41,748 --> 01:12:43,209 get a bound state with a kink. 1695 01:12:43,209 --> 01:12:44,500 But let's check that intuition. 1696 01:12:44,500 --> 01:12:47,300 We want to actually solve this problem. 1697 01:12:47,300 --> 01:12:49,170 So we'll do the same thing we did before. 1698 01:12:49,170 --> 01:12:55,390 We now write the general solution in the places 1699 01:12:55,390 --> 01:12:58,610 where the potential is constant, which is on the left 1700 01:12:58,610 --> 01:12:59,920 and on the right. 1701 01:12:59,920 --> 01:13:02,460 And then we want to impose appropriate boundary conditions 1702 01:13:02,460 --> 01:13:05,562 at the interface and at infinity, where these are going 1703 01:13:05,562 --> 01:13:08,020 to be normalizable, and this is whatever the right boundary 1704 01:13:08,020 --> 01:13:08,590 conditions are. 1705 01:13:08,590 --> 01:13:10,050 So we are going to have to derive the appropriate boundary 1706 01:13:10,050 --> 01:13:10,700 conditions. 1707 01:13:10,700 --> 01:13:12,020 So let's just do that quickly. 1708 01:13:12,020 --> 01:13:17,710 So phi with definite e is equal to a-- 1709 01:13:17,710 --> 01:13:20,020 So in this region in the left, it's 1710 01:13:20,020 --> 01:13:22,270 either growing or decreasing exponential. 1711 01:13:22,270 --> 01:13:27,010 So ae to the alpha x plus be to the minus alpha x. 1712 01:13:27,010 --> 01:13:29,370 And this is x less than 0. 1713 01:13:29,370 --> 01:13:34,130 And ce to the alpha x plus de to the minus 1714 01:13:34,130 --> 01:13:36,970 alpha x for x greater than 0. 1715 01:13:36,970 --> 01:13:39,047 So first off, let's hit normalizability. 1716 01:13:39,047 --> 01:13:40,630 What must be true for normalizability? 1717 01:13:43,255 --> 01:13:45,740 Yeah, they'd better be converging to zero here 1718 01:13:45,740 --> 01:13:48,840 and converging to zero here, which means that c had better 1719 01:13:48,840 --> 01:13:52,381 vanish and b had better vanish. 1720 01:13:52,381 --> 01:13:52,880 OK. 1721 01:13:52,880 --> 01:13:56,260 So those guys are gone from normalizability. 1722 01:13:56,260 --> 01:13:59,170 And meanwhile, if this is symmetric, 1723 01:13:59,170 --> 01:14:01,790 what is going to be true of the ground state? 1724 01:14:01,790 --> 01:14:03,090 It's going to be symmetric. 1725 01:14:03,090 --> 01:14:07,214 So a must be equal to d. 1726 01:14:07,214 --> 01:14:08,490 Great. 1727 01:14:08,490 --> 01:14:12,220 So a is now just some overall normalization constant, 1728 01:14:12,220 --> 01:14:13,720 which we can fix from normalization. 1729 01:14:13,720 --> 01:14:15,595 So it looks like this should be the solution. 1730 01:14:15,595 --> 01:14:17,060 We have an exponential. 1731 01:14:17,060 --> 01:14:18,420 We have an exponential. 1732 01:14:18,420 --> 01:14:23,080 But there's one more boundary condition to fix. 1733 01:14:23,080 --> 01:14:25,440 We have to satisfy some matching. 1734 01:14:25,440 --> 01:14:27,190 We have to satisfy the boundary conditions 1735 01:14:27,190 --> 01:14:28,106 at the delta function. 1736 01:14:28,106 --> 01:14:30,310 So what are those? 1737 01:14:30,310 --> 01:14:32,770 What are those matching conditions? 1738 01:14:32,770 --> 01:14:36,820 So we can get that from the energy eigenvalue equation, 1739 01:14:36,820 --> 01:14:39,850 which says that phi prime prime is 1740 01:14:39,850 --> 01:14:41,851 equal to h bar squared upon 2m. 1741 01:14:41,851 --> 01:14:42,350 Sorry. 1742 01:14:45,027 --> 01:14:46,110 Get your dimensions right. 1743 01:14:46,110 --> 01:14:49,680 So it's 2m over h bar squared v minus e. 1744 01:14:49,680 --> 01:14:54,470 In this case, v is equal to minus v0 delta function. 1745 01:14:58,690 --> 01:14:59,910 That's very strange. 1746 01:14:59,910 --> 01:15:05,392 So minus 2m over h bar squared v0 delta of x minus e-- 1747 01:15:05,392 --> 01:15:06,350 I pulled out the minus. 1748 01:15:06,350 --> 01:15:09,360 --so plus e phi. 1749 01:15:12,090 --> 01:15:13,860 So this must be true at every point. 1750 01:15:13,860 --> 01:15:15,318 This of course, is zero everywhere, 1751 01:15:15,318 --> 01:15:20,460 except for at the origin. 1752 01:15:20,460 --> 01:15:22,420 So what we want to do is we want to turn this 1753 01:15:22,420 --> 01:15:24,029 into a boundary condition. 1754 01:15:24,029 --> 01:15:25,820 And we know what the boundary condition is. 1755 01:15:25,820 --> 01:15:28,604 If v is a delta function, that means that phi prime prime 1756 01:15:28,604 --> 01:15:30,270 is also a delta function or proportional 1757 01:15:30,270 --> 01:15:31,390 to a delta function. 1758 01:15:31,390 --> 01:15:33,350 That means that phi prime is a step function. 1759 01:15:33,350 --> 01:15:34,308 And how did I get that? 1760 01:15:34,308 --> 01:15:36,622 I got that by integrating phi prime prime. 1761 01:15:36,622 --> 01:15:38,955 You integrate a delta function, you get a step function. 1762 01:15:38,955 --> 01:15:39,960 Well, that's cool. 1763 01:15:39,960 --> 01:15:42,700 How do we figure out what step function discontinuity 1764 01:15:42,700 --> 01:15:43,750 gives us? 1765 01:15:43,750 --> 01:15:44,530 Let's integrate. 1766 01:15:44,530 --> 01:15:46,488 Let's integrate right across the delta function 1767 01:15:46,488 --> 01:15:48,730 and figure out what the discontinuity is. 1768 01:15:48,730 --> 01:15:50,880 So let's take this equation, integrate it 1769 01:15:50,880 --> 01:15:52,649 from minus epsilon to epsilon, where 1770 01:15:52,649 --> 01:15:53,940 epsilon is a very small number. 1771 01:15:56,780 --> 01:15:58,150 And that's epsilon to epsilon. 1772 01:15:58,150 --> 01:16:00,025 So what is this going to give us on the left? 1773 01:16:00,025 --> 01:16:01,570 Well, integral of a total derivative 1774 01:16:01,570 --> 01:16:04,680 is just the value of the thing at a value to the point. 1775 01:16:04,680 --> 01:16:09,040 So this is going to be phi prime at epsilon 1776 01:16:09,040 --> 01:16:11,620 minus phi prime and minus epsilon. 1777 01:16:11,620 --> 01:16:12,460 What does that mean? 1778 01:16:12,460 --> 01:16:15,230 The difference between the derivative 1779 01:16:15,230 --> 01:16:17,440 just after the origin and just before the origin. 1780 01:16:17,440 --> 01:16:20,350 This is the discontinuity for very small epsilon. 1781 01:16:20,350 --> 01:16:22,220 This is the discontinuity of the derivative 1782 01:16:22,220 --> 01:16:23,720 at the origin at the delta function. 1783 01:16:23,720 --> 01:16:26,053 And we already expected it to have a step discontinuity. 1784 01:16:26,053 --> 01:16:26,980 And there it is. 1785 01:16:26,980 --> 01:16:28,242 And how big is it? 1786 01:16:28,242 --> 01:16:29,700 Well on the right hand side we have 1787 01:16:29,700 --> 01:16:32,475 2m minus 2m upon h bar squared. 1788 01:16:32,475 --> 01:16:33,850 And we're going to get two terms. 1789 01:16:33,850 --> 01:16:36,050 We get a term from integrating the first term. 1790 01:16:36,050 --> 01:16:39,700 But over this narrow window, around, 1791 01:16:39,700 --> 01:16:43,130 let's say epsilon was here, over this narrow window we can treat 1792 01:16:43,130 --> 01:16:47,160 the wave function as being more or less constant. 1793 01:16:47,160 --> 01:16:51,042 But in any case, it's continuous. 1794 01:16:51,042 --> 01:16:52,250 And this is a delta function. 1795 01:16:52,250 --> 01:16:54,874 So we know what we get from the integral of the delta function. 1796 01:16:54,874 --> 01:16:58,870 We just get the value v0 phi at the delta function. 1797 01:16:58,870 --> 01:17:01,660 Phi of the zero at the delta functions, so phi 0. 1798 01:17:01,660 --> 01:17:05,230 We get a second term, which is plus the energy integrated 1799 01:17:05,230 --> 01:17:06,491 against phi. 1800 01:17:06,491 --> 01:17:07,490 The energy's a constant. 1801 01:17:07,490 --> 01:17:09,150 And phi is continuous. 1802 01:17:09,150 --> 01:17:11,350 So this, whatever else you can say about it, 1803 01:17:11,350 --> 01:17:12,995 is roughly the constant value of phi 1804 01:17:12,995 --> 01:17:15,220 at the origin times the energy times the width, which 1805 01:17:15,220 --> 01:17:17,070 is epsilon. 1806 01:17:17,070 --> 01:17:20,466 So plus-order epsilon terms. 1807 01:17:20,466 --> 01:17:21,984 Everyone cool with that? 1808 01:17:21,984 --> 01:17:23,400 So now what I'm going to do is I'm 1809 01:17:23,400 --> 01:17:25,399 going to take the limit as epsilon goes to zero. 1810 01:17:28,830 --> 01:17:31,540 So I'm just going to take that the discontinuity just 1811 01:17:31,540 --> 01:17:32,040 across zero. 1812 01:17:32,040 --> 01:17:37,590 So this is going to give me, of this gives me 1813 01:17:37,590 --> 01:17:41,240 the change in the slope at the origin. 1814 01:17:41,240 --> 01:17:42,090 OK. 1815 01:17:42,090 --> 01:17:43,590 The derivative just after the origin 1816 01:17:43,590 --> 01:17:46,170 minus the derivative just before the origin is equal to-- 1817 01:17:46,170 --> 01:17:48,490 These order epsilon terms go away. 1818 01:17:48,490 --> 01:17:54,800 --minus 2m upon h bar squared v0 phi. 1819 01:17:54,800 --> 01:17:59,060 So that's my continuity. 1820 01:17:59,060 --> 01:18:00,680 That's the condition for continuity 1821 01:18:00,680 --> 01:18:03,410 of the derivative and appropriate discontinuity 1822 01:18:03,410 --> 01:18:07,540 of the first derivative at the origin. 1823 01:18:07,540 --> 01:18:10,510 And so this, when we plug-in these values 1824 01:18:10,510 --> 01:18:12,390 of this form for the wave function, 1825 01:18:12,390 --> 01:18:14,390 when we take a derivative, all we're going to do 1826 01:18:14,390 --> 01:18:16,919 is we're going to pick up an alpha. 1827 01:18:16,919 --> 01:18:18,460 And so when we work all of this out-- 1828 01:18:18,460 --> 01:18:20,879 I'm not going to go through the algebra. 1829 01:18:20,879 --> 01:18:22,920 You're going to go through it on the problem set. 1830 01:18:22,920 --> 01:18:25,450 --when we take this condition, when we impose this condition 1831 01:18:25,450 --> 01:18:27,270 with this wave function, it gives us 1832 01:18:27,270 --> 01:18:28,870 a very specific value for alpha. 1833 01:18:28,870 --> 01:18:33,560 This is only solvable if alpha is equal to mv0 1834 01:18:33,560 --> 01:18:34,640 upon h bar squared. 1835 01:18:38,840 --> 01:18:39,830 Good. 1836 01:18:39,830 --> 01:18:41,310 So let's just check the units. 1837 01:18:41,310 --> 01:18:44,870 So this is momentum times length, momentum times length. 1838 01:18:44,870 --> 01:18:46,600 This is mass. 1839 01:18:46,600 --> 01:18:48,940 This is an energy times a length. 1840 01:18:48,940 --> 01:18:52,770 So this has overall units of, p squared over m, 1841 01:18:52,770 --> 01:18:55,300 overall units of 1 upon length, which is what we wan. 1842 01:18:55,300 --> 01:18:56,900 So that's good. 1843 01:18:56,900 --> 01:18:59,030 So we get alpha is equal to mv0 upon h bar. 1844 01:18:59,030 --> 01:19:00,890 And that gives us the form of the potential. 1845 01:19:00,890 --> 01:19:03,640 And it also tells us that the energy, plugging this back in, 1846 01:19:03,640 --> 01:19:08,530 is equal to minus h bar squared alpha squared upon 2m, 1847 01:19:08,530 --> 01:19:10,770 which we could then plug-in the value of alpha 1848 01:19:10,770 --> 01:19:12,110 and solve for v0. 1849 01:19:12,110 --> 01:19:15,487 So what we found is that there is a single bound state 1850 01:19:15,487 --> 01:19:17,070 of the delta function potential, which 1851 01:19:17,070 --> 01:19:19,028 we could have gotten by just taking this limit. 1852 01:19:19,028 --> 01:19:20,960 It's a fun way to rederive the same result. 1853 01:19:20,960 --> 01:19:22,740 It's a nice check on your understanding. 1854 01:19:22,740 --> 01:19:25,370 So we find that there's a single bound state 1855 01:19:25,370 --> 01:19:27,110 of the delta function potential. 1856 01:19:27,110 --> 01:19:29,980 Now, what about an odd bound state? 1857 01:19:29,980 --> 01:19:32,180 We assumed at an important point that this was even. 1858 01:19:32,180 --> 01:19:35,610 What if I assume that it was odd? 1859 01:19:35,610 --> 01:19:38,120 One node, what if we had assumed that it was odd? 1860 01:19:38,120 --> 01:19:40,020 What would be true of the wave function? 1861 01:19:40,020 --> 01:19:42,860 Well for odd, this would be a, and this would be minus a. 1862 01:19:42,860 --> 01:19:46,100 So the value of the wave function at the origin is what? 1863 01:19:46,100 --> 01:19:46,700 Zero. 1864 01:19:46,700 --> 01:19:49,430 So that tells us the value of the wave function is zero. 1865 01:19:49,430 --> 01:19:51,780 What's the discontinuity? 1866 01:19:51,780 --> 01:19:52,340 Zero. 1867 01:19:52,340 --> 01:19:54,760 So it's as if there's no potential 1868 01:19:54,760 --> 01:19:57,480 because it has a zero right at the delta function. 1869 01:19:57,480 --> 01:19:59,430 Yeah? 1870 01:19:59,430 --> 01:20:03,160 But that means that this wave function, an odd wave function, 1871 01:20:03,160 --> 01:20:06,980 doesn't notice the delta function potential. 1872 01:20:06,980 --> 01:20:09,430 So is there a bound state? 1873 01:20:09,430 --> 01:20:11,100 No. 1874 01:20:11,100 --> 01:20:13,630 So how many bound states are there? 1875 01:20:13,630 --> 01:20:17,180 Always exactly one for the single isolated delta function. 1876 01:20:17,180 --> 01:20:18,560 On your problem set, you're going 1877 01:20:18,560 --> 01:20:21,390 to use the result of the single isolated delta function, 1878 01:20:21,390 --> 01:20:24,520 and more broadly you're going to derive the results 1879 01:20:24,520 --> 01:20:26,050 for two delta functions. 1880 01:20:26,050 --> 01:20:28,310 So you might say, why two delta functions? 1881 01:20:28,310 --> 01:20:30,470 And the answer is, the two delta function 1882 01:20:30,470 --> 01:20:33,080 problem, which involves no math-- 1883 01:20:33,080 --> 01:20:33,580 Right? 1884 01:20:33,580 --> 01:20:35,670 It's a totally straightforward, simple problem. 1885 01:20:35,670 --> 01:20:38,400 You can all do it right now on a piece of paper. 1886 01:20:38,400 --> 01:20:39,780 The two delta function problem is 1887 01:20:39,780 --> 01:20:42,370 going to turn out to be an awesome model 1888 01:20:42,370 --> 01:20:44,515 for the binding of atoms. 1889 01:20:44,515 --> 01:20:47,810 And we're going to use it as intuition on your problem set 1890 01:20:47,810 --> 01:20:50,890 to explain how quantum mechanical effects can lead 1891 01:20:50,890 --> 01:20:54,390 to an attractive force between two atoms. 1892 01:20:54,390 --> 01:20:55,760 See you next time. 1893 01:20:55,760 --> 01:20:57,310 [APPLAUSE]