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7:38
Transcript
0:06
so in quantum
0:08
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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0:32
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
0:47
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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1:08
and with A and B belonging to the real
1:13
numbers and we say that the real part of
1:16
Z is a and the imaginary part of Z is B
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we also Define the complex
1:26
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
1:48
numbers here and the imaginary Parts
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here so um you can think of this as i b
1:55
or B but uh this is the complex number
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maybe IB would be a better way to write
2:00
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
2:34
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
2:48
number and uh that's uh pretty important
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so there's one other identity that is is
3:00
very useful I might as well mention it
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here as we're going to be working with
3:06
complex numbers and uh for more practice
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and complex numbers you'll see the
3:13
homework so
3:16
suppose I have in the complex plane an
3:19
angle
3:21
Theta and I want to figure out what is
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this complex number Z here at unit
3:28
radius
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so I would
3:33
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
4:02
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
4:14
amazing is that this is equal to e to
4:17
the I
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Theta and that is very
4:22
non-trivial to prove it you have to work
4:25
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
4:41
electromagnetism you sometimes use them
4:44
in classical mechanics but you always
4:46
used it in an auxiliary way it was not
4:50
directly relevant because the electric
4:55
field is real the position is real the
4:57
velocity is real everything is real and
5:00
the equations are real on the other hand
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in quantum mechanics the equation
5:06
already has an I so in quantum
5:13
mechanics p is a complex number
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necessarily it has to be in fact if it
5:22
would be
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real you could have a you would have a
5:26
contradiction because if s is real turns
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out for all physical systems we're
5:32
interested in h on py real gives you a
5:37
real thing and here if SI is real the
5:40
derivative is real and this is imaginary
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and you have a contradiction so there
5:44
are no solutions that are
5:47
real so you need complex numbers they're
5:51
not auxiliary on the other hand you can
5:54
never measure a complex number complex
5:57
you measure real numbers Dieter
6:01
of position weight anything that you
6:04
really measure at the end of the day is
6:06
a real number so if the wave function
6:09
was a complex number was the issue of
6:12
what is the physical interpretation and
6:14
maxb had the idea that you have to
6:18
calculate the real number called the
6:21
norm of this square and this is
6:24
proportional to probabilities
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so uh that was a great discovery and had
6:37
a lot to do with the development of
6:39
quantum mechanics many people hated this
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uh uh in fact shinger himself hated it
6:46
um and uh his invention of the shringer
6:49
cat was an attempt to show how
6:51
ridiculous was the idea of thinking of
6:54
these things as
6:55
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
7:16
at this at some moment in your Quantum
7:19
sequence at MIT Einstein Podolski Rosen
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was a attempt to show that quantum
7:26
mechanics was wrong and led to amazing
7:29
discovery
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it was the epr paper itself was wrong
7:34
but it brought up ideas that turned out
7:37
to be very important
— end of transcript —
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