1 00:00:00,726 --> 00:00:07,120 PROFESSOR: The left hand side here is 0 because H0 and N0 2 00:00:07,120 --> 00:00:10,480 by taking the eigenstate equation is 3 00:00:10,480 --> 00:00:13,570 equal to En0 times N0. 4 00:00:13,570 --> 00:00:20,520 So this really evaluates to En0 and therefore the two cancel, 5 00:00:20,520 --> 00:00:21,670 so this thing is 0. 6 00:00:24,640 --> 00:00:30,010 The right hand side, this is a number, En1. 7 00:00:30,010 --> 00:00:30,880 It's a number. 8 00:00:30,880 --> 00:00:32,630 It's in an expectation value. 9 00:00:32,630 --> 00:00:36,910 N0 has a unit expectation value, so this is En1. 10 00:00:40,020 --> 00:00:41,970 That's the number. 11 00:00:41,970 --> 00:00:44,500 And then I have an operator here. 12 00:00:44,500 --> 00:00:50,550 So this is minus N0 delta H N0. 13 00:00:53,200 --> 00:00:58,660 So we get a very famous and important equation 14 00:00:58,660 --> 00:01:07,210 that En1 is equal to N0 delta H N0. 15 00:01:12,240 --> 00:01:15,990 The first correction to the energy 16 00:01:15,990 --> 00:01:20,850 is obtained by finding the expectation 17 00:01:20,850 --> 00:01:29,160 value of the perturbation in the unperturbed state. 18 00:01:29,160 --> 00:01:33,090 So you should be happy to hear that. 19 00:01:33,090 --> 00:01:37,500 It says that to find the first correction in the energy, 20 00:01:37,500 --> 00:01:41,490 you don't need to know what happens to the state. 21 00:01:41,490 --> 00:01:47,340 I didn't need to know what N1 was, how the state changes, 22 00:01:47,340 --> 00:01:49,530 to know how the energy changes. 23 00:01:49,530 --> 00:01:55,740 The energy changed before, and I just don't need it. 24 00:01:55,740 --> 00:02:00,720 I just need the original state and the perturbation. 25 00:02:00,720 --> 00:02:09,330 So it's a very nice result, and it has a simple generalization. 26 00:02:09,330 --> 00:02:14,270 The simple generalization is that I'm 27 00:02:14,270 --> 00:02:20,780 going to use another blackboard, but let's look at it. 28 00:02:20,780 --> 00:02:26,730 The simple generalization comes from doing the same exact thing 29 00:02:26,730 --> 00:02:29,620 with the order k equation. 30 00:02:29,620 --> 00:02:30,740 So let's do that. 31 00:02:33,610 --> 00:02:41,710 I put an N0 to the left of this equation. 32 00:02:41,710 --> 00:02:43,810 Now what do we get? 33 00:02:43,810 --> 00:02:47,750 From this term, we get 0 for the same reason. 34 00:02:47,750 --> 00:02:53,200 There's an N0 now here, and the H0 35 00:02:53,200 --> 00:02:55,990 is killed by this term, gives you 0. 36 00:02:58,740 --> 00:03:01,800 And then we're going to continue with this 37 00:03:01,800 --> 00:03:05,250 and see what happens with this various terms. 38 00:03:05,250 --> 00:03:06,330 Let's look at it. 39 00:03:06,330 --> 00:03:12,770 So I'll put the N0-- 40 00:03:17,800 --> 00:03:27,000 0 on the left hand side was equal to 0. 41 00:03:27,000 --> 00:03:31,660 And on the right hand side, what do we get? 42 00:03:31,660 --> 00:03:34,770 Well, let me do a couple of terms. 43 00:03:34,770 --> 00:03:50,950 N0 En1 minus delta H and k minus 1 plus all these other terms. 44 00:03:59,480 --> 00:04:02,250 So look what happens here. 45 00:04:02,250 --> 00:04:04,450 Let's do the next term, for example. 46 00:04:04,450 --> 00:04:11,610 En2, N0, and k minus 2. 47 00:04:16,480 --> 00:04:21,089 Well, from here, this is a number, 48 00:04:21,089 --> 00:04:27,900 so I have here this goes out the overlap of N0 with nk minus 1. 49 00:04:27,900 --> 00:04:31,320 But we said that all the higher corrections 50 00:04:31,320 --> 00:04:34,410 have no component along N0. 51 00:04:34,410 --> 00:04:38,060 So this thing will give you 0. 52 00:04:38,060 --> 00:04:40,230 On the other hand, here is an operator, 53 00:04:40,230 --> 00:04:42,200 so there's nothing I can say. 54 00:04:42,200 --> 00:04:50,630 So I'll write it minus N0 delta H and k minus 1. 55 00:04:54,420 --> 00:04:56,100 And then what else? 56 00:04:56,100 --> 00:04:59,860 Well, you have N0 then k minus 2 here. 57 00:04:59,860 --> 00:05:04,440 That's 0 because this is a higher state 58 00:05:04,440 --> 00:05:08,520 and all the terms give you 0 until you get here where 59 00:05:08,520 --> 00:05:11,580 the N0 with the N0 give you 1. 60 00:05:11,580 --> 00:05:16,110 So the only term that survives is the last one, 61 00:05:16,110 --> 00:05:17,670 and we get enk. 62 00:05:24,290 --> 00:05:29,810 So this gives you the result that enk 63 00:05:29,810 --> 00:05:37,880 is equal to N0 delta H and k minus 1. 64 00:05:42,710 --> 00:05:47,480 I box it because it's another nice formula. 65 00:05:47,480 --> 00:05:51,020 It tells you that the kth order energy 66 00:05:51,020 --> 00:05:58,100 is given if you know the k minus 1 state. 67 00:05:58,100 --> 00:06:01,820 If you have figured out the k minus 1 correction 68 00:06:01,820 --> 00:06:08,450 to the state, then you know the energy of the kth correction. 69 00:06:08,450 --> 00:06:11,360 This formula certainly works when 70 00:06:11,360 --> 00:06:17,630 k is equal to 1 in which case it reproduces the formula we had 71 00:06:17,630 --> 00:06:19,250 on the blackboard to the right. 72 00:06:19,250 --> 00:06:22,790 When k is equal to 1, you get the expectation value 73 00:06:22,790 --> 00:06:26,240 of delta H around zero.