Okay, so in this video we are going to trying to understand intuitively how the
form of Schrodinger's equation for a free particle on a line. Okay, so, so remember
what we have with the free particle on a line, so we have. The particle is free to
be any where on the real line and, and it's described, it state is described by wave
function. Which might look like this. It's um, it's psi of x at time t. So, this is
maybe, this is maybe what it looks like at time t=0, at general time t, and we want
to understand how this state evolves as t increases. Okay so, what we are going to do is
we're gonna assume, you know, there's this one general principle that we're gonna
assume that the equation of motion is described by it's, it's a local
differential equation, meaning it's this very important principle in physics, which
is that of locality, that every point in space, sort of, minds its own business.
Well. Well, it, it minds it's own business, but it, it sorts of look around
at an infinitesimal neighborhood around it. So it, it, it, um, well. It, it's
mind, it minds its own business, but also the business of sort of a small
neighborhood around itself. Okay. So, so, so now, what, what, what, what do we want
to say? Well, we want to say that, that the change in the state. So if you want to
write D Psi XT by DT. Right, the, the way this is described is by, by the following
kind of differential equation, the, the, the weight of change in the state at the
point X. So if this was the point X, this is the time T, that's what the super
position looks like and now, what, what this point X does is it looks in its
neighbourhood and it looks at a point, sort of, uh, uh, at x plus delta-x, and at
x minus delta-x. So it looks in its tiny neighborhood, and so, and it looks at its
own amplitude, so this is what psi of x is, and it compares it to psi of x plus
delta x, and psi of x minus delta x. And it does this in a particularly simple way,
so what it does is, it looks at the average of psi of x plus delta x, and psi
of x minus delta x. So, so what's this average of these two points? Well, the
average of these two points is, is that. Right, so, so what it's doing is it's
comparing this point, which is psi of x minus delta x, oops. Plus psi of X, plus
delta X divided by two and it's comparing it to, its own value which is psi of X.
And basically what this, what this equation says is that if you look at this
difference, the difference between these two quantities. So, this value. So let's
call this Y. What Schrodinger's equation says is that the rate of change is proportional to Y.
So the rate of change of psi at this point X is proportional to the difference
between psi at this point X minus the average of the neighbors. And the neighbors
infinite to the left and to the right. Okay, so, so now, can you, you know, as
delta X turns to zero. Can you describe what this, what y is in terms of psi of x?
So, so we can ask, can you describe Y in terms of psi of X? Some of you might be
guessing maybe Y is, Y is just a, you know, Y is something like the derivative
of the psi of X, with respect to X. Well, that's a pretty good guess, but let's,
let's look at whether that's true. So, so suppose that, um, that psi of x happen to
look like the straight line in this region. So, so this is psi of x. There's,
there's x. Okay and there's X + delta X, X - delta X. So, what's the average of psi
of X - delta of X and psi of X + delta of X. Well the average of these two points is
exactly that. And so Y is exactly zero because psi of X is exactly, exactly the
average of psi of X minus delta X and psi of X plus delta X. So there's something else
going on here and so actually if you think about it a little bit more it turns out
that Y is proportional to the second derivative of psi of X with respect to X.
Okay, the, the reason, you might, you know, the way you might see this is that
this difference psi of X minus delta X plus psi of X, plus delta X, minus psi of
X, so this divide by two minus I X. This is the same as psi of x + delta x - psi
x. I'm just putting this and this together, minus another psi X. And then I can
subtract off psi of X minus delta X. Okay? So this first piece is proportional to the
derivative of x of psi x at x. And this, at, at, um, and this is proportional to
the derivative of psi x as, at x-delta x. So when you subtract these two, it's
proportional to the second derivative. And so, what's equation says is that the
rate of change of psi with respect to T. It's proportional to. So, since I don't
care about constants in this particular lecture, I'm just being very imprecise,
I'll just say it's equal to the second derivative of psi with respect to, X. Well
not quite. So, you see if, if this is all that was happening, then, this would not
be very good because what you know, what Schrodinger's equation would say is that
every point then tries to look like the average of its neighbors, and no matter
where you start from, your wave function would, would eventually become completely
flat. So to keep it from going flat, but still to keep the change proportional to
this difference, what you do is move in the orthogonal direction in the complex
plane. Okay so, so let's see what, what this means. So, so remember, our, our
equation now is I D psi of x, t by dt is d2 psi x, t by dx squared. Okay why, why I
times this? So, let's say we are working on the complex plane, this is real, this
is imaginary. And let's say we have a, we have complex number, which is like a
vector in this complex plane. Okay, so. So now um, if you take this, this complex
number and you instead, you know if, you, you, you could move it in some direction.
So, for example, you could, you could change it. But if you change it. In a
direction which is I times itself. Then you're, then you're changing it in like
this. And what you're doing is you're, you're basically rotating the complex
number around, and it leaves its magnitude unchanged. And so, this is the sort of
thing that Schrodinger's equation is doing. It's saying, Well, the rate of
change is proportional to the secondary derivative, but we wan t to move it in the
orthog, you know, orthogonal direction in the complex plane. So that we actually
keep the normalization of the wave function, so that it actually ends up
still being a unit wave function. Okay, so, so, you know lets, lets state back and
see what this is told us about Schrodinger's equation. So, what it's telling us is, if
you have a free particle on a line. And there's some superposition that describes
it. So, you know, it's, it's some superposition that, that, that describes
it. Now, what's, what's happening to this, this superposition? Well, basically, every
point looks around. It looks at its, you know, it looks, uh, every point X. Looks
at its small neighborhood to the right, to the left and it compares itself to
its, to its neighbors on, the average of its neighbors on the left to the right.
And it changes it's amplitude at a rate which is proportional to this, this, the
difference between itself and the average of its neighbors. Another way of saying it
is, it changes its amplitude. At a rate that depends upon the second derivative of
the wave function with respect to X. Okay, so what it means is, so now, now think of,
think of at every point, you know the, the amplitude is not real it, its actually a
complex number. So what you really have here. At every point is a complex plane
and your amplitude is, is some complex number in this, in this complex plane. So
that's real, that's imaginary. And lets say the, the amplitude happened to be
imaginary right now. But now, what happens is. The amplitude starts precessing. Right
it, you know, it acquires a phase. It, it starts rotating in this, in this complex
plane, but at what rate does it rotate? Well it rotates at a rate which is
proportional to the second derivative at this point. So, at the points where, where
the wave function changes very rapidly uh, the, the second derivative is, is
large. Well the, it's going to rotate quickly. But where, where the second
derivative is zero, it's just not going to rotate very much at all. And that's what
Schrodinger's equation tells us, about how, the way function changes in time. Okay, so now,
as I said, um you know, in this form of Schrodinger's equation, I've, I've been very sloppy.
I've, I've just ignored, um, uh, constant signs, etcetera. If, if you want to see
what Schrodinger's equation for a free particle really looks like, well, it looks like this it,
it has a form I H bar, where H bar is the reduced planck-, plancks constant D psi by
DT is minus H bar squared over 2M, and it's mass of the particle, D2 psi by D x
squared. Okay so we're, we're going to look at this precise form of Schrodinger's
equation in the next lecture. But for now I just wanted to give you some intuitive
feel about what it might be telling us about the evolution of the system.