1 00:00:00,499 --> 00:00:03,980 PROFESSOR: We have two equations now relating 2 00:00:03,980 --> 00:00:06,680 this number of states. 3 00:00:06,680 --> 00:00:13,300 And now you can say, oh, OK, so I look at the k line. 4 00:00:13,300 --> 00:00:15,470 And I look at the little piece of the k 5 00:00:15,470 --> 00:00:17,600 and say, oh, how many states were there 6 00:00:17,600 --> 00:00:21,470 with 0 potential, some number, first blackboard. 7 00:00:21,470 --> 00:00:25,010 How many states are there now with some potential, 8 00:00:25,010 --> 00:00:26,760 some other number? 9 00:00:26,760 --> 00:00:28,230 It has changed. 10 00:00:28,230 --> 00:00:33,890 For every-- because these two equations, the n for equal dk, 11 00:00:33,890 --> 00:00:36,710 the n is not equal to the dn prime. 12 00:00:36,710 --> 00:00:41,660 In one case, the energy levels or the momentum levels 13 00:00:41,660 --> 00:00:44,930 are more compressed or more separated, 14 00:00:44,930 --> 00:00:47,920 but whatever it is, whatever the sign is, 15 00:00:47,920 --> 00:00:51,500 there is a little discrepancy. 16 00:00:51,500 --> 00:00:55,490 So both of them are giving me the total number 17 00:00:55,490 --> 00:00:59,920 of positive energy states in the little dk. 18 00:00:59,920 --> 00:01:02,650 Case So if I take the difference, 19 00:01:02,650 --> 00:01:05,470 I will get some information. 20 00:01:05,470 --> 00:01:08,260 So I would say the following, if I 21 00:01:08,260 --> 00:01:13,930 want to calculate the number of positive energy solutions 22 00:01:13,930 --> 00:01:18,550 and now I think the following, I take the potential V equals 0 23 00:01:18,550 --> 00:01:23,710 and slowly but surely deepen it, push it, and do things 24 00:01:23,710 --> 00:01:31,750 and create the potential V of x slowly, slow the formation. 25 00:01:31,750 --> 00:01:38,170 In this process, I can look at a little interval dk 26 00:01:38,170 --> 00:01:44,780 and tell how many states are positive energy states I lost. 27 00:01:44,780 --> 00:01:51,460 So if, for example, dn is bigger than dn prime, dn equal 5 28 00:01:51,460 --> 00:01:56,020 and dn prime is equal to 3, I started with 5 positive energy 29 00:01:56,020 --> 00:01:59,020 states in this little interval and by the time 30 00:01:59,020 --> 00:02:02,140 I change the potential I ended up with 3. 31 00:02:02,140 --> 00:02:03,730 So I lost 2. 32 00:02:03,730 --> 00:02:14,740 So let me write here the number of positive energy 33 00:02:14,740 --> 00:02:29,420 solutions lost in the interval dk 34 00:02:29,420 --> 00:02:46,690 as the potential is turned on is dn the original number 35 00:02:46,690 --> 00:02:48,990 minus the dn prime. 36 00:02:52,480 --> 00:02:55,690 If that's positive, I've lost state. 37 00:02:55,690 --> 00:03:01,340 If it's negative, I gained state, positive energy states. 38 00:03:01,340 --> 00:03:04,570 In this number, we can calculate the difference. 39 00:03:04,570 --> 00:03:16,290 This is minus 1 over pi d delta dk dk. 40 00:03:16,290 --> 00:03:17,467 I'll put it here. 41 00:03:35,720 --> 00:03:39,650 We'll we're not far. 42 00:03:39,650 --> 00:03:41,690 We'll this is what you lost. 43 00:03:41,690 --> 00:03:43,910 The number of positive energy eigenstates 44 00:03:43,910 --> 00:03:46,580 that you lost in little dk. 45 00:03:46,580 --> 00:03:51,950 To see how many positive energy states you lost over all, 46 00:03:51,950 --> 00:03:55,460 you must integrate over all the dk's and see how much 47 00:03:55,460 --> 00:03:58,010 you lost in every little piece. 48 00:03:58,010 --> 00:04:14,470 So the number of positive energy solutions lost, not in the dk, 49 00:04:14,470 --> 00:04:26,210 but lost as the potential is turned on 50 00:04:26,210 --> 00:04:35,940 is equal to the integral over k from 0 to infinity of minus 1 51 00:04:35,940 --> 00:04:45,700 over pi d delta dk is in the way of that expression 52 00:04:45,700 --> 00:04:49,370 of that right coincide. 53 00:04:49,370 --> 00:04:53,300 But this is a total derivative. 54 00:04:53,300 --> 00:05:02,280 So this is minus 1 over pi delta of k evaluated 55 00:05:02,280 --> 00:05:05,490 between infinity and 0. 56 00:05:05,490 --> 00:05:11,580 And therefore, the number of states lost is 1 over pi, 57 00:05:11,580 --> 00:05:17,170 because of the sign down to 0 minus delta of infinity. 58 00:05:24,920 --> 00:05:28,150 So we're almost there. 59 00:05:28,150 --> 00:05:34,410 This is the number of positive energy solutions lost. 60 00:05:34,410 --> 00:05:39,210 Now I want to emphasize that the situation is quite interesting. 61 00:05:39,210 --> 00:05:41,250 Let me make a little drawing here. 62 00:05:53,470 --> 00:05:55,700 So suppose here is the case where 63 00:05:55,700 --> 00:05:58,070 you have the potential equal to 0 64 00:05:58,070 --> 00:06:01,820 and here is energy equal to 0. 65 00:06:01,820 --> 00:06:03,800 Then you have all these states. 66 00:06:10,200 --> 00:06:15,780 Now even though we've put the wall, 67 00:06:15,780 --> 00:06:18,870 the wall allows us to count the states, 68 00:06:18,870 --> 00:06:21,790 but there are still going to be an infinite number of states. 69 00:06:21,790 --> 00:06:26,920 The infinite square wall has an infinite number of states. 70 00:06:26,920 --> 00:06:36,620 So that thing really continues, but what happens by the time v 71 00:06:36,620 --> 00:06:39,200 is deferred from 0? 72 00:06:39,200 --> 00:06:45,288 Here is that the E equals 0 line and here is 73 00:06:45,288 --> 00:06:48,510 the E equals 0 line. 74 00:06:48,510 --> 00:06:51,910 As we've discussed, as you change 75 00:06:51,910 --> 00:07:00,180 the potential slowly, this are going to shift a little 76 00:07:00,180 --> 00:07:03,160 and some are going to go down here, 77 00:07:03,160 --> 00:07:06,130 are going to become bound states. 78 00:07:06,130 --> 00:07:15,912 They're going to be a number of bound states, N bound states, 79 00:07:15,912 --> 00:07:21,370 number of bound states equal N. And then 80 00:07:21,370 --> 00:07:24,400 there's going to be still sub states here 81 00:07:24,400 --> 00:07:26,860 that's also go to infinity. 82 00:07:31,770 --> 00:07:38,210 So you cannot quite say so easily, well, 83 00:07:38,210 --> 00:07:42,380 the number of states here minus the number of states here is 84 00:07:42,380 --> 00:07:43,970 the number lost. 85 00:07:43,970 --> 00:07:47,020 That's not true, because that's infinite, that's infinite, 86 00:07:47,020 --> 00:07:50,000 and subtracting infinity is bad. 87 00:07:50,000 --> 00:07:56,420 But you know that you've lost a number of finite number 88 00:07:56,420 --> 00:08:00,830 of positive energy solutions. 89 00:08:00,830 --> 00:08:06,710 So as you track here, the number of states must-- 90 00:08:06,710 --> 00:08:11,140 the states must go into each other. 91 00:08:11,140 --> 00:08:15,880 And therefore, if these four states are now here, 92 00:08:15,880 --> 00:08:20,140 before they were here, and those were the positive energy 93 00:08:20,140 --> 00:08:24,370 solutions that were lost, in going from here to here, 94 00:08:24,370 --> 00:08:32,650 you lost positive energy solutions. 95 00:08:32,650 --> 00:08:37,270 You lost a finite number of positive energy solutions. 96 00:08:37,270 --> 00:08:39,610 Even though there's infinite here and infinite here, 97 00:08:39,610 --> 00:08:40,820 you lost some. 98 00:08:40,820 --> 00:08:43,960 And you did that by keeping track at any place 99 00:08:43,960 --> 00:08:47,140 how much you lost. 100 00:08:47,140 --> 00:08:51,700 And therefore the states lost are never really lost. 101 00:08:51,700 --> 00:08:56,440 They are the ones that became the band states here. 102 00:08:56,440 --> 00:08:59,380 So the positive energy states that got lost 103 00:08:59,380 --> 00:09:00,700 are the bound states. 104 00:09:00,700 --> 00:09:05,260 So the number bound states is equal to the number 105 00:09:05,260 --> 00:09:08,230 of positive energy solutions, because there 106 00:09:08,230 --> 00:09:09,550 are no lost states. 107 00:09:09,550 --> 00:09:14,800 So this is equal to a number of bound states, 108 00:09:14,800 --> 00:09:27,990 because there are overall no lost states.