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PROFESSOR: interpretation
of the wave function.

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--pretation--

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the wave function.

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So you should look at
what the inventor said.

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So what did Schrodinger say?

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Schrodinger thought
that psi represents

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particles that disintegrate.

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You have a wave function.

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And the wave function is
spread all over space,

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so the particle has
disintegrated completely.

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And wherever you find more psi,
more of the particle is there.

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That was his interpretation.

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Then came Max Born.

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He said, that doesn't
look right to me.

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If I have a particle, but I
solve the Schrodinger equation.

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Everybody started solving
the Schrodinger equation.

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So they solved it for a particle
that hits a Coulomb potential.

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And they find that the
wave function falls off

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like 1 over r.

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OK, the wave function
falls off like 2 over r.

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So is the particle
disintegrating?

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And if you measure, you get
a little bit of the particle

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here?

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No.

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Max Born said, we've
done this experiment.

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The particle chooses
some way to go.

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And it goes one way,
and when you measure,

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you get the full particle.

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The particle never
disintegrates.

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So Schrodinger hated
what Max Born said.

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Einstein hated it.

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But never mind.

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Max Born was right.

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Max Born said, it
represents probabilities.

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And why did they hate it?

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Because suddenly you
lose determinism.

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You can just talk
about probability.

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So that was sort of funny.

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And in fact, neither
Einstein nor Schrodinger

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ever reconciled themselves
with the probabilistic

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interpretation.

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They never quite liked it.

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It's probably said that
the whole Schrodinger cat

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experiment was a
way of Schrodinger

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to try to say how ridiculous the
probability interpretation was.

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Of course, it's not ridiculous.

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It's right.

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And the important
thing is summarized,

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I think, with one sentence here.

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I'll write it.

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Psi of x and t does not tell
how much of the particle--

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is at x at time t.

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But rather--

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what is the probability--

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probability--

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--bility--

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to find it--

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at x at time t.

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So in one sentence,
the first clause

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is what Schrodinger
said, and it's not that.

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It's not what fraction
of the particle you get,

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how much of the
particle you get.

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It's the probability of getting.

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But that requires--

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a little more precision.

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Because if a particle
can be anywhere,

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the probability of being at one
point, typically, will be 0.

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It's a continuous
probability distribution.

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So the way we think of this
is we say, we have a point x.

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Around that point x, we
construct a little cube.

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d cube x.

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And the probability--
probability dp,

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the little probability to find
the particle at xt in the cube,

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within the cube--

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the cube--

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is equal to the value of the
wave function at that point.

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Norm squared times
the volume d cube x.

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So that's the probability
to find the particle

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at that little cube.

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You must find the square
of the wave function

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and multiply by the
little element of volume.

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So that gives you the
probability distribution.

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And that's, really, what
the interpretation means.

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So it better be, if you
have a single particle--

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particle, it better be that
the integral all over space--

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all over space-- of psi
squared of x and t squared

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must be equal to 1.

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Because that particle
must be found somewhere.

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And the sum of the probabilities
to be found everywhere

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must add up to 1.

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So it better be
that this is true.

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And this poses a
set of difficulties

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that we have to explore.

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Because you wrote the
Schrodinger equation.

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And this Schrodinger
equation tells you

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how psi evolve in time.

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Now, a point I want to emphasize
is that the Schrodinger

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equation says, suppose
you know the wave function

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all over space.

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You know it's here
at some time t0.

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The Schrodinger equation
implies that that determines

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the wave function for any time.

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Why?

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Because if you know the
wave function throughout x,

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you can calculate
the right hand side

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of this equation for any x.

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And then you know how
psi changes in time.

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And therefore, you can
integrate with your computer

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the differential equation
and find the wave function

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at a later time all over space,
and then at a later time.

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So knowing the wave
function at one time

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determines the wave
function at all times.

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So we could run into a
big problem, which is--

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suppose your wave
function at some time t0

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satisfies this at
the initial time.

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Well, you cannot force the wave
function to satisfy it at any

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time.

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Because the wave function now
is determined by the Schrodinger

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equation.

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So you have the possibility that
you normalize the wave function

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well.

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It makes sense at some time.

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But the Schrodinger equation
later, by time evolution,

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gives you another
thing that doesn't

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satisfy this for all times.

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So what we will have
to understand next time

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is how the Schrodinger
equation does the right thing

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and manages to make
this consistent.

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If it's a probability at
some time, at a later time

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it will still be a
probability distribution.