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..