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Three-dimensional case.
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Now, in the future
homework, you will
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be doing the equivalent
of this calculation
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here with the Laplacians--
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it's not complicated--
so that you will
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derive with the current is.
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And the current must be a very
similar formula as this one.
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And indeed, I'll
just write it here.
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The current is h bar over m,
the imaginary part of psi star.
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And instead of ddx, you
expect the gradient of psi.
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That is the current
for the probability
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in three dimensions.
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And the analog of this equation,
d rho dt plus dj dx equals 0,
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is d rho dt plus divergence
of j is equal to 0.
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That is current conservation.
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Perhaps you do remember
that from your study
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of electromagnetism.
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That's how Maxwell discovered
the displacement current
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when he tried to figure
out how everything
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was compatible with
current conservation.
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Anyway, that argument
I'll do in a second
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so that it will become clearer.
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So one last thing here--
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it's something also-- you
can check the units here of j
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is 1 over l squared times 1 over
t, so probability per unit area
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and unit time.
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So what did we have?
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We were doing the integral of
the derivative of the integral
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given by n.
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It was over here, dn dt.
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We worked hard on it.
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And dn dt was the
integral of d rho dt.
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00:02:49,260 --> 00:02:56,070
So it was the integral
of d rho dt dx.
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00:02:59,120 --> 00:03:06,530
But we showed now that
d rho dt is minus dj dx.
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00:03:09,750 --> 00:03:14,280
So here you have integral
from minus infinity
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00:03:14,280 --> 00:03:20,250
to infinity dx of dj dx.
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And therefore, this is--
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I should have a minus sign,
because it was minus dj dx.
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This is minus the current
of x equals infinity times
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p minus the current at x
equals minus infinity nt.
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And as we more or
less hinted before,
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since the current is equal to
h over 2im psi star [INAUDIBLE]
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00:04:01,450 --> 00:04:10,910
psi dx minus psi [INAUDIBLE]
psi star dx, as you
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go to plus infinity
or minus infinity,
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these things go to
0 given the boundary
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conditions that we put.
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Because psi or psi star
go to 0 to infinity,
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and the derivates are
bound at the infinity.
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00:04:31,470 --> 00:04:39,440
So this is 0, dn dt 0.
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All is good.
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And two things happened.
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In the way of doing
this, we realized
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that the computation we
have done pretty much
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established that this is equal
to that, because dn dt is
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the difference of
these two integrals,
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and we showed it's 0.
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So this is true.
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And therefore, we suspect
h is a Hermitian operator.
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And the thing that we should
do in order to make sure it is
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is put two different functions
here, not two equal functions.
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This worked for
two equal function,
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but for two different
functions, and check it as well.
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And we'll leave
it as an exercise.
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It's a good exercise.
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So this shows the consistency.
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But we discover two
important ideas--
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one, the existence of a current
for probability, and two,
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h is a Hermitian operator.
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So last thing is to
explain the analogy
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with current conservation.
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I think this should
help as well.
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So the interpretation
that we'll have
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is the same as we have
in electromagnetism.
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00:06:59,470 --> 00:07:04,030
And there's a complete
analogy for everything here.
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So not for the wave function,
but for all these charge
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densities and current densities.
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So we have electromagnetism
and quantum mechanics.
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We have rho.
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Here is the charge density.
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And here is the
probability density.
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If you have a total
charge q in a volume,
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here is the probability to
find the particle in a volume.
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There is a j in Maxwell's
equations as well,
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and that's a current density.
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Amber's law has that current.
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It generates the curl of b.
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And here is a probability
current density.
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So that's the table.
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So what I want to make
sure is that you understand
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why these equations,
like this or that,
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are more powerful than just
showing that dn dt is 0.
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They imply a local
conservation of probability.
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You see, there has to be
physics of this thing.
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So the total
probability must be 1.
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But suppose you have the
probability distributed
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over space.
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There must be some
relation between the way
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the probabilities are varying
at one point and varying
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in other points so that
everything is consistent.
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And those are these
differential relations
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that say that whenever you
see a probability density
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change anywhere, it's because
there is some current.
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And that makes sense.
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If you see the charge density
in some point in space changing,
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it's because there
must be a current.
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So thanks to the
current, you can
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learn how to interpret
the probability much more
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physically.
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Because if you ask what is the
probability that the particle
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is from this distance
to that distance,
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you can look at what the
currents are at the edges
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and see whether that probability
is increasing or decreasing.
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So let's see that.
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Suppose you have a
volume, and define
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the charge inside the volume.
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Then you say OK, does this
charge change in time?
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Sure, it could.
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So dq dt is equal
to integral d rho dt
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d cube x over the volume.
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But d rho dt, by the current
conservation equation--
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that's the equation
we're trying to make sure
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your intuition is clear about--
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this is equal to minus
the integral of j--
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no, of divergence of j d
cube x over the volume.
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OK.
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But then Gauss's law.
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Gauss's Lot tells you that
you can relate this divergence
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to a surface integral.
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dq dt is therefore minus
the surface integral,
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the area of the
current times that.
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So I'll write it as minus
jda, the flux of the current,
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over the surface that bounds--
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this is the volume, and there's
the surface bounding it.
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So by the divergence
theorem, it becomes this.
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And this is how you understand
current conservation.
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You say, look, charge is
never created or destroyed.
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So if you see the charge
inside the volume changing,
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it's because there's
some current escaping
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through the surface.
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So that's the physical
interpretation
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of that differential equation,
of that d rho dt plus
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divergence of j is equal to 0.
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This is current conservation.
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And many people look at
this equation and say, what?
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Current conservation?
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I don't see anything.
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But when you look at this
equation, you say, oh, yes.
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The charge changes only because
it escapes the volume, not
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created nor destroyed.
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So the same thing happens
for the probability.
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Now, let me close up
with this statement
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in one dimension, which is the
one you care, at this moment.
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And on the line, you
would have points a and b.
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And you would say
the probability
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to be within a and b is the
integral from a to b dx of rho.
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That's your probability.
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That's the integral of
psi squared from a to b.
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Now, what is the time
derivative of it?
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00:14:15,616 --> 00:14:24,990
dp ab dt would be integral
from a to b dx of d rho dt.
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But again, for that case,
d rho dt is minus dj dx.
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So this is minus dx
dj dx between b and a.
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And what is that?
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Well, you get the
j at the boundary.
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So this is minus j at x equals
bt minus j at x equal a, t.
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So simplifying it,
you get dp ab dt
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is equal to minus j at x
equals bt plus j of a, t.
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Let's see if that makes sense.
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You have been looking
for the particle
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and decided to look at
this range from a to b.
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That's the probability
to find it there.
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00:15:53,970 --> 00:15:57,390
We learned already that
the total probability
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to find it anywhere
is going to be 1,
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and that's going
to be conserved,
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00:16:01,825 --> 00:16:03,440
and it's going to
be no problems.
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But now let's just
ask given what happens
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to this probability in time.
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Well, it could change,
because the wave
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00:16:10,290 --> 00:16:12,000
function could go up and down.
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00:16:12,000 --> 00:16:15,300
Maybe the wave
function was big here
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and a little later is small
so there's less probability
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00:16:18,330 --> 00:16:20,850
to find it here.
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00:16:20,850 --> 00:16:23,580
But now you have another
physical variable
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00:16:23,580 --> 00:16:26,790
to help you understand it,
and that's the current.
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00:16:26,790 --> 00:16:31,020
That formula we found
there for j of x and t
185
00:16:31,020 --> 00:16:33,420
in the upper
blackboard box formula
186
00:16:33,420 --> 00:16:36,480
is a current that
can be computed.
187
00:16:36,480 --> 00:16:42,580
And here you see
if the probability
188
00:16:42,580 --> 00:16:46,540
to find the particle
in this region changes,
189
00:16:46,540 --> 00:16:52,690
it's because some current must
be escaping from the edges.
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00:16:52,690 --> 00:16:57,130
And let's see if the
formula gives it right.
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00:16:57,130 --> 00:17:00,790
Well, we're assuming
quantities are
192
00:17:00,790 --> 00:17:06,400
positive if they
have plus components
193
00:17:06,400 --> 00:17:08,280
in the direction of x.
194
00:17:08,280 --> 00:17:12,880
So this current is the current
component in the x direction.
195
00:17:12,880 --> 00:17:14,829
And it should not
be lost-- maybe
196
00:17:14,829 --> 00:17:18,339
I didn't quite say it--
that if you are dealing
197
00:17:18,339 --> 00:17:28,300
with a divergence of j, this
is dj x dx plus dj y dy plus dj
198
00:17:28,300 --> 00:17:30,460
z dz.
199
00:17:30,460 --> 00:17:34,330
And in the case of one
dimension, you will have those,
200
00:17:34,330 --> 00:17:36,110
and you get this equation.
201
00:17:36,110 --> 00:17:39,190
So it's certainly the reduction.
202
00:17:39,190 --> 00:17:43,030
But here you see indeed,
if the currents are
203
00:17:43,030 --> 00:17:47,100
positive-- if the
current at b is positive,
204
00:17:47,100 --> 00:17:49,040
there is a current going out.
205
00:17:49,040 --> 00:17:51,770
So that tends to
reduce the probability.
206
00:17:51,770 --> 00:17:55,570
That's why the sign
came out with a minus.
207
00:17:55,570 --> 00:17:59,140
On the other hand, if
there is a current in a,
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00:17:59,140 --> 00:18:01,900
that tends to send
in probability,
209
00:18:01,900 --> 00:18:04,520
and that's why it
increases it here.
210
00:18:04,520 --> 00:18:07,740
So the difference between
these two currents
211
00:18:07,740 --> 00:18:10,820
determines whether
the probability here
212
00:18:10,820 --> 00:18:13,470
increases or decreases.