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Okay. So, so now, in, in the next video,
we'll talk about how to solve

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Schroedinger's equation for a free
[inaudible], well. For, not a free

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particle, but a particle in a box, which
is something we'll, we'll, we'll look at

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soon. Finally, there's one other thing we
might we might take a look at. Which is,

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let's look at a different way of, deriving
this uncertainty relation. Which is in

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terms of commutators. Okay, so remember
that in one of the early. Lectures. It

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derives the, an uncertainty principle for
a q bit, where there were two quantities

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we were interested in, there were, there
were a bit. And there was a sign. And the

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bit correspondent to a, to a measurement
in the Z basis which is given by one minus

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one. And sine corresponded to a
measurement in the x spaces, which was

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the, the x observable was given by zero,
one, one, zero. In this case the

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eignvalues were zero and one. With eigen
val-, eigen values one and -one. In this

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case, the eigen vectors were + and - with
eigen values one and -one. Now, why was

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there uncertainty relation between these
two, between these two, these two

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quantities? Well, the reason is that these
two operators have very different eigen

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vectors. So the eigen vectors of the one
operator look like this. And the

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eigenvectors of the other look like this,
plus and minus. So these are as far apart

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as you can expect these basis vectors to
be. You, you, you make these basis vectors

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look as far apart as possible. Another way
of saying this is, well actually these,

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these two operators have different
eigenvectors, if and only if, they don't

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commute. So, meaning that. X times Z is
not equal to Z times X. Okay so well

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what's, what's X times Z equal to? Well
that's 0110 times one minus 100. Is what?

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Is minus 1100, whereas. The z times x,
which is one minus 100 times 1100 is what?

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It's a. >> One minus one equals zero. >>
So, another way you can write this is by

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saying well the commutator, between x and
z which is defined to be x times z minus z

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times x. I t's non zero and in this case,
x times z minus three times x is this

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minus that, which is. Which is that
matrix. Okay, so it turns out there's a

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very elegant formulation of the
uncertainty principle in terms of

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commentators. So, for the for position on
momentum, That is the principle. Well, it

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turns out that the, this formula by
looking at commentator between x hat and v

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hat. Which is x hat p hat minus p hat x
hat. It turns out that this is Actually,

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IHR. And now, what we are interested in,
of course, is. Is the, is the spread

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delta. X hat Delta P hat Okay. So, it
turns out that what you can show, is that

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this is at least. The absolute value of
the cometator divided by two. Meaning it's

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ugliest, H [inaudible] over two. Yeah, so
this, this ends up being a, a, you know,

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this is a, this is a actually, this is a
more general theorem. The, the theorem

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actually says, this is true for any two
operators. The Delta A times Delta, B is

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at least. The. [inaudible] between a and b
Divided by two. And so, so if you use X

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hat and V hat for A and B you get this
result. So, Now lets, lets try to

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understand, at least as I wont, I won't
prove this for you now. Maybe I'll make up

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a separate video, to, to prove this, but,
lets try to understand at least why the

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commutative X hat and P hat are
[inaudible]. This will help us understand

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how to deal with these operators. So, how
do you evaluate this, this commutator?

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Well what is, what is X hat P hat, minus P
hat, X hat? What? To, to evaluate it, we

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apply to some, some. Wave function psi X.
And what do we get? Well, X hat. You know,

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this is X hat, P hat, psi X- P hat, X hat,
psi of X. Okay. So, so what is, what is X

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hat applied to psi of X? It's just X  psi
of X. And what's P hat? Well P hat is. Ih

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by D by DX And what about this? Well, this
is I h var, d by dx, psi of x, and then

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whatever this is as a function of x. What
happens when you apply x hat to and we

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just get x times that, minus. So, what's
this whole thing? It's IH bar times X D

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[inaudible] . By DX minus Dy, dx of x, 5x.
Okay, so remember what d by d, x of x psi

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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

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of psi of x. Okay, so. So you sub,
substitute this in there. You, you cancel.

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Cancel this and this. And, so, you get I H
bar Psi of X. So what's the operator? It's

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just IH bar. Okay. So that's the position
momentum uncertainty relation. It says

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that the spread in, in, in the position 
the spread in the momentum is at least,

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the uncertainty in this, in the, in the
position  the uncertainty in the

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momentum. The product is at least H bar
over two.