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PROFESSOR: Let's
do a work check.
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So main check.
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If integral psi star x t0,
psi x t0 dx is equal to 1
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at t equal to t0,
as we say there,
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then it must hold for later
times, t greater than t0.
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This is what we want to
check, or verify, or prove.
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Now, to do it, we're
going to take our time.
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So it's not going to happen in
five minutes, not 10 minutes,
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maybe not even half an hour.
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Not because it's
so difficult. It's
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because there's so
many things that one
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can say in between
that teach you
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a lot about quantum mechanics.
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So we're going to
take our time here.
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So we're going to first rewrite
it with better notation.
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So we'll define rho
of x and t, which
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is going to be called
the probability density.
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And it's nothing else than
what you would expect, psi star
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of x and t, psi of x and t.
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It's a probability density.
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You know that has the
right interpretation,
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it's psi squared.
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And that's the kind of thing
that integrated over space
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gives you the total probability.
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So this is a
positive number given
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by this quantity is called
the probability density.
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Fine.
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What do we know about
this probability density
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that we're trying to
find about its integral?
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So define next N of t to be the
integral of rho of x and t dx.
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Integrate this
probability density
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throughout space, and that's
going to give you N of t.
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Now, what do we know?
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We know that N of t, or
let's assume that N of t0
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is equal to 1.
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N is that normalization.
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It's that total integral
of the probability
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what had to be equal to 1.
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Well, let's assume N
at t0 is equal to 1.
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That's good.
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The question is, will
the Schrodinger equation
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guarantee that--
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and here's the claim--
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dN dt is equal to 0?
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Will the Schrodinger
equation guarantee this?
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If the Schrodinger
equation guarantees
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that this derivative
is, indeed, zero,
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then we're in good business.
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Because the derivative
is zero, the value's 1,
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will remain 1 forever.
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Yes?
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AUDIENCE: May I ask
why you specified
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for t greater than t0?
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Well, I don't have to specify
for t greater than t naught.
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I could do it for all t
different than t naught.
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But if I say this way, as
imagining that somebody
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prepares the system
at some time,
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t naught, and maybe the system
didn't exist for other times
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below.
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Now, if a system
existed for long time
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and you look at it at t naught,
then certainly the Schrodinger
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equation should imply
that it works later
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and it works before.
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So it's not really necessary,
but no loss of generality.
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OK, so that's it.
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Will it guarantee that?
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Well, that's our thing to do.
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So let's begin the work by doing
a little bit of a calculation.
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And so what do we need to do?
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We need to find the
derivative of this quantity.
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So what is this derivative of N
dN dt will be the integral d dt
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of rho of x and t dx.
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So I went here and
brought in the d dt, which
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became a partial derivative.
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Because this is just a
function of t, but inside here,
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there's a function of
t and a function of x.
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So I must make clear that
I'm just differentiating t.
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So is d dt of rho.
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And now we can write
it as integral dx.
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What this rho?
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Psi star psi.
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So we would have d dt of psi
star times psi plus psi star
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d dt of psi.
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OK.
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And here you see, if you
were waiting for that,
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that the Schrodinger
equation has to be necessary.
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Because we have the psi dt.
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And that information is there
with Schrodinger's equation.
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So let's do that.
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So what do we have?
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ih bar d psi dt equal h psi.
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We'll write it like
that for the time being
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without copying all what h is.
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That would take a lot of time.
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And from this equation,
you can find immediately
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that d psi dt is minus
i over h bar h hat psi.
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Now we need to complex
conjugate this equation,
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and that is always
a little more scary.
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Actually, the way
to do this in a way
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that you never get into
scary or strange things.
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So let me take the complex
conjugate of this equation.
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Here I would have i
goes to minus i h bar,
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and now I would have--
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we can go very slow--
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d psi dt star equals, and then
I'll be simple minded here.
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I think it's the best.
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I'll just start the
right hand side.
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I start the left hand side
and start the right hand side.
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Now here, the complex conjugate
of a derivative, in this case
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I want to clarify what it is.
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It's just the derivative
of the complex conjugate.
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So this is minus ih
bar d/dt of psi star
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equals h hat psi
star, that's fine.
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And from here, if I multiply
again by i divided by h bar,
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we get d psi star dt is equal
to i over h star h hat psi star.
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We obtain this useful formula
and this useful formula,
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and both go into our
calculation of dN dt.
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So what do we have here?
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dN dt equals integral dx, and
I will put an i over h bar,
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I think, here.
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Yes.
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i over h bar.
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Look at this term first.
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We have i over h
bar, h psi star psi.
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And the second term
involves a d psi dt that
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comes with an opposite sign.
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Same factor of i over h bar,
so minus psi star h psi.
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So the virtue of what
we've done so far
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is that it doesn't
look so bad yet.
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And looks relatively clean, and
it's very suggestive, actually.
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So what's happening?
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We want to show that
dN dt is equal to 0.
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Now, are we going to be
able to show that simply
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that to do a lot of algebra
and say, oh, it's 0?
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Well, it's kind of
going to work that way,
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but we're going to
do the work and we're
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going to get to dN dt being
an integral of something.
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And it's just not
going to look like 0,
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but it will be
manipulated in such a way
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that you can argue it's 0
using the boundary condition.
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So it's kind of interesting
how it's going to work.
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But here structurally,
you see what
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must happen for this
calculation to succeed.
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So we need for this to be 0.
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We need the following
thing to happen.
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The integral of h
hat psi star psi
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be equal to the integral
of psi star h psi.
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And I should write the dx's.
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They are there.
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So this would guarantee
that dN dt is equal to 0.
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So that's a very nice
statement, and it's kind of nice
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is that you have one
function starred,
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one function non-starred.
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The h is where the function
needs to be starred,
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but on the other
side of the equation,
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the h is on the other side.
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So you've kind of moved the
h from the complex conjugated
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function to the non-complex
conjugated function.
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From the first function
to this second function.
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And that's a very nice thing
to demand of the Hamiltonian.
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So actually what
seems to be happening
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is that this conservation
of probability
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will work if your
Hamiltonian is good enough
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to do something like this.
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And this is a nice formula,
it's a famous formula.
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This is true if H is
a Hermitian operator.
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It's a very interesting
new name that
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shows up that an
operator being Hermitian.
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So this is what I
was promising you,
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that we're going to
do this, and we're
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00:14:35,420 --> 00:14:40,500
going to be learning all kinds
of funny things as it happens.
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So what is it for a
Hermitian operator?
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Well, a Hermitian
operator, H, would actually
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satisfy the following.
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That the integral,
H psi 1 star psi 2
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is equal to the integral
of psi 1 star H psi 2.
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So an operator is said to be
Hermitian if you can move it
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from the first part to the
second part in this sense,
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and with two
different functions.
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So this should be possible
to do if an operator is
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to be called Hermitian.
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Now, of course, if it holds
for two arbitrary functions,
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it holds when the two functions
are the same, in this case.
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So what we need is
a particular case
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of the condition of hermiticity.
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Hermiticity simply means that
the operator does this thing.
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Any two functions that you put
here, this equality is true.
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Now if you ask yourself, how
do I even understand that?
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What allows me to move the H
from one side to the other?
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We'll see it very soon.
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But it's the fact that H
has second derivatives,
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00:16:41,380 --> 00:16:44,240
and maybe you can
integrate them by parts
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and move the derivatives
from the psi 1 to the psi 2,
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and do all kinds of things.
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But you should try to think at
this moment structurally, what
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00:16:55,220 --> 00:16:58,610
kind of objects you have, what
kind of properties you have.
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00:16:58,610 --> 00:17:02,840
And the objects
are this operator
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00:17:02,840 --> 00:17:05,540
that controls the
time evolution, called
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00:17:05,540 --> 00:17:07,170
the Hamiltonian.
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00:17:07,170 --> 00:17:12,420
And if I want probability
interpretation to make sense,
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00:17:12,420 --> 00:17:17,420
we need this equality, which is
a consequence of hermiticity.
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00:17:17,420 --> 00:17:23,170
Now, I'll maybe use a
little of this blackboard.
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00:17:23,170 --> 00:17:28,280
I haven't used it much before.
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00:17:28,280 --> 00:17:31,460
In terms of Hermitian
operators, I'm
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00:17:31,460 --> 00:17:35,540
almost there with a definition
of a Hermitian operator.
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00:17:35,540 --> 00:17:40,550
I haven't quite given it to
you, but let's let state it,
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00:17:40,550 --> 00:17:46,750
given that we're already in
this discussion of hermiticity.
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00:17:46,750 --> 00:17:53,690
So this is what is called the
Hermitian operator, does that.
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00:17:53,690 --> 00:18:11,580
But in general, rho,
given an operator T,
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00:18:11,580 --> 00:18:29,980
one defines its hermitian
conjugate P dagger as follows.
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00:18:29,980 --> 00:18:37,135
So you have the
integral of psi 1 star T
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00:18:37,135 --> 00:18:44,990
psi 2, and that must be
rearranged until it looks
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00:18:44,990 --> 00:18:56,000
like T dagger psi 1 star psi 2.
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00:18:56,000 --> 00:18:59,270
Now, these things
are the beginning
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00:18:59,270 --> 00:19:04,970
of a whole set of ideas that are
terribly important in quantum
216
00:19:04,970 --> 00:19:05,910
mechanics.
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00:19:05,910 --> 00:19:09,960
Hermitian operators, or
eigenvalues and eigenvectors.
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00:19:09,960 --> 00:19:12,170
So it's going to take
a little time for you
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00:19:12,170 --> 00:19:14,040
to get accustomed to them.
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00:19:14,040 --> 00:19:15,770
But this is the beginning.
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00:19:15,770 --> 00:19:17,990
You will explore a little
bit of these things
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00:19:17,990 --> 00:19:20,690
in future homework, and
start getting familiar.
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00:19:20,690 --> 00:19:24,950
For now, it looks very
strange and unmotivated.
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00:19:24,950 --> 00:19:28,550
Maybe you will see that
that will change soon, even
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00:19:28,550 --> 00:19:32,190
throughout today's lecture.
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00:19:32,190 --> 00:19:35,900
So this is the
Hermitian conjugate.
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00:19:35,900 --> 00:19:39,320
So if you want to calculate
the Hermitian conjugate,
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00:19:39,320 --> 00:19:42,710
you must start with this thing,
and start doing manipulations
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00:19:42,710 --> 00:19:48,410
to clean up the psi 2,
have nothing at the psi 2,
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00:19:48,410 --> 00:19:51,680
everything acting on
psi 1, and that thing
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00:19:51,680 --> 00:19:53,780
is called the dagger.
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00:19:53,780 --> 00:20:06,830
And then finally, T is Hermitian
if T dagger is equal to T.
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00:20:06,830 --> 00:20:10,880
So its Hermitian
conjugate is itself.
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00:20:10,880 --> 00:20:13,730
It's almost like people
say a real number is
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00:20:13,730 --> 00:20:17,300
a number whose complex
conjugate is equal to itself.
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00:20:17,300 --> 00:20:23,240
So a Hermitian operator is
one whose Hermitian conjugate
237
00:20:23,240 --> 00:20:28,400
is equal to itself, and
you see if T is Hermitian,
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00:20:28,400 --> 00:20:34,370
well then it's back to T
and T in both places, which
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00:20:34,370 --> 00:20:36,800
is what we've been saying here.
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00:20:36,800 --> 00:20:39,910
This is a Hermitian operator.