Necessity of complex numbers

so in quantum mechanics you see this I appearing here and it's a complex number square root of minus1 and that shows that somehow complex numbers are very important well it's difficult to overemphasize their importance so I is the square root of minus1 was invented by people in order to solve equations equations like x² = -1 and it so happens that once you invent I you don't need to invent more numbers and you can solve every polom 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 um 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 and the imaginary part of Z is B we also Define the complex conjugate of Z which is a minus IB and we picture the complex number uh Z by putting a on the x axis B on the Y AIS and we think of the complex number Z here kind

of like putting the real numbers here and the imaginary Parts here so um you can think of this as i b or B but uh this is the complex number maybe IB would be a better way to write it here so with complex numbers there's one uh more uh useful identity you define the norm of the complex number to be square root of a 2 + b^ S and then this results in the norm squared being a 2 + b^ 2 and it's actually equal to Z * Z star a very fundamental equation Z *

Z Star if you multiply Z * Z Star you get a squ + b squ so the norm squared the norm of this thing is a real number and uh that's uh pretty important so there's one other identity that is is very useful I might as well mention it here as we're going to be working with complex numbers and uh for more practice and 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 its real part would be cosine Theta and its imaginary part would be S Theta it's a circle of radius one so that must be the complex number Z must be equal to cosine theta plus I sin Theta because the real part of it is cosine Theta it's indeed that horizontal part 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 number so uh complex numbers you use them in electromagnetism you sometimes use them in classical mechanics but you always used 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 p is a complex number necessarily it has to be in fact if it would be real

you could have a you would have a contradiction because if s is real turns out for all physical systems we're interested in h on py real gives you a real thing and here if SI is real the derivative 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 complex you measure real numbers Dieter of 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 was the issue of what is the physical interpretation and maxb had the idea that you have to calculate the real number called the norm of this square and this is proportional to probabilities so uh that was a great discovery and had a lot to do with the development of quantum mechanics many people hated this uh uh in fact shinger himself hated it um and uh his invention of the shringer 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 wrong in that way but when very good physicists are wrong uh they are not wrong for silly reasons they are wrong for good reasons and we can learn a lot from their thinking and uh this epr things that we will discuss at this at some moment in your Quantum sequence at MIT Einstein Podolski Rosen was a attempt to show that quantum mechanics was wrong and led to amazing discovery it was the epr paper itself was wrong but it brought up ideas that turned out to be

very important