1 00:00:00,570 --> 00:00:04,925 In the first lecture, we guessed in, in some sense the form of the Schrodinger 2 00:00:04,925 --> 00:00:09,670 equation relying on a few very basic, experimental facts, such as wave particle 3 00:00:09,670 --> 00:00:15,599 duality and the dispersion relation of a free electron. 4 00:00:16,600 --> 00:00:20,758 But we didn't talk too much about the meaning of the main object that appears 5 00:00:20,758 --> 00:00:26,922 in this equation the wavefunction, which was introduced in an ad hoc fashion. 6 00:00:26,922 --> 00:00:31,537 So today, I'm going to talk about the actual physical interpretation of quantum 7 00:00:31,537 --> 00:00:36,909 theory formulated this week. And we are also going to introduce the, a 8 00:00:36,909 --> 00:00:41,240 few very important standard notations and conventions that are going to be used 9 00:00:41,240 --> 00:00:45,449 throughout the course and also that are routinely used in the standard literature 10 00:00:45,449 --> 00:00:50,668 on the subject. So, here is the Schrodinger equation, 11 00:00:50,668 --> 00:00:54,890 once again we're going to see it very often in this course. 12 00:00:54,890 --> 00:00:58,538 And I'm also showing here the, a part of the first page of the original paper by 13 00:00:58,538 --> 00:01:02,232 Schrodinger, published in December of 1926. 14 00:01:02,232 --> 00:01:06,912 And it's actually very interesting how Schrodinger came up with this work and 15 00:01:06,912 --> 00:01:11,388 this equation. So, apparently the story started back in 16 00:01:11,388 --> 00:01:15,944 1925 or so, when he was working under Debye, and Debye had just read a paper 17 00:01:15,944 --> 00:01:20,364 about de Broglie, where de Broglie was introducing his wave particle duality 18 00:01:20,364 --> 00:01:26,563 ideas. So, he got interested in these ideas and 19 00:01:26,563 --> 00:01:32,480 suggested Schrodinger to give a seminar on de Broglie's work. 20 00:01:32,480 --> 00:01:35,600 So apparently, Schrodinger actually dismissed this at first, saying that he 21 00:01:35,600 --> 00:01:38,720 didn't even want to think about such a silly theory. 22 00:01:38,720 --> 00:01:45,90 But he had to give in because well, Debye was effectively his supervisor. 23 00:01:45,90 --> 00:01:49,306 And so, he was looking into de Broglie's work, trying to present it in a more 24 00:01:49,306 --> 00:01:54,570 mathematically sophisticated form. And in doing so, he came up with the 25 00:01:54,570 --> 00:01:58,850 Schrodinger equation essentially, as we now know it. 26 00:01:58,850 --> 00:02:03,258 And which brought him the worldwide recognition and a Nobel Prize in Physics 27 00:02:03,258 --> 00:02:07,718 in 1933. So, another important thing that 28 00:02:07,718 --> 00:02:11,500 Schrodinger did was that he used his equation to solve a very important 29 00:02:11,500 --> 00:02:15,648 problem of the charged particle and the cool-down potential, which essentially 30 00:02:15,648 --> 00:02:22,698 describes a quantum hydrogen atom. And he found the energy level structure 31 00:02:22,698 --> 00:02:28,590 which is which was consistent with the with de Broglie's atom. 32 00:02:28,590 --> 00:02:32,910 And so, actually, Charles Clark is going to talk about the the solution later in 33 00:02:32,910 --> 00:02:37,170 this course but I'm just going to mention here that this was indeed very important 34 00:02:37,170 --> 00:02:43,440 and it was a clear smoking gun that Schrodinger was on the right track. 35 00:02:43,440 --> 00:02:47,140 But according to Debye at least quite interestingly that Schrodinger didn't 36 00:02:47,140 --> 00:02:50,855 really quite understand the true meaning of his own work. 37 00:02:50,855 --> 00:02:54,420 He actually dismissed about the importance of it at first. 38 00:02:54,420 --> 00:02:58,110 So, I don't know if it was just Debye being jealous or it, it was true. 39 00:02:58,110 --> 00:03:01,960 But certainly if you actually read the original paper by Schrodinger, you don't 40 00:03:01,960 --> 00:03:05,315 see too much insight into the true physical interpretation of his own 41 00:03:05,315 --> 00:03:08,995 equation. And the correct interpretation of 42 00:03:08,995 --> 00:03:12,917 Schrodinger's equation was developed very shortly after Schrodinger's work by Max 43 00:03:12,917 --> 00:03:17,670 Born. perhaps, actually, I should say and note 44 00:03:17,670 --> 00:03:22,269 the correct interpretation but the commonly accepted interpretation because 45 00:03:22,269 --> 00:03:26,301 scientists actually have been arguing about the interpretation up to these 46 00:03:26,301 --> 00:03:31,390 days. And some are still not convinced that the 47 00:03:31,390 --> 00:03:35,111 Born interpretation and the Born, Born rule that I'm going to present is the 48 00:03:35,111 --> 00:03:41,772 only a correct view of Quantum Physics. But certainly, this Born rule is the 49 00:03:41,772 --> 00:03:47,248 cornerstone of a, a standard quantum theory, and it is indeed consistent with 50 00:03:47,248 --> 00:03:53,460 all experimental data as we know it at this stage. 51 00:03:53,460 --> 00:03:58,580 Now before formulating this rule I would like to make a few general comments about 52 00:03:58,580 --> 00:04:04,530 Quantum Physics and make, make them in contrast to classical Physics. 53 00:04:04,530 --> 00:04:07,764 So, in classical Physics, if we have a classical system, let's say, this is a 54 00:04:07,764 --> 00:04:13,60 closed classical system with some closed classical particles moving around. 55 00:04:13,60 --> 00:04:16,651 And if we know everything about the system, let's say,we know all the 56 00:04:16,651 --> 00:04:20,71 coordinates of all the particles of a certain moment of time, and all the 57 00:04:20,71 --> 00:04:26,334 velocities at a certain moment of time. Then classical theory predicts with 58 00:04:26,334 --> 00:04:31,234 absolute certainty, every, the result of any conceivable experiment that is going 59 00:04:31,234 --> 00:04:35,574 to happen in the future, which is determined by, let's say trajectories of 60 00:04:35,574 --> 00:04:41,388 particles at any subsequent moment of time. 61 00:04:41,388 --> 00:04:46,720 And these deter, determine all possible outcomes of all possible experiments. 62 00:04:46,720 --> 00:04:51,282 Now, this is the classical system. So, the truth about the quantum system, 63 00:04:51,282 --> 00:04:55,902 so if you look at the, the actual association with quantum systems, so if 64 00:04:55,902 --> 00:05:03,174 you have a quantum system like that, say, with some wavefunction of psi. 65 00:05:03,174 --> 00:05:07,864 So, even if we know everything possible about this quantum system, that is, if we 66 00:05:07,864 --> 00:05:12,219 know psi, and if we know all the forces, everything about this system, we still 67 00:05:12,219 --> 00:05:20,140 cannot possibly predict with certainty, the outcomes of well-posed experiments. 68 00:05:20,140 --> 00:05:24,968 Let's say, if we have some detectors here, and they measure well, let's say 69 00:05:24,968 --> 00:05:30,521 detect electrons or some other particles. So, even if we know everything about the 70 00:05:30,521 --> 00:05:33,176 quantum electrons, we still cannot predict for sure which detector is 71 00:05:33,176 --> 00:05:36,550 going to pick up electrons at certain moment of time. 72 00:05:36,550 --> 00:05:41,302 And so, this uncertainty and is intrinsic to Quantum Physics and there is no way 73 00:05:41,302 --> 00:05:45,565 around it. And so, Born, Born realized it and he 74 00:05:45,565 --> 00:05:51,740 also found a way to quantify his uncertainty using the wavefunction. 75 00:05:51,740 --> 00:05:57,356 And here's what we know now is the Born rule, is written here, so basically he 76 00:05:57,356 --> 00:06:02,298 proposed, and afterwards it was confirmed. 77 00:06:02,298 --> 00:06:06,330 again, by comparing with experimental data, that the absolute value of the 78 00:06:06,330 --> 00:06:09,914 wavefunction, as related as certain position in space, and a certain moment 79 00:06:09,914 --> 00:06:13,946 of time gives a probability density of finding a quantum particle described by 80 00:06:13,946 --> 00:06:20,593 this wavefunction in this this position and at this moment of time t. 81 00:06:20,593 --> 00:06:24,949 Well, the probability itself is going to of psi, the absolute value of psi squared 82 00:06:24,949 --> 00:06:31,850 times the volume the elementary volume in the vicinity of a certain point. 83 00:06:31,850 --> 00:06:37,380 This work of Max Born turned out to be extremely influential and groundbreaking 84 00:06:37,380 --> 00:06:44,920 and he eventually received a Nobel Prize for these ideas in part in 1954. 85 00:06:44,920 --> 00:06:48,400 As a matter of fact, he should have received this award much earlier. 86 00:06:48,400 --> 00:06:52,529 He was nominated by Albert Einstein actually back in the 30s. 87 00:06:52,529 --> 00:06:55,966 but he didn't get it due to some political reasons. 88 00:06:55,966 --> 00:07:00,706 In but in any case, this probabilistic interpretation was and actually remains 89 00:07:00,706 --> 00:07:04,366 very remarkable feature of Quantum Physics, which is also a source of 90 00:07:04,366 --> 00:07:10,176 confusion often times. And I'm actually supposed to tell you, 91 00:07:10,176 --> 00:07:16,540 now about this probabilistic property being a mystery of Quantum Physics and 92 00:07:16,540 --> 00:07:22,509 something which we will never be able to understand. 93 00:07:22,509 --> 00:07:26,60 But I would actually argue otherwise that there is, in fact, nothing mysterious 94 00:07:26,60 --> 00:07:29,470 about it. If, by mysterious, we mean something that 95 00:07:29,470 --> 00:07:32,947 is not consistent with our everyday intuition, everyday experience, so 96 00:07:32,947 --> 00:07:37,222 actually uncertainty is a part of our everyday life whether we are dealing with 97 00:07:37,222 --> 00:07:40,870 a sporting event or any other event in our life, we can predict with certainty 98 00:07:40,870 --> 00:07:46,850 if we know everything how it's going to turn out. 99 00:07:46,850 --> 00:07:52,920 And so I don't see a reason to demand from size deterministic answers. 100 00:07:52,920 --> 00:07:56,365 And nature just doesn't work in this way, and so we should just accept this fact 101 00:07:56,365 --> 00:07:58,233 and get used to it.