1 00:00:00,012 --> 00:00:06,446 Okay, welcome back everybody. Today we're going to actually use the 2 00:00:06,446 --> 00:00:12,886 Feynman path integral to obtain a few very interesting results and physical 3 00:00:12,886 --> 00:00:16,988 phenomena. And in the first video today, we're going 4 00:00:16,988 --> 00:00:21,722 to actually derive a very unexpected surprising result. 5 00:00:21,722 --> 00:00:28,622 Namely, we're going to see how the Newton equation second Newton law of classical 6 00:00:28,622 --> 00:00:35,397 physics appears in the so-called classical limit of the Feynman path integral. 7 00:00:35,398 --> 00:00:40,210 So more precisely what I'm going to, what I'm going to do, I'm going to start with 8 00:00:40,210 --> 00:00:45,158 the with an expression that we discussed in the previous lecture. 9 00:00:45,158 --> 00:00:50,719 So which tells of that the probability of a quantum particle to go from an initial 10 00:00:50,719 --> 00:00:56,280 point R sub i to find a point R sub f can be written as the sum absolute value of a, 11 00:00:56,280 --> 00:01:02,163 a sum over all classical trajectories. Which sort of symbolically represents what 12 00:01:02,163 --> 00:01:07,784 we call the path integral of this individual quantum mechanical transition 13 00:01:07,784 --> 00:01:10,890 amplitudes. And each of these amplitudes is written as 14 00:01:10,890 --> 00:01:15,498 the exponential of this symmetric constant i, times the classical action, divided by 15 00:01:15,498 --> 00:01:19,998 the Planck Constant. And surprisingly, what we're going to see 16 00:01:19,998 --> 00:01:27,446 is that the familiar equations of motion of the classical physics the, Newton's 17 00:01:27,446 --> 00:01:34,831 second law, appears as a result of taking the classical limit in this expression. 18 00:01:34,831 --> 00:01:38,927 And I'm going to formulate precisely what I mean by the classical limit, but at this 19 00:01:38,927 --> 00:01:43,075 stage let me just tell you that the classical limit will imply suppressing the 20 00:01:43,075 --> 00:01:46,709 essence of quantum mechanics. The interference phenomena, which is, 21 00:01:46,709 --> 00:01:49,620 which are controlled by this, this Planck constant H bar. 22 00:01:49,620 --> 00:01:54,834 Now, just as a side comment here, sort of the historical comment, let me mention 23 00:01:54,834 --> 00:01:58,744 something about this, equation. So, of course, this is I expect. 24 00:01:58,745 --> 00:02:04,371 All of you, or the majority of you to know this equation is something you heard of 25 00:02:04,371 --> 00:02:09,269 sometimes in high school which is the second Newton law that mass times 26 00:02:09,269 --> 00:02:14,562 acceleration of a classical particle is equal to the sum of all the forces acting 27 00:02:14,562 --> 00:02:20,425 on the particle. So, apparently if you read actually the, 28 00:02:20,426 --> 00:02:28,198 the original scientific work by Newton. The first version, the first edition of 29 00:02:28,198 --> 00:02:32,892 this Pincipia was published back in 1687, a long time ago. 30 00:02:32,892 --> 00:02:37,970 So we're not actually see this equation in this modern form. 31 00:02:37,970 --> 00:02:42,593 The closest formulation to what we currently call Newton's second law of 32 00:02:42,593 --> 00:02:45,556 motion. Appeared in his original work in the 33 00:02:45,556 --> 00:02:48,770 following form. The change in motion is proportional to 34 00:02:48,770 --> 00:02:53,250 the motive forced impressed and takes place along the straight line in which 35 00:02:53,250 --> 00:02:57,190 that force is impressed. It probably requires some imagination to 36 00:02:57,190 --> 00:03:00,672 connect this statement toward to well to this equation. 37 00:03:00,673 --> 00:03:06,300 But in any case obviously Newton not only understood, the basic principles of 38 00:03:06,300 --> 00:03:09,680 classical mechanics, he was the one who created it. 39 00:03:09,680 --> 00:03:15,174 So there's no question about that. So this comment, I just wanted to make to 40 00:03:15,174 --> 00:03:20,534 show you that sometimes the original discoveries evolve very strongly from 41 00:03:20,534 --> 00:03:24,695 their original form. And become even though we still give 42 00:03:24,695 --> 00:03:30,599 credit to people who were the fi rst to put them together the final form may be 43 00:03:30,599 --> 00:03:34,540 quite different from what they were envisioning. 44 00:03:34,540 --> 00:03:40,312 And this is a perfect example of that. Now, going back to the main subject of, 45 00:03:40,313 --> 00:03:44,596 this video. So now I'm going to actually, show you how 46 00:03:44,596 --> 00:03:50,756 mathematically this, old and well known Newton, second Newton Law appears from 47 00:03:50,756 --> 00:03:56,820 this modern, more sophisticated quantum theory, seemingly unrelated theory. 48 00:03:56,820 --> 00:04:00,968 In order for me to show you how it happens, I need to present a mathematical 49 00:04:00,968 --> 00:04:05,694 trick or method ih, called Laplace's method or saddle point approximation which 50 00:04:05,694 --> 00:04:10,828 allow us to calculate certain integrals that otherwise cannot be calculated really 51 00:04:10,828 --> 00:04:13,435 quickly. And this method as you all see sort of 52 00:04:13,435 --> 00:04:18,661 propagates to more complicated theories including our ability to calculate certain 53 00:04:18,661 --> 00:04:24,215 path integrals approximately. Now we have seen already the Gaussian 54 00:04:24,215 --> 00:04:29,915 integral, which is an integral of e to the power minus x squared and between minus 55 00:04:29,915 --> 00:04:33,660 infinity and to plus infinity. And we know the result is equal to the 56 00:04:33,660 --> 00:04:37,536 square root of, of pi. So of course what this integral is, Is the 57 00:04:37,536 --> 00:04:43,276 area enclosed by this curve which is nothing but the plot of this Gaussian 58 00:04:43,276 --> 00:04:46,664 function, e minus x squared as a function of x. 59 00:04:46,664 --> 00:04:51,894 Now, of course, this function is special, well, not very, nothing particularly 60 00:04:51,894 --> 00:04:57,120 special about this function but it does appear in many fields of math and physics 61 00:04:57,120 --> 00:05:01,956 and one property of this function is that it, It is very, very fast when the 62 00:05:01,956 --> 00:05:06,592 argument x becomes large, either very negative or very positive. 63 00:05:06,592 --> 00:05:12,900 Now in general, however, if we want to calculate the integral of let's see, from 64 00:05:12,900 --> 00:05:18,444 minus infinity to plus infinity or some other limits of a function e to minus f of 65 00:05:18,444 --> 00:05:22,187 x where f of x is a relatively arbitrary function. 66 00:05:22,188 --> 00:05:27,177 There is no close mathematical expression typically for a Gaussian integral. 67 00:05:27,177 --> 00:05:31,602 But it turns out that if there's a small parameter in the exponential, if this 68 00:05:31,602 --> 00:05:35,892 function actually involves, so not just f of x but f of x divided by some epsilon 69 00:05:35,892 --> 00:05:40,078 and this epsilon is very small Then, 1, in many cases, can simplify things. 70 00:05:40,078 --> 00:05:45,626 And I will explain the logic, behind this simplification on a particular example of 71 00:05:45,626 --> 00:05:50,725 this function of effects. Which is x squared plus, 1 over x squared. 72 00:05:50,725 --> 00:05:54,598 And so I want, I want to integrate the exponential of minus this function, from 73 00:05:54,598 --> 00:05:59,767 my minus infinity to plus infinity. So there's no closed analytical expression 74 00:05:59,767 --> 00:06:04,311 for the integral of this word but it turns out that if indeed we have this small 75 00:06:04,311 --> 00:06:08,571 parameter in the denominator of the exponential things can actually be 76 00:06:08,571 --> 00:06:12,434 simplified. In order to see how it happens, that we 77 00:06:12,434 --> 00:06:18,647 pull up this function, f of x and Soo if we do so we say that there Minimum but of 78 00:06:18,647 --> 00:06:22,677 course due to the minus sign in the exponential. 79 00:06:22,678 --> 00:06:27,087 The smaller the f of x, the larger the exponential itself. 80 00:06:27,088 --> 00:06:30,029 And vice versa. So if f of x on the other hand becomes 81 00:06:30,029 --> 00:06:34,521 very large and positive, e to the power minus f of x becomes negligible. 82 00:06:34,521 --> 00:06:39,157 So if we now plot the function that we actually want to integrate. 83 00:06:39,158 --> 00:06:45,063 It's going to look approximately like this so as my artistic expression of what it's 84 00:06:45,063 --> 00:06:51,039 good look like the point here is that it will have a maximum around where around f 85 00:06:51,039 --> 00:06:56,351 of x is minimal and is going to have the stales which becomes smaller and smaller 86 00:06:56,351 --> 00:07:00,067 as f of x becomes larger. And so what we want of course 87 00:07:00,067 --> 00:07:04,969 geometrically, we want to collect the area enclosed by this curve. 88 00:07:04,970 --> 00:07:08,900 And so some area will come from the original, this maximum, and some area 89 00:07:08,900 --> 00:07:13,432 become, will come from these tails. So, but as epsilon this parameter here, 90 00:07:13,432 --> 00:07:18,546 becomes smaller and smaller, let's say, when it's 0.1 or something like that. 91 00:07:18,546 --> 00:07:22,954 So then it implies that I will exponentiate either by one or 0.1, which 92 00:07:22,954 --> 00:07:27,250 means either power of 10. Of whatever these tails r. 93 00:07:27,250 --> 00:07:32,237 And so these tails will become less and less relevant on the background of this 94 00:07:32,238 --> 00:07:35,716 peak. So the, for instance, for smaller epsilon, 95 00:07:35,716 --> 00:07:40,723 so let's say my f of x were smaller, epsilon is going to look like this. 96 00:07:40,724 --> 00:07:47,730 So, and, and in some sense, it's going to become closer and closer to the Gaussian 97 00:07:47,730 --> 00:07:52,148 form. So the reason for that because if I expand 98 00:07:52,148 --> 00:07:59,415 my function f of x in Taylor series in the vicinity of this point f of x equals 1. 99 00:07:59,415 --> 00:08:06,807 So what I'm going to get so its the first one which is just the value of my function 100 00:08:06,807 --> 00:08:15,076 at h f at f's equal 1 plus f prime of f equals 1 x minus 1 plus f 2 primes, 101 00:08:15,076 --> 00:08:20,842 divided by 2 x minus 1 squared. But since this is a minimum of my 102 00:08:20,842 --> 00:08:25,770 function, so the derivative, the first derivative of my function here of n, I 103 00:08:25,770 --> 00:08:30,221 should say becomes equal to 0. And so the only terms we sh-, sort of in 104 00:08:30,221 --> 00:08:35,071 the leading order that matter i-, are this, this guy and the quadratiture. 105 00:08:35,071 --> 00:08:39,171 So essentially which means the integral that I'm dealing with, Gaussian. 106 00:08:39,172 --> 00:08:42,329 And, as I mentioned, we know how to calculate Gaussian integral. 107 00:08:42,330 --> 00:08:47,410 So without going through the algebra, let me just, sort of present, the results. 108 00:08:47,410 --> 00:08:54,625 So here is, I first present sort of a general result for a, almost arbitrary 109 00:08:54,625 --> 00:08:59,161 function, f or x. And the first factor here accounts 110 00:08:59,161 --> 00:09:04,856 essentially from the just simply calculating the value of this exponential 111 00:09:04,856 --> 00:09:09,098 in the minimum point of f. And the second term is the result of 112 00:09:09,098 --> 00:09:13,982 integrating the Gaussian part much like this Gaussian integral. 113 00:09:13,982 --> 00:09:17,365 So for, for the particular function we consider well we don't, we are not going 114 00:09:17,365 --> 00:09:20,290 to need this result at all. So it is just, it was just example but 115 00:09:20,290 --> 00:09:22,510 nevertheless, here is the, here is the answer. 116 00:09:22,510 --> 00:09:27,502 Let me also mention another fact, which I'm not going to prove it but it's 117 00:09:27,502 --> 00:09:33,274 important for the following, that even if we have a slightly different integral when 118 00:09:33,274 --> 00:09:37,892 we have an exponential of i, the measuring constant i over some. 119 00:09:37,892 --> 00:09:44,042 Small parameter and sum f of x. So this integral often times can too be 120 00:09:44,042 --> 00:09:48,464 simplified using this, very similar saddle-point approximation which we just 121 00:09:48,464 --> 00:09:51,568 discussed. And the reason for that is because e to 122 00:09:51,568 --> 00:09:55,626 power i f of x is, can be written as a bunch of cosines and sines. 123 00:09:55,627 --> 00:10:00,319 And the if epsilon becomes very small they become stronger oscillating functions. 124 00:10:00,320 --> 00:10:05,190 Which on the average give us 0 if epsilon is small, because we have a, a plus, a 125 00:10:05,190 --> 00:10:09,805 positive part of the cosine and negative part of the cosine and then in some sense, 126 00:10:09,805 --> 00:10:13,005 they balance each other out and give on the average, 0. 127 00:10:13,005 --> 00:10:17,746 And there is a similar argument that you can actually write the expression of this 128 00:10:17,746 --> 00:10:22,319 integral in a very similar form and focus only on the minimum of this function. 129 00:10:22,320 --> 00:10:27,969 Now, let me go back to quantum physics and we're going to see how the previous 130 00:10:27,970 --> 00:10:31,825 mathematical discussion is going to become relevant. 131 00:10:31,826 --> 00:10:38,692 Now, remember, so I've, my motivation in the very beginning, I said that I want to 132 00:10:38,692 --> 00:10:43,437 take the classical limit of my quantum mechanical theory. 133 00:10:43,438 --> 00:10:46,587 So what does it mean to take a classical limit? 134 00:10:46,588 --> 00:10:51,640 So let us recall that the most interesting quantum phenomenon, the way how quantum 135 00:10:51,640 --> 00:10:57,310 mechanics was actually discovered Implied interference between different waves, sort 136 00:10:57,310 --> 00:11:00,630 of describing wave functions is describing a particle. 137 00:11:00,630 --> 00:11:05,040 So if we want to kill in some sense the quantum mechanical effects, we want to 138 00:11:05,040 --> 00:11:09,940 suppress the interference phenomenon, or we want to make the wavelength as small as 139 00:11:09,940 --> 00:11:13,470 possible. So let's say if we have a particle with a 140 00:11:13,470 --> 00:11:17,407 momentum p. So the corresponding wavelength is going 141 00:11:17,407 --> 00:11:21,264 to be per, the argument is going to be h bar over 2pi p. 142 00:11:21,265 --> 00:11:26,670 So if you want to set a lambda to 0, that is completely suppress the interference 143 00:11:26,670 --> 00:11:31,995 effect effects of the particlelization wave lengths, we want to set in some sense 144 00:11:31,995 --> 00:11:35,164 h to 0. So the Planck constant, which controls the 145 00:11:35,164 --> 00:11:40,122 quantum effects, which is sort of the essence of quantum mechanics, so it should 146 00:11:40,122 --> 00:11:43,280 be set to zero in order to take the classical limit. 147 00:11:43,280 --> 00:11:47,084 So now, recall the expression for the Feynman path integral. 148 00:11:47,085 --> 00:11:51,880 So we see that in the quasi-classical limit, in the classical limit, we have an 149 00:11:51,880 --> 00:11:55,546 H four always taken to zero. The function that we are actually or 150 00:11:55,546 --> 00:11:59,511 function I'll better say, that we're integrating, becomes in some sense very 151 00:11:59,511 --> 00:12:03,010 similar to the types of functions that we saw in the previous slide. 152 00:12:03,010 --> 00:12:08,844 With the h bar here at applying constant, being the coolant in a sense with the h. 153 00:12:08,844 --> 00:12:14,374 To the epsilon, parameter epsilon that control the applicability of the settle 154 00:12:14,374 --> 00:12:18,247 point approximation. And also in the full analogy with this 155 00:12:18,248 --> 00:12:22,310 previous slide with the settle point approximation we can see that a, as h 156 00:12:22,310 --> 00:12:26,860 becomes smaller and smaller so we can only focus on the trajectories in the vicinity 157 00:12:26,860 --> 00:12:29,789 of the trajectory for which the action is minimal. 158 00:12:31,490 --> 00:12:37,898 So basically the limit of h going to zero therefore reproduces the principle of 159 00:12:37,898 --> 00:12:43,820 least action, which is another sort of cornerstone of classical physics. 160 00:12:43,820 --> 00:12:49,080 Now I should comment here that of course h bar the Planck constant, is a fundamental 161 00:12:49,080 --> 00:12:52,237 physics constant. We cannot take it to zero, we cannot take 162 00:12:52,237 --> 00:12:54,649 the limit physically of h bar going to zero. 163 00:12:54,650 --> 00:12:57,960 What it actually means, when I'm saying that h bar goes to zero. 164 00:12:57,960 --> 00:13:00,875 And this also, you're going to see it in textbooks on quasi classical 165 00:13:00,875 --> 00:13:04,139 approximation. It means that, we have, essentially, two 166 00:13:04,139 --> 00:13:09,634 types of length scales in our problem. 1 type of length scale is, the wavelengths 167 00:13:09,634 --> 00:13:13,275 of our quantum particles. And the other type of length scales is 168 00:13:13,275 --> 00:13:17,679 the, typical length scales in our system. And if with the wave length of our quantum 169 00:13:17,679 --> 00:13:22,365 particles are much more than everything else, this effectively means this limit 170 00:13:22,365 --> 00:13:26,193 and this effectively means the applicability of the quasi classical 171 00:13:26,193 --> 00:13:29,599 approximation. So for those of you who are familiar with 172 00:13:29,599 --> 00:13:34,782 the Lagrangian classical mechanics it should be clear now that a principle of 173 00:13:34,782 --> 00:13:39,527 least action already demands that we essentially are going to reproduce 174 00:13:39,527 --> 00:13:42,732 Newton's equations. As was advertised in the beginning of the 175 00:13:42,732 --> 00:13:45,778 lecture. So you can stop listening to the lecture 176 00:13:45,778 --> 00:13:50,590 right here, and move on to the next part. But for the sake of completeness. 177 00:13:50,590 --> 00:13:56,284 Let me, nevertheless present or remind you how these equations would come about. 178 00:13:56,285 --> 00:14:01,136 So again, how the question we're now asking, okay we have found that while if 179 00:14:01,136 --> 00:14:06,106 we take the Planck constant to zero, or if we consider the wavelengths which are much 180 00:14:06,106 --> 00:14:10,581 more that everything else. We essentially extract we must extract the 181 00:14:10,581 --> 00:14:15,411 minimum of the classical action from the path integral and this minimum would occur 182 00:14:15,411 --> 00:14:18,800 on a particular trajectory that particle would follow. 183 00:14:18,800 --> 00:14:24,488 So how to find this special trajectory? So we can sort of, motivate the standard 184 00:14:24,488 --> 00:14:29,180 what is know as a variational analysis by this again simple analogy using the usual 185 00:14:29,180 --> 00:14:32,558 functions. So, suppose we have an arbitrary function 186 00:14:32,558 --> 00:14:37,114 f of x, which has a minimum or maximum principal too, somewhere, but let's assume 187 00:14:37,114 --> 00:14:41,534 it's a minimum, and we want to determine the point where this minimum actually 188 00:14:41,534 --> 00:14:44,521 occurs. And for this I can again use the Taylor 189 00:14:44,521 --> 00:14:48,669 expansion for the wavefunction, I'm sorry, for this function f in the vicinity of 190 00:14:48,669 --> 00:14:51,100 this point x naught, which I am trying to find. 191 00:14:51,100 --> 00:14:54,748 And so it's going to have the value of the function at x naught, the first 192 00:14:54,748 --> 00:14:59,114 derivative, the second derivitive, etc... And the minimum of this function, see if I 193 00:14:59,114 --> 00:15:03,822 have a functin, so this minumum. Is going to occur where this derivative or 194 00:15:03,822 --> 00:15:09,582 the certain the tangent to this plot f of x as a function of x is going to be 195 00:15:09,582 --> 00:15:14,000 horizontal to the x direction or the first derivative vanishes. 196 00:15:14,000 --> 00:15:20,076 Now, an analogy to this if we have now a function or in particular, our action f of 197 00:15:20,076 --> 00:15:23,208 x. So to determine a particular trajectory, 198 00:15:23,208 --> 00:15:26,750 which we will identify the classical trajectory. 199 00:15:26,750 --> 00:15:31,690 So we have to demand that if we calculate this action in the vicinity of this 200 00:15:31,690 --> 00:15:35,366 classical trajectory in which the action is miminal. 201 00:15:35,367 --> 00:15:40,404 The first sort of variation, the analogy well the anal-, analogous part of this 202 00:15:40,404 --> 00:15:45,640 first derivative is going to vanish. And this symbol delta s equals zero. 203 00:15:45,640 --> 00:15:52,915 So this equation is sort of mathematical expression for this principle of the list 204 00:15:52,915 --> 00:15:56,433 action. And in some sense, all of classical 205 00:15:56,433 --> 00:16:02,559 physics Is contained in this equation which is actually quite remarkable. 206 00:16:02,559 --> 00:16:07,461 Now, if we now go back to, well recall a particular action where study now which is 207 00:16:07,461 --> 00:16:12,498 just the kinetic energy, mv seqared over 2 minus the potential energy for a certain 208 00:16:12,498 --> 00:16:16,308 quantum particle. Similar to find the classical trajectory, 209 00:16:16,308 --> 00:16:20,980 as we know we want to calculate the first variation and we do so by writing this 210 00:16:20,980 --> 00:16:25,460 action in the vicinity of the classical trajectory that we want to find. 211 00:16:25,460 --> 00:16:29,960 So in some sense, what we should think about is that we have this initial point. 212 00:16:29,960 --> 00:16:36,653 And we have this final point and there is some unique trajectory that we're looking 213 00:16:36,653 --> 00:16:40,817 for, but there's also trajectory very close to it. 214 00:16:40,818 --> 00:16:45,976 And this division from this so this black trajectory here is the classical 215 00:16:45,976 --> 00:16:49,970 trajectory and the red ones are, are r classical plus d r. 216 00:16:49,970 --> 00:16:54,487 This d r is a small deviation from the classical path. 217 00:16:54,488 --> 00:16:59,115 So if we now just simply plug this expression into this a expression for the 218 00:16:59,115 --> 00:17:02,582 action. So we're going to have the kinetic energy 219 00:17:02,582 --> 00:17:07,000 true, so of course the velocity is, just d r over d t. 220 00:17:07,000 --> 00:17:12,170 Or for brevity we can write as r dot. So we're gonave have are the two terms and 221 00:17:12,170 --> 00:17:16,017 we're going to have two terms as an argument of the potential energy. 222 00:17:16,018 --> 00:17:20,940 And since dr is very small we're going to keep track only of the linear terms here. 223 00:17:20,940 --> 00:17:25,660 So for example in the kinetic energy part. So here we're going to have. 224 00:17:25,660 --> 00:17:34,220 So M over 2 plus equal velocitiy squared. Plus, the linear term, which, comes from, 225 00:17:34,220 --> 00:17:41,473 2r dot dr. So we're going to have m or plus equal dot 226 00:17:41,473 --> 00:17:45,210 plus dr dot. And this guy, what we're going to do. 227 00:17:45,210 --> 00:17:50,065 We're going to. Expand this function in Taylor series up 228 00:17:50,065 --> 00:17:54,991 to linear order. And again we can do that because this d r 229 00:17:54,991 --> 00:18:01,134 is very small. And, so, we're going to have the classical 230 00:18:01,134 --> 00:18:08,470 sorry our classical minus gv over gr the gradient, dot e r. 231 00:18:08,470 --> 00:18:16,317 And everything integrated over, time. So we have four terms here. 232 00:18:16,318 --> 00:18:20,558 So this term is simple enough. And these two terms actually this guy and 233 00:18:20,558 --> 00:18:24,477 this guy, together they represent a the classical action. 234 00:18:24,477 --> 00:18:27,150 So the one which occurs on the classical trajectory. 235 00:18:27,150 --> 00:18:31,246 Remember what we want is not the classical action itself but in some sense the 236 00:18:31,246 --> 00:18:35,278 derivative of the action, which we, something which is proportional to the 237 00:18:35,278 --> 00:18:37,560 d,r. So the only term which is sort of 238 00:18:37,560 --> 00:18:43,892 non-trivial is this guy. Okay, and it can be further simplified by 239 00:18:43,892 --> 00:18:50,642 using the integration by parts. So just to remind you, I'm sure you have 240 00:18:50,643 --> 00:18:55,311 many of you have seen it before, so if we have an integral. 241 00:18:55,311 --> 00:18:59,935 Let's say from a to b of, of a function f times g prime, dx. 242 00:18:59,935 --> 00:19:05,880 So we can write it as a derivative of everything, of g prime minus f prime g. 243 00:19:05,880 --> 00:19:13,349 And so here I have the full derivative, and I basically calculate, The, the value 244 00:19:13,349 --> 00:19:19,654 of the functions, and the limits and essentially move the derivative from 245 00:19:19,654 --> 00:19:25,011 function g to the function f. Now neuro keys they're all of the function 246 00:19:25,011 --> 00:19:30,135 g is played by this d r dot, and the role of the function f is played by this m r 247 00:19:30,135 --> 00:19:34,276 dot, so we want to move the derivative from here to here. 248 00:19:34,277 --> 00:19:40,667 But the this full derivative theorem. So, if we're going to calculate the, 249 00:19:40,667 --> 00:19:45,847 values of, this function dr in the endpoints at time zero and at time t. 250 00:19:45,848 --> 00:19:49,587 They could respond to the endpoints of, of this trajectory. 251 00:19:49,587 --> 00:19:52,105 So, and our trajectory, remember, is pinned. 252 00:19:52,105 --> 00:19:56,940 So we have to go from the prescribed initial point or prescribed final point. 253 00:19:56,940 --> 00:20:02,736 So there is no freedom for this sort of fluctuation dv from these initial and 254 00:20:02,736 --> 00:20:06,055 final points. They can only it can only give the h from 255 00:20:06,055 --> 00:20:09,810 the classical trajectory in between. So what I'm saying here is that this 256 00:20:09,810 --> 00:20:15,130 theorem, the, the full derivative theorem in this particular case, vanishes, and we 257 00:20:15,130 --> 00:20:19,052 can simply move the derivative from here to here with a minus sign. 258 00:20:19,052 --> 00:20:25,202 So if we put everything together, based on this discussion, so essentially, if we 259 00:20:25,202 --> 00:20:31,090 integrate by parts and replace this in this term with, with classical action. 260 00:20:31,090 --> 00:20:36,436 We're going to have this classical action minus because there isn't going to be a 261 00:20:36,436 --> 00:20:41,222 minus here and there is already minus here in the integral from 0 to r. 262 00:20:41,223 --> 00:20:47,287 And there are two dots second derivative because we moved it plus d v over d r. 263 00:20:47,287 --> 00:20:52,637 And everything is multiplied by this, this thing, delta r. 264 00:20:52,637 --> 00:20:59,471 Now and remember, this is exactly our first derivation, and this is something we 265 00:20:59,471 --> 00:21:03,420 want to set to zero. And this guy might as well be equal to 266 00:21:03,420 --> 00:21:07,638 zero for every possible sort of fluctuation away from the classical 267 00:21:07,638 --> 00:21:12,360 trajectory. And this implies that this Expression in 268 00:21:12,360 --> 00:21:18,055 the curly brackets, so this expression must be identically equal to zero for the 269 00:21:18,055 --> 00:21:23,627 first variation to vanish, for us to be able to find the minimum of the actions. 270 00:21:23,628 --> 00:21:30,780 If we're going to , if we going to derive it we're going to get m r two dots is 271 00:21:30,780 --> 00:21:36,208 equal to minus d v over d r. Well, the second derivative of the 272 00:21:36,209 --> 00:21:42,904 coordinate is known as the acceleration so the left hand side is simply m times a and 273 00:21:42,905 --> 00:21:49,554 the right hand side is the gradient of the potential energy which is the force acting 274 00:21:49,554 --> 00:21:55,251 on the particle. So we see that we indeed have derived the 275 00:21:55,251 --> 00:22:02,450 Newton's second law from the principle of the action being minimal. 276 00:22:02,450 --> 00:22:08,858 And this principle, itself, followed from us taking the formal limit of h bar going 277 00:22:08,858 --> 00:22:12,983 to 0, from setting to 0 the wavelengths of our particle.