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so in quantum

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mechanics you see this I appearing here

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and it's a complex number square root of

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minus1 and that shows that somehow

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complex numbers are very important well

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it's difficult to overemphasize their

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importance so I is the square root of

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minus1 was invented by people in order

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to solve equations equations like x² =

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-1 and it so happens that once you

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invent I you don't need to invent more

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numbers and you can solve every polom

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equation with just I and square root of

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I well square root of I can be written

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in terms of I and other numbers so um if

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you have a complex number

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Z 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

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numbers and we say that the real part of

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Z is a and the imaginary part of Z is B

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we also Define the complex

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conjugate of Z which is a minus

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IB and we picture the complex number uh

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Z by putting a on the x axis B on the Y

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AIS and we think of the complex number Z

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here kind of like putting the real

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numbers here and the imaginary Parts

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here so um you can think of this as i b

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or B but uh this is the complex number

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maybe IB would be a better way to write

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it

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here so with complex numbers there's one

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uh more uh useful identity you define

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the norm of the complex number to be

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square root of a 2 + b^

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S and then this results in the norm

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squared being a 2 + b^ 2 and it's

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actually equal to Z * Z star a very

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fundamental equation Z * Z Star if you

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multiply Z * Z

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Star you get a squ + b squ so the norm

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squared the norm of this thing is a real

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number and uh that's uh pretty important

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so there's one other identity that is is

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very useful I might as well mention it

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here as we're going to be working with

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complex numbers and uh for more practice

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and complex numbers you'll see the

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homework so

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suppose I have in the complex plane an

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angle

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Theta and I want to figure out what is

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this complex number Z here at unit

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radius

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so I would

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know that its real part would be cosine

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Theta and its imaginary part would be S

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Theta it's a circle of radius

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one so that must be the complex number Z

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must be equal to cosine theta plus I sin

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Theta because the real part of it is

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cosine Theta it's indeed that horizontal

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part projection and the imaginary part

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is the vertical

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projection well the thing that is very

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amazing is that this is equal to e to

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the I

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Theta and that is very

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non-trivial to prove it you have to work

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a bit but it's a very famous result and

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we'll use it

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so that is complex number so

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uh complex numbers you use them in

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electromagnetism you sometimes use them

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in classical mechanics but you always

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used it in an auxiliary way it was not

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directly relevant because the electric

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field is real the position is real the

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velocity is real everything is real and

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the equations are real on the other hand

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in quantum mechanics the equation

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already has an I so in quantum

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mechanics p is a complex number

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necessarily it has to be in fact if it

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would be

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real you could have a you would have a

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contradiction because if s is real turns

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out for all physical systems we're

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interested in h on py real gives you a

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real thing and here if SI is real the

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derivative is real and this is imaginary

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and you have a contradiction so there

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are no solutions that are

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real so you need complex numbers they're

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not auxiliary on the other hand you can

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never measure a complex number complex

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you measure real numbers Dieter

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of position weight anything that you

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really measure at the end of the day is

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a real number so if the wave function

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was a complex number was the issue of

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what is the physical interpretation and

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maxb had the idea that you have to

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calculate the real number called the

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norm of this square and this is

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proportional to probabilities

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so uh that was a great discovery and had

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a lot to do with the development of

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quantum mechanics many people hated this

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uh uh in fact shinger himself hated it

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um and uh his invention of the shringer

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cat was an attempt to show how

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ridiculous was the idea of thinking of

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these things as

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probabilities but he was wrong and

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Einstein was wrong wrong in that way but

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when very good physicists are wrong uh

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they are not wrong for silly reasons

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they are wrong for good reasons and we

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can learn a lot from their thinking and

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uh this epr things that we will discuss

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at this at some moment in your Quantum

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sequence at MIT Einstein Podolski Rosen

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was a attempt to show that quantum

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mechanics was wrong and led to amazing

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discovery

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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