1 00:00:04,740 --> 00:00:08,002 Welcome back everyone. I'm Charles Clark, and these next few 2 00:00:08,002 --> 00:00:12,630 lectures we're going to looking at the problem solving the Schrödinger Equation. 3 00:00:13,930 --> 00:00:18,410 Now in the recent Broadway stage musical, "Curtains", a character says Putting on a 4 00:00:18,410 --> 00:00:23,570 musical must be the most fulfilling thing a person could ever do. 5 00:00:23,570 --> 00:00:27,523 Well, here at exploring quantum physics, we think that solving the Schrodinger 6 00:00:27,523 --> 00:00:32,900 equation is one of the most fulfilling things a person can ever do. 7 00:00:32,900 --> 00:00:37,770 and today you will be that person. We are going to start by. 8 00:00:37,770 --> 00:00:41,340 Actually solving the Schrodinger equation or guiding you to solve the Schrodinger 9 00:00:41,340 --> 00:00:44,604 equation this is tailored to people who've never solved that equation in 10 00:00:44,604 --> 00:00:49,070 their lives before. But we hope there will always be some 11 00:00:49,070 --> 00:00:52,400 information that's useful to people with much experience. 12 00:00:52,400 --> 00:00:56,570 Well let's see. Well here is the Schrodinger equation. 13 00:00:56,570 --> 00:01:00,130 You've seen it before. I h. 14 00:01:00,130 --> 00:01:05,227 Bar decided to use h psi. And let me emphasize again according to 15 00:01:05,227 --> 00:01:10,882 quantum mechanics, all the information that we are able to use to describe the 16 00:01:10,882 --> 00:01:19,843 system is contained in the wave function. And so, this The equation is first order 17 00:01:19,843 --> 00:01:25,954 in time because it, it, tells us how to determine the state at some later time 18 00:01:25,954 --> 00:01:34,052 from the state at a given time. So this equation should be thought of as 19 00:01:34,052 --> 00:01:38,667 an initial value problem in which we are given psi of t at some initial time t And 20 00:01:38,667 --> 00:01:45,518 then, we integrate this equation to find[UNKNOWN] of T plus[UNKNOWN]. 21 00:01:45,518 --> 00:01:50,410 Well, you might ask, what could possibly go wrong in such a simple scheme. 22 00:01:51,590 --> 00:01:58,480 And here is a little in line video to test your understanding. 23 00:01:58,480 --> 00:02:04,140 Well I guess you see from this in line video that 24 00:02:04,140 --> 00:02:07,350 The mechanics of the Schrodinger equation is apparently quite straightforward. 25 00:02:07,350 --> 00:02:11,430 If we have a wave function and a Hamiltonian operator we can just advance 26 00:02:11,430 --> 00:02:17,590 the equation gradually in time by repeated application of that operator. 27 00:02:20,790 --> 00:02:26,710 Well like much in life The difficulty really emerges with the details. 28 00:02:26,710 --> 00:02:30,076 So here's a recent paper, I, showing you my own work cause I do know something 29 00:02:30,076 --> 00:02:35,004 about it. an it's a calculation relevant to some 30 00:02:35,004 --> 00:02:40,000 atomic clock research involving the Ytterbium atom. 31 00:02:40,000 --> 00:02:45,066 you can go look at your periodic table. Ytterbium is at the end of the rare earth 32 00:02:45,066 --> 00:02:50,525 period has 70 electrons. Now let's just say, supposing we were 33 00:02:50,525 --> 00:02:55,550 doing a, a, you know, a calculation that took very explicit account of all the 34 00:02:55,550 --> 00:03:01,940 electrons in the turban atom, there are 70 of those. 35 00:03:01,940 --> 00:03:06,854 And if we represented them on a, a spacial grid that was just ten points on 36 00:03:06,854 --> 00:03:11,768 a side, which is not a lot to represent the motion of a particle we get this 37 00:03:11,768 --> 00:03:21,830 impossibly large number to deal with. Now, it can be reduced by quite a bit. 38 00:03:21,830 --> 00:03:28,288 Due to symmetry, considerations. But it's still just totally infeasible. 39 00:03:28,288 --> 00:03:31,870 So, Much of the, the practical issues of 40 00:03:31,870 --> 00:03:37,380 solving the Schrodinger equation have to do with finding feasible schemes. 41 00:03:41,460 --> 00:03:45,996 Now let me emphasize that the full Schrodinger equation is explicitly time 42 00:03:45,996 --> 00:03:49,598 dependence. It's, the equation for solving the, for 43 00:03:49,598 --> 00:03:55,570 the evolution of quantum system time. And, there are many cases in which. 44 00:03:55,570 --> 00:03:57,600 You need to pursue it from that standpoint. 45 00:03:57,600 --> 00:04:00,744 Here, here's an example. It's actually a solution of a, of a 46 00:04:00,744 --> 00:04:05,640 non-linear Schrödinger equation that describes Bose-Einstein condensate, and 47 00:04:05,640 --> 00:04:13,220 it shows the breakup of a soliton in that condensate into a series of vortex rings. 48 00:04:13,220 --> 00:04:18,236 So to l-, to understand things like this You need to have a full, time-dependent 49 00:04:18,236 --> 00:04:22,262 description. On the other hand, as we have seen over 50 00:04:22,262 --> 00:04:25,982 an over again, there're a lot of interesting phenomena, such as the, you 51 00:04:25,982 --> 00:04:30,198 know, my favorite, the Balmer's spectrum of the hydrogen atom, that seem to be 52 00:04:30,198 --> 00:04:36,766 described by stationary quantum states. That is, these are states that have, a 53 00:04:36,766 --> 00:04:41,610 very simple time evolution. And they're persistent. 54 00:04:41,610 --> 00:04:45,898 They lead to, to sharp structures in, in spectra such as the transitions between 55 00:04:45,898 --> 00:04:49,484 levels. so for the rest of this, these lectures 56 00:04:49,484 --> 00:04:53,196 we're going to focus on the stationary state problem in which we take the time 57 00:04:53,196 --> 00:04:56,966 dependent Schrodinger equation and then look at the special case when the 58 00:04:56,966 --> 00:05:02,660 solution is an eigenfunction of the Hamiltonian operator. 59 00:05:02,660 --> 00:05:09,626 Okay, so just to remember the terminology, this is, this is an 60 00:05:09,626 --> 00:05:18,269 eigenvalue equation, or eigenfunction, and the The Hamiltonian operator is 61 00:05:18,269 --> 00:05:28,250 given, this is sort of like input to the calculation. 62 00:05:29,330 --> 00:05:34,489 Psi and E are unknown, Psi is a so called eigenfunction or eigenvector, we tend to 63 00:05:34,489 --> 00:05:41,225 use those terms exchangeably. And E, the energy of the system is the 64 00:05:41,225 --> 00:05:45,735 eigenvalue. Okay, so what does it mean to actually 65 00:05:45,735 --> 00:05:50,922 solve this equation? Well, the Schrodinger equation is 66 00:05:50,922 --> 00:05:57,010 implemented to describe a wide variety of physical systems. 67 00:05:57,010 --> 00:06:02,122 so our lectures are just going to focus on things that are described in terms of 68 00:06:02,122 --> 00:06:09,680 a, a potential that affects to motion of an electro-, a particle of mass m. 69 00:06:09,680 --> 00:06:12,293 So the, the general form that we're going to consider, which is very rich in terms 70 00:06:12,293 --> 00:06:15,218 of its applications Is an equation, it's a partial differential equation in three 71 00:06:15,218 --> 00:06:17,424 dimensions. we'll occasionally, we will of course for 72 00:06:17,424 --> 00:06:18,514 illustrative purposes consider the problems in lower dimensions as well. 73 00:06:18,514 --> 00:06:20,164 And so it has the form of the kinetic operator here minus h[UNKNOWN] squared 74 00:06:20,164 --> 00:06:41,327 over 2M times del squared. just so you remember, delta squared is d 75 00:06:41,327 --> 00:06:49,260 squared, dx squared plus d squared, dy squared plus d squared, dz squared. 76 00:06:49,260 --> 00:06:54,506 this is the Laplacian operator, widely used throughout physics, 77 00:06:54,506 --> 00:07:02,220 electromagnetism, acoustics. fluid dynamics, and quantum mechanics. 78 00:07:03,390 --> 00:07:10,070 V, is the potential soul[/g] for, for the hydrogen atom . 79 00:07:10,070 --> 00:07:15,470 V would be equal to minus Z E squared over R. 80 00:07:15,470 --> 00:07:22,508 Or for the harmonic oscillator 1/2 M mega squared R squared. 81 00:07:24,400 --> 00:07:29,341 So, for the usual type of problem, we know what the potential is, and we're 82 00:07:29,341 --> 00:07:35,280 kind of solved for the unknown wave function and energy. 83 00:07:37,430 --> 00:07:41,758 But we're going to start here with a, a constructive approach. 84 00:07:41,758 --> 00:07:47,342 Which is based on the idea. I mean, how many actual functions do you 85 00:07:47,342 --> 00:07:51,300 know? of course, you can say, okay. 86 00:07:51,300 --> 00:07:55,620 A function is X, or X plus X to the 5th over 7. 87 00:07:55,620 --> 00:07:59,460 You can generate a large number of ad hoc functions. 88 00:07:59,460 --> 00:08:03,030 but perhaps many of you have a fairly limited. 89 00:08:03,030 --> 00:08:07,854 Vocabulary in terms of generality that would be expanded during this course so 90 00:08:07,854 --> 00:08:12,276 lets just say we are going to start with functions that we recognize and then see 91 00:08:12,276 --> 00:08:16,698 if we can find a potential and an energy For which those functions are solutions 92 00:08:16,698 --> 00:08:26,036 of the Schrodinger equation. This is actually quite a straightforward 93 00:08:26,036 --> 00:08:31,400 thing to do. So here's the Schrodinger equation, now 94 00:08:31,400 --> 00:08:39,423 what we do is we take this term here and move it over to the right hand side. 95 00:08:39,423 --> 00:08:42,591 Then we take the Common factors of, of the wave function and put them over on, 96 00:08:42,591 --> 00:08:46,344 onto the far left here, and we get this very simple equation. 97 00:08:46,344 --> 00:08:54,024 Which means, I mean all you need to do to solve this, all you need to do is be able 98 00:08:54,024 --> 00:09:02,064 to take derivatives So you take the trial wave function, apply the Laplacian 99 00:09:02,064 --> 00:09:14,610 operator to it, divide by the wave function, and see what you get. 100 00:09:15,850 --> 00:09:23,710 So here's a little in-line quiz. To, help walk you through, to what must 101 00:09:23,710 --> 00:09:30,105 be the simplest possible example. So I hope you found that fairly 102 00:09:30,105 --> 00:09:33,330 straightforward. Now, the function that we c-, chose, 103 00:09:33,330 --> 00:09:37,554 which was, s-, size equal to a, a constant, is, you might say, it's not the 104 00:09:37,554 --> 00:09:42,322 most useful solution that you've ever seen. 105 00:09:42,322 --> 00:09:48,088 It does in fact have an application in scattering theory, but it's hardly a 106 00:09:48,088 --> 00:09:57,436 generic function. Our message for the moment is that any 107 00:09:57,436 --> 00:10:02,350 function will provide a solution to a Schrodinger equation. 108 00:10:02,350 --> 00:10:04,400 But not necessarily for physically interesting potential. 109 00:10:05,780 --> 00:10:10,492 And we'll see that in the next part that for any given value of the energy and 110 00:10:10,492 --> 00:10:15,888 whatever the potential may be, there are infinitely many solutions of this, this 111 00:10:15,888 --> 00:10:23,690 Schrodinger equation. So we need more information. 112 00:10:23,690 --> 00:10:28,314 Then is just, evident in the mathematical statement of the equation of this level 113 00:10:28,314 --> 00:10:33,100 in order to perform the solutions that we need to find. 114 00:10:35,120 --> 00:10:39,735 Having said that, however, we're going to conclude this part by construction of E 115 00:10:39,735 --> 00:10:43,765 and V for I say one of the most widely used functions in quantum mechanics, I 116 00:10:43,765 --> 00:10:48,055 think I'll argue in the next segment is the most widely used function in quantum 117 00:10:48,055 --> 00:10:52,150 mechanics, and you've seen it before but let's, let's look at it from the 118 00:10:52,150 --> 00:11:03,616 standpoint of the constructive approach. As act at this point switch to the online 119 00:11:03,616 --> 00:11:12,048 video, and online quiz and then we'll conclude with the a few-, the last few 120 00:11:12,048 --> 00:11:20,940 words I say, in a moment. So that was a demonstration of recovering 121 00:11:20,940 --> 00:11:25,880 the harmonic oscillator potential. From the Gaussian wave function, which is 122 00:11:25,880 --> 00:11:30,750 solution, and we're going to take that up in more detail in the next part.