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.