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PROFESSOR: So in
quantum mechanics,
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you see this i appearing here,
and it's a complex number--
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the square root of minus 1.
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And that shows that
somehow complex numbers
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are very important.
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Well it's difficult
to overemphasize
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the importance of i--
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is the square root of minus 1
was invented by people in order
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to solve equations.
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Equations like x
squared equals minus 1.
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And it so happens
that once you invent i
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you need to invent
more numbers, and you
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can solve every polynomial
equation with just i.
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And square root of i--
well square root of i
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can be written in terms
of i and other numbers.
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So if you have a
complex number z--
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we sometimes write
it this way, and we
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say it belongs to
the complex numbers,
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and with a and b belonging
to the real numbers.
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And we say that
the real part of z
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is a, the imaginary
part of z is b.
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We also define the
complex conjugate of z,
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which is a minus i b and we
picture the complex number z
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by putting a on the
x-axis b on the y-axis,
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and we think of the
complex number z here--
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kind of like putting
the real numbers here
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and the imaginary parts here.
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So you can think
of this as ib or b,
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but this is the complex number--
maybe ib would be a better way
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to write it here.
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So with complex numbers, there
is one more useful identity.
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You define the norm
of the complex number
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to be square root of a
squared plus b squared
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and then this
results in the norm
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squared being a
squared plus b squared.
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And it's actually equal
to z times z star.
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A very fundamental equation--
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z times z star--
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if you multiply z
times z star, you
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get a squared plus b squared.
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So the norm squared--
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the norm of this thing
is a real number.
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And that's pretty important.
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So there is one other
identity that is very useful
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and I might well
mention it here as we're
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going to be working
with complex numbers.
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And for more practice
on complex numbers,
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you'll see the homework.
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So suppose I have in the
complex plane an angle theta,
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and I want to figure out what
is this complex number z here
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at unit radius.
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So I would know that it's real
part would be cosine theta.
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And its imaginary part
would be sine theta.
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It's a circle of radius 1.
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So that must be
the complex number.
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z must be equal to cosine
theta plus i sine theta.
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Because the real part
of it is cosine theta.
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It's in that horizontal
part's projection.
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And the imaginary part is
the vertical projection.
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Well the thing that
is very amazing
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is that this is equal
to e to the i theta.
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And that is very non-trivial.
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To prove it, you
have to work a bit,
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but it's a very famous
result and we'll use it.
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So that is complex numbers.
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So complex numbers you use
them in electromagnetism.
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You sometimes use them
in classical mechanics,
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but you always use it
in an auxiliary way.
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It was not directly relevant
because the electric field
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is real, the position is,
real the velocity is real--
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everything is real and
the equations are real.
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On the other hand,
in quantum mechanics,
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the equation already has an i.
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So in quantum mechanics, psi is
a complex number necessarily.
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It has to be.
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In fact, if it would be real,
you would have a contradiction
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because if psi is
real, turns out
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for all physical systems we're
interested in, H on psi real
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gives you a real thing.
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And here, if psi is real
then the relative is real,
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and this is imaginary and
you have a contradiction.
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So there are no
solutions that are real.
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So you need complex numbers.
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They're not auxiliary.
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On the other hand, you can
never measure a complex number.
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You measure real numbers--
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ammeter, position,
weight, anything
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that you really measure
at the end of the day
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is a real number.
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So if the wave function
was a complex number,
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it was the issue of what is
the physical interpretation.
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And Max Born had
the idea that you
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have to calculate the
real number called
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the norm of this
square, and this is
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proportional to probabilities.
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So that was a great
discovery and had a lot
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to do with the development
of quantum mechanics.
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Many people hated this.
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In fact, Schrodinger
himself hated it,
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and his invention of
the Schrodinger cat
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was an attempt to
show how ridiculous
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was the idea of thinking of
these things as probabilities.
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But he was wrong, and Einstein
was wrong in that way.
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But when very good
physicists are wrong,
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they are not wrong
for silly reasons,
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they are wrong for good
reasons, and we can learn a lot
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from their thinking.
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And this EPR are
things that we will
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discuss at some moment in
your quantum sequence at MIT.
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Einstein-Podolski-Rosen
was an attempt
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to show that quantum
mechanics was wrong
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and led to amazing discoveries.
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It was the EPR paper
itself was wrong,
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but it brought up
ideas that turned out
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to be very important.