1 00:00:00,560 --> 00:00:04,528 Now we're going to rely on wave particle duality to deduce the form of the main 2 00:00:04,528 --> 00:00:09,80 equation of quantum physics and the Schrodinger equation. 3 00:00:09,80 --> 00:00:13,900 But before going to this equation I first would like to ask the following question. 4 00:00:13,900 --> 00:00:17,368 Which picture is more fundamental, the particle-based picture or the wave based 5 00:00:17,368 --> 00:00:20,578 picture? So here I have a simple example of what 6 00:00:20,578 --> 00:00:25,437 we usually mean by a particle. In this example, here is the baseball and 7 00:00:25,437 --> 00:00:29,280 so this is an object which has a well-defined velocity and position at any 8 00:00:29,280 --> 00:00:33,348 given time. We can easy to identify position just by 9 00:00:33,348 --> 00:00:37,975 looking at it and measure the velocity. On contrast, when we're talking about 10 00:00:37,975 --> 00:00:42,545 waves, it doesn't really make sense to us the question where it located. 11 00:00:42,545 --> 00:00:47,355 So here for instance, we have an example of wave which is generated using a rope 12 00:00:47,355 --> 00:00:52,980 and well, we cannot define the position of the wave, okay? 13 00:00:52,980 --> 00:00:58,50 Probably what we can do, we can define its phase velocity and the wavelength and 14 00:00:58,50 --> 00:01:02,990 the corresponding wave vectors, so which is so wavelength lambda is equal to 2 pi 15 00:01:02,990 --> 00:01:07,610 divided by k. k is the wavelength, right? 16 00:01:07,610 --> 00:01:11,130 Omega is the frequency. So the velocity of the wave is the 17 00:01:11,130 --> 00:01:15,461 coefficient between the frequency and the, coefficient of proportionality 18 00:01:15,461 --> 00:01:21,133 between the frequency and the wavelength. Now there are two common expressions 19 00:01:21,133 --> 00:01:24,73 which are two types of common expressions that I use to describe a simple 20 00:01:24,73 --> 00:01:27,900 sinusoidal wave that we will see pretty often. 21 00:01:27,900 --> 00:01:32,730 So one is a just using the sine or cosine function such as here, so in this example 22 00:01:32,730 --> 00:01:37,980 u is the vertical displacement, which is a function of x position and time, and x 23 00:01:37,980 --> 00:01:43,676 in this case, in his horizontal direction. 24 00:01:43,676 --> 00:01:47,763 So another way to describe a wave is using this exponential function of an 25 00:01:47,763 --> 00:01:52,960 imaginary constant i. So i carries the square root of minus 1. 26 00:01:52,960 --> 00:01:56,265 and, they're equivalent to one another. And most commonly we're actually going to 27 00:01:56,265 --> 00:01:59,854 use the latter expression. Now going back to the main question what, 28 00:01:59,854 --> 00:02:02,834 what is more fundamental, particles or waves? 29 00:02:02,834 --> 00:02:07,68 I would argue, is that waves are in fact much more fundamental in that we can't 30 00:02:07,68 --> 00:02:11,84 really represent the wave in terms of a particle. 31 00:02:11,84 --> 00:02:15,494 But we can surely decompose a localized particle into waves. 32 00:02:15,494 --> 00:02:19,659 And, this decomposition goes by the name of Fourier transform. 33 00:02:19,659 --> 00:02:24,741 So, here I have an example of a generic Fourier transform, which is a wave, which 34 00:02:24,741 --> 00:02:29,438 is represented in an arbitrary reasonable function F of X in terms of this 35 00:02:29,438 --> 00:02:37,98 exponentials E to the power of ikx. We are sure transform to this wave 36 00:02:37,98 --> 00:02:43,763 describing wave at any given time. And, the corresponding efficiency here, 37 00:02:43,763 --> 00:02:48,727 which multiply these exponential are both the upper mornings of the function f of 38 00:02:48,727 --> 00:02:53,808 x. In particular we can consider a function 39 00:02:53,808 --> 00:02:59,531 f of x which shows a sharp peak, see, with a narrow, spread out, let's say 40 00:02:59,531 --> 00:03:04,398 delta x. And we can consider this function as a 41 00:03:04,398 --> 00:03:08,990 mathematical description of an entity which is a particle like entity. 42 00:03:08,990 --> 00:03:13,392 And by decomposing it into a Fourier transform, we represent it as a linear 43 00:03:13,392 --> 00:03:17,929 combination of waves. So finally I would like to use the wave 44 00:03:17,929 --> 00:03:22,153 particle duality for electrons now in order to guess, if you wondered, to 45 00:03:22,153 --> 00:03:29,740 derive in quotes the wave equation that governs their quantum wave properties. 46 00:03:29,740 --> 00:03:33,480 So here we're relying on on experiments of the type performed by, let's say 47 00:03:33,480 --> 00:03:37,605 Dennison and Germer, lacking the fraction where we clearly saw that the data can be 48 00:03:37,605 --> 00:03:43,638 on the studio but we assume that the electrons behaves behave as waves. 49 00:03:43,638 --> 00:03:47,298 And so let's assume that this is so, let's assume that the free electron is 50 00:03:47,298 --> 00:03:51,198 described by some wave function, psi, so the precise physical meaning of this 51 00:03:51,198 --> 00:03:55,218 function will be discussed later about the course, but this stage let us just 52 00:03:55,218 --> 00:04:01,102 assume that this is so. So just like in the case of photons we 53 00:04:01,102 --> 00:04:04,900 have an electric field which had the form of a plain wave. 54 00:04:04,900 --> 00:04:08,290 Here we have some electron wave function which has the form of a plain wave. 55 00:04:08,290 --> 00:04:14,456 With some momentum P and energy epsilon. But, unlike photons where energy scales 56 00:04:14,456 --> 00:04:18,544 with momentum linearly, so, we know that for that non-relativistic electrons, the 57 00:04:18,544 --> 00:04:23,640 kinetic energy is basically m v squared over 2 or a p squared over 2m. 58 00:04:23,640 --> 00:04:28,491 So, what we, what we have to demand is whatever equation governs the quantum 59 00:04:28,491 --> 00:04:34,824 properties of electrons. It must give us this functional form to 60 00:04:34,824 --> 00:04:41,240 describe a free electron. And this spectrum as a constraint on this 61 00:04:41,240 --> 00:04:44,771 solution. And so, basically we can construct such 62 00:04:44,771 --> 00:04:48,740 an equation by hand, that gives that has this properties. 63 00:04:48,740 --> 00:04:52,900 And this is what we can do, so we can write this equation. 64 00:04:52,900 --> 00:04:56,794 Now here if we plug in this guy into this equation, so the first derivative is 65 00:04:56,794 --> 00:05:01,920 going to give us minus h squared over 2 m, m here is the mass of course. 66 00:05:01,920 --> 00:05:08,600 The second derivative is going to give us minus 1 over h bar squared p squared. 67 00:05:08,600 --> 00:05:13,760 And the second term here is going to give us, minus i h bar d over d t, is minus i 68 00:05:13,760 --> 00:05:19,52 h bar epsilon. And we will have the same playing with 69 00:05:19,52 --> 00:05:26,82 multiplying both terms, so we must demand that this that this guys are equal to 0. 70 00:05:26,82 --> 00:05:32,472 Okay, so a lot of things cancel out here, so we have P squared over 2M here, and we 71 00:05:32,472 --> 00:05:38,860 have a minus minus, so basically minus X on here. 72 00:05:38,860 --> 00:05:43,414 And indeed, were, we reproduce this desired spec. 73 00:05:43,414 --> 00:05:48,560 Now, this constructional course is nothing but a guess. 74 00:05:48,560 --> 00:05:51,992 But we can, it's a very convenient guess in that we can generalize this equation 75 00:05:51,992 --> 00:05:55,400 just by writing it in a slightly different form. 76 00:05:55,400 --> 00:05:58,344 So let me do so. so what we're going to do, we are 77 00:05:58,344 --> 00:06:02,690 going to put the, time derivative in the left hand side. 78 00:06:02,690 --> 00:06:08,66 So basically this time derivative is going to be on the left hand side, i h 79 00:06:08,66 --> 00:06:12,618 bar d over d t psi. And eh, this guy, we're going to 80 00:06:12,618 --> 00:06:16,317 interpret it as a energy function. So after all, p squared over 2m is just 81 00:06:16,317 --> 00:06:19,184 the energy of a free electron but what we are going to do, we're going to put here 82 00:06:19,184 --> 00:06:22,990 some energy function, otherwise known as Hamiltonian. 83 00:06:22,990 --> 00:06:29,370 So this guy is very important object, it's a Hamiltonian. 84 00:06:29,370 --> 00:06:33,914 And we're going to postulate that this Hamiltonian is essentially kinetic 85 00:06:33,914 --> 00:06:39,320 energy, P squared over 2m plus some potential energy, P of r. 86 00:06:39,320 --> 00:06:45,116 So our P here is an operator minus i h bar radiant acting on whatever it is that 87 00:06:45,116 --> 00:06:52,910 we have here so this wave function. So and this the canonical Schrodinger 88 00:06:52,910 --> 00:07:00,140 equation that basically contains most if not all non-relativistic physics. 89 00:07:00,140 --> 00:07:03,251 So of course the way we derived it, in quotes, is not really a proper 90 00:07:03,251 --> 00:07:05,780 derivation. It was more like a guess. 91 00:07:05,780 --> 00:07:08,86 And in some sense this is what Schrodinger did. 92 00:07:08,86 --> 00:07:13,300 well, almost hundred years ago now. And, the reason we actually believe that 93 00:07:13,300 --> 00:07:17,253 this equation is a true equation that describes quantum physics, is because 94 00:07:17,253 --> 00:07:20,852 when we use it, we actually solve problems where, apart from just free 95 00:07:20,852 --> 00:07:27,210 particles, where who have In non-trivial[INAUDIBLE] potential lensky. 96 00:07:27,210 --> 00:07:32,270 Let's say in the case of an atom, or in, in the case of sketrinkle particles. 97 00:07:32,270 --> 00:07:36,156 So this equation produces theoretical results which are perfectly consistent 98 00:07:36,156 --> 00:07:40,546 with the experimental observations. So there is no question now, now these 99 00:07:40,546 --> 00:07:44,630 that this equation indeed, it describes quantum reality. 100 00:07:44,630 --> 00:07:49,529 But the way that people in the early days that got to this equation, was very very 101 00:07:49,529 --> 00:07:51,745 non-trivial.