1 00:00:00,520 --> 00:00:08,340 PROFESSOR: Expectation values of operators. 2 00:00:17,630 --> 00:00:22,150 So this is, in a sense, one of our first steps 3 00:00:22,150 --> 00:00:26,290 that we're going to take towards the interpretation of quantum 4 00:00:26,290 --> 00:00:27,280 mechanics. 5 00:00:27,280 --> 00:00:30,250 We've had already that the wave function tells you 6 00:00:30,250 --> 00:00:31,540 about probabilities. 7 00:00:31,540 --> 00:00:36,010 But that's not quite enough to have the full interpretation 8 00:00:36,010 --> 00:00:37,110 of what we're doing. 9 00:00:37,110 --> 00:00:42,910 So let's think of operators and expectation values 10 00:00:42,910 --> 00:00:44,410 that we can motivate. 11 00:00:44,410 --> 00:00:54,440 So for example, if you have a random variable q, 12 00:00:54,440 --> 00:00:55,935 that can take values-- 13 00:01:02,000 --> 00:01:05,810 so this could be a coin that can take values heads and tails. 14 00:01:05,810 --> 00:01:11,510 It could be a pair of dice that takes many values-- 15 00:01:11,510 --> 00:01:29,580 can take values in the set q1 up to qn with probabilities 16 00:01:29,580 --> 00:01:44,360 p1 up to pn, then in statistics, or 8044, 17 00:01:44,360 --> 00:01:47,750 you would say that this variable, this random variable 18 00:01:47,750 --> 00:01:50,450 has an expectation value. 19 00:01:50,450 --> 00:01:54,470 And the expectation value-- 20 00:01:54,470 --> 00:02:03,590 denoted by this angular symbols over here, left and right-- 21 00:02:03,590 --> 00:02:09,880 it's given by the sum over i, i equal 1 to n, 22 00:02:09,880 --> 00:02:13,907 of the possible values the random variable can 23 00:02:13,907 --> 00:02:16,720 take times the probabilities. 24 00:02:16,720 --> 00:02:21,080 It's a definition that makes sense. 25 00:02:21,080 --> 00:02:26,350 And it's thought to be, this expectation value is, 26 00:02:26,350 --> 00:02:30,340 the expected value, or average value, 27 00:02:30,340 --> 00:02:34,570 that you would obtain if you did the experiment of tossing 28 00:02:34,570 --> 00:02:37,150 the random variable many times. 29 00:02:37,150 --> 00:02:39,820 For each value of the random variable, 30 00:02:39,820 --> 00:02:42,610 you multiply by the probability. 31 00:02:42,610 --> 00:02:46,130 And that's the number you expect to get. 32 00:02:46,130 --> 00:02:50,960 So in a quantum system, we follow this analogy 33 00:02:50,960 --> 00:02:52,100 very closely. 34 00:02:52,100 --> 00:02:55,250 So what do we have in a quantum system? 35 00:02:55,250 --> 00:03:09,260 In the quantum system you have that psi star of x and t. 36 00:03:13,190 --> 00:03:29,220 The x is the probability that the particle 37 00:03:29,220 --> 00:03:34,290 is in x, x plus dx. 38 00:03:36,900 --> 00:03:39,900 So that's the probability that the particle 39 00:03:39,900 --> 00:03:44,820 is going to be found between x and x plus dx. 40 00:03:44,820 --> 00:03:48,584 The position of this particle is like a random variable. 41 00:03:48,584 --> 00:03:50,500 You never know where you are going to find it. 42 00:03:50,500 --> 00:03:53,910 But he has different probabilities to find it. 43 00:03:53,910 --> 00:04:00,630 So we could now define in complete analogy to here, 44 00:04:00,630 --> 00:04:05,400 the expectation value of the position operator, 45 00:04:05,400 --> 00:04:11,570 or the expectation value of the position, expectation 46 00:04:11,570 --> 00:04:17,959 value of x hat, or the position, and say, well, I'm 47 00:04:17,959 --> 00:04:21,019 going to do exactly what I have here. 48 00:04:21,019 --> 00:04:26,930 I will sum the products of the position times the probability 49 00:04:26,930 --> 00:04:27,980 for the position. 50 00:04:27,980 --> 00:04:30,200 So I have to do it as an integral. 51 00:04:33,580 --> 00:04:36,650 And in this integral, I have to multiply 52 00:04:36,650 --> 00:04:44,920 the position times the probability for the position. 53 00:04:49,430 --> 00:04:56,480 So the probability that the disc takes a value of x, basically 54 00:04:56,480 --> 00:05:04,640 all that is in the interval dx about x is this quantity. 55 00:05:04,640 --> 00:05:06,860 And that's the position that you get when 56 00:05:06,860 --> 00:05:09,030 you estimate this probability. 57 00:05:09,030 --> 00:05:13,790 So you must sum the values of the random variable 58 00:05:13,790 --> 00:05:16,430 times its probability. 59 00:05:16,430 --> 00:05:22,190 And that is taken in quantum mechanics to be a definition. 60 00:05:29,260 --> 00:05:36,140 We can define the expectation value of x by this quantity. 61 00:05:36,140 --> 00:05:38,570 And what does it mean experimentally? 62 00:05:38,570 --> 00:05:41,810 It means that in quantum mechanics, if you 63 00:05:41,810 --> 00:05:45,170 have a system represented by a wave function, 64 00:05:45,170 --> 00:05:49,640 you should build many copies of the system, 100 65 00:05:49,640 --> 00:05:51,530 copies of the system. 66 00:05:51,530 --> 00:05:55,520 In all of these copies, you measure the position. 67 00:05:55,520 --> 00:05:57,890 And you make a table of the values 68 00:05:57,890 --> 00:05:59,750 that you measure the position. 69 00:05:59,750 --> 00:06:04,820 And you measure them at the same time in the 100 copies. 70 00:06:04,820 --> 00:06:06,950 There's an experimentalist on each one, 71 00:06:06,950 --> 00:06:09,470 and it measures the position of x. 72 00:06:09,470 --> 00:06:12,800 You construct the table, take the average, 73 00:06:12,800 --> 00:06:18,070 and that's what this quantity should be telling you. 74 00:06:18,070 --> 00:06:22,450 So this quantity, as you can see, may depend on time. 75 00:06:31,410 --> 00:06:36,110 But it does give you the interpretation 76 00:06:36,110 --> 00:06:41,900 of expected value coinciding with a system, 77 00:06:41,900 --> 00:06:46,100 now the quantum mechanical system, for which the position 78 00:06:46,100 --> 00:06:49,760 is not anymore a quantity that is well defined 79 00:06:49,760 --> 00:06:51,260 and it's always the same. 80 00:06:51,260 --> 00:06:54,020 It's a random variable, and each measurement 81 00:06:54,020 --> 00:06:56,810 can give you a different value of the position. 82 00:06:56,810 --> 00:07:01,150 Quantum mechanically, this is the expected value. 83 00:07:01,150 --> 00:07:04,400 And the interpretation is, again if you 84 00:07:04,400 --> 00:07:09,170 measure many times, that is the value, the average value, 85 00:07:09,170 --> 00:07:11,360 you will observe. 86 00:07:11,360 --> 00:07:18,210 But now we can do the same thing to understand 87 00:07:18,210 --> 00:07:19,470 expectation values. 88 00:07:19,470 --> 00:07:21,620 We can do it with the momentum. 89 00:07:21,620 --> 00:07:24,670 And this is a little more non-trivial. 90 00:07:27,390 --> 00:07:34,410 So we have also, just like we said here, that psi star psi 91 00:07:34,410 --> 00:07:37,890 dx is the probability that the particle is in there, 92 00:07:37,890 --> 00:07:46,460 you also have that phi of p squared. 93 00:07:46,460 --> 00:08:04,390 dp is the probability to find the particle with momentum 94 00:08:04,390 --> 00:08:12,470 in the range p p plus dp. 95 00:08:17,790 --> 00:08:26,650 So how do we define the expected value of the momentum? 96 00:08:26,650 --> 00:08:30,960 The expected value of the momentum 97 00:08:30,960 --> 00:08:37,350 would be given by, again, the sum of the random variable, 98 00:08:37,350 --> 00:08:41,190 which is the momentum, times the probability 99 00:08:41,190 --> 00:08:42,495 that you get that value. 100 00:08:47,840 --> 00:08:49,000 So this is it. 101 00:08:49,000 --> 00:08:53,290 It's very analogous to this expression. 102 00:08:57,480 --> 00:08:58,975 But it's now with momentum. 103 00:09:05,360 --> 00:09:07,630 Well, this is a pretty nice thing. 104 00:09:07,630 --> 00:09:13,330 But we can learn more about it by pushing the analogy more. 105 00:09:13,330 --> 00:09:17,860 And you could say, look, this is perfect. 106 00:09:17,860 --> 00:09:22,450 But it's all done in momentum space. 107 00:09:22,450 --> 00:09:24,340 What would happen if you would try 108 00:09:24,340 --> 00:09:27,820 to do this in position space? 109 00:09:27,820 --> 00:09:32,670 That is, you know how 5p is related to psi of x. 110 00:09:32,670 --> 00:09:35,050 So write everything in terms of x. 111 00:09:35,050 --> 00:09:39,850 I would like to see this formula in terms of x. 112 00:09:39,850 --> 00:09:41,980 It would be a very good thing to have. 113 00:09:41,980 --> 00:09:45,470 So let's try to do that. 114 00:09:45,470 --> 00:09:52,030 So we have to do a little bit of work here with integrals. 115 00:09:52,030 --> 00:09:54,790 So it's not so bad. 116 00:09:54,790 --> 00:10:04,120 p phi star of p phi of p dp. 117 00:10:04,120 --> 00:10:07,270 And for this one, you have to write it 118 00:10:07,270 --> 00:10:10,400 as an integral over some position. 119 00:10:10,400 --> 00:10:14,030 So let me call it over position x prime. 120 00:10:14,030 --> 00:10:17,845 This you will write this an integral over some position x. 121 00:10:22,030 --> 00:10:26,020 And then we're going to try to rewrite the whole thing 122 00:10:26,020 --> 00:10:32,470 in terms of coordinate space. 123 00:10:32,470 --> 00:10:33,920 So what do we have here? 124 00:10:33,920 --> 00:10:36,063 We have integral pdp. 125 00:10:39,970 --> 00:10:46,180 And the first phi star would be the integral over dx prime. 126 00:10:46,180 --> 00:10:49,300 We said there is the square root 2 pi h 127 00:10:49,300 --> 00:10:52,990 bar that we can't forget. 128 00:10:52,990 --> 00:11:04,710 5p, it would have an e to the ip x prime over h bar, 129 00:11:04,710 --> 00:11:11,580 and a psi star of x prime. 130 00:11:11,580 --> 00:11:15,780 So I did conjugate this phi star of p. 131 00:11:15,780 --> 00:11:17,670 I may have it here. 132 00:11:17,670 --> 00:11:21,680 Yes, it's here. 133 00:11:21,680 --> 00:11:28,750 I conjugated it and did the integral over x prime. 134 00:11:28,750 --> 00:11:37,300 And now we have another one, integral vx over 2 pi h bar e 135 00:11:37,300 --> 00:11:44,800 to the minus ipx over h bar, and you have psi of x. 136 00:11:49,930 --> 00:11:51,940 Now there's a lot of integrals there, 137 00:11:51,940 --> 00:11:58,890 and let's try to get them simplified. 138 00:12:19,590 --> 00:12:26,280 So we're going to try to do first the p integral. 139 00:12:26,280 --> 00:12:29,660 So let's try to clean up everything in such a way 140 00:12:29,660 --> 00:12:34,590 that we have only p done first. 141 00:12:34,590 --> 00:12:41,915 So we'll have a 1 over 2 pi h bar from the two square roots. 142 00:12:41,915 --> 00:12:46,460 And I'll have the two integrals dx prime psi 143 00:12:46,460 --> 00:12:53,840 of x prime star the x psi of x. 144 00:12:57,420 --> 00:13:02,420 So again, as we said, these integrals 145 00:13:02,420 --> 00:13:03,800 we just wrote them out. 146 00:13:03,800 --> 00:13:05,300 They cannot be done. 147 00:13:05,300 --> 00:13:11,670 So our only hope is to simplify first the pdp integral. 148 00:13:11,670 --> 00:13:20,120 So here we would have integral of dp times p times 149 00:13:20,120 --> 00:13:24,690 e to the ipx prime over h bar. 150 00:13:24,690 --> 00:13:30,344 And e to the minus ipx over h bar. 151 00:13:41,550 --> 00:13:45,210 Now we need a little bit of-- probably 152 00:13:45,210 --> 00:13:51,120 if you were doing this, it would not be obvious what to do, 153 00:13:51,120 --> 00:13:54,720 unless you have some intuition of what the momentum 154 00:13:54,720 --> 00:13:57,630 operator used to be. 155 00:13:57,630 --> 00:14:05,100 The momentum operator used to be dvx, basically. 156 00:14:05,100 --> 00:14:08,930 Now this integral would be a delta function 157 00:14:08,930 --> 00:14:10,890 if the p was not here. 158 00:14:10,890 --> 00:14:13,080 But here is the p. 159 00:14:13,080 --> 00:14:16,350 So what I should try to do is get rid of that p 160 00:14:16,350 --> 00:14:18,900 in order to understand what we have. 161 00:14:18,900 --> 00:14:26,430 So here we'll do integral dp. 162 00:14:26,430 --> 00:14:31,382 And look, output here minus h bar over idvx. 163 00:14:35,720 --> 00:14:38,260 And leave everything here to the right, 164 00:14:38,260 --> 00:14:45,713 e to the ipx prime over h bar p to the minus ipx over h bar. 165 00:14:48,890 --> 00:14:50,630 I claim this is the same. 166 00:14:50,630 --> 00:14:55,070 Because this operator, h over iddx, 167 00:14:55,070 --> 00:14:57,930 well, it doesn't act on x prime. 168 00:14:57,930 --> 00:14:59,930 But it acts here. 169 00:14:59,930 --> 00:15:04,760 And when it does, it will produce just the factor of p 170 00:15:04,760 --> 00:15:05,750 that you have. 171 00:15:05,750 --> 00:15:09,410 Because the minus i and the minus i will cancel. 172 00:15:09,410 --> 00:15:11,110 The h bar will cancel. 173 00:15:11,110 --> 00:15:14,240 And the ddh will just bring down a p. 174 00:15:14,240 --> 00:15:20,420 So this is the way to have this work out quite nicely. 175 00:15:24,230 --> 00:15:29,510 Now this thing is inside the integral. 176 00:15:29,510 --> 00:15:33,810 But it could as well be outside the integral. 177 00:15:33,810 --> 00:15:36,500 It has nothing to do with dp. 178 00:15:36,500 --> 00:15:40,670 So I'll rewrite this again. 179 00:15:40,670 --> 00:15:50,170 I'll write it as dx prime psi of x prime star integral dx 180 00:15:50,170 --> 00:15:51,130 psi of x. 181 00:15:53,830 --> 00:16:01,780 And I'll put this here, minus h bar over iddx, 182 00:16:01,780 --> 00:16:03,900 in front of the integral. 183 00:16:03,900 --> 00:16:08,670 The 1 over 2 pi h bar here. 184 00:16:08,670 --> 00:16:17,218 Integral dp e to the ipx prime minus x over h bar. 185 00:16:20,820 --> 00:16:22,890 So I simply did a couple of things. 186 00:16:22,890 --> 00:16:27,920 I moved that 1 over 2 pi h to the right. 187 00:16:27,920 --> 00:16:34,210 And then I said this derivative could be outside the integral. 188 00:16:34,210 --> 00:16:36,140 Because it's an integral over p. 189 00:16:36,140 --> 00:16:39,405 It doesn't interfere with x derivative, so I took it out. 190 00:16:42,910 --> 00:16:46,240 Now the final two steps, we're almost there. 191 00:16:50,170 --> 00:16:54,790 The first step is to say, with this it's a ddx. 192 00:16:54,790 --> 00:16:58,070 And yes, this is a function of x and x prime. 193 00:16:58,070 --> 00:17:00,190 But I don't want to take that derivative. 194 00:17:00,190 --> 00:17:02,950 Because I'm going to complicate things. 195 00:17:02,950 --> 00:17:07,190 In fact, this is already looking like a delta function. 196 00:17:07,190 --> 00:17:09,190 There's a dp dp. 197 00:17:09,190 --> 00:17:12,730 And the h bars that actually would cancel. 198 00:17:12,730 --> 00:17:16,630 So this is a perfectly nice delta function. 199 00:17:16,630 --> 00:17:18,589 You can change variables. 200 00:17:18,589 --> 00:17:25,310 Do p equal u times h bar and see that actually the h bar 201 00:17:25,310 --> 00:17:26,869 doesn't matter. 202 00:17:26,869 --> 00:17:31,030 And this is just delta of x prime minus x. 203 00:17:35,160 --> 00:17:40,340 And in here, you could act on the delta function. 204 00:17:40,340 --> 00:17:44,180 But you could say, no, let me do integration by parts 205 00:17:44,180 --> 00:17:45,390 and act on this one. 206 00:17:52,170 --> 00:17:54,670 When you do integration by parts, 207 00:17:54,670 --> 00:17:58,560 you have to worry about the term at the boundary. 208 00:17:58,560 --> 00:18:03,070 But if your wave functions vanish sufficiently fast 209 00:18:03,070 --> 00:18:05,530 at infinity, there's no problem. 210 00:18:05,530 --> 00:18:08,350 So let's assume we're in that case. 211 00:18:08,350 --> 00:18:11,980 We will integrate by parts and then do the delta function. 212 00:18:11,980 --> 00:18:13,320 So what do we have here? 213 00:18:16,260 --> 00:18:26,203 I have integral dx prime psi star of x prime integral dx. 214 00:18:26,203 --> 00:18:29,940 And now I have, because of the sign of integration 215 00:18:29,940 --> 00:18:37,420 by parts, h over iddx of psi. 216 00:18:37,420 --> 00:18:42,470 And then we have the delta function of x minus x prime. 217 00:18:49,400 --> 00:18:52,880 It's probably better still to write 218 00:18:52,880 --> 00:19:02,540 the integral like this, dx h bar over iddx of psi times 219 00:19:02,540 --> 00:19:11,890 integral dx prime psi star of x prime delta of x minus x prime. 220 00:19:16,096 --> 00:19:20,720 And we're almost done, so that's good. 221 00:19:25,590 --> 00:19:26,700 We're almost done. 222 00:19:26,700 --> 00:19:30,480 We can do the integral over x prime. 223 00:19:30,480 --> 00:19:35,270 And it will elevate that wave function at x. 224 00:19:38,590 --> 00:19:43,850 So at the end of the day, what have we found? 225 00:19:43,850 --> 00:19:47,950 We found that p, the expectation value of p, 226 00:19:47,950 --> 00:19:58,210 equal integral of p phi of p squared dp is equal to-- 227 00:19:58,210 --> 00:19:59,670 we do this integral. 228 00:19:59,670 --> 00:20:05,630 So we have integral dx, I'll write it two times, 229 00:20:05,630 --> 00:20:15,275 h over i d psi dx of x and t and psi star of x and t. 230 00:20:18,170 --> 00:20:21,550 I'm not sure I carried that times. 231 00:20:21,550 --> 00:20:23,500 I didn't put the time anywhere. 232 00:20:23,500 --> 00:20:28,860 So maybe I shouldn't put it here yet. 233 00:20:28,860 --> 00:20:30,780 This is what we did. 234 00:20:30,780 --> 00:20:35,160 And might as well write it in the standard order, 235 00:20:35,160 --> 00:20:39,060 where the complex conjugate function appears first. 236 00:20:45,300 --> 00:20:46,375 This is what we found. 237 00:20:48,880 --> 00:20:53,280 So this is actually very neat. 238 00:20:53,280 --> 00:20:57,810 Let me put the time back everywhere you could put time. 239 00:20:57,810 --> 00:21:00,210 Because this is a time dependent thing. 240 00:21:00,210 --> 00:21:09,480 So p, expectation value is p phi of p and t 241 00:21:09,480 --> 00:21:21,960 squared dp is equal to integral dx psi star of x. 242 00:21:21,960 --> 00:21:26,430 And now we have, if you wish, p hat psi 243 00:21:26,430 --> 00:21:31,230 of x, where p hat is what we used 244 00:21:31,230 --> 00:21:33,450 to call the momentum operator. 245 00:21:43,360 --> 00:21:46,000 So look what has happened. 246 00:21:46,000 --> 00:21:51,430 We started with this expression for the expectation 247 00:21:51,430 --> 00:21:55,630 value of the momentum justified by the probabilistic 248 00:21:55,630 --> 00:21:58,710 interpretation of phi. 249 00:21:58,710 --> 00:22:02,670 And we were led to this expression, which 250 00:22:02,670 --> 00:22:05,620 is very similar to this one. 251 00:22:05,620 --> 00:22:10,440 You see, you have the psi star, the psi, and the x there. 252 00:22:10,440 --> 00:22:15,050 But here, the momentum appeared at this position, 253 00:22:15,050 --> 00:22:21,020 acting on the wave function psi, not on psi star. 254 00:22:21,020 --> 00:22:24,360 And that's the way, in quantum mechanics, 255 00:22:24,360 --> 00:22:30,270 people define expectation values of operators in general. 256 00:22:30,270 --> 00:22:45,040 So in general, for an operator q, 257 00:22:45,040 --> 00:22:50,680 we'll define the expectation value of q 258 00:22:50,680 --> 00:23:05,440 to be integral dx psi star of x and t q acting 259 00:23:05,440 --> 00:23:08,250 on the psi of x and t. 260 00:23:14,050 --> 00:23:22,280 So you will always do this of putting the operator 261 00:23:22,280 --> 00:23:25,550 to act on the second part of the wave function, 262 00:23:25,550 --> 00:23:28,190 on the second appearance of the wave function. 263 00:23:28,190 --> 00:23:31,020 Not on the psi star, but on the psi. 264 00:23:36,550 --> 00:23:41,950 We can do other examples of this and our final theorem. 265 00:23:45,160 --> 00:23:47,380 This is, of course, time dependent. 266 00:23:51,870 --> 00:23:58,190 So let me do one example and our final time dependence 267 00:23:58,190 --> 00:24:00,565 analysis of this quantity. 268 00:24:22,410 --> 00:24:27,450 So for example, if you would think of the kinetic operator 269 00:24:27,450 --> 00:24:30,110 example. 270 00:24:30,110 --> 00:24:42,030 Kinetic operator t is p squared over 2m is a kinetic operator. 271 00:24:42,030 --> 00:24:45,900 How would you compute its expectation value? 272 00:24:50,260 --> 00:24:55,015 Expectation value of the kinetic operator is what? 273 00:24:57,630 --> 00:25:04,400 Well, I could do the position space calculation, 274 00:25:04,400 --> 00:25:11,170 in which I think of the kinetic operator as an operator that 275 00:25:11,170 --> 00:25:15,830 acts in position space where the momentum is h bar over iddx. 276 00:25:15,830 --> 00:25:23,950 So then I would have integral dx psi star of x and t. 277 00:25:23,950 --> 00:25:31,070 And then I would have minus h squared over 2m d second dx 278 00:25:31,070 --> 00:25:35,090 squared of psi of x and t. 279 00:25:41,180 --> 00:25:44,330 So here I did exactly what I was supposed 280 00:25:44,330 --> 00:25:47,790 to do given this formula. 281 00:25:47,790 --> 00:25:51,890 But you could do another thing if you wished. 282 00:25:51,890 --> 00:25:57,020 You could say, look, I can work in momentum space. 283 00:25:57,020 --> 00:26:02,360 This is a momentums operator p. 284 00:26:02,360 --> 00:26:07,860 Just like I defined the expectation value of p, 285 00:26:07,860 --> 00:26:11,070 I could have the expectation value of p squared. 286 00:26:11,070 --> 00:26:13,820 So the other possibility is that you 287 00:26:13,820 --> 00:26:26,160 have t is equal to the integral dp of p squared 288 00:26:26,160 --> 00:26:32,460 over 2m times phi of p squared. 289 00:26:35,390 --> 00:26:37,230 This is the operator. 290 00:26:37,230 --> 00:26:39,140 And this is the probability. 291 00:26:39,140 --> 00:26:43,820 Or you could write it more elegantly perhaps. 292 00:26:43,820 --> 00:26:53,900 dp phi star of p t squared over 2m phi of p. 293 00:26:59,450 --> 00:27:02,030 These are just integrals of numbers. 294 00:27:02,030 --> 00:27:05,710 All these are numbers already. 295 00:27:05,710 --> 00:27:08,440 So in momentum space, it's easier 296 00:27:08,440 --> 00:27:13,310 to find the expectation value of the kinetic operator. 297 00:27:13,310 --> 00:27:16,420 In coordinate space, you have to do this. 298 00:27:16,420 --> 00:27:22,950 You might even say, look, this thing looks positive. 299 00:27:27,600 --> 00:27:30,090 Because it's p squared of the number squared. 300 00:27:30,090 --> 00:27:33,000 In the center here, it looks negative. 301 00:27:33,000 --> 00:27:36,920 But that's an illusion. 302 00:27:36,920 --> 00:27:41,020 The second derivative can be partially integrated. 303 00:27:41,020 --> 00:27:42,970 One of the two derivatives can be 304 00:27:42,970 --> 00:27:47,380 integrated to act on this one. 305 00:27:47,380 --> 00:27:51,700 So if you do partial integration by parts, 306 00:27:51,700 --> 00:27:58,030 you would have integral dx h squared over 2m. 307 00:27:58,030 --> 00:28:06,820 And then you would have d psi dx squared by integration 308 00:28:06,820 --> 00:28:09,420 by parts. 309 00:28:09,420 --> 00:28:12,970 And that's clearly positive as well. 310 00:28:12,970 --> 00:28:15,770 So it's similar to this.