WEBVTT
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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.