1 00:00:00,240 --> 00:00:04,200 In the first lecture I introduced the basics of the Schrodinger formulation of 2 00:00:04,200 --> 00:00:07,480 quantum mechanics, including the key notion of a wave function. 3 00:00:07,480 --> 00:00:12,648 This is the most commonly used formulation of quantum theory, and we're going to use 4 00:00:12,648 --> 00:00:16,152 it throughout the course. We're going to discuss in great detail the 5 00:00:16,152 --> 00:00:19,981 interpretation of the wave function and properties of the Schrodinger equation. 6 00:00:19,981 --> 00:00:25,133 But today, I would like to give you an idea about an alternative formulation of 7 00:00:25,133 --> 00:00:28,922 quantum theory using so-called Feynman path integral. 8 00:00:28,923 --> 00:00:33,876 Which in my opinion, is a very beautiful formulation of quantum mechanics. 9 00:00:33,877 --> 00:00:38,705 If you define I should mention that this material is almost never taught, at least 10 00:00:38,705 --> 00:00:43,397 at the undergraduate level, and when it is taught it's usually said towards the end 11 00:00:43,397 --> 00:00:46,413 of the course. But today I will experiment a little bit 12 00:00:46,413 --> 00:00:49,584 with this material, and will introduce it right away. 13 00:00:49,585 --> 00:00:54,814 And one of the goals here is for me to tell you that what you read in regular 14 00:00:54,814 --> 00:01:00,196 text books and actually here in this course is just one way to think about 15 00:01:00,196 --> 00:01:05,422 quantum physics, the most commonly accepted way to describe quantum mechanics 16 00:01:05,422 --> 00:01:09,770 but there are many other ways, actually. Some of them don't even include the notion 17 00:01:09,770 --> 00:01:12,822 of the Boolean function. And you should be aware of their 18 00:01:12,822 --> 00:01:18,212 existence, at least, and keep an open mind here and actually in general whenever you 19 00:01:18,212 --> 00:01:22,184 study science. The derivation that I'm going to present 20 00:01:22,184 --> 00:01:29,084 later in this segment follows very closely this paper by Richard Feynman well a part 21 00:01:29,084 --> 00:01:35,248 of the paper, the paper is much more detailed written back in 1948 and this 22 00:01:35,248 --> 00:01:40,052 paper is a typical Feyman. So if you read the abstract the first 23 00:01:40,052 --> 00:01:44,888 sentence of the abstract says non-relativistic quantum mechanics is 24 00:01:44,888 --> 00:01:49,688 formulated here in a different way. So he suggest to the completely new a way 25 00:01:49,688 --> 00:01:55,422 to and to think about things here. Now I should mention that well, notice the 26 00:01:55,422 --> 00:01:59,356 date, 1948. So this time was actually very difficult 27 00:01:59,356 --> 00:02:06,476 time for Feynman, so there is this book which I would recommend for you to take a 28 00:02:06,476 --> 00:02:13,507 look if you're interested in Feynman as a person Perfectly Reasonable, it's called 29 00:02:13,507 --> 00:02:18,158 Perfectly Reasonable Derivations From the Beaten Track. 30 00:02:18,158 --> 00:02:23,666 So, this book is really, there is no shortage of books about Feynman, but this 31 00:02:23,666 --> 00:02:29,342 book is pretty much a collection of letters that Feynman wrote through, 32 00:02:29,342 --> 00:02:35,448 throughout his life and when you read, when you read this book this literature 33 00:02:35,448 --> 00:02:41,210 going back to this period tho the 40s, you see that it was a very, very painful time 34 00:02:41,210 --> 00:02:47,058 for Feynman because his wife Arleen died, the school, the high school sweetheart 35 00:02:47,058 --> 00:02:52,562 whom he married died in 1945, in June of 1945 of tuberculosis and he was, it was 36 00:02:52,562 --> 00:02:57,894 very different and it was not easy for him, had to deal with it so the ladders 37 00:02:57,894 --> 00:03:01,443 would sort of bared this feeling very closely. 38 00:03:01,443 --> 00:03:07,533 And despite this difficult time and maybe suffering some sort of inspiration, so he 39 00:03:07,533 --> 00:03:12,753 came up with the number of very influential very, very unusual ideas and 40 00:03:12,753 --> 00:03:16,436 one of those ideas is this path integral relation. 41 00:03:16,436 --> 00:03:21,178 Naniki is going back to the actual physics, so I'm going to present the 42 00:03:21,178 --> 00:03:26,284 derivation but and this I mentioned the derivation is rarely introduced in the 43 00:03:26,284 --> 00:03:31,084 very beginning of quantum mechanics. And one of the reasons here, of course, is 44 00:03:31,084 --> 00:03:33,390 that technically this issue is quite involved. 45 00:03:33,390 --> 00:03:38,574 And furthermore the end result of this calculation is actually a new mathematical 46 00:03:38,574 --> 00:03:42,749 object that hadn't even existed before Feynman wrote it. 47 00:03:42,750 --> 00:03:48,080 And after he did, mathematicians have been arguing to this day about its precise 48 00:03:48,080 --> 00:03:51,379 meaning. So, there is some technical issues, there 49 00:03:51,379 --> 00:03:56,538 are some technical issues which exist, but for those of you who are not really 50 00:03:56,538 --> 00:04:01,928 interested in these technicalities, again I just would like to briefly tell you the 51 00:04:01,928 --> 00:04:07,322 main ideas without going into, into math. So let us consider a quantum particle 52 00:04:07,323 --> 00:04:12,790 localized to the initial moment of time equals zero in the vicinity of a certain 53 00:04:12,790 --> 00:04:18,516 point, initial point, R sub i. And let's ask the question of what is the 54 00:04:18,516 --> 00:04:24,872 probability for this particle to reach a final point, R sub f in a time t. 55 00:04:24,872 --> 00:04:30,858 So if it were a classical particle it would have followed a unique well defined 56 00:04:30,858 --> 00:04:34,913 classical trajectory. Let's say if it were a free particle it 57 00:04:34,913 --> 00:04:39,585 would have been just a straight line connecting two points, but in general it 58 00:04:39,585 --> 00:04:44,841 would have been a solution of the, the second Newton equation or the Lagrange 59 00:04:44,841 --> 00:04:49,281 equations, but it would have been a unique classical trajectory. 60 00:04:49,281 --> 00:04:56,201 So the main result of Feynman in this paper is that in quantum mechanics, the 61 00:04:56,201 --> 00:05:02,153 particle goes over all possible trajectories at the same time, and In some 62 00:05:02,153 --> 00:05:05,418 sense. So it falls into classical trajectory, you 63 00:05:05,418 --> 00:05:10,966 know, this trajectory, any trajectory you can imagine and there is a weight complex 64 00:05:10,966 --> 00:05:16,003 weight associated with each trajectory which is the exponential of i times 65 00:05:16,003 --> 00:05:19,138 classical action divided by the Planck constant. 66 00:05:20,440 --> 00:05:26,390 So classical action here we're going to discuss it in more detail later, but just 67 00:05:26,390 --> 00:05:32,425 to remind you that classical action is an integral from 0 to t of the Lagrange which 68 00:05:32,425 --> 00:05:37,950 is the kinetic energy, basically mb squared over 2 minus the potential energy 69 00:05:37,950 --> 00:05:42,370 times et. So this in classical physics the minimum 70 00:05:42,370 --> 00:05:46,949 of this action gives rise. To Newton equations and to classical 71 00:05:46,949 --> 00:05:51,830 equations of motion. In quantum mechanics there's no principal 72 00:05:51,830 --> 00:05:58,765 of least action as Feynman showed but all actions are allowed and all of them give 73 00:05:58,765 --> 00:06:05,951 rise to some terms in quantum theory. Now going back to the probability of going 74 00:06:05,951 --> 00:06:12,611 from the initial point to the final is some sense this probability can be 75 00:06:12,611 --> 00:06:19,937 represented as a sum over all well, the absolute value squared of the sum, of all 76 00:06:19,937 --> 00:06:25,405 possible classical actions. Overall fads that I labeled here by an 77 00:06:25,405 --> 00:06:31,411 index L and this sum itself is essentially symbolically represents what we're going 78 00:06:31,411 --> 00:06:36,260 to see as a path integral which will come out of the theory naturally. 79 00:06:36,260 --> 00:06:42,287 And it's really a very remarkable result. Now it turns out that again so the 80 00:06:42,287 --> 00:06:47,257 mathematical part of it is quite subtle but one can actually solve some problems 81 00:06:47,257 --> 00:06:51,750 just by using this is as sort of a cartoon picture of what a path integral is. 82 00:06:51,750 --> 00:06:56,400 And we're going to I'm going to give you an example such a solution So, even if you 83 00:06:56,400 --> 00:07:01,987 don't follow, very closely and carefully, the derivation of the, mathematical 84 00:07:01,988 --> 00:07:07,657 formalism. You can still follow the, main results, 85 00:07:07,658 --> 00:07:10,913 The main qualitative results later on.