L1.3 Necessity of complex numbers.

PROFESSOR: So in
quantum mechanics, you see this i appearing here,
and it's a complex number– the square root of minus 1. And that shows that
somehow complex numbers are very important. Well it's difficult
to overemphasize the importance of i– is the square root of minus 1
was invented by people in order to solve equations. Equations like x
squared equals minus 1. And it so happens
that once you invent i you need to invent
more numbers, and you can solve every polynomial
equation with just i. And square root of i–
well square root of i can be written in terms
of i and other numbers.

So if you have a
complex number z– we sometimes write
it this way, and we say it belongs to
the complex numbers, and with a and b belonging
to the real numbers. And we say that
the real part of z is a, the imaginary
part of z is b. We also define the
complex conjugate of z, which is a minus i b and we
picture the complex number z by putting a on the
x-axis b on the y-axis, and we think of the
complex number z here– kind of like putting
the real numbers here and the imaginary parts here. So you can think
of this as ib or b, but this is the complex number–
maybe ib would be a better way to write it here. So with complex numbers, there
is one more useful identity. You define the norm
of the complex number to be square root of a
squared plus b squared and then this
results in the norm squared being a
squared plus b squared.

And it's actually equal
to z times z star. A very fundamental equation– z times z star– if you multiply z
times z star, you get a squared plus b squared. So the norm squared– the norm of this thing
is a real number. And that's pretty important. So there is one other
identity that is very useful and I might well
mention it here as we're going to be working
with complex numbers.

And for more practice
on complex numbers, you'll see the homework. So suppose I have in the
complex plane an angle theta, and I want to figure out what
is this complex number z here at unit radius. So I would know that it's real
part would be cosine theta. And its imaginary part
would be sine theta. It's a circle of radius 1. So that must be
the complex number. z must be equal to cosine
theta plus i sine theta. Because the real part
of it is cosine theta. It's in that horizontal
part's projection. And the imaginary part is
the vertical projection. Well the thing that
is very amazing is that this is equal
to e to the i theta. And that is very non-trivial. To prove it, you
have to work a bit, but it's a very famous
result and we'll use it. So that is complex numbers. So complex numbers you use
them in electromagnetism. You sometimes use them
in classical mechanics, but you always use it
in an auxiliary way.

It was not directly relevant
because the electric field is real, the position is,
real the velocity is real– everything is real and
the equations are real. On the other hand,
in quantum mechanics, the equation already has an i. So in quantum mechanics, psi is
a complex number necessarily. It has to be. In fact, if it would be real,
you would have a contradiction because if psi is
real, turns out for all physical systems we're
interested in, H on psi real gives you a real thing.

And here, if psi is real
then the relative is real, and this is imaginary and
you have a contradiction. So there are no
solutions that are real. So you need complex numbers. They're not auxiliary. On the other hand, you can
never measure a complex number. You measure real numbers– ammeter, position,
weight, anything that you really measure
at the end of the day is a real number. So if the wave function
was a complex number, it was the issue of what is
the physical interpretation. And Max Born had
the idea that you have to calculate the
real number called the norm of this
square, and this is proportional to probabilities.

So that was a great
discovery and had a lot to do with the development
of quantum mechanics. Many people hated this. In fact, Schrodinger
himself hated it, and his invention of
the Schrodinger cat was an attempt to
show how ridiculous was the idea of thinking of
these things as probabilities. But he was wrong, and Einstein
was wrong in that way. But when very good
physicists are wrong, they are not wrong
for silly reasons, they are wrong for good
reasons, and we can learn a lot from their thinking. And this EPR are
things that we will discuss at some moment in
your quantum sequence at MIT.

Einstein-Podolski-Rosen
was an attempt to show that quantum
mechanics was wrong and led to amazing discoveries. It was the EPR paper
itself was wrong, but it brought up
ideas that turned out to be very important..

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