1 00:00:00,000 --> 00:00:03,780 PROFESSOR: A new way of thinking about this 2 00:00:03,780 --> 00:00:07,485 is based on integral equations. 3 00:00:07,485 --> 00:00:11,070 So it's a nice method. 4 00:00:11,070 --> 00:00:15,540 It's less complicated than it seems, suddenly leads 5 00:00:15,540 --> 00:00:21,060 to some expressions for the scattering amplitude. 6 00:00:21,060 --> 00:00:26,625 We'll find another formula for this quantity, f of k of theta, 7 00:00:26,625 --> 00:00:32,056 when we cannot calculate it with phase shifts. 8 00:00:32,056 --> 00:00:34,380 But phase shifts is very powerful 9 00:00:34,380 --> 00:00:36,755 if your problem has spherical symmetry. 10 00:00:36,755 --> 00:00:39,825 It's probably the first thing you try, 11 00:00:39,825 --> 00:00:42,960 because there's nothing to prevent you from finding 12 00:00:42,960 --> 00:00:46,656 a solution, in that case. 13 00:00:46,656 --> 00:00:51,720 OK, so integral equations, so integral scattering 14 00:00:51,720 --> 00:01:05,630 equation, OK, integral scattering equation. 15 00:01:12,038 --> 00:01:13,810 So what are we solving? 16 00:01:13,810 --> 00:01:22,588 We're solving minus h squared over 2m Laplacian, 17 00:01:22,588 --> 00:01:32,170 plus v of r of potential, that is not necessarily spherical, 18 00:01:32,170 --> 00:01:39,810 equals or acts on a psi of r, to give you h squared, 19 00:01:39,810 --> 00:01:46,546 k squared over 2m, psi of r. 20 00:01:46,546 --> 00:01:49,905 So it's an equation for eigenstates, the Schrodinger 21 00:01:49,905 --> 00:01:51,830 equation. 22 00:01:51,830 --> 00:01:52,860 It's good. 23 00:01:52,860 --> 00:01:53,940 It's nice. 24 00:01:53,940 --> 00:01:56,100 And this is what we've been solving. 25 00:01:56,100 --> 00:02:00,170 We've been solving if for some particular cases. 26 00:02:00,170 --> 00:02:02,640 The case of physical interest for us 27 00:02:02,640 --> 00:02:05,370 was when there was a plane wave coming in 28 00:02:05,370 --> 00:02:11,505 from infinity and this spherical wave going out at the infinity. 29 00:02:11,505 --> 00:02:13,710 Now, at this moment, it's probably 30 00:02:13,710 --> 00:02:19,600 convenient to clean up the units from this equation a little. 31 00:02:19,600 --> 00:02:23,930 So I'll use a notation in which V of r 32 00:02:23,930 --> 00:02:33,535 is equal to h squared over 2m, U of r-- 33 00:02:33,535 --> 00:02:36,040 U still reminds you of a potential, I think. 34 00:02:36,040 --> 00:02:37,720 And it's a good thing. 35 00:02:37,720 --> 00:02:41,720 It's a definition of U of r. 36 00:02:41,720 --> 00:02:42,760 It's a definition. 37 00:02:46,012 --> 00:02:49,635 And in that way, I can get rid of the h 38 00:02:49,635 --> 00:02:52,875 bars, in this scattering equation. 39 00:02:52,875 --> 00:02:54,170 And what do we get? 40 00:02:54,170 --> 00:03:07,331 We get minus Laplacian plus U of r, acting on psi of r, 41 00:03:07,331 --> 00:03:12,420 is equal to k squared, acting on psi of r. 42 00:03:21,968 --> 00:03:24,450 OK, that looks nice. 43 00:03:24,450 --> 00:03:28,200 It probably is nicer if you put it in a way 44 00:03:28,200 --> 00:03:34,665 that the left hand side is kind of a nice, simple operator 45 00:03:34,665 --> 00:03:38,570 and the right hand side involves the potential. 46 00:03:38,570 --> 00:03:42,030 So I'll move it along, so that I can 47 00:03:42,030 --> 00:03:45,035 pass the now left squared to the right, 48 00:03:45,035 --> 00:03:47,420 but then make that the left hand side. 49 00:03:47,420 --> 00:03:50,700 So this will be Laplacian squared plus k 50 00:03:50,700 --> 00:04:03,180 squared on psi of r is equal to U of r, psi of r. 51 00:04:06,714 --> 00:04:11,815 OK, so we've done nothing so far, 52 00:04:11,815 --> 00:04:18,301 except put the equation in a way that stimulates our thinking. 53 00:04:18,301 --> 00:04:21,550 We kind of think of the right hand 54 00:04:21,550 --> 00:04:26,125 side as a source and the left hand side 55 00:04:26,125 --> 00:04:29,865 as kind of the equation you want to solve. 56 00:04:29,865 --> 00:04:33,450 Sometimes you have an equation like that. 57 00:04:33,450 --> 00:04:35,160 Then you can get this zero here. 58 00:04:35,160 --> 00:04:37,410 Well that's when the potential is zero. 59 00:04:37,410 --> 00:04:39,420 But sometimes there's a bit of potential, 60 00:04:39,420 --> 00:04:44,430 so the solution feeds back into this. 61 00:04:44,430 --> 00:04:50,540 Whenever you have an equation of this form, some nice operator-- 62 00:04:50,540 --> 00:04:52,730 so this we call nice operator. 63 00:04:56,944 --> 00:05:03,330 Acting on psi, giving your something that depends on psi-- 64 00:05:03,330 --> 00:05:08,170 maybe it's not that nice or not that simple. 65 00:05:08,170 --> 00:05:14,380 We can try to solve this using Green's function. 66 00:05:14,380 --> 00:05:17,380 That's what Green's functions are good for. 67 00:05:17,380 --> 00:05:20,265 So let's try to use a Green's function. 68 00:05:20,265 --> 00:05:23,146 So what is a Green's function? 69 00:05:23,146 --> 00:05:27,840 It is basically a way of understanding 70 00:05:27,840 --> 00:05:29,780 what this operator is. 71 00:05:29,780 --> 00:05:35,450 So nice operator must have a Green's function. 72 00:05:39,724 --> 00:05:41,890 So what is your Green's function here? 73 00:05:41,890 --> 00:05:50,780 It's called G. And we'll say it's depends 74 00:05:50,780 --> 00:05:52,810 on r minus r prime. 75 00:05:52,810 --> 00:05:59,400 r prime is an arbitrary point so far, but here is what it does. 76 00:05:59,400 --> 00:06:02,520 It's basically a solution. 77 00:06:13,400 --> 00:06:18,420 You want this Green's function to be almost zero, 78 00:06:18,420 --> 00:06:20,920 except they're not quite equal to zero. 79 00:06:20,920 --> 00:06:24,760 You want it to be equal to a delta function. 80 00:06:31,786 --> 00:06:34,790 So that's the definition of the Green's function, 81 00:06:34,790 --> 00:06:41,565 is that thing which is the solution of a similar equation, 82 00:06:41,565 --> 00:06:45,005 where you have the nice operator acting on the Green's function 83 00:06:45,005 --> 00:06:47,830 being just a delta function. 84 00:06:47,830 --> 00:06:50,150 It's almost saying that the Green's function 85 00:06:50,150 --> 00:06:54,080 is the thing that solves that equation with zero 86 00:06:54,080 --> 00:06:56,810 on the right hand side. 87 00:06:56,810 --> 00:06:58,220 Except that it's not really zero, 88 00:06:58,220 --> 00:07:00,305 it's a source in the right hand side, 89 00:07:00,305 --> 00:07:03,406 it's a delta function of this point. 90 00:07:03,406 --> 00:07:08,690 So you would say maybe, OK, I don't know. 91 00:07:08,690 --> 00:07:11,805 Why would I care about this equation? 92 00:07:11,805 --> 00:07:14,566 You care for two reasons. 93 00:07:14,566 --> 00:07:18,600 The first reason is that this equation is easier 94 00:07:18,600 --> 00:07:21,150 to solve than this one, doesn't involve 95 00:07:21,150 --> 00:07:25,116 the potential, which is complicated. 96 00:07:25,116 --> 00:07:27,420 It just involves a delta function. 97 00:07:30,570 --> 00:07:35,630 So we can have this G and solve it once, 98 00:07:35,630 --> 00:07:37,830 because it doesn't involve the potential. 99 00:07:37,830 --> 00:07:40,645 And then the great thing about this equation 100 00:07:40,645 --> 00:07:43,135 is that it allows you to write the solution 101 00:07:43,135 --> 00:07:48,025 for the top equation without having to do any work anymore. 102 00:07:48,025 --> 00:07:52,510 Once you have the Green's function, you're done. 103 00:07:52,510 --> 00:07:56,340 So let's assume you have the Green's function. 104 00:07:56,340 --> 00:07:59,650 How would you write the solution for this equation? 105 00:08:02,902 --> 00:08:09,350 So maybe a superposition, basically, and we'll 106 00:08:09,350 --> 00:08:12,100 do it using that. 107 00:08:20,382 --> 00:08:24,140 So our aim is to use this Green's function, 108 00:08:24,140 --> 00:08:28,105 if we had it, to write a solution for this equation. 109 00:08:28,105 --> 00:08:30,010 So here is the claim. 110 00:08:30,010 --> 00:08:32,600 You write psi of x. 111 00:08:35,215 --> 00:08:42,954 It's going to be given by a beginning one, psi zero of x. 112 00:08:42,954 --> 00:08:44,770 That is going to be a funny one. 113 00:08:44,770 --> 00:08:58,068 This one solves Laplacian plus k squared on psi zero of x or r 114 00:08:58,068 --> 00:08:59,330 is equal to zero. 115 00:08:59,330 --> 00:09:00,170 0. 116 00:09:00,170 --> 00:09:04,400 So whenever you have an equation of this form, 117 00:09:04,400 --> 00:09:07,490 you can add any solution. 118 00:09:07,490 --> 00:09:11,810 Anything that is killed by this, can be added 119 00:09:11,810 --> 00:09:14,555 to whatever solution you have. 120 00:09:14,555 --> 00:09:25,200 So let's assume that we have a solution of this form psi zero, 121 00:09:25,200 --> 00:09:28,044 of this form. 122 00:09:28,044 --> 00:09:34,605 Now, here I'll add one more thing, the important part. 123 00:09:34,605 --> 00:09:38,505 It's going to be an integral over r prime, 124 00:09:38,505 --> 00:09:40,460 here is the superposition. 125 00:09:40,460 --> 00:09:44,280 I'm going to think of this potential, U 126 00:09:44,280 --> 00:09:50,490 of r, as kind of existing at every point r prime. 127 00:09:50,490 --> 00:09:53,335 So I'm going go write the solution 128 00:09:53,335 --> 00:09:57,390 as a superposition that involves the Green's function. 129 00:09:57,390 --> 00:10:06,940 So this will be G of r minus r prime, times U of r prime, 130 00:10:06,940 --> 00:10:09,130 times psi of r prime. 131 00:10:15,442 --> 00:10:23,350 I claim that this is equivalent to this equation that we have. 132 00:10:23,350 --> 00:10:27,475 That this psi provides-- 133 00:10:27,475 --> 00:10:28,900 more than equivalent. 134 00:10:28,900 --> 00:10:31,060 I think the precise way to say, this 135 00:10:31,060 --> 00:10:36,440 provides a solution of this equation, the way I've written. 136 00:10:36,440 --> 00:10:38,345 So let's try. 137 00:10:38,345 --> 00:10:42,430 Suppose I calculate Laplacian plus k 138 00:10:42,430 --> 00:10:47,815 squared on psi of x here? 139 00:10:47,815 --> 00:10:52,735 OK, the first term, it's already zero. 140 00:10:52,735 --> 00:11:00,240 So Laplacian plus k squared on psi zero, it's already zero. 141 00:11:00,240 --> 00:11:04,930 And then I come here and I say, OK, I'm a Laplacian. 142 00:11:04,930 --> 00:11:08,270 I care about r, because I'm a Laplacian. 143 00:11:08,270 --> 00:11:10,660 I don't care about r prime. 144 00:11:10,660 --> 00:11:14,146 So I come in here. 145 00:11:14,146 --> 00:11:17,560 And I ignore r prime, ignore r prime. 146 00:11:17,560 --> 00:11:20,620 I cannot ignore this thing. 147 00:11:20,620 --> 00:11:29,590 So we have plus integral dr prime, Laplacian plus k 148 00:11:29,590 --> 00:11:44,330 squared, acting on G of r minus r prime, times U of r prime, 149 00:11:44,330 --> 00:11:45,760 psi of r prime. 150 00:11:51,070 --> 00:11:57,210 And now because the Green's function 151 00:11:57,210 --> 00:12:05,940 was designed to give you a delta of r minus r prime, 152 00:12:05,940 --> 00:12:11,950 this is an integral that can be done, the integral over r 153 00:12:11,950 --> 00:12:20,530 prime, and just set the rest of the integrand at the point r, 154 00:12:20,530 --> 00:12:22,855 because you integrate over r prime. 155 00:12:22,855 --> 00:12:27,950 And this delta function fires when r prime is equal to r. 156 00:12:27,950 --> 00:12:32,950 So this gives me U of r, psi of r. 157 00:12:36,876 --> 00:12:41,420 And that is the equation I wanted to solve, 158 00:12:41,420 --> 00:12:43,570 the equation that we have here. 159 00:12:43,570 --> 00:12:48,306 So we have turned the problem. 160 00:12:48,306 --> 00:12:52,100 This cannot be called the solution, 161 00:12:52,100 --> 00:12:55,125 because we have not solved it. 162 00:12:55,125 --> 00:12:58,680 We have turned the problem into a different kind of problem. 163 00:13:01,340 --> 00:13:04,010 It might even seem that we've made 164 00:13:04,010 --> 00:13:10,430 the problem worse by turning this into an integral equation. 165 00:13:10,430 --> 00:13:12,830 There's no derivatives here. 166 00:13:12,830 --> 00:13:17,495 But the function that we're looking for 167 00:13:17,495 --> 00:13:21,025 appears outside and inside the integral. 168 00:13:21,025 --> 00:13:24,530 So these things are called integral equations. 169 00:13:24,530 --> 00:13:29,945 And the power of an integral equation 170 00:13:29,945 --> 00:13:39,000 is the insight it can give you once you have an idea of what 171 00:13:39,000 --> 00:13:41,325 the Green's function is. 172 00:13:41,325 --> 00:13:47,550 And also it's a good place to do recursive approximations. 173 00:13:47,550 --> 00:13:50,625 In that, you can essentially begin 174 00:13:50,625 --> 00:13:57,285 and say, OK, I know the wave function is this plus that. 175 00:13:57,285 --> 00:14:01,890 But maybe in some sense, I can think of this thing 176 00:14:01,890 --> 00:14:04,730 as the leading solution. 177 00:14:04,730 --> 00:14:07,835 I could substitute the leading solution in here 178 00:14:07,835 --> 00:14:09,675 and try to make an approximation. 179 00:14:09,675 --> 00:14:12,330 That's going to give you a nice approximation, 180 00:14:12,330 --> 00:14:13,970 the Born approximation. 181 00:14:13,970 --> 00:14:15,460 We'll see that soon. 182 00:14:15,460 --> 00:14:18,030 That should be all psi r's. 183 00:14:18,030 --> 00:14:22,791 I think I'm using r's, so I please-- 184 00:14:22,791 --> 00:14:33,796 so r's and r's everywhere, yeah, no difference at this moment. 185 00:14:33,796 --> 00:14:40,901 OK, so the next step is solving for the Green's function. 186 00:14:40,901 --> 00:14:42,565 We need the Green's function. 187 00:14:42,565 --> 00:14:45,700 Otherwise, we can't make progress with this equation. 188 00:14:45,700 --> 00:14:48,740 So I'm going to do a simple solution of the Green's 189 00:14:48,740 --> 00:14:53,900 function, basically by doing a couple of checks 190 00:14:53,900 --> 00:14:59,696 and saying that is the answer we're interested in. 191 00:14:59,696 --> 00:15:06,495 And there are several possible Green's functions. 192 00:15:06,495 --> 00:15:09,360 And depending on the problem you're solving, 193 00:15:09,360 --> 00:15:11,985 you choose the right Green's function. 194 00:15:11,985 --> 00:15:16,875 And we'll choose the one that is suitable for us now. 195 00:15:16,875 --> 00:15:19,230 This is something that can be done. 196 00:15:19,230 --> 00:15:21,615 The general solutions can be obtained 197 00:15:21,615 --> 00:15:24,380 by counter integration. 198 00:15:24,380 --> 00:15:28,020 And there's all kinds of nice methods to do this. 199 00:15:28,020 --> 00:15:31,670 But in fact, in this case, it's really simple. 200 00:15:31,670 --> 00:15:34,230 You don't need any of those complicated things. 201 00:15:34,230 --> 00:15:38,800 You can just pretty much write the solution. 202 00:15:38,800 --> 00:15:40,230 So that's what I'm going to do. 203 00:15:45,400 --> 00:15:48,200 So what do we need for the Green's function? 204 00:15:48,200 --> 00:15:50,520 So we have a Green's function that 205 00:15:50,520 --> 00:15:53,280 depends on r minus r prime. 206 00:15:59,385 --> 00:16:04,330 And so it depends on a vector. 207 00:16:04,330 --> 00:16:07,710 So let me simplify the matter by saying, 208 00:16:07,710 --> 00:16:12,375 OK, since it depends on a vector, 209 00:16:12,375 --> 00:16:15,855 I'll just first calculate what this G 210 00:16:15,855 --> 00:16:21,810 of r, the Green's function of r, when the vector is r. 211 00:16:21,810 --> 00:16:25,380 Or you can think it's when r prime is equal to zero. 212 00:16:25,380 --> 00:16:29,910 Whatever I find for G of r, this one 213 00:16:29,910 --> 00:16:32,370 is obtained by whatever I find here, 214 00:16:32,370 --> 00:16:35,130 put in instead r minus r prime. 215 00:16:35,130 --> 00:16:40,176 So G of r is enough for what we want to do. 216 00:16:40,176 --> 00:16:49,500 So it should have Laplacian plus k squared. 217 00:16:49,500 --> 00:16:58,540 And G of r should be for, not for, but it's delta of r. 218 00:16:58,540 --> 00:17:04,349 So we've looked at that in fact at the beginning 219 00:17:04,349 --> 00:17:07,335 of this course, not of this course, of the discussion 220 00:17:07,335 --> 00:17:08,700 of scattering. 221 00:17:08,700 --> 00:17:11,314 We looked at this equation. 222 00:17:11,314 --> 00:17:15,030 In fact, we wanted to solve it when the right hand 223 00:17:15,030 --> 00:17:19,440 side was zero, to find solutions of this scattering equation. 224 00:17:19,440 --> 00:17:25,140 And we found that these G's could be of the form 225 00:17:25,140 --> 00:17:30,420 e to the ikr plus, minus ikr over r. 226 00:17:33,979 --> 00:17:39,980 Those were the spherical waves that solved this equation 227 00:17:39,980 --> 00:17:41,375 with zero potential. 228 00:17:41,375 --> 00:17:44,200 Those are our solutions. 229 00:17:44,200 --> 00:17:52,565 Now, it's easy to see that Laplacian plus k squared 230 00:17:52,565 --> 00:17:58,664 on this G, that has a plus or a minus, 231 00:17:58,664 --> 00:18:03,546 is equal to zero for r different from zero. 232 00:18:03,546 --> 00:18:07,220 The formulas for the Laplacian that you can us for r 233 00:18:07,220 --> 00:18:11,080 different from zero, you can check that is true. 234 00:18:11,080 --> 00:18:12,740 And it's easy. 235 00:18:12,740 --> 00:18:14,690 You can use that formula, for example, 236 00:18:14,690 --> 00:18:19,620 that the Laplacian is one over r the second-- well, 237 00:18:19,620 --> 00:18:20,965 it should be partial. 238 00:18:20,965 --> 00:18:28,550 The second dr squared r, so psi, psi, like that. 239 00:18:28,550 --> 00:18:32,190 That formula in one 30 seconds, you 240 00:18:32,190 --> 00:18:36,222 can see that that works for r different from zero. 241 00:18:36,222 --> 00:18:41,300 But then there's also a formula that you 242 00:18:41,300 --> 00:18:46,850 know that Laplacian of one over r from electromagnetism 243 00:18:46,850 --> 00:18:50,550 is minus 4pi pi times the delta function of r. 244 00:18:53,558 --> 00:18:58,960 It comes from Poisson's equation in electromagnetism. 245 00:18:58,960 --> 00:19:03,885 So that's saying Laplacian of the potential 246 00:19:03,885 --> 00:19:05,100 is the charge density. 247 00:19:05,100 --> 00:19:07,590 This is the potential for a charge. 248 00:19:07,590 --> 00:19:11,430 The charge that r equals zero is a singularity. 249 00:19:11,430 --> 00:19:13,500 It's something you study. 250 00:19:13,500 --> 00:19:22,570 So here if I put a minus one over 4pi for G plus, 251 00:19:22,570 --> 00:19:30,160 minus, a minus one over 4pi, the Laplacian of this whole thing 252 00:19:30,160 --> 00:19:30,835 is zero. 253 00:19:30,835 --> 00:19:36,475 But when you approach zero, the Laplacian of this function, 254 00:19:36,475 --> 00:19:39,340 this function approaches just one over r. 255 00:19:39,340 --> 00:19:40,570 That's how it goes to zero. 256 00:19:40,570 --> 00:19:42,250 This goes to one. 257 00:19:42,250 --> 00:19:45,580 And the Laplacian of one over r gives it 258 00:19:45,580 --> 00:19:48,150 this minus 4pi cancels with that. 259 00:19:48,150 --> 00:19:52,065 And it will give you a delta function. 260 00:19:52,065 --> 00:19:58,510 So the delta function will arise correctly from this quantity. 261 00:20:01,924 --> 00:20:05,580 So I've argued without solving this equation, 262 00:20:05,580 --> 00:20:10,890 that this does give you a solution for the Green's 263 00:20:10,890 --> 00:20:13,146 function. 264 00:20:13,146 --> 00:20:17,770 It's Laplacian plus k squared is zero away from zero. 265 00:20:17,770 --> 00:20:22,150 Unless you approach zero, it has a right singularity 266 00:20:22,150 --> 00:20:25,285 to give a delta function. 267 00:20:25,285 --> 00:20:27,880 So this is your Green's function. 268 00:20:27,880 --> 00:20:30,550 As I said many ways to derive it. 269 00:20:30,550 --> 00:20:35,290 If you want to see griffiths, see other ways 270 00:20:35,290 --> 00:20:38,020 in which it can be done, then you can also 271 00:20:38,020 --> 00:20:43,850 check this by doing properly the vector calculus. 272 00:20:43,850 --> 00:20:47,815 You can think of Laplacian as divergence of the gradient 273 00:20:47,815 --> 00:20:49,780 and calculate every step. 274 00:20:49,780 --> 00:20:53,515 And it's fun to do it as well. 275 00:20:53,515 --> 00:20:55,095 You check that this works. 276 00:20:55,095 --> 00:20:57,190 So this works. 277 00:20:57,190 --> 00:20:58,900 And now we have to make choices. 278 00:21:02,716 --> 00:21:07,310 And our choices are going to be adjusted to the problem we're 279 00:21:07,310 --> 00:21:08,845 trying to solve. 280 00:21:08,845 --> 00:21:11,690 We're trying to solve a scattering problem. 281 00:21:11,690 --> 00:21:15,650 And psi zero represents a solution. 282 00:21:15,650 --> 00:21:20,495 And from the way we thought of our waves, 283 00:21:20,495 --> 00:21:27,920 we had psi equal e to the ikz plus f of theta and phi, e 284 00:21:27,920 --> 00:21:31,670 to the ikr over r. 285 00:21:31,670 --> 00:21:36,390 So it's reasonable to try to set. 286 00:21:36,390 --> 00:21:38,120 And we're going to find solutions 287 00:21:38,120 --> 00:21:44,120 by setting psi zero to be e to the ikz. 288 00:21:44,120 --> 00:21:48,560 It does solve the equation Laplacian squared plus k 289 00:21:48,560 --> 00:21:51,652 squared equals zero. 290 00:21:51,652 --> 00:21:58,910 And then we're going to set the Green's function to be G plus. 291 00:21:58,910 --> 00:22:02,540 With a plus here, because we want solutions of the type 292 00:22:02,540 --> 00:22:03,410 e to the ikr. 293 00:22:05,912 --> 00:22:09,710 We've already decided those were the solutions we need. 294 00:22:09,710 --> 00:22:12,410 And a Green's function with a plus in there 295 00:22:12,410 --> 00:22:15,988 is going to generate that. 296 00:22:15,988 --> 00:22:21,455 So those are fine. 297 00:22:21,455 --> 00:22:24,410 You could have chosen another thing here, an arbitrary 298 00:22:24,410 --> 00:22:25,115 solution. 299 00:22:25,115 --> 00:22:27,950 And you could have chosen a G minus as well. 300 00:22:27,950 --> 00:22:31,910 And you would get a solution of the scattering equation. 301 00:22:31,910 --> 00:22:35,420 It would not have much to do with the solution we're 302 00:22:35,420 --> 00:22:37,556 trying to get. 303 00:22:37,556 --> 00:22:42,710 So let's show that this gives us this kind of solution we want 304 00:22:42,710 --> 00:22:46,010 to get and gives us already a formula 305 00:22:46,010 --> 00:22:48,965 for the scattering amplitude. 306 00:22:48,965 --> 00:22:51,290 So let's do that. 307 00:22:51,290 --> 00:22:57,710 For that we just need to write what the Green's function is 308 00:22:57,710 --> 00:23:00,020 and approximate. 309 00:23:05,965 --> 00:23:07,620 So it will be a first step. 310 00:23:10,752 --> 00:23:13,665 OK, so what do we have? 311 00:23:13,665 --> 00:23:19,485 Psi of r is equal to e to the ikz. 312 00:23:19,485 --> 00:23:23,140 I'm writing up this stop equation, 313 00:23:23,140 --> 00:23:28,140 but we now write it with the right boundary conditions, 314 00:23:28,140 --> 00:23:32,580 which correspond to psi zero on a particular choice 315 00:23:32,580 --> 00:23:40,740 of the Green's function, v cubed r prime G of r minus r 316 00:23:40,740 --> 00:23:48,255 prime, U of r prime, all those are vectors, 317 00:23:48,255 --> 00:23:52,530 and psi of r prime. 318 00:23:52,530 --> 00:23:55,350 All those are vectors, except G, which 319 00:23:55,350 --> 00:23:59,655 is pretty in that it is very spherically symmetric. 320 00:23:59,655 --> 00:24:07,165 This G plus, minus of r just depends on the magnitude 321 00:24:07,165 --> 00:24:09,655 of r, which is very nice. 322 00:24:09,655 --> 00:24:14,080 OK, so a couple of things that we can do here 323 00:24:14,080 --> 00:24:16,975 and most everybody would do. 324 00:24:16,975 --> 00:24:22,760 Let's G plus of r minus r prime is 325 00:24:22,760 --> 00:24:28,945 equal to minus one over 4pi, e to the ik, 326 00:24:28,945 --> 00:24:31,735 length of r minus r prime-- 327 00:24:31,735 --> 00:24:34,420 that's what this is supposed to be. 328 00:24:34,420 --> 00:24:37,910 Over the length of r minus r prime. 329 00:24:37,910 --> 00:24:40,780 Remember I said to you that whenever 330 00:24:40,780 --> 00:24:45,460 we have G of r, then to get the one we need, 331 00:24:45,460 --> 00:24:49,840 we'll just have to replace the r by r minus r prime. 332 00:24:49,840 --> 00:24:56,110 And here this little r or r without the vector 333 00:24:56,110 --> 00:24:58,080 is the magnitude of this vector. 334 00:24:58,080 --> 00:25:03,088 So we've put the magnitude of those vectors there. 335 00:25:03,088 --> 00:25:06,715 Now, people look for approximations in here. 336 00:25:06,715 --> 00:25:08,440 So here is what we need to do. 337 00:25:08,440 --> 00:25:10,390 Imagine you are the origin. 338 00:25:10,390 --> 00:25:12,460 Here is the potential. 339 00:25:12,460 --> 00:25:15,325 And you have your system of coordinates. 340 00:25:15,325 --> 00:25:23,080 You are integrating over places where the potential exists. 341 00:25:23,080 --> 00:25:27,190 So the r prime that you're integrating 342 00:25:27,190 --> 00:25:31,000 remains inside the range of the potential. 343 00:25:31,000 --> 00:25:33,326 Here is zero. 344 00:25:33,326 --> 00:25:39,530 And then there is a r, which is where you are looking at. 345 00:25:42,100 --> 00:25:44,880 And there is an angle between these two. 346 00:25:44,880 --> 00:25:52,191 Maybe I should make the angle a little bigger, smaller, I mean, 347 00:25:52,191 --> 00:25:52,690 r. 348 00:25:55,580 --> 00:26:01,115 And in general, we look always far enough. 349 00:26:01,115 --> 00:26:06,570 This is an exact solution, but we might as well 350 00:26:06,570 --> 00:26:10,506 consider looking at psi's that are 351 00:26:10,506 --> 00:26:13,860 located far from the potential. 352 00:26:13,860 --> 00:26:19,585 So we have the idea of what should we do with this terms. 353 00:26:19,585 --> 00:26:24,735 One thing you can do is to say that in the denominator here, 354 00:26:24,735 --> 00:26:30,990 r minus r prime is good enough to set it 355 00:26:30,990 --> 00:26:34,620 to r in the denominator. 356 00:26:37,455 --> 00:26:42,390 And for the numerator, however, it's a lot more sensitive, 357 00:26:42,390 --> 00:26:45,660 because you have a k here. 358 00:26:45,660 --> 00:26:50,560 And depending on whether k is large or small-- 359 00:26:50,560 --> 00:26:53,370 a small difference in r minus r prime, 360 00:26:53,370 --> 00:26:55,350 if you're estimating this and you 361 00:26:55,350 --> 00:26:58,380 make an error comparable with the wavelength 362 00:26:58,380 --> 00:27:00,780 of your particle, you make a big mistake. 363 00:27:00,780 --> 00:27:06,150 If you're far way, you're 100 times farther than the range 364 00:27:06,150 --> 00:27:10,685 of the potential, here you're making a 1% error, no problem. 365 00:27:10,685 --> 00:27:16,560 But if you're making a 1% error in the estimate of this, 366 00:27:16,560 --> 00:27:20,750 that might be still comparable to the wavelength of this k. 367 00:27:20,750 --> 00:27:23,910 So you have to be a lot more careful in the phase, 368 00:27:23,910 --> 00:27:26,595 than you have to be on the other one. 369 00:27:26,595 --> 00:27:34,200 So for the phase, phase, we will take 370 00:27:34,200 --> 00:27:44,940 r minus r prime equal to r minus n, r prime. 371 00:27:47,592 --> 00:27:53,870 So what is n? n is a unit vector in this direction. 372 00:27:53,870 --> 00:28:00,630 And indeed, approximately, this distance 373 00:28:00,630 --> 00:28:05,370 here is approximately equal to the distance 374 00:28:05,370 --> 00:28:09,720 r minus the projection of r prime into this one. 375 00:28:09,720 --> 00:28:12,390 So that is your approximate distance. 376 00:28:12,390 --> 00:28:15,970 So here is the final formula we're going to write today. 377 00:28:15,970 --> 00:28:20,076 I'm running out of time. 378 00:28:20,076 --> 00:28:27,315 But that's what we wanted to end up with. 379 00:28:27,315 --> 00:28:31,590 So here is what we get. 380 00:28:31,590 --> 00:28:42,000 So G plus of r minus r prime has become minus one over 4pi r. 381 00:28:42,000 --> 00:28:43,980 The denominator is r. 382 00:28:43,980 --> 00:28:56,200 There's an e to the ikr and then e to the minus ikn dot r prime. 383 00:28:56,200 --> 00:29:00,950 So that's-- and the arrows are crucial here. 384 00:29:00,950 --> 00:29:03,710 If I don't put an arrow somewhere, 385 00:29:03,710 --> 00:29:05,540 it probably means something. 386 00:29:05,540 --> 00:29:15,935 So psi of r is equal to e to the ikz plus minus one over 4pi r. 387 00:29:15,935 --> 00:29:18,380 The integral over r prime doesn't 388 00:29:18,380 --> 00:29:25,940 care about this, the cubed r prime, e to the minus ikn dot 389 00:29:25,940 --> 00:29:34,970 r prime, times U of r prime, times psi of r prime, all 390 00:29:34,970 --> 00:29:39,476 that multiplied by-- 391 00:29:39,476 --> 00:29:41,630 let me take the r out. 392 00:29:41,630 --> 00:29:45,750 e to the ikr over r. 393 00:29:45,750 --> 00:29:48,260 So look at what we obtain. 394 00:29:48,260 --> 00:29:54,065 I just rewrote, finally, this expression 395 00:29:54,065 --> 00:29:55,970 for the Green's function over there, 396 00:29:55,970 --> 00:30:00,260 with the integral over r prime and the Green's function here. 397 00:30:00,260 --> 00:30:03,455 You have this quantity in brackets. 398 00:30:03,455 --> 00:30:06,770 This quantity in brackets here is 399 00:30:06,770 --> 00:30:13,295 nothing but the f of theta phi of the scattering amplitude. 400 00:30:13,295 --> 00:30:18,465 You have this equal to that, plus e to the ikr over r. 401 00:30:18,465 --> 00:30:23,180 So we'll continue that next time and finish up 402 00:30:23,180 --> 00:30:26,980 with scattering probably in the first half of the next lecture.