1 00:00:00,700 --> 00:00:04,164 Welcome back, everybody. And this week we're going to go back to 2 00:00:04,164 --> 00:00:09,686 more traditional quantum mechanics. That is quantum mechanics described in 3 00:00:09,686 --> 00:00:14,186 terms of the Schrodinger equation. And today I'm going to solve the 4 00:00:14,186 --> 00:00:18,550 Schrodinger equation on a number of very important examples. 5 00:00:18,550 --> 00:00:23,104 Where we'll show you how quantum potentials can capture quantum particles 6 00:00:23,104 --> 00:00:28,223 that results in simple balance states. So even though the solutions that we are 7 00:00:28,223 --> 00:00:31,487 going to see, the examples that we are going to see are rather elementary from 8 00:00:31,487 --> 00:00:34,802 the quantum mechanical point of view, well otherwise, they will require some 9 00:00:34,802 --> 00:00:38,750 thinking. So, these examples are going to give us 10 00:00:38,750 --> 00:00:42,780 insight into some rather complicated and fascinating problems. 11 00:00:42,780 --> 00:00:45,325 Such as, for instance theory of superconductivities. 12 00:00:45,325 --> 00:00:49,798 So, at the end of the lecture today, we're going to discuss how our solutions 13 00:00:49,798 --> 00:00:55,620 are relevant to the key phenomenon that is responsible for super-conductivity 14 00:00:55,620 --> 00:01:02,684 that is so called Cooper pairing. But before we get to this point, we need 15 00:01:02,684 --> 00:01:07,264 to go over the basics. So let me start first, with the problem 16 00:01:07,264 --> 00:01:11,110 with electron in a box, that we'll define in a second. 17 00:01:11,110 --> 00:01:14,954 But before that, let me just talk about what kinds of problems we want to, we 18 00:01:14,954 --> 00:01:19,18 can't possibly study with a Schrodinger equation. 19 00:01:19,18 --> 00:01:21,409 So, here is the canonical Schrodinger equation. 20 00:01:21,409 --> 00:01:25,150 we have seen it already, many times. It's a time-dependent Schrodinger 21 00:01:25,150 --> 00:01:30,1 equation as we will see later today. So, actually, if the potential does not 22 00:01:30,1 --> 00:01:33,501 depend on time, there is really no need to study the time-dependent Schrodinger 23 00:01:33,501 --> 00:01:38,332 equation. So we'll see, it can be transformed into 24 00:01:38,332 --> 00:01:43,290 an Eigenvalue problem that we will actually solve. 25 00:01:43,290 --> 00:01:47,382 And in any case, so the first term in this in the single particle, again, 26 00:01:47,382 --> 00:01:50,924 insinuates. So this first term in this Hamiltonian is 27 00:01:50,924 --> 00:01:53,741 just kinetic energy. So there is essentially, it's a non 28 00:01:53,741 --> 00:01:56,714 negotiable term. And the second term is something which is 29 00:01:56,714 --> 00:02:00,618 sort of a problem, dependent, we should determine the potential in which our 30 00:02:00,618 --> 00:02:06,680 particle, quantum particle, moves. So, amazingly the variety of all possible 31 00:02:06,680 --> 00:02:11,860 problems In single particle quantum mechanics, are contained in this equation 32 00:02:11,860 --> 00:02:18,196 in a compact way, and they differ simply by the choice of B of R. 33 00:02:18,196 --> 00:02:22,344 But, this compactness of basically quantum mechanics, is a bit deceptive 34 00:02:22,344 --> 00:02:26,65 because it can give you the impression that it can just solve all possible 35 00:02:26,65 --> 00:02:31,786 quantum mechanics problems in one goals by solving this equation. 36 00:02:31,786 --> 00:02:35,251 Well, this equation is rather each equation, and depending on the choice of 37 00:02:35,251 --> 00:02:38,661 your far of the potential which is particle moves, so we can get completely 38 00:02:38,661 --> 00:02:43,561 different physical situations. And understanding these physical 39 00:02:43,561 --> 00:02:48,350 situations would require dramatically different mathematical approach is two. 40 00:02:48,350 --> 00:02:52,244 However, very roughly, what we can do, we can classify our quantum potential by 41 00:02:52,244 --> 00:02:56,20 putting it in one of two categories of either an attractive or a repulsive 42 00:02:56,20 --> 00:02:59,973 potential, which in the context of quantum physics are called the former are 43 00:02:59,973 --> 00:03:06,768 called potential wills and the latter are called potential barriers. 44 00:03:06,768 --> 00:03:11,927 So here, let me present a, well, sort of simple illustration of what a typical 45 00:03:11,927 --> 00:03:17,82 potential well looks like. So this is v of x, seen 1D quantum 46 00:03:17,82 --> 00:03:20,833 mechanics. And this is a 1D caught, coordinate, and 47 00:03:20,833 --> 00:03:25,164 a potential barrier would look sort of opposite to it, it would look like a 48 00:03:25,164 --> 00:03:31,625 hill. And today we are going to focus our 49 00:03:31,625 --> 00:03:35,680 attention on the physics of quantum potential wells. 50 00:03:35,680 --> 00:03:39,610 So and basically to give you sort of the result right from the outset. 51 00:03:39,610 --> 00:03:42,46 And maybe most of you probably already know this. 52 00:03:42,46 --> 00:03:46,600 in potential wells we're going to see that the available energy levels for the 53 00:03:46,600 --> 00:03:50,570 particle to occupy with the negative energy. 54 00:03:50,570 --> 00:03:54,490 So they're going to be quantized, that is, the, the particle wouldn't be able to 55 00:03:54,490 --> 00:03:58,531 have any arbitrary energy. It's just like in classical physics, 56 00:03:58,531 --> 00:04:01,999 where we could assign any energy, we could, we could put let's say, a ball at 57 00:04:01,999 --> 00:04:06,620 any level and let it oscillate between the turning points. 58 00:04:06,620 --> 00:04:10,292 In quantum mechanics lets say if we put electron in a potential well with this 59 00:04:10,292 --> 00:04:13,970 landscape we cannot assign it in arbitrary energy. 60 00:04:13,970 --> 00:04:17,220 So the available energies are going to be quantized, and finding this quantized 61 00:04:17,220 --> 00:04:20,170 energy levels is one of the sort of canonical problems that we're going to 62 00:04:20,170 --> 00:04:24,60 solve. But this business about finding the 63 00:04:24,60 --> 00:04:29,0 quantized energy levels in an arbitrary quantum well is in principle, rather 64 00:04:29,0 --> 00:04:34,448 complicated technical. And the complexity of this, sort of, 65 00:04:34,448 --> 00:04:38,480 exercise might depends on the particular form of the potential of what we're 66 00:04:38,480 --> 00:04:42,557 studying. however to see, the appearance of this 67 00:04:42,557 --> 00:04:46,333 quantum levels[/g] in general we can focus on the simplest of examples of the 68 00:04:46,333 --> 00:04:50,220 effects. And that's what we're going to do now, 69 00:04:50,220 --> 00:04:55,330 and perhaps the very simplest example is potential which has a so called hard wall 70 00:04:55,330 --> 00:05:02,536 boundaries which implies the full length. So it basically means a potential where 71 00:05:02,536 --> 00:05:07,296 beyond certain points, let's say x equals 0 and x equals l so the value of the 72 00:05:07,296 --> 00:05:12,550 potential of the effects is equal to infinity. 73 00:05:12,550 --> 00:05:16,140 So there is no way the particles can move beyond this point. 74 00:05:16,140 --> 00:05:19,595 These are infinite walls. And in between these two, these two 75 00:05:19,595 --> 00:05:23,60 points, the value of the potential is exactly equal to 0, so which means that 76 00:05:23,60 --> 00:05:26,470 the particle is sort of free to move between these two walls but it cannot go 77 00:05:26,470 --> 00:05:32,232 outside. And this is what is known as a problem of 78 00:05:32,232 --> 00:05:37,588 an electron in the box. So as you can probably guess the solution 79 00:05:37,588 --> 00:05:41,68 of this problem we're going to complete in the remainder of this segment is 80 00:05:41,68 --> 00:05:44,954 going to involved certain energy levels which are going to form a discreet 81 00:05:44,954 --> 00:05:49,160 series. So it turns out to understand the 82 00:05:49,160 --> 00:05:54,346 phenomena of quantization one really doesn't need at quantum mechanics at all. 83 00:05:54,346 --> 00:05:59,420 the quantization occurs in classical physics all over the place. 84 00:05:59,420 --> 00:06:03,512 And so here, for instance, we have an example of a very familiar object a 85 00:06:03,512 --> 00:06:07,604 guitar, which relies on quantization in some sense of the wavelengths and 86 00:06:07,604 --> 00:06:12,985 frequencies available in its oscillating strings. 87 00:06:14,190 --> 00:06:18,610 So here we have, strings sort of which are free to oscillate between two points 88 00:06:18,610 --> 00:06:23,324 where they're pinned down. And that, those two points, in some sense 89 00:06:23,324 --> 00:06:27,120 correspond to the hard world boundaries in the corresponding, quantum mechanical 90 00:06:27,120 --> 00:06:30,659 problem that we're going to study a little later. 91 00:06:31,900 --> 00:06:36,660 And the available wavelengths much depend on the distance between these points, 92 00:06:36,660 --> 00:06:40,650 between these hard walls. Lets call it L. 93 00:06:40,650 --> 00:06:44,790 And to corresponding frequencies are related through the available wavelengths 94 00:06:44,790 --> 00:06:49,202 Y the speed of sound. Let us now think about, what is the 95 00:06:49,202 --> 00:06:53,710 longest possible wavelength that we can induce in a finance strings. 96 00:06:53,710 --> 00:06:58,110 So these are our basic, our hard walls. So this is our 0. 97 00:06:58,110 --> 00:07:01,766 And this is our l. And the hard walls in this context 98 00:07:01,766 --> 00:07:06,560 essentially mean that the the strings cannot oscillate here. 99 00:07:06,560 --> 00:07:11,696 So the displacement let's call it u. from this horizontal axis is exactly 100 00:07:11,696 --> 00:07:16,254 zero. It, x equals zero, and x equals l. 101 00:07:16,254 --> 00:07:22,214 in other words, so we must have news, at these end points. 102 00:07:22,214 --> 00:07:30,510 And, the longest possible wavelength that achieves that is, lambda equals 2l. 103 00:07:30,510 --> 00:07:34,410 And this is sort of the fundamental oscillation that we can, 104 00:07:34,410 --> 00:07:37,318 The longest possible wavelength that we can induce in this stream. 105 00:07:37,318 --> 00:07:41,422 Because, if we try to make the wavelength even longer it would imply, essentially, 106 00:07:41,422 --> 00:07:46,600 that we would have either no node in this point, or no node in this point. 107 00:07:46,600 --> 00:07:49,650 And, this will violate our boundary condition. 108 00:07:49,650 --> 00:07:53,957 But there of course many more wavelengths that are possible, that would satisfy the 109 00:07:53,957 --> 00:07:58,645 proper boundary conditions. And we can achieve we can sort of find 110 00:07:58,645 --> 00:08:03,260 these additional wavelengths, or higher harmonics, as they are called, by find, 111 00:08:03,260 --> 00:08:08,360 by putting additional notes in between these endpoints. 112 00:08:08,360 --> 00:08:13,244 And so for instance this example Gives us a wavelength which is exactly equal to 113 00:08:13,244 --> 00:08:17,550 the L, to the distance between the inputs. 114 00:08:17,550 --> 00:08:22,200 And we can continue this proceedure and generate even more wavelength. 115 00:08:22,200 --> 00:08:26,375 And this wavelength, I'm going to follow this quantization rule, if you want. 116 00:08:26,375 --> 00:08:32,190 with the corresponding. The solution while if you look at the 117 00:08:32,190 --> 00:08:37,90 snapshot of oscillating string. at a certain time. 118 00:08:37,90 --> 00:08:40,70 So the solution is simply going to be given by this sign. 119 00:08:40,70 --> 00:08:42,500 Well some amplicude which is not really important. 120 00:08:42,500 --> 00:08:48,450 Sign of two pie X or along the sub N. So and of course if X is equal to zero. 121 00:08:48,450 --> 00:08:50,517 We're going to have the bond and we're going to set aside the boundary 122 00:08:50,517 --> 00:08:52,580 condition. X equals zero. 123 00:08:52,580 --> 00:08:57,270 And if x is going to be equal to L this quantization rule is going to enforce the 124 00:08:57,270 --> 00:09:02,590 other boundary conditions namely that the displacement vanishes at x equals L so we 125 00:09:02,590 --> 00:09:08,468 have a note here. Now if we look at the corresponding 126 00:09:08,468 --> 00:09:12,820 quantum mechanical problem, so it is essential is very similar to the problem 127 00:09:12,820 --> 00:09:18,170 with this oscillating string. With the only difference being that 128 00:09:18,170 --> 00:09:21,754 instead of putting in the less extreme in between the two points, we put an 129 00:09:21,754 --> 00:09:25,446 electron wave in between these two points. 130 00:09:25,446 --> 00:09:29,156 But the wavelengths that are available for the electron, so the electron cannot 131 00:09:29,156 --> 00:09:33,485 really move beyond this hard wall. Falls right where the probability of 132 00:09:33,485 --> 00:09:36,220 finding the electron area is equal to zero. 133 00:09:36,220 --> 00:09:41,404 Therefore we must demand that, well the probability of, side square of x equals 134 00:09:41,404 --> 00:09:46,60 zero and l is identically equal to zero. Okay. 135 00:09:46,60 --> 00:09:49,781 So or another words side itself is equal to zero and this is much similar to 136 00:09:49,781 --> 00:09:54,10 having a node. At the, at the end point of this string. 137 00:09:54,10 --> 00:09:57,349 So if we solve this problem for the electron, well which involve, which will 138 00:09:57,349 --> 00:10:00,953 involves in this case actually solving the Schrodinger equation, we're going to 139 00:10:00,953 --> 00:10:05,714 find exactly the same wavelength. And the wave function is actually going 140 00:10:05,714 --> 00:10:09,780 to look exactly the same as the, of the displacement of the strings. 141 00:10:09,780 --> 00:10:13,93 So there's actually no difference. I just copied pasted exactly the same 142 00:10:13,93 --> 00:10:16,378 equations. So we're only replacing use of M which is 143 00:10:16,378 --> 00:10:21,450 a displacement by side of M which is the wave function of electron. 144 00:10:21,450 --> 00:10:24,465 So this shows you that there's actually there is a lot of analogy in quantization 145 00:10:24,465 --> 00:10:29,720 of electric wavelength. Wavelength and quantization of wavelength 146 00:10:29,720 --> 00:10:34,850 of classical objects, classical objects. So the only difference, and the important 147 00:10:34,850 --> 00:10:38,388 difference here, would be in how the frequency or the energy in the case of 148 00:10:38,388 --> 00:10:43,697 electron scale with the wavelength. So here we discuss that the frequency of 149 00:10:43,697 --> 00:10:47,296 the oscillation, which by the way determines the sound you actually hear 150 00:10:47,296 --> 00:10:53,410 scales linearly with the wavelength. On the other hand we know that the energy 151 00:10:53,410 --> 00:10:58,130 of a quantum particle of just well, kinetic energy, essentially a free 152 00:10:58,130 --> 00:11:03,100 particle is going to be p squared over 2 m. 153 00:11:03,100 --> 00:11:08,941 And if we recall the, deployed relation between the momentum 154 00:11:08,941 --> 00:11:14,800 and the wavelength, which is 2 pi h bar over lambda is equal to p. 155 00:11:14,800 --> 00:11:18,582 So then we can readily combine these two results, the quantization of the 156 00:11:18,582 --> 00:11:22,582 wavelength. And the scaling of the energy to get the 157 00:11:22,582 --> 00:11:26,500 quantization of energy. So if you put everything together, the 158 00:11:26,500 --> 00:11:30,490 quantization rule and the debroyal relation, we're going to get the 159 00:11:30,490 --> 00:11:36,390 quantized energy levels. e sub n, where n is a positive integer 160 00:11:36,390 --> 00:11:42,240 equal to pi squared. H squared, n squared divided by the l 161 00:11:42,240 --> 00:11:47,830 squared, the distance here. This corresponds to the momentum squared 162 00:11:47,830 --> 00:11:52,993 times 1 over twice the mass. So and notice that interestingly we 163 00:11:52,993 --> 00:11:57,963 solved the Schrodinger equation without writing it down, the only thing we used 164 00:11:57,963 --> 00:12:02,386 is Basically was the boundary conditions and 165 00:12:02,386 --> 00:12:07,0 the some analogy with classical physics along with the, broad relation. 166 00:12:07,0 --> 00:12:10,315 But of course one can solve it formally and we're going to show, later how it 167 00:12:10,315 --> 00:12:14,18 works. So, sadly one cannot always solve the 168 00:12:14,18 --> 00:12:19,800 Schrodinger equation in such a simplistic way without writing it down. 169 00:12:19,800 --> 00:12:24,503 So and to understand. Why it happens, we can again, use this 170 00:12:24,503 --> 00:12:29,640 guitar sort of string analogy. So, if we push down on the string, so we 171 00:12:29,640 --> 00:12:34,539 effectively shorten the distance between these end points and the effective end 172 00:12:34,539 --> 00:12:39,689 points that we enforce here. And by doing so, we, of course, change 173 00:12:39,689 --> 00:12:43,730 the fundamental wavelength. And the quantization of the wave length 174 00:12:43,730 --> 00:12:48,50 and the frequencies, this sort of results in different sounds that we produce this 175 00:12:48,50 --> 00:12:51,833 way. But, if we push, not very hard, but if we 176 00:12:51,833 --> 00:12:55,865 just touch the string here, so this creates instead of an infinite wall, sort 177 00:12:55,865 --> 00:13:02,490 of, the, the impenetrable wall for the string to propagate beyond this point. 178 00:13:02,490 --> 00:13:06,702 It creates a finite barrier. And to solve, the wave equation for this 179 00:13:06,702 --> 00:13:09,614 string, the classical wave equation in the presence of such a small 180 00:13:09,614 --> 00:13:12,994 perturbation, is actually more complicated than solving, this equation 181 00:13:12,994 --> 00:13:16,936 for 2 hard walls. And likewise, we're going to see, 182 00:13:16,936 --> 00:13:20,792 actually, that, the, problem of a quantum mechanical particle. 183 00:13:20,792 --> 00:13:23,918 When, instead of the hard walls. We have, let's see, finite barrier here, 184 00:13:23,918 --> 00:13:27,794 is slightly more complicated, well more complicated and it requires, actually a 185 00:13:27,794 --> 00:13:31,727 serious mathematical calculations in solving the actual Schrodinger equation, 186 00:13:31,727 --> 00:13:37,167 which is a differential equation. We're going to discuss it in the 187 00:13:37,167 --> 00:13:38,843 following segment.