Okay. So, so now, in, in the next video, we'll talk about how to solve
Schroedinger's equation for a free [inaudible], well. For, not a free
particle, but a particle in a box, which is something we'll, we'll, we'll look at
soon. Finally, there's one other thing we might we might take a look at. Which is,
let's look at a different way of, deriving this uncertainty relation. Which is in
terms of commutators. Okay, so remember that in one of the early. Lectures. It
derives the, an uncertainty principle for a q bit, where there were two quantities
we were interested in, there were, there were a bit. And there was a sign. And the
bit correspondent to a, to a measurement in the Z basis which is given by one minus
one. And sine corresponded to a measurement in the x spaces, which was
the, the x observable was given by zero, one, one, zero. In this case the
eignvalues were zero and one. With eigen val-, eigen values one and -one. In this
case, the eigen vectors were + and - with eigen values one and -one. Now, why was
there uncertainty relation between these two, between these two, these two
quantities? Well, the reason is that these two operators have very different eigen
vectors. So the eigen vectors of the one operator look like this. And the
eigenvectors of the other look like this, plus and minus. So these are as far apart
as you can expect these basis vectors to be. You, you, you make these basis vectors
look as far apart as possible. Another way of saying this is, well actually these,
these two operators have different eigenvectors, if and only if, they don't
commute. So, meaning that. X times Z is not equal to Z times X. Okay so well
what's, what's X times Z equal to? Well that's 0110 times one minus 100. Is what?
Is minus 1100, whereas. The z times x, which is one minus 100 times 1100 is what?
It's a. >> One minus one equals zero. >> So, another way you can write this is by
saying well the commutator, between x and z which is defined to be x times z minus z
times x. I t's non zero and in this case, x times z minus three times x is this
minus that, which is. Which is that matrix. Okay, so it turns out there's a
very elegant formulation of the uncertainty principle in terms of
commentators. So, for the for position on momentum, That is the principle. Well, it
turns out that the, this formula by looking at commentator between x hat and v
hat. Which is x hat p hat minus p hat x hat. It turns out that this is Actually,
IHR. And now, what we are interested in, of course, is. Is the, is the spread
delta. X hat Delta P hat Okay. So, it turns out that what you can show, is that
this is at least. The absolute value of the cometator divided by two. Meaning it's
ugliest, H [inaudible] over two. Yeah, so this, this ends up being a, a, you know,
this is a, this is a actually, this is a more general theorem. The, the theorem
actually says, this is true for any two operators. The Delta A times Delta, B is
at least. The. [inaudible] between a and b Divided by two. And so, so if you use X
hat and V hat for A and B you get this result. So, Now lets, lets try to
understand, at least as I wont, I won't prove this for you now. Maybe I'll make up
a separate video, to, to prove this, but, lets try to understand at least why the
commutative X hat and P hat are [inaudible]. This will help us understand
how to deal with these operators. So, how do you evaluate this, this commutator?
Well what is, what is X hat P hat, minus P hat, X hat? What? To, to evaluate it, we
apply to some, some. Wave function psi X. And what do we get? Well, X hat. You know,
this is X hat, P hat, psi X- P hat, X hat, psi of X. Okay. So, so what is, what is X
hat applied to psi of X? It's just X psi of X. And what's P hat? Well P hat is. Ih
by D by DX And what about this? Well, this is I h var, d by dx, psi of x, and then
whatever this is as a function of x. What happens when you apply x hat to and we
just get x times that, minus. So, what's this whole thing? It's IH bar times X D
[inaudible] . By DX minus Dy, dx of x, 5x. Okay, so remember what d by d, x of x psi
of x is? This is d by d x of x, times psi of x then psi of x, plus x times d by dx
of psi of x. Okay, so. So you sub, substitute this in there. You, you cancel.
Cancel this and this. And, so, you get I H bar Psi of X. So what's the operator? It's
just IH bar. Okay. So that's the position momentum uncertainty relation. It says
that the spread in, in, in the position the spread in the momentum is at least,
the uncertainty in this, in the, in the position the uncertainty in the
momentum. The product is at least H bar over two.