WEBVTT
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PROFESSOR: Uncertainty.
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When you talk about random
variables, random variable Q,
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we've said that it has
values Q1 up to, say, Qn,
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and probabilities
P1 up to Pn, we
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speak of a standard
deviation, delta Q,
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as the uncertainty,
the standard deviation.
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And how is that standard
deviation defined?
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Well you begin by
making sure you
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know what is the
expectation value of the--
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or the average value of
this random variable,
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which was defined, last
time, I think I put braces,
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but bar is kind
of nice sometimes
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too, at least for
random variables,
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and it's the sum of
the Pi times the Qi.
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The uncertainty is also
some expectation value.
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And expectation
value of deviation.
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So the uncertainty squared
is the expectation value,
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sum over i, of deviations of the
random variable from the mean.
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So you calculate
the expected value
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of the difference of your random
variable and the mean squared,
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and that is the square of
the standard deviation.
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Now this is the definition.
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And it's a very nice
definition because it
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makes a few things clear.
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For example, the left hand
side is delta Q squared, which
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means it's a positive number.
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And the right hand side
is also a positive number,
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because you have probabilities
times differences of quantities
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squared.
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So this is all greater
and equal to zero.
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And moreover, you can
actually say the following.
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If the uncertainty, or the
standard deviation, is zero,
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the random variable
is not that random.
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Because if this whole thing
is 0, this delta squared,
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delta Q squared must be
0 and this must be 0.
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But each term here is positive.
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So each term must be 0,
because of any one of them
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was not equal to zero, you would
get a non-zero contribution.
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So any possible Qi that must
have a Pi different from 0
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must be equal to Qbar.
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So if delta cubed
is equal to 0, Qi
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is equal to Q as
not random anymore.
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OK, now we can simplify
this expression.
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Do the following.
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By simplifying, I mean
expand the right-hand side.
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So sum over i, Pi Qi
squared, minus 2 sum over i,
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Pi Qi Q bar plus sum
over i, Pi Q bar squared.
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This kind of thing
shows up all the time,
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shows up in quantum mechanic as
well, as we'll see in a second.
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And you need to be able
to see what's happenening.
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Here, you're having
the expectation value
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of Qi squared.
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That's the definition of
a bar of some variable,
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you'd multiply with variable
by the exponent of [INAUDIBLE].
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What is this?
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This a little more funny.
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First, you should know
that Q bar is a number,
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so it can go out.
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So it's minus 2 Q bar.
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And then all that is left is
this, but that's another Q bar.
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So it's another Q bar.
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And here, you take this one
out because it's a number,
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and the sum of the
probabilities is 1,
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so it's Q bar squared as well.
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And it always comes out
that way, this minus 2
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Q bar squared plus
Q bar squared.
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So at the end, Delta Q, it's
another famous property,
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is the mean of the square
minus the square of the mean.
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And from this, since this
is greater or equal than 0,
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you always conclude that
the mean of the square
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is always bigger than the--
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maybe I shouldn't
have the i here,
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I think it's a random
variable Q squared.
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So the mean, the square of
this is greater or equal
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than Q bar squared.
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OK.
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Well, what happens
in quantum mechanics,
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let give you the definition and
a couple of ways of writing it.
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So here comes the definition.
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It's inspired by this thing.
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So in quantum mechanics,
permission operator Q
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will define the uncertainty
of Q in the state,
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Psi O squared as the
expectation value
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of Q squared minus the
expectation value of Q squared.
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Those are things that you
know in quantum mechanics,
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how you're supposed to compute.
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Because you know what
an expectation value
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is in any state Psi.
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You so Psi star,
the operator, Psi.
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And here you do this
thing, so it's all clear.
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So it's a perfectly
good definition.
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Maybe it doesn't give
you too much insight yet,
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but let me say two
things, and we'll leave
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them to complete for next time.
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Which is claim one, one,
that Delta Q squared
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Psi can be written
as the expectation
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value of Q minus absolute
expectation value of Q squared.
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Like that.
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Look.
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It looks funny, and
we'll elaborate this,
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but the first claim is that
this is a possible re-writing.
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You can write this uncertainty
as a single expectation value.
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This is the analog of this
equation in quantum mechanics.
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Claim two is another re-writing.
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Delta Q squared on Psi
can be re-written as this.
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That's an integral.
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Q minus Q and Psi.
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Look at that.
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You act on Psi with the
operator, Q, and multiplication
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by the expectation value
of Q. This is an operator,
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this is a number
multiplied by Psi.
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You can add to this on
the [? wave ?] function,
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you can square it,
and then integrate.
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And that is also
the uncertainty.
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We'll show these
two things next time
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and show one more thing that
the uncertainty vanishes
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if and only if the state
is an ideal state of Q.
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So If the state that
you are looking for
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is an ideal state of Q,
you have no uncertainty.
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And if you have no
uncertainty, the state
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must be an ideal state
of Q. So those all things
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will come from this planes, that
we'll elaborate on next time.