[00:06] so in quantum
[00:08] mechanics you see this I appearing here
[00:12] and it's a complex number square root of
[00:16] minus1 and that shows that somehow
[00:19] complex numbers are very important well
[00:22] it's difficult to overemphasize their
[00:25] importance so I is the square root of
[00:28] minus1 was invented by people in order
[00:32] to solve equations equations like x² =
[00:37] -1 and it so happens that once you
[00:41] invent I you don't need to invent more
[00:43] numbers and you can solve every polom
[00:47] equation with just I and square root of
[00:50] I well square root of I can be written
[00:53] in terms of I and other numbers so um if
[00:58] you have a complex number
[01:01] Z we sometimes write it this way and we
[01:06] say it belongs to the complex numbers
[01:08] and with A and B belonging to the real
[01:13] numbers and we say that the real part of
[01:16] Z is a and the imaginary part of Z is B
[01:23] we also Define the complex
[01:26] conjugate of Z which is a minus
[01:31] IB and we picture the complex number uh
[01:36] Z by putting a on the x axis B on the Y
[01:41] AIS and we think of the complex number Z
[01:45] here kind of like putting the real
[01:48] numbers here and the imaginary Parts
[01:51] here so um you can think of this as i b
[01:55] or B but uh this is the complex number
[01:58] maybe IB would be a better way to write
[02:00] it
[02:01] here so with complex numbers there's one
[02:05] uh more uh useful identity you define
[02:09] the norm of the complex number to be
[02:13] square root of a 2 + b^
[02:16] S and then this results in the norm
[02:21] squared being a 2 + b^ 2 and it's
[02:26] actually equal to Z * Z star a very
[02:30] fundamental equation Z * Z Star if you
[02:34] multiply Z * Z
[02:37] Star you get a squ + b squ so the norm
[02:42] squared the norm of this thing is a real
[02:48] number and uh that's uh pretty important
[02:55] so there's one other identity that is is
[03:00] very useful I might as well mention it
[03:04] here as we're going to be working with
[03:06] complex numbers and uh for more practice
[03:09] and complex numbers you'll see the
[03:13] homework so
[03:16] suppose I have in the complex plane an
[03:19] angle
[03:21] Theta and I want to figure out what is
[03:24] this complex number Z here at unit
[03:28] radius
[03:30] so I would
[03:33] know that its real part would be cosine
[03:40] Theta and its imaginary part would be S
[03:44] Theta it's a circle of radius
[03:50] one so that must be the complex number Z
[03:55] must be equal to cosine theta plus I sin
[03:58] Theta because the real part of it is
[04:02] cosine Theta it's indeed that horizontal
[04:05] part projection and the imaginary part
[04:09] is the vertical
[04:11] projection well the thing that is very
[04:14] amazing is that this is equal to e to
[04:17] the I
[04:19] Theta and that is very
[04:22] non-trivial to prove it you have to work
[04:25] a bit but it's a very famous result and
[04:28] we'll use it
[04:33] so that is complex number so
[04:37] uh complex numbers you use them in
[04:41] electromagnetism you sometimes use them
[04:44] in classical mechanics but you always
[04:46] used it in an auxiliary way it was not
[04:50] directly relevant because the electric
[04:55] field is real the position is real the
[04:57] velocity is real everything is real and
[05:00] the equations are real on the other hand
[05:04] in quantum mechanics the equation
[05:06] already has an I so in quantum
[05:13] mechanics p is a complex number
[05:18] necessarily it has to be in fact if it
[05:22] would be
[05:23] real you could have a you would have a
[05:26] contradiction because if s is real turns
[05:30] out for all physical systems we're
[05:32] interested in h on py real gives you a
[05:37] real thing and here if SI is real the
[05:40] derivative is real and this is imaginary
[05:42] and you have a contradiction so there
[05:44] are no solutions that are
[05:47] real so you need complex numbers they're
[05:51] not auxiliary on the other hand you can
[05:54] never measure a complex number complex
[05:57] you measure real numbers Dieter
[06:01] of position weight anything that you
[06:04] really measure at the end of the day is
[06:06] a real number so if the wave function
[06:09] was a complex number was the issue of
[06:12] what is the physical interpretation and
[06:14] maxb had the idea that you have to
[06:18] calculate the real number called the
[06:21] norm of this square and this is
[06:24] proportional to probabilities
[06:33] so uh that was a great discovery and had
[06:37] a lot to do with the development of
[06:39] quantum mechanics many people hated this
[06:42] uh uh in fact shinger himself hated it
[06:46] um and uh his invention of the shringer
[06:49] cat was an attempt to show how
[06:51] ridiculous was the idea of thinking of
[06:54] these things as
[06:55] probabilities but he was wrong and
[06:58] Einstein was wrong wrong in that way but
[07:01] when very good physicists are wrong uh
[07:05] they are not wrong for silly reasons
[07:08] they are wrong for good reasons and we
[07:09] can learn a lot from their thinking and
[07:13] uh this epr things that we will discuss
[07:16] at this at some moment in your Quantum
[07:19] sequence at MIT Einstein Podolski Rosen
[07:24] was a attempt to show that quantum
[07:26] mechanics was wrong and led to amazing
[07:29] discovery
[07:30] it was the epr paper itself was wrong
[07:34] but it brought up ideas that turned out
[07:37] to be very important