1 00:00:04,212 --> 00:00:09,790 PROFESSOR: Square well. 2 00:00:09,790 --> 00:00:11,260 So what is this problem? 3 00:00:11,260 --> 00:00:17,110 This is the problem of having a particle that can actually just 4 00:00:17,110 --> 00:00:23,590 move on a segment, like it can move on this eraser, 5 00:00:23,590 --> 00:00:25,180 just from the left to the right. 6 00:00:25,180 --> 00:00:27,110 It cannot escape here. 7 00:00:27,110 --> 00:00:31,860 So the way we represent it is the interval 0 8 00:00:31,860 --> 00:00:36,790 to a on the x-axis. 9 00:00:36,790 --> 00:00:44,164 And there's going to be two walls, one wall to the left 10 00:00:44,164 --> 00:00:49,510 and one wall to the right, and no potential in between. 11 00:00:49,510 --> 00:01:00,837 That is, I write the potential V of x as 0, for x in between a 12 00:01:00,837 --> 00:01:09,190 and 0, and infinity for x less than or equal to 0, 13 00:01:09,190 --> 00:01:13,330 and x greater than or equal to a. 14 00:01:16,300 --> 00:01:20,120 So basically the particle can move from 0 to a, 15 00:01:20,120 --> 00:01:21,260 and nowhere else. 16 00:01:21,260 --> 00:01:23,270 The potential is infinity. 17 00:01:27,730 --> 00:01:35,680 Now, this problem, meaning that the wave function-- 18 00:01:35,680 --> 00:01:38,470 the particle cannot be outside the interval, 19 00:01:38,470 --> 00:01:41,560 means that the wave function must vanish outside 20 00:01:41,560 --> 00:01:42,400 the interval. 21 00:01:42,400 --> 00:01:44,530 And you could say, how do you know? 22 00:01:44,530 --> 00:01:49,800 Well, if the potential is close to infinite amount of energy 23 00:01:49,800 --> 00:01:55,150 to be there, so the particle cannot really be there 24 00:01:55,150 --> 00:01:57,585 if it's really infinite energy that you need. 25 00:01:57,585 --> 00:02:02,530 You will see in the finite square well that the particle 26 00:02:02,530 --> 00:02:07,870 has probability to be in regions where it classically cannot be. 27 00:02:07,870 --> 00:02:12,970 But that probability will go to 0 if the potential is infinite. 28 00:02:12,970 --> 00:02:18,370 So we can think of it as a limit and we will reconfirm that. 29 00:02:18,370 --> 00:02:21,710 But in fact, if the potential is infinity, 30 00:02:21,710 --> 00:02:29,020 we will take it to mean that psi of x is equal to 0 for x 31 00:02:29,020 --> 00:02:33,715 less than 0 and for x greater than a. 32 00:02:37,330 --> 00:02:42,330 I am putting this equals or-- 33 00:02:42,330 --> 00:02:45,560 there are many ways of doing this. 34 00:02:45,560 --> 00:02:47,250 If this function, as this continues, 35 00:02:47,250 --> 00:02:50,550 you have a wall at a, is the potential 0 at a 36 00:02:50,550 --> 00:02:51,820 or is it infinity? 37 00:02:51,820 --> 00:02:53,860 Well it doesn't quite matter. 38 00:02:53,860 --> 00:02:59,560 The issue is that the wave function is 0 here, 39 00:02:59,560 --> 00:03:03,620 is 0 there, and we've said that the wave 40 00:03:03,620 --> 00:03:05,720 function must be continuous. 41 00:03:05,720 --> 00:03:11,650 So it should be 0 by that time you're at 0 or at a. 42 00:03:11,650 --> 00:03:17,645 So therefore we will take psi of 0 to be 0, 43 00:03:17,645 --> 00:03:22,600 and psi of a to be 0 by continuity. 44 00:03:30,680 --> 00:03:38,570 So we discuss why the wave function has to be continuous. 45 00:03:38,570 --> 00:03:42,140 If the wave function is not continuous, 46 00:03:42,140 --> 00:03:44,480 the second derivative of the wave function 47 00:03:44,480 --> 00:03:47,510 is terribly singular. 48 00:03:47,510 --> 00:03:50,490 It's like a derivative of a delta function, which 49 00:03:50,490 --> 00:03:52,910 is an impossible situation. 50 00:03:52,910 --> 00:03:55,070 So the wave function, we will take 51 00:03:55,070 --> 00:03:57,860 it to vanish at these two places, 52 00:03:57,860 --> 00:04:00,260 and this is what is called a hard wall. 53 00:04:06,970 --> 00:04:11,960 So what is the Schrodinger equation? 54 00:04:11,960 --> 00:04:14,745 The Schrodinger equation is, again, a free Schrodinger 55 00:04:14,745 --> 00:04:15,460 equation. 56 00:04:15,460 --> 00:04:19,300 Nothing, no potential here, so it's the same Schrodinger 57 00:04:19,300 --> 00:04:29,790 equation we had there, psi double prime equals minus 2mE 58 00:04:29,790 --> 00:04:35,180 over h squared, psi of x. 59 00:04:35,180 --> 00:04:41,870 Or, again, minus k squared psi of x. 60 00:04:45,300 --> 00:04:47,890 Let's solve this. 61 00:04:47,890 --> 00:04:51,070 So how do we do it? 62 00:04:51,070 --> 00:04:54,130 Well it's, again, a very simple equation, 63 00:04:54,130 --> 00:04:56,900 but this time it's conveniences-- 64 00:04:56,900 --> 00:04:59,150 we don't have a circle or periodicity 65 00:04:59,150 --> 00:05:00,890 to use sines and cosines. 66 00:05:00,890 --> 00:05:08,050 So I'll take psi of x to be c1 cosine of kx 67 00:05:08,050 --> 00:05:10,420 plus c2 sine of kx. 68 00:05:14,640 --> 00:05:18,310 But the wave function must vanish at 0. 69 00:05:21,950 --> 00:05:26,640 And at 0, the cosine is 1, so you get c1. 70 00:05:26,640 --> 00:05:32,830 And the sine is 0, so this must be 0, so c1 is gone. 71 00:05:32,830 --> 00:05:38,250 There's no c1 contribution to the solution. 72 00:05:38,250 --> 00:05:48,420 So psi of x is c2 sine of kx. 73 00:05:51,400 --> 00:05:52,730 But we're not done. 74 00:05:52,730 --> 00:05:58,130 We need this function to vanish at the other side. 75 00:05:58,130 --> 00:06:08,380 So psi of x equals a must be 0, and that c2 sine of ka 76 00:06:08,380 --> 00:06:10,366 must be 0. 77 00:06:10,366 --> 00:06:16,980 And therefore we realize that ka must be equal 78 00:06:16,980 --> 00:06:27,970 to a multiple of pi because sine vanishes for 0, pi, 2 pi, 3 pi, 79 00:06:27,970 --> 00:06:33,445 minus pi, minus 2 pi, minus 3 pi, all the multiples of pi. 80 00:06:36,160 --> 00:06:44,820 And therefore we will write kn equals 2 pi n-- 81 00:06:44,820 --> 00:06:46,120 not 2 pi n. 82 00:06:46,120 --> 00:06:50,750 Pi n over a. 83 00:06:55,760 --> 00:07:04,550 OK, well, let me ask you, what should we take for n? 84 00:07:04,550 --> 00:07:06,900 All integers? 85 00:07:06,900 --> 00:07:10,140 Should we skip some? 86 00:07:10,140 --> 00:07:12,980 We took all integers for the circle, 87 00:07:12,980 --> 00:07:16,970 but should we take all integers here? 88 00:07:16,970 --> 00:07:19,400 So what happens here, n equals 0. 89 00:07:19,400 --> 00:07:21,980 What's the problem with n equals 0? 90 00:07:21,980 --> 00:07:26,640 n equals 0, k equals 0, the wave function vanishes. 91 00:07:26,640 --> 00:07:30,260 Well, wave function vanishing is really bad 92 00:07:30,260 --> 00:07:31,960 because there is no particle then. 93 00:07:31,960 --> 00:07:33,890 There is nowhere in the probability 94 00:07:33,890 --> 00:07:35,150 to find the particle. 95 00:07:35,150 --> 00:07:39,302 So n equals 0 is not allowed, for sure. 96 00:07:39,302 --> 00:07:44,290 n equals 0, no. 97 00:07:44,290 --> 00:07:48,185 So why did we allow it, n equals 0, in the circle? 98 00:07:48,185 --> 00:07:53,590 In the circle for n equals 0, exponential doesn't vanish. 99 00:07:53,590 --> 00:07:55,605 It's a constant and that constant 100 00:07:55,605 --> 00:07:58,240 is a fine wave function. 101 00:07:58,240 --> 00:08:01,360 0 is not fine, but the constant is good. 102 00:08:01,360 --> 00:08:03,150 But n equals 0 is not. 103 00:08:03,150 --> 00:08:09,310 So how about positive ends or negative ends. 104 00:08:09,310 --> 00:08:13,690 And here comes the problem, see we're getting to it. 105 00:08:13,690 --> 00:08:21,820 For n equals minus 2 or for n equals 2. 106 00:08:21,820 --> 00:08:25,050 So in one case, k is a number. 107 00:08:25,050 --> 00:08:30,430 And in the other case, k is the opposite sign number. 108 00:08:30,430 --> 00:08:37,350 And sine of a number, or minus a number, that number goes out. 109 00:08:37,350 --> 00:08:47,350 So if you have a sine of minus kx, that's minus sine of kx. 110 00:08:47,350 --> 00:08:50,560 And two wave functions that differ by a sign 111 00:08:50,560 --> 00:08:54,040 are the same wave function, physically. 112 00:08:54,040 --> 00:08:55,240 There's nothing different. 113 00:08:55,240 --> 00:08:58,030 They could differ by an i and other things. 114 00:08:58,030 --> 00:09:02,740 So when you pick negative n minus 1, 115 00:09:02,740 --> 00:09:08,090 or pick n equals plus 1, you get the same wave function, 116 00:09:08,090 --> 00:09:09,790 but just different by a sign. 117 00:09:09,790 --> 00:09:11,360 So it's not new. 118 00:09:11,360 --> 00:09:14,620 So in this case, it's very interesting 119 00:09:14,620 --> 00:09:18,060 that we must restrict ourselves. 120 00:09:18,060 --> 00:09:23,930 We can correct all this and just say n equals 1, 2, 121 00:09:23,930 --> 00:09:27,238 3, all the way to infinity. 122 00:09:31,420 --> 00:09:36,880 The wave function, then, is psi n 123 00:09:36,880 --> 00:09:44,140 of x, is proportional to sine of n pi x over a. 124 00:09:44,140 --> 00:09:48,250 And you look at it and you say, yes, that looks nice. 125 00:09:48,250 --> 00:09:50,170 For x equals 0, it vanishes. 126 00:09:50,170 --> 00:09:53,800 For x equals a, it vanishes. 127 00:09:53,800 --> 00:09:56,380 n and minus n would give me the same wave 128 00:09:56,380 --> 00:09:58,120 function up to a sine. 129 00:09:58,120 --> 00:09:59,240 So this is good. 130 00:09:59,240 --> 00:10:01,740 I just have to normalize it. 131 00:10:01,740 --> 00:10:07,268 And normalizing it would be done by putting an n here. 132 00:10:07,268 --> 00:10:15,500 And then the integral psi n squared dx from 0 to a only 133 00:10:15,500 --> 00:10:24,130 would be n squared integral from 0 to a dx of sine squared n 134 00:10:24,130 --> 00:10:26,590 pi x over a. 135 00:10:32,490 --> 00:10:38,160 Now, you can do this integral by calculation. 136 00:10:38,160 --> 00:10:40,950 And our sine squared is written in terms 137 00:10:40,950 --> 00:10:47,730 of a double angle cosine of double angle plus a 1/2. 138 00:10:47,730 --> 00:10:49,790 The intuition with these things are 139 00:10:49,790 --> 00:10:55,710 that if you're integrating over the right interval that 140 00:10:55,710 --> 00:11:04,710 contains an integer number of cycles of the sine squared, 141 00:11:04,710 --> 00:11:09,120 then the sine squared has average 1/2. 142 00:11:09,120 --> 00:11:12,240 Because sine squared plus cosine squared is equal to 1. 143 00:11:12,240 --> 00:11:15,040 So you don't have to do the interval in general. 144 00:11:15,040 --> 00:11:19,920 This is n squared times 1/2 times the length 145 00:11:19,920 --> 00:11:21,790 of the interval, which is a. 146 00:11:24,610 --> 00:11:27,885 And therefore n squared, this is equal to 1, 147 00:11:27,885 --> 00:11:32,820 and therefore n is equal to square root of 2/a 148 00:11:32,820 --> 00:11:37,320 and we can write now our solutions. 149 00:11:40,730 --> 00:11:48,770 Our solutions are m psi n of x equals the square root 150 00:11:48,770 --> 00:11:57,617 of 2/a sine n pi x over a. 151 00:11:57,617 --> 00:12:03,030 And n equals 1, 2, up to infinity. 152 00:12:03,030 --> 00:12:15,770 And En is equal to h bar squared, k squared, 153 00:12:15,770 --> 00:12:22,238 so pi squared, n squared, a squared, to m. 154 00:12:28,050 --> 00:12:34,080 That's it for the solutions of-- 155 00:12:34,080 --> 00:12:36,520 are there degeneracies? 156 00:12:36,520 --> 00:12:38,140 No. 157 00:12:38,140 --> 00:12:42,630 Every energy state is different because there's 158 00:12:42,630 --> 00:12:45,630 any single 1, 2, 3, infinity, each one 159 00:12:45,630 --> 00:12:48,930 has more energy than the next. 160 00:12:48,930 --> 00:12:54,240 No, I'm sorry, the energy increases as you increase n. 161 00:12:54,240 --> 00:12:56,880 The energy levels actually become more and more 162 00:12:56,880 --> 00:12:59,780 spaced out. 163 00:12:59,780 --> 00:13:04,230 And the last thing I want to do with this box 164 00:13:04,230 --> 00:13:09,420 is to look at the states and see how they look and gather 165 00:13:09,420 --> 00:13:12,540 some important properties that are 166 00:13:12,540 --> 00:13:16,130 going to be very relevant soon.