1 00:00:03,610 --> 00:00:06,940 Hello everyone, welcome back to Exploring Quantum Physics. 2 00:00:06,940 --> 00:00:08,500 I'm Charles Clark. 3 00:00:08,500 --> 00:00:10,410 We're now going to start to apply some of 4 00:00:10,410 --> 00:00:14,320 the greatest tools that mathematicians have ever developed. 5 00:00:14,320 --> 00:00:16,530 The special functions and apply them to 6 00:00:16,530 --> 00:00:20,140 solving problems of Quantum mechanics of simple systems. 7 00:00:22,360 --> 00:00:23,600 Now special functions. 8 00:00:23,600 --> 00:00:27,500 There are many good sources of information on special functions available. 9 00:00:27,500 --> 00:00:30,970 but it's nice, that they have a common reference for, so 10 00:00:30,970 --> 00:00:32,590 that everyone in this course will be using the same one. 11 00:00:34,020 --> 00:00:37,760 So we're using this the digital library mathematical functions, 12 00:00:37,760 --> 00:00:40,960 from my employer, the National Institute of Standards and Technology. 13 00:00:40,960 --> 00:00:44,050 It's a free App, freely available worldwide. 14 00:00:44,050 --> 00:00:47,780 And it contains information on, 15 00:00:47,780 --> 00:00:50,300 on all the functions that we'll be using in this course and in 16 00:00:50,300 --> 00:00:53,860 many more, and, well, there are many other things to be said about it. 17 00:00:53,860 --> 00:00:57,920 But when I make a reference to something like the Airy function, which 18 00:00:57,920 --> 00:01:03,070 will be done later in this part, then what I mean are the, the 19 00:01:03,070 --> 00:01:06,360 specific representations that are provided there, which 20 00:01:06,360 --> 00:01:08,290 they're very much standard are the same 21 00:01:08,290 --> 00:01:11,720 that you'll find in Wikipedia, but I encourage you to look at this anyway. 22 00:01:14,830 --> 00:01:21,360 Two several other resources for for you. short crash course on mathematics. 23 00:01:21,360 --> 00:01:23,580 that's something certainly worth looking at because it 24 00:01:23,580 --> 00:01:27,190 does provide a constructive by Zack Lains and his 25 00:01:27,190 --> 00:01:29,990 collaborators, and, gives you pretty good idea of 26 00:01:29,990 --> 00:01:34,700 the level of mathematical treatment that started this course. 27 00:01:34,700 --> 00:01:40,060 another one page guide to how, to the 28 00:01:40,060 --> 00:01:43,800 solutions, known solutions of the Schrodinger 29 00:01:43,800 --> 00:01:46,110 equation in terms of special functions. 30 00:01:48,210 --> 00:01:54,733 and what else, oh, a sample of a program written in spreadsheet form for 31 00:01:54,733 --> 00:01:59,826 solving the Schrodinger equation by numerical integration. 32 00:01:59,826 --> 00:02:03,140 So that's something we won't, I won't be using 33 00:02:03,140 --> 00:02:05,530 that in this part, but something worth looking at. 34 00:02:05,530 --> 00:02:07,800 There's a discussion discussion there. 35 00:02:07,800 --> 00:02:10,490 And it'll give you some simple guides as to 36 00:02:10,490 --> 00:02:13,450 how to numerically integrate the Schrodinger equation in cases 37 00:02:13,450 --> 00:02:15,700 when you're dealing with a a complex 38 00:02:15,700 --> 00:02:17,430 problem for which there is no known solution. 39 00:02:23,600 --> 00:02:27,080 So, who was the first person to use 40 00:02:27,080 --> 00:02:29,989 this approach, solving a Schrodinger equation using special functions? 41 00:02:31,380 --> 00:02:33,660 no prizes for getting this right, it was, in fact, Schrodinger. 42 00:02:33,660 --> 00:02:39,159 And you might say that his first work, first papers 43 00:02:39,159 --> 00:02:44,307 on the, on the wave equation, were influential for 44 00:02:44,307 --> 00:02:50,100 several reasons. But one, one lasting 45 00:02:50,100 --> 00:02:55,630 development was that it, it, it cast the equation, the Schrodinger equation of 46 00:02:55,630 --> 00:02:58,310 Quantum mechanics, into a form that had 47 00:02:58,310 --> 00:03:03,540 been well understood by in classical mechanics. 48 00:03:03,540 --> 00:03:07,000 And that was to re-, reduced 49 00:03:07,000 --> 00:03:09,360 the Schrodinger equation, differential equation that 50 00:03:09,360 --> 00:03:12,600 had solutions in terms of functions that were known at the time. 51 00:03:14,080 --> 00:03:15,353 So what Schrodinger, 52 00:03:15,353 --> 00:03:18,301 what Schrodinger did, of course, his, his work, 53 00:03:18,301 --> 00:03:20,920 the importance of the work stands on its own. 54 00:03:20,920 --> 00:03:24,456 He explained the spectrum of a hydrogen atom, calculated the shifts of spectral 55 00:03:24,456 --> 00:03:27,056 lines, he invented, invented perturbation theory, all 56 00:03:27,056 --> 00:03:29,710 in this series of four remarkable papers. 57 00:03:29,710 --> 00:03:34,930 But certainly is very important for the, the wider 58 00:03:34,930 --> 00:03:40,760 implementation of his ideas, that he developed a framework that mapped 59 00:03:40,760 --> 00:03:47,660 on to well-developed problems of classical mathematical physics, 60 00:03:47,660 --> 00:03:49,460 and so we're following the same approach here. 61 00:03:53,510 --> 00:03:58,938 So in this part of the lecture, we're going to look at 62 00:03:58,938 --> 00:04:04,760 three, we're going to stick with one-dimensional examples. 63 00:04:04,760 --> 00:04:11,740 And we'll look at a constant potential, a linear potential, and, I 64 00:04:11,740 --> 00:04:14,400 guess, it'll be in the next part what are this quadratic potential. 65 00:04:14,400 --> 00:04:18,830 But you might say that these are the, the three simplest possible potentials in 66 00:04:18,830 --> 00:04:24,290 terms of development as, as polynomials. And 67 00:04:24,290 --> 00:04:29,210 the solutions of the Schrodinger Equation for these cases are the exponential 68 00:04:29,210 --> 00:04:34,280 function, for the linear, for the constant potential, the Airy function for 69 00:04:34,280 --> 00:04:39,277 the linear potential, so this is exponential Airy, 70 00:04:39,277 --> 00:04:44,040 and parabolic cylinder 71 00:04:44,040 --> 00:04:45,000 for the quadratic. 72 00:04:45,000 --> 00:04:47,170 Now, the harmonic oscillator's a special case of that. 73 00:04:48,310 --> 00:04:51,080 but we, we're going to go through in this, in this 74 00:04:51,080 --> 00:04:55,720 part we're going to go through these two cases, and, and try 75 00:04:55,720 --> 00:05:00,670 to infer from them some general principles that are relevant both for the analytical 76 00:05:00,670 --> 00:05:06,840 and numerical solution of one-dimensional Schrodinger equations. 77 00:05:06,840 --> 00:05:09,020 Right, so let's start with the linear potential. 78 00:05:09,020 --> 00:05:14,750 We just take V of x to be a constant. And so here's the Schrodinger equation. 79 00:05:14,750 --> 00:05:22,040 Now, the point is we want to find solutions for any value of the energy. 80 00:05:22,040 --> 00:05:24,130 We want to find a general, general solution 81 00:05:24,130 --> 00:05:27,860 valid for all, for all energies because then 82 00:05:27,860 --> 00:05:30,780 with, with such solutions in hand we can 83 00:05:30,780 --> 00:05:34,360 find the particular solutions that we need to satisfy 84 00:05:34,360 --> 00:05:35,720 boundary conditions. 85 00:05:35,720 --> 00:05:38,200 You'll see why this is important in a moment. 86 00:05:38,200 --> 00:05:42,720 But for example, if you just consider a case where the, the potential is 87 00:05:44,570 --> 00:05:48,560 is, is constant over different regions 88 00:05:51,020 --> 00:05:53,080 but consider different values in different regions. 89 00:05:53,080 --> 00:05:56,880 And you want to find a wave function, a solution to wave equation, that 90 00:05:56,880 --> 00:06:01,710 is valid it, it'll be, it'll be a sufficient to have this information. 91 00:06:01,710 --> 00:06:05,120 So you can piece, you an stitch the various pieces together. 92 00:06:07,010 --> 00:06:16,100 So the approach we take is to transform the Schrodinger equation up here into 93 00:06:16,100 --> 00:06:20,530 a standard type. in this case the standard 94 00:06:20,530 --> 00:06:25,370 form that we see, that we see is here, where rho is a 95 00:06:26,700 --> 00:06:31,374 scaled variable, dimensionless variable, and and 96 00:06:31,374 --> 00:06:37,290 then the plus or minus sign has to do with whether 97 00:06:37,290 --> 00:06:45,790 energy, positive or negative. 98 00:06:45,790 --> 00:06:49,930 We know that energy's can, can be any number on the real line. 99 00:06:49,930 --> 00:06:52,030 so we need to account for both of those cases. 100 00:06:55,970 --> 00:07:00,590 Okay, so we just take that system. We 101 00:07:03,510 --> 00:07:06,130 transform, we find that dimensionless variable row is 102 00:07:06,130 --> 00:07:10,230 kx, k is the wave vector and it's 103 00:07:13,190 --> 00:07:16,850 depending on whether the energy is greater than or less than 104 00:07:18,860 --> 00:07:22,760 the potential energy, well this, the value of k is, is 105 00:07:25,280 --> 00:07:29,720 determined by that. And so there's two classes of solutions. 106 00:07:29,720 --> 00:07:35,810 So, when E is greater than V0, we have two independent solutions. 107 00:07:35,810 --> 00:07:39,690 So, you know, let me just say again, as a second order differential equation, 108 00:07:41,350 --> 00:07:43,330 any second order differential equation like this 109 00:07:43,330 --> 00:07:47,009 one, has at least two independent solutions. 110 00:07:48,010 --> 00:07:50,780 there are many different ways to choose them but 111 00:07:50,780 --> 00:07:53,690 here I've just, I've, I've given you two independent 112 00:07:53,690 --> 00:07:57,220 solutions sin a rho, and the cosine in a rho. 113 00:07:57,220 --> 00:08:02,122 we'll say a little bit more about the choice momentarily. 114 00:08:02,122 --> 00:08:06,230 And then in the region where E is 115 00:08:06,230 --> 00:08:11,540 less than V0, the two inde-, the two independent 116 00:08:11,540 --> 00:08:16,890 solutions can be chosen of this form, E to the rho and E to the minus rho. 117 00:08:16,890 --> 00:08:20,410 So any other solution that exists in this 118 00:08:20,410 --> 00:08:26,040 region can, is going to be a linear combination of these two functions. 119 00:08:26,040 --> 00:08:31,220 But that means the generic behavior is divergence, 120 00:08:31,220 --> 00:08:36,572 because if you just take you just take a random choice of these two functions, 121 00:08:36,572 --> 00:08:42,720 then the resulting solution is going to diverge 122 00:08:42,720 --> 00:08:45,010 as rho goes to infinity. 123 00:08:47,480 --> 00:08:51,270 And in the exceptional case where you just choose this function, 124 00:08:51,270 --> 00:08:55,716 then it diverges going to and rho go to the minus infinity. 125 00:08:55,716 --> 00:08:58,420 Now that can be, you know that can be changed by another boundary condition. 126 00:08:58,420 --> 00:09:00,050 You have to be aware of that. 127 00:09:00,050 --> 00:09:06,780 In this, for this particular circumsance, the generic behavior is divergence. 128 00:09:06,780 --> 00:09:10,035 So that, that's when that means that you, 129 00:09:10,035 --> 00:09:12,650 you have difficult choices that have to be made. 130 00:09:16,420 --> 00:09:21,280 So I'll make two points about this,, this, this, you know, subsets 131 00:09:21,280 --> 00:09:27,820 trivial Schrodinger equation, which either has oscillatory or exponential solutions. 132 00:09:27,820 --> 00:09:29,350 And that is if you have a 133 00:09:29,350 --> 00:09:32,420 general understanding of the solutions, then you could 134 00:09:32,420 --> 00:09:38,020 do a, a piecewise approximate solution, to to 135 00:09:38,020 --> 00:09:41,460 a real potential, get a real potential that 136 00:09:41,460 --> 00:09:42,960 has some arbitrary shape. 137 00:09:42,960 --> 00:09:47,270 You can approximate it on a piecewise basis, and then by knowing how to solve 138 00:09:47,270 --> 00:09:50,450 the equation regions of constant potential, you can 139 00:09:50,450 --> 00:09:55,120 sort of, propagate solutions between different such regions. 140 00:09:55,120 --> 00:09:59,150 This is something that's used in modeling semiconductor heterostructures and so on, 141 00:09:59,150 --> 00:10:06,080 or the photonic band-gap materials are well understood in terms of having 142 00:10:10,220 --> 00:10:12,830 techniques for, to propagate a wave function 143 00:10:12,830 --> 00:10:15,860 across a constant region and then to stitch 144 00:10:15,860 --> 00:10:19,270 the, you know, stitch the transition matrix 145 00:10:19,270 --> 00:10:22,820 together, to see what the net result is. 146 00:10:22,820 --> 00:10:28,770 So actually why don't we just now apply that approach to the simplest possible 147 00:10:28,770 --> 00:10:32,580 problem of crossing an interface, and we're 148 00:10:32,580 --> 00:10:35,500 going to look at an example that you, 149 00:10:35,500 --> 00:10:36,590 you see every day. 150 00:10:37,850 --> 00:10:40,030 if you look at a, look through a window 151 00:10:40,030 --> 00:10:42,660 you see a, you see a reflection of yourself. 152 00:10:42,660 --> 00:10:46,220 I mean even if the glass is transparent there is some reflection. 153 00:10:46,220 --> 00:10:54,540 And so it turns out that this is a phenomena that's an example 154 00:10:54,540 --> 00:11:00,200 of this, the sort of the the potential step function in Quantum mechanics. 155 00:11:00,200 --> 00:11:00,780 So we have 156 00:11:01,994 --> 00:11:10,920 V of x as a function of x. this is air, this is glass. 157 00:11:10,920 --> 00:11:17,070 So, the propagation of a light-wave as it, enters a window is the same very 158 00:11:17,070 --> 00:11:22,840 similar mathematically to the the problem of a particle moving in a potential. 159 00:11:22,840 --> 00:11:25,140 So the way that, that is framed is that the 160 00:11:27,960 --> 00:11:33,270 we have Y incident from a large distance on the negative axis. 161 00:11:33,270 --> 00:11:36,820 That wave function is a form of e to the ikx. 162 00:11:36,820 --> 00:11:41,980 Then there's a reflected component, which is of the type R, the reflection 163 00:11:41,980 --> 00:11:48,250 coefficient times e to the mi-, minus ikx, and then the the 164 00:11:48,250 --> 00:11:53,150 transmitted coefficient here. Now, how do we, 165 00:11:54,270 --> 00:11:58,530 how do we determine the effect of glass on light? 166 00:11:59,754 --> 00:12:03,590 let me refer you to an inline quiz. 167 00:12:06,220 --> 00:12:12,980 So when light goes into glass, it's the, the 168 00:12:12,980 --> 00:12:21,020 refractive index of glass is about 1.5, that means that its wave length decreases. 169 00:12:21,020 --> 00:12:25,440 And so that is that means that the, the wave, the, the wave 170 00:12:25,440 --> 00:12:30,450 equation, spacial part of the wave equation in the interior as this form. 171 00:12:30,450 --> 00:12:31,100 And so now, 172 00:12:31,100 --> 00:12:36,340 now it's very easy to it's very easy to determine the reflection coefficient. 173 00:12:36,340 --> 00:12:38,210 So what we, we have, we have, we have a wave 174 00:12:38,210 --> 00:12:45,080 function on two sides of the interface, the air and the glass. 175 00:12:45,080 --> 00:12:47,510 So we have to match the values of the wave function to the boundary. 176 00:12:47,510 --> 00:12:50,920 So for x equals 0, that's very easy, it's 1 plus R here. 177 00:12:50,920 --> 00:12:53,130 It's T here, so we have to have 1 plus R is equal to T. 178 00:12:54,590 --> 00:12:56,645 And then, we match the derivatives, 179 00:12:56,645 --> 00:13:01,200 that's is the equation of continuity that's required in Quantum mechanics. 180 00:13:01,200 --> 00:13:06,720 So, once again, it's easy to differentiate this side and the answer's ik 1 181 00:13:06,720 --> 00:13:12,540 minus R, and this side is just iknT. And so now 182 00:13:18,090 --> 00:13:19,860 what do we do? Well we divide 183 00:13:22,620 --> 00:13:23,570 we, we 184 00:13:25,610 --> 00:13:28,360 we, we, we take this side, 185 00:13:33,020 --> 00:13:38,020 and divide it by, by the other one. So, you see, I'm just 186 00:13:38,020 --> 00:13:43,610 taking, this divided by that, this divided by that, equals that divided by that. 187 00:13:43,610 --> 00:13:49,110 Get this simple equation for the reflective's coefficient. 188 00:13:49,110 --> 00:13:51,490 So, you see that, there, the k, the 189 00:13:51,490 --> 00:13:54,860 wave vector disappears as does the transmission coefficient. 190 00:13:54,860 --> 00:13:58,900 And then it's easy to invert this equation, to get this. 191 00:13:58,900 --> 00:14:04,132 So R is equal to minus 0.5 over 2.5, which is 1 192 00:14:04,132 --> 00:14:08,880 fifth. So the actual reflected intensity is R 193 00:14:08,880 --> 00:14:15,591 squared. Coefficient is 0.04, so it's 4%. 194 00:14:15,591 --> 00:14:21,270 This is, this is about right. When you put light through a window that 195 00:14:21,270 --> 00:14:26,990 normal instance about 4% is reflected. 196 00:14:26,990 --> 00:14:31,988 So on the next part of this lecture we're going to explore the next 197 00:14:31,988 --> 00:14:37,880 sequence, next in the sequence of complexity, the linear potential.