WEBVTT
00:00:04.610 --> 00:00:09.900
PROFESSOR: So in
quantum mechanics,
00:00:09.900 --> 00:00:14.030
you see this i appearing here,
and it's a complex number--
00:00:14.030 --> 00:00:17.030
the square root of minus 1.
00:00:17.030 --> 00:00:20.390
And that shows that
somehow complex numbers
00:00:20.390 --> 00:00:21.530
are very important.
00:00:21.530 --> 00:00:25.160
Well it's difficult
to overemphasize
00:00:25.160 --> 00:00:26.810
the importance of i--
00:00:26.810 --> 00:00:32.299
is the square root of minus 1
was invented by people in order
00:00:32.299 --> 00:00:33.950
to solve equations.
00:00:33.950 --> 00:00:38.735
Equations like x
squared equals minus 1.
00:00:38.735 --> 00:00:42.150
And it so happens
that once you invent i
00:00:42.150 --> 00:00:45.100
you need to invent
more numbers, and you
00:00:45.100 --> 00:00:50.070
can solve every polynomial
equation with just i.
00:00:50.070 --> 00:00:52.660
And square root of i--
well square root of i
00:00:52.660 --> 00:00:56.340
can be written in terms
of i and other numbers.
00:00:56.340 --> 00:01:02.980
So if you have a
complex number z--
00:01:02.980 --> 00:01:06.040
we sometimes write
it this way, and we
00:01:06.040 --> 00:01:08.670
say it belongs to
the complex numbers,
00:01:08.670 --> 00:01:14.450
and with a and b belonging
to the real numbers.
00:01:14.450 --> 00:01:17.260
And we say that
the real part of z
00:01:17.260 --> 00:01:23.770
is a, the imaginary
part of z is b.
00:01:23.770 --> 00:01:28.520
We also define the
complex conjugate of z,
00:01:28.520 --> 00:01:37.090
which is a minus i b and we
picture the complex number z
00:01:37.090 --> 00:01:42.040
by putting a on the
x-axis b on the y-axis,
00:01:42.040 --> 00:01:46.720
and we think of the
complex number z here--
00:01:46.720 --> 00:01:49.670
kind of like putting
the real numbers here
00:01:49.670 --> 00:01:51.790
and the imaginary parts here.
00:01:51.790 --> 00:01:56.140
So you can think
of this as ib or b,
00:01:56.140 --> 00:01:59.800
but this is the complex number--
maybe ib would be a better way
00:01:59.800 --> 00:02:02.110
to write it here.
00:02:02.110 --> 00:02:09.110
So with complex numbers, there
is one more useful identity.
00:02:09.110 --> 00:02:13.100
You define the norm
of the complex number
00:02:13.100 --> 00:02:17.870
to be square root of a
squared plus b squared
00:02:17.870 --> 00:02:21.410
and then this
results in the norm
00:02:21.410 --> 00:02:25.700
squared being a
squared plus b squared.
00:02:25.700 --> 00:02:30.230
And it's actually equal
to z times z star.
00:02:30.230 --> 00:02:32.940
A very fundamental equation--
00:02:32.940 --> 00:02:34.640
z times z star--
00:02:34.640 --> 00:02:39.170
if you multiply z
times z star, you
00:02:39.170 --> 00:02:40.940
get a squared plus b squared.
00:02:40.940 --> 00:02:43.340
So the norm squared--
00:02:43.340 --> 00:02:45.830
the norm of this thing
is a real number.
00:02:49.160 --> 00:02:52.020
And that's pretty important.
00:02:52.020 --> 00:03:01.320
So there is one other
identity that is very useful
00:03:01.320 --> 00:03:05.590
and I might well
mention it here as we're
00:03:05.590 --> 00:03:08.010
going to be working
with complex numbers.
00:03:08.010 --> 00:03:11.250
And for more practice
on complex numbers,
00:03:11.250 --> 00:03:14.070
you'll see the homework.
00:03:14.070 --> 00:03:22.460
So suppose I have in the
complex plane an angle theta,
00:03:22.460 --> 00:03:26.910
and I want to figure out what
is this complex number z here
00:03:26.910 --> 00:03:30.620
at unit radius.
00:03:30.620 --> 00:03:37.290
So I would know that it's real
part would be cosine theta.
00:03:41.736 --> 00:03:45.220
And its imaginary part
would be sine theta.
00:03:45.220 --> 00:03:47.480
It's a circle of radius 1.
00:03:51.240 --> 00:03:54.780
So that must be
the complex number.
00:03:54.780 --> 00:03:59.880
z must be equal to cosine
theta plus i sine theta.
00:03:59.880 --> 00:04:03.350
Because the real part
of it is cosine theta.
00:04:03.350 --> 00:04:07.790
It's in that horizontal
part's projection.
00:04:07.790 --> 00:04:12.440
And the imaginary part is
the vertical projection.
00:04:12.440 --> 00:04:15.110
Well the thing that
is very amazing
00:04:15.110 --> 00:04:20.060
is that this is equal
to e to the i theta.
00:04:20.060 --> 00:04:23.480
And that is very non-trivial.
00:04:23.480 --> 00:04:26.150
To prove it, you
have to work a bit,
00:04:26.150 --> 00:04:29.030
but it's a very famous
result and we'll use it.
00:04:32.420 --> 00:04:35.970
So that is complex numbers.
00:04:35.970 --> 00:04:42.690
So complex numbers you use
them in electromagnetism.
00:04:42.690 --> 00:04:45.750
You sometimes use them
in classical mechanics,
00:04:45.750 --> 00:04:49.200
but you always use it
in an auxiliary way.
00:04:49.200 --> 00:04:55.440
It was not directly relevant
because the electric field
00:04:55.440 --> 00:04:59.430
is real, the position is,
real the velocity is real--
00:04:59.430 --> 00:05:03.030
everything is real and
the equations are real.
00:05:03.030 --> 00:05:05.820
On the other hand,
in quantum mechanics,
00:05:05.820 --> 00:05:08.820
the equation already has an i.
00:05:08.820 --> 00:05:19.500
So in quantum mechanics, psi is
a complex number necessarily.
00:05:19.500 --> 00:05:21.470
It has to be.
00:05:21.470 --> 00:05:27.620
In fact, if it would be real,
you would have a contradiction
00:05:27.620 --> 00:05:30.410
because if psi is
real, turns out
00:05:30.410 --> 00:05:36.410
for all physical systems we're
interested in, H on psi real
00:05:36.410 --> 00:05:37.930
gives you a real thing.
00:05:37.930 --> 00:05:41.490
And here, if psi is real
then the relative is real,
00:05:41.490 --> 00:05:44.030
and this is imaginary and
you have a contradiction.
00:05:44.030 --> 00:05:48.830
So there are no
solutions that are real.
00:05:48.830 --> 00:05:50.890
So you need complex numbers.
00:05:50.890 --> 00:05:52.750
They're not auxiliary.
00:05:52.750 --> 00:05:57.420
On the other hand, you can
never measure a complex number.
00:05:57.420 --> 00:05:59.600
You measure real numbers--
00:05:59.600 --> 00:06:03.780
ammeter, position,
weight, anything
00:06:03.780 --> 00:06:06.030
that you really measure
at the end of the day
00:06:06.030 --> 00:06:07.690
is a real number.
00:06:07.690 --> 00:06:11.190
So if the wave function
was a complex number,
00:06:11.190 --> 00:06:14.310
it was the issue of what is
the physical interpretation.
00:06:14.310 --> 00:06:17.910
And Max Born had
the idea that you
00:06:17.910 --> 00:06:21.660
have to calculate the
real number called
00:06:21.660 --> 00:06:24.360
the norm of this
square, and this is
00:06:24.360 --> 00:06:26.240
proportional to probabilities.
00:06:33.040 --> 00:06:37.810
So that was a great
discovery and had a lot
00:06:37.810 --> 00:06:40.460
to do with the development
of quantum mechanics.
00:06:40.460 --> 00:06:43.850
Many people hated this.
00:06:43.850 --> 00:06:46.420
In fact, Schrodinger
himself hated it,
00:06:46.420 --> 00:06:50.440
and his invention of
the Schrodinger cat
00:06:50.440 --> 00:06:52.630
was an attempt to
show how ridiculous
00:06:52.630 --> 00:06:56.470
was the idea of thinking of
these things as probabilities.
00:06:56.470 --> 00:07:00.490
But he was wrong, and Einstein
was wrong in that way.
00:07:00.490 --> 00:07:05.260
But when very good
physicists are wrong,
00:07:05.260 --> 00:07:08.050
they are not wrong
for silly reasons,
00:07:08.050 --> 00:07:11.140
they are wrong for good
reasons, and we can learn a lot
00:07:11.140 --> 00:07:12.820
from their thinking.
00:07:12.820 --> 00:07:16.300
And this EPR are
things that we will
00:07:16.300 --> 00:07:21.640
discuss at some moment in
your quantum sequence at MIT.
00:07:21.640 --> 00:07:25.090
Einstein-Podolski-Rosen
was an attempt
00:07:25.090 --> 00:07:27.280
to show that quantum
mechanics was wrong
00:07:27.280 --> 00:07:30.250
and led to amazing discoveries.
00:07:30.250 --> 00:07:34.070
It was the EPR paper
itself was wrong,
00:07:34.070 --> 00:07:37.300
but it brought up
ideas that turned out
00:07:37.300 --> 00:07:39.120
to be very important.