1 00:00:01,001 --> 00:00:06,008 Okay, so in this video we are going to trying to understand intuitively how the 2 00:00:06,008 --> 00:00:14,004 form of Schrodinger's equation for a free particle on a line. Okay, so, so remember 3 00:00:14,004 --> 00:00:23,003 what we have with the free particle on a line, so we have. The particle is free to 4 00:00:23,003 --> 00:00:31,008 be any where on the real line and, and it's described, it state is described by wave 5 00:00:31,008 --> 00:00:44,008 function. Which might look like this. It's um, it's psi of x at time t. So, this is 6 00:00:44,008 --> 00:00:50,004 maybe, this is maybe what it looks like at time t=0, at general time t, and we want 7 00:00:50,004 --> 00:01:01,003 to understand how this state evolves as t increases. Okay so, what we are going to do is 8 00:01:01,003 --> 00:01:06,000 we're gonna assume, you know, there's this one general principle that we're gonna 9 00:01:06,000 --> 00:01:10,009 assume that the equation of motion is described by it's, it's a local 10 00:01:10,009 --> 00:01:16,009 differential equation, meaning it's this very important principle in physics, which 11 00:01:16,009 --> 00:01:22,008 is that of locality, that every point in space, sort of, minds its own business. 12 00:01:22,008 --> 00:01:27,009 Well. Well, it, it minds it's own business, but it, it sorts of look around 13 00:01:27,009 --> 00:01:35,000 at an infinitesimal neighborhood around it. So it, it, it, um, well. It, it's 14 00:01:35,000 --> 00:01:39,007 mind, it minds its own business, but also the business of sort of a small 15 00:01:39,007 --> 00:01:47,004 neighborhood around itself. Okay. So, so, so now, what, what, what, what do we want 16 00:01:47,004 --> 00:01:52,006 to say? Well, we want to say that, that the change in the state. So if you want to 17 00:01:52,006 --> 00:02:02,007 write D Psi XT by DT. Right, the, the way this is described is by, by the following 18 00:02:02,007 --> 00:02:06,008 kind of differential equation, the, the, the weight of change in the state at the 19 00:02:06,008 --> 00:02:11,003 point X. So if this was the point X, this is the time T, that's what the super 20 00:02:11,003 --> 00:02:18,006 position looks like and now, what, what this point X does is it looks in its 21 00:02:18,006 --> 00:02:30,003 neighbourhood and it looks at a point, sort of, uh, uh, at x plus delta-x, and at 22 00:02:30,003 --> 00:02:41,000 x minus delta-x. So it looks in its tiny neighborhood, and so, and it looks at its 23 00:02:41,000 --> 00:02:50,008 own amplitude, so this is what psi of x is, and it compares it to psi of x plus 24 00:02:50,008 --> 00:02:56,003 delta x, and psi of x minus delta x. And it does this in a particularly simple way, 25 00:02:56,003 --> 00:03:03,005 so what it does is, it looks at the average of psi of x plus delta x, and psi 26 00:03:03,005 --> 00:03:10,008 of x minus delta x. So, so what's this average of these two points? Well, the 27 00:03:10,008 --> 00:03:19,005 average of these two points is, is that. Right, so, so what it's doing is it's 28 00:03:19,005 --> 00:03:35,000 comparing this point, which is psi of x minus delta x, oops. Plus psi of X, plus 29 00:03:35,000 --> 00:03:43,008 delta X divided by two and it's comparing it to, its own value which is psi of X. 30 00:03:45,000 --> 00:03:53,004 And basically what this, what this equation says is that if you look at this 31 00:03:53,004 --> 00:04:00,001 difference, the difference between these two quantities. So, this value. So let's 32 00:04:00,001 --> 00:04:10,009 call this Y. What Schrodinger's equation says is that the rate of change is proportional to Y. 33 00:04:13,007 --> 00:04:19,006 So the rate of change of psi at this point X is proportional to the difference 34 00:04:19,006 --> 00:04:26,000 between psi at this point X minus the average of the neighbors. And the neighbors 35 00:04:26,000 --> 00:04:34,004 infinite to the left and to the right. Okay, so, so now, can you, you know, as 36 00:04:34,004 --> 00:04:42,004 delta X turns to zero. Can you describe what this, what y is in terms of psi of x? 37 00:04:43,002 --> 00:04:58,002 So, so we can ask, can you describe Y in terms of psi of X? Some of you might be 38 00:04:58,002 --> 00:05:08,002 guessing maybe Y is, Y is just a, you know, Y is something like the derivative 39 00:05:08,002 --> 00:05:21,006 of the psi of X, with respect to X. Well, that's a pretty good guess, but let's, 40 00:05:21,006 --> 00:05:30,007 let's look at whether that's true. So, so suppose that, um, that psi of x happen to 41 00:05:30,007 --> 00:05:38,003 look like the straight line in this region. So, so this is psi of x. There's, 42 00:05:38,003 --> 00:05:51,005 there's x. Okay and there's X + delta X, X - delta X. So, what's the average of psi 43 00:05:51,005 --> 00:05:57,006 of X - delta of X and psi of X + delta of X. Well the average of these two points is 44 00:05:57,006 --> 00:06:04,008 exactly that. And so Y is exactly zero because psi of X is exactly, exactly the 45 00:06:04,008 --> 00:06:12,004 average of psi of X minus delta X and psi of X plus delta X. So there's something else 46 00:06:12,004 --> 00:06:18,002 going on here and so actually if you think about it a little bit more it turns out 47 00:06:18,002 --> 00:06:24,004 that Y is proportional to the second derivative of psi of X with respect to X. 48 00:06:25,006 --> 00:06:32,001 Okay, the, the reason, you might, you know, the way you might see this is that 49 00:06:32,001 --> 00:06:47,005 this difference psi of X minus delta X plus psi of X, plus delta X, minus psi of 50 00:06:47,005 --> 00:07:04,001 X, so this divide by two minus I X. This is the same as psi of x + delta x - psi 51 00:07:04,001 --> 00:07:19,008 x. I'm just putting this and this together, minus another psi X. And then I can 52 00:07:19,008 --> 00:07:29,003 subtract off psi of X minus delta X. Okay? So this first piece is proportional to the 53 00:07:29,003 --> 00:07:39,005 derivative of x of psi x at x. And this, at, at, um, and this is proportional to 54 00:07:39,005 --> 00:07:44,005 the derivative of psi x as, at x-delta x. So when you subtract these two, it's 55 00:07:44,005 --> 00:07:54,007 proportional to the second derivative. And so, what's equation says is that the 56 00:07:54,007 --> 00:08:02,004 rate of change of psi with respect to T. It's proportional to. So, since I don't 57 00:08:02,004 --> 00:08:06,007 care about constants in this particular lecture, I'm just being very imprecise, 58 00:08:06,007 --> 00:08:18,006 I'll just say it's equal to the second derivative of psi with respect to, X. Well 59 00:08:18,006 --> 00:08:25,009 not quite. So, you see if, if this is all that was happening, then, this would not 60 00:08:25,009 --> 00:08:32,000 be very good because what you know, what Schrodinger's equation would say is that 61 00:08:32,000 --> 00:08:36,000 every point then tries to look like the average of its neighbors, and no matter 62 00:08:36,000 --> 00:08:40,008 where you start from, your wave function would, would eventually become completely 63 00:08:40,008 --> 00:08:47,000 flat. So to keep it from going flat, but still to keep the change proportional to 64 00:08:47,000 --> 00:08:54,004 this difference, what you do is move in the orthogonal direction in the complex 65 00:08:54,004 --> 00:09:04,005 plane. Okay so, so let's see what, what this means. So, so remember, our, our 66 00:09:04,005 --> 00:09:19,005 equation now is I D psi of x, t by dt is d2 psi x, t by dx squared. Okay why, why I 67 00:09:19,005 --> 00:09:27,000 times this? So, let's say we are working on the complex plane, this is real, this 68 00:09:27,000 --> 00:09:33,002 is imaginary. And let's say we have a, we have complex number, which is like a 69 00:09:33,002 --> 00:09:44,009 vector in this complex plane. Okay, so. So now um, if you take this, this complex 70 00:09:44,009 --> 00:09:49,004 number and you instead, you know if, you, you, you could move it in some direction. 71 00:09:49,004 --> 00:09:55,005 So, for example, you could, you could change it. But if you change it. In a 72 00:09:55,005 --> 00:10:02,005 direction which is I times itself. Then you're, then you're changing it in like 73 00:10:02,005 --> 00:10:06,006 this. And what you're doing is you're, you're basically rotating the complex 74 00:10:06,006 --> 00:10:12,000 number around, and it leaves its magnitude unchanged. And so, this is the sort of 75 00:10:12,000 --> 00:10:15,007 thing that Schrodinger's equation is doing. It's saying, Well, the rate of 76 00:10:15,007 --> 00:10:19,005 change is proportional to the secondary derivative, but we wan t to move it in the 77 00:10:19,005 --> 00:10:24,003 orthog, you know, orthogonal direction in the complex plane. So that we actually 78 00:10:24,003 --> 00:10:28,004 keep the normalization of the wave function, so that it actually ends up 79 00:10:28,004 --> 00:10:36,006 still being a unit wave function. Okay, so, so, you know lets, lets state back and 80 00:10:36,006 --> 00:10:41,006 see what this is told us about Schrodinger's equation. So, what it's telling us is, if 81 00:10:41,006 --> 00:10:48,003 you have a free particle on a line. And there's some superposition that describes 82 00:10:48,003 --> 00:10:54,001 it. So, you know, it's, it's some superposition that, that, that describes 83 00:10:54,001 --> 00:11:02,001 it. Now, what's, what's happening to this, this superposition? Well, basically, every 84 00:11:02,001 --> 00:11:10,002 point looks around. It looks at its, you know, it looks, uh, every point X. Looks 85 00:11:10,002 --> 00:11:18,000 at its small neighborhood to the right, to the left and it compares itself to 86 00:11:18,000 --> 00:11:23,005 its, to its neighbors on, the average of its neighbors on the left to the right. 87 00:11:23,005 --> 00:11:31,006 And it changes it's amplitude at a rate which is proportional to this, this, the 88 00:11:31,006 --> 00:11:36,005 difference between itself and the average of its neighbors. Another way of saying it 89 00:11:36,005 --> 00:11:44,004 is, it changes its amplitude. At a rate that depends upon the second derivative of 90 00:11:44,004 --> 00:11:51,007 the wave function with respect to X. Okay, so what it means is, so now, now think of, 91 00:11:51,007 --> 00:11:57,005 think of at every point, you know the, the amplitude is not real it, its actually a 92 00:11:57,005 --> 00:12:05,008 complex number. So what you really have here. At every point is a complex plane 93 00:12:05,008 --> 00:12:12,005 and your amplitude is, is some complex number in this, in this complex plane. So 94 00:12:12,005 --> 00:12:18,009 that's real, that's imaginary. And lets say the, the amplitude happened to be 95 00:12:18,009 --> 00:12:25,008 imaginary right now. But now, what happens is. The amplitude starts precessing. Right 96 00:12:25,008 --> 00:12:31,000 it, you know, it acquires a phase. It, it starts rotating in this, in this complex 97 00:12:31,000 --> 00:12:35,008 plane, but at what rate does it rotate? Well it rotates at a rate which is 98 00:12:35,008 --> 00:12:42,007 proportional to the second derivative at this point. So, at the points where, where 99 00:12:42,007 --> 00:12:47,009 the wave function changes very rapidly uh, the, the second derivative is, is 100 00:12:47,009 --> 00:12:52,002 large. Well the, it's going to rotate quickly. But where, where the second 101 00:12:52,002 --> 00:12:56,008 derivative is zero, it's just not going to rotate very much at all. And that's what 102 00:12:56,008 --> 00:13:04,004 Schrodinger's equation tells us, about how, the way function changes in time. Okay, so now, 103 00:13:04,004 --> 00:13:11,007 as I said, um you know, in this form of Schrodinger's equation, I've, I've been very sloppy. 104 00:13:11,007 --> 00:13:18,001 I've, I've just ignored, um, uh, constant signs, etcetera. If, if you want to see 105 00:13:18,001 --> 00:13:22,002 what Schrodinger's equation for a free particle really looks like, well, it looks like this it, 106 00:13:22,002 --> 00:13:29,007 it has a form I H bar, where H bar is the reduced planck-, plancks constant D psi by 107 00:13:29,007 --> 00:13:42,001 DT is minus H bar squared over 2M, and it's mass of the particle, D2 psi by D x 108 00:13:42,001 --> 00:13:46,000 squared. Okay so we're, we're going to look at this precise form of Schrodinger's 109 00:13:46,000 --> 00:13:52,002 equation in the next lecture. But for now I just wanted to give you some intuitive 110 00:13:52,002 --> 00:13:57,006 feel about what it might be telling us about the evolution of the system.