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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.