1 00:00:00,300 --> 00:00:04,387 In this video, I'm going to present the rather technical derivation of an 2 00:00:04,387 --> 00:00:09,574 important equation of quantum theory. The continuity equation for probability, 3 00:00:09,574 --> 00:00:14,515 which both relies on the Borne interpretation of quantum theory and also 4 00:00:14,515 --> 00:00:21,216 serves as a sort of safety check for it. Because it establishes the conservation of 5 00:00:21,216 --> 00:00:24,310 probability. The idea here is that, even though we 6 00:00:24,310 --> 00:00:28,732 cannot see with certainty where exactly our quantum particle is located, we can 7 00:00:28,732 --> 00:00:32,659 certainly, we can be sure that it's located somewhere in space so if we 8 00:00:32,659 --> 00:00:37,048 perform a measurement and look for the particle everywhere, we're going to find 9 00:00:37,048 --> 00:00:41,480 it with a probability equal to one. And this total probability never becomes 10 00:00:41,480 --> 00:00:44,844 smaller than one and certainly cannot become larger than one, so it is 11 00:00:44,844 --> 00:00:48,210 conserved. And this simple conservation law gives 12 00:00:48,210 --> 00:00:51,914 rise to this continuity equation that we're going to derive. 13 00:00:51,914 --> 00:00:56,474 Now, the form of this equation is actually not specific to quantum theory, and it 14 00:00:56,474 --> 00:00:59,614 appears in many different fields of physics. 15 00:00:59,615 --> 00:01:04,113 So raw here is the density of a conservative quantity. 16 00:01:04,114 --> 00:01:09,928 In our case this is the probability density, which per the born rule is equal 17 00:01:09,928 --> 00:01:13,769 to the absolute value of the wave function squared. 18 00:01:14,780 --> 00:01:20,608 And g here is the probability current, which we actually are going to derive. 19 00:01:20,608 --> 00:01:25,248 So to see that this equation indeed describes some sort of a conservation 20 00:01:25,248 --> 00:01:28,114 law,. Let me consider an arbitrary volume in 21 00:01:28,114 --> 00:01:33,096 space. So let's say this is a volume V, and let 22 00:01:33,096 --> 00:01:40,438 me also denote the surface encircling this volume as dV. 23 00:01:40,438 --> 00:01:46,110 And let me be interested in the probability of finding the particle of my 24 00:01:46,110 --> 00:01:51,786 quantum particle inside this volume. So this probability, let me call it P sub 25 00:01:51,786 --> 00:01:56,952 v, is going to be an integral of the probability density, which appears here 26 00:01:56,952 --> 00:02:00,924 over this volume. Now to see to look at the dynamics of this 27 00:02:00,924 --> 00:02:06,244 probability with time, let me integrate both sides of this equation over the 28 00:02:06,244 --> 00:02:10,002 volume. Well, the right hand side here is zero, 29 00:02:10,002 --> 00:02:16,802 there is not much to integrate, but the first term is going to give me just dp 30 00:02:16,802 --> 00:02:23,742 over dt, and the second term is going to be an integral of the diversions of g. 31 00:02:23,742 --> 00:02:29,622 And everything, the sum of these two terms is equal to 0. 32 00:02:29,623 --> 00:02:35,978 Now at this stage, I can use the Gauss's theorem to handle the second term, which 33 00:02:35,978 --> 00:02:42,234 tells me that the integral of a full diversion, so for a volume, is equal to a 34 00:02:42,234 --> 00:02:47,846 flux of the vector of which I calculated diversions, through the surface 35 00:02:47,846 --> 00:02:52,283 surrounding my volume. So this is my dv. 36 00:02:52,283 --> 00:02:58,890 So in ds here is an elementary An elementary surface element, with the 37 00:02:58,890 --> 00:03:04,553 vector pointing, outwards. So I'm sure, some of you have already 38 00:03:04,553 --> 00:03:09,807 seen, the Guassian theorem, let's say, in the theory of electromagnetism. 39 00:03:09,808 --> 00:03:12,700 But if, you forgot about it, or you have never seen it. 40 00:03:12,700 --> 00:03:19,360 So let me just mention that this Gauss, theorem is in some sense, similar to the 41 00:03:19,360 --> 00:03:25,660 simple identity that we oftentimes use for usual integral, so let's see if we have an 42 00:03:25,660 --> 00:03:31,240 integral between a and b or the full derivative of a function df dx, so we can 43 00:03:31,240 --> 00:03:37,270 write the result in the result as f of b minus f of So essentially in this case we 44 00:03:37,270 --> 00:03:43,840 have a one-dimensional segment from a to b, or we, we integrate and of we have a 45 00:03:43,840 --> 00:03:48,014 full derivative we can only focus on the end points. 46 00:03:48,014 --> 00:03:51,510 So that's the only thing which answer is the final result. 47 00:03:51,510 --> 00:03:56,550 So likewise if we have more complicated integral now, an integral over, over three 48 00:03:56,550 --> 00:04:00,609 dimensional volume but of a full derivative, which it divergence. 49 00:04:00,610 --> 00:04:05,350 E is, so instead of having two boundary values we have a, a surface integral going 50 00:04:05,350 --> 00:04:10,722 through this surface through this dv. So in some sense this dv is similar to a 51 00:04:10,722 --> 00:04:17,250 and b, so the volume is similar to the segment of between a and b and the full 52 00:04:17,250 --> 00:04:21,637 derivative df. Over the X, is similar to the divergence 53 00:04:21,637 --> 00:04:24,745 of G. So what we actually have established here 54 00:04:24,745 --> 00:04:29,959 is the derivative, what we call it P dot of the probability of finding a particle 55 00:04:29,959 --> 00:04:35,252 in the volume V is equal to minus the flux of the probability current again, we're 56 00:04:35,252 --> 00:04:40,632 going to derive it a little later. Through the surface surrounding this 57 00:04:40,632 --> 00:04:45,724 volume and let's see if we have a current say going outwards here luckily, so that 58 00:04:45,724 --> 00:04:50,816 only so that there is forms as small angle with the ds so then this g.ds is positive 59 00:04:50,816 --> 00:04:55,984 which corresponds to the negative change in the probability so and it makes sense 60 00:04:55,984 --> 00:05:00,550 because, it means that. The current carries away the probability, 61 00:05:00,550 --> 00:05:04,474 and the probability of finding a particle inside decreases. 62 00:05:04,474 --> 00:05:08,512 So if we, on the other hand, have the current going inside, the probability is 63 00:05:08,512 --> 00:05:11,994 going to increase. So in order for me to prove the continuity 64 00:05:11,994 --> 00:05:17,238 equation in the form I just formulated, let me calculate directly the probability 65 00:05:17,238 --> 00:05:22,558 of finding my particle in the volume v over time using the boron, rule and, the 66 00:05:22,558 --> 00:05:27,574 boron rule basically implies that I calculate the derivative of this integral 67 00:05:27,574 --> 00:05:32,666 of the volume of the absolute value of my wave function squared, and the absolute 68 00:05:32,666 --> 00:05:37,834 value can vary in, of course by definition as a product of the wave function and its 69 00:05:37,834 --> 00:05:42,329 complex integrated everything and degraded over the volume. 70 00:05:42,330 --> 00:05:52,030 So now if I apply this derivative to this product, I can write it as psi star dot 71 00:05:52,030 --> 00:05:58,136 psi plus psi star psi dot. So in order for me to simplify it further 72 00:05:58,136 --> 00:06:03,640 Let me just use the Schrodinger equation in it's center form, which actually 73 00:06:03,640 --> 00:06:11,689 appears on the logo of our, of our course. So, and express the psi dot from here as 74 00:06:11,689 --> 00:06:17,560 simply minus i over H bar doing on, acting on sine. 75 00:06:17,560 --> 00:06:23,721 So I just divided it by H bar, and multiplied both sides with the imaginary 76 00:06:23,721 --> 00:06:27,850 constant I. So I can also drive the same equation 77 00:06:27,850 --> 00:06:34,339 with, for psi star dot and the since Hamiltonian is sort of real cause the 78 00:06:34,339 --> 00:06:40,161 kinetic potential is just going to be plus I over H bar, H acting on psi. 79 00:06:40,161 --> 00:06:45,024 Now my H here is a combination of kinetic energy and potential energy. 80 00:06:45,025 --> 00:06:50,392 So kinetic energy is sort of an involved operator which is the momentum squared and 81 00:06:50,392 --> 00:06:55,330 potential energy is just multiplication, it just multiplies my wave function. 82 00:06:55,330 --> 00:07:02,062 So if I put everything together, what I will find is, the following, for P V dot, 83 00:07:02,062 --> 00:07:08,002 so I'm going to have here instead of psi star dot, I'm going to have this guy, so 84 00:07:08,002 --> 00:07:13,408 I. H bar h psi, and here we have minus i h 85 00:07:13,408 --> 00:07:20,386 bar psi star h psi. So if h were a number so these 2 terms 86 00:07:20,386 --> 00:07:27,195 would cancel each other out. And it does happen, as a matter of fact 87 00:07:27,195 --> 00:07:33,638 for the potential energy draw So if, so this H can actually be expre-, replaced 88 00:07:33,638 --> 00:07:38,040 with a kinetic energy. But there's no consolation necessarily for 89 00:07:38,040 --> 00:07:41,810 the kinetic energy because again, this is an operator which acts on different 90 00:07:41,810 --> 00:07:47,408 functions here and here. And, this operator is equal to just p 91 00:07:47,408 --> 00:07:53,818 squared over 2m, or minus h squared over 2m Laplacian. 92 00:07:53,818 --> 00:07:55,409 So we should write as delta number squared. 93 00:07:55,409 --> 00:08:02,614 So again, putting everything together, so what I have is minus i h bar 2 m, which I 94 00:08:02,614 --> 00:08:11,705 can factor out outside the integral. And in the brackets I'm going to have the 95 00:08:11,705 --> 00:08:20,885 Laplacian number squared psi start and psi minus psi star number squared, squared 96 00:08:20,885 --> 00:08:24,002 psi. So the last step here is to integrate this 97 00:08:24,002 --> 00:08:28,893 expression by parts, and if I do so essentially by moving this delta from here 98 00:08:28,893 --> 00:08:33,711 to here and from here back to here, so I see that these terms become identical and 99 00:08:33,711 --> 00:08:41,386 cancel each other out. And the only term which survives is, the 100 00:08:41,386 --> 00:08:51,302 full derivative of delta up side star. Psi minus complex conjugated. 101 00:08:51,302 --> 00:08:57,641 So, And this whole thing, well, times this coefficient, is exactly the current that 102 00:08:57,641 --> 00:09:02,609 we have been looking for because, again, we ended up with the full derivative of a 103 00:09:02,609 --> 00:09:09,436 vector integrated over the volume. So we can write it as minus an integral of 104 00:09:09,436 --> 00:09:18,548 sum G over DS over the distribute, this sort of encircling, encircling our volume. 105 00:09:18,548 --> 00:09:24,911 So finally, we can just collect everything from this expression and write the final 106 00:09:24,911 --> 00:09:30,914 expression for the current, which I will arrive in the full length forum, which is, 107 00:09:30,914 --> 00:09:35,520 1/2 side star, p, over m, edging on side, plus complex country. 108 00:09:35,520 --> 00:09:41,792 So, the reason I can write like this is because P is an operator, which is equal 109 00:09:41,792 --> 00:09:46,414 to minus I H bar. Nebla and this minus i h and r, hr nebla 110 00:09:46,414 --> 00:09:52,126 appears here, and so if we look at this 5 expressions, we see that the symmetric 111 00:09:52,126 --> 00:09:57,754 effect makes sort of intuitive sense, because in classical physics, a classical 112 00:09:57,754 --> 00:10:03,634 current associated with the density, rho, is simply the density times the velocity 113 00:10:03,634 --> 00:10:07,446 with which it moves. Now, here in quantum mechanics the 114 00:10:07,446 --> 00:10:11,982 momentum is an operator so velocity which is momentum divided by m is also an 115 00:10:11,982 --> 00:10:15,883 operator. And this operator sort of acts on the 116 00:10:15,883 --> 00:10:20,793 density which is which is rho is psi star psi. 117 00:10:20,794 --> 00:10:28,398 So and so this is the final result which connects the change in the probability of 118 00:10:28,398 --> 00:10:35,698 finding a particle in the volume v with the flux of a certain probability current 119 00:10:35,698 --> 00:10:38,263 flowing through the surface.