WEBVTT

00:00:00.690 --> 00:00:04.770
PROFESSOR: We have
found in this solution

00:00:04.770 --> 00:00:08.660
with some energy like this,
that there's a decaying

00:00:08.660 --> 00:00:11.590
exponential over this side.

00:00:11.590 --> 00:00:17.700
And the question is
often asked, well what

00:00:17.700 --> 00:00:20.610
happens if you tried
to measure the particle

00:00:20.610 --> 00:00:22.410
in the forbidden region?

00:00:22.410 --> 00:00:24.120
Must be a problem.

00:00:24.120 --> 00:00:27.200
If you find the particle
in the forbidden region,

00:00:27.200 --> 00:00:30.280
it has energy E that
is less than v0,

00:00:30.280 --> 00:00:32.820
so you you have
found the particle

00:00:32.820 --> 00:00:34.710
with negative kinetic energy.

00:00:34.710 --> 00:00:35.620
How does it look?

00:00:35.620 --> 00:00:36.570
How can it happen?

00:00:36.570 --> 00:00:37.890
What's going on?

00:00:37.890 --> 00:00:41.700
Can you really find the particle
in the forbidden region?

00:00:41.700 --> 00:00:47.630
And then how does this negative
kinetic energy look like?

00:00:47.630 --> 00:00:53.400
The answer is that it's kind
of funny what happens here.

00:00:53.400 --> 00:00:56.510
You can make two statements.

00:00:56.510 --> 00:01:03.150
It would be contradictory,
contradictory

00:01:03.150 --> 00:01:04.005
if you could make--

00:01:06.740 --> 00:01:10.370
could say the following things.

00:01:10.370 --> 00:01:19.530
One, that the particle is
in the forbidden region,

00:01:19.530 --> 00:01:21.690
forbidden region.

00:01:21.690 --> 00:01:29.250
And two, that the particle
has energy less than v0.

00:01:29.250 --> 00:01:36.390
Because then it would mean
negative kinetic energy.

00:01:36.390 --> 00:01:42.920
So if you can say these two
things, it seems contradictory.

00:01:42.920 --> 00:01:48.140
So quantum mechanics
evades this problem.

00:01:48.140 --> 00:01:50.510
Now, this is not
discussed as far

00:01:50.510 --> 00:01:56.780
as I can see, except in
some lecture notes of Gordon

00:01:56.780 --> 00:02:03.750
[? Boehme. ?] And because the
argument is not 100% precise,

00:02:03.750 --> 00:02:06.280
but they think the spirit
of the argument is clear.

00:02:06.280 --> 00:02:08.979
So I want to share it with you.

00:02:08.979 --> 00:02:09.914
So here is the catch.

00:02:12.690 --> 00:02:16.080
This particle,
remember it's governed

00:02:16.080 --> 00:02:20.910
by e to the minus kappa x
is the forbidden region.

00:02:20.910 --> 00:02:26.400
So the length scale here where
you can find it, the particle.

00:02:26.400 --> 00:02:29.640
The length scale is,
this forbidden region

00:02:29.640 --> 00:02:36.860
stretches to about x of
the order 1 over kappa.

00:02:39.500 --> 00:02:43.460
If you are going to find it, it
is in the region of a distance

00:02:43.460 --> 00:02:44.860
1 over kappa.

00:02:44.860 --> 00:02:47.830
At 10 1 over kappa you're
not going to find it.

00:02:47.830 --> 00:02:50.340
The exponential is too small.

00:02:50.340 --> 00:02:52.820
But remember, what was kappa?

00:02:52.820 --> 00:03:03.620
Kappa squared was 2m v0
minus E over h squared.

00:03:03.620 --> 00:03:06.120
That's what it was.

00:03:06.120 --> 00:03:10.060
Now if you want
to see and declare

00:03:10.060 --> 00:03:12.700
that you have this
particle, you would

00:03:12.700 --> 00:03:17.740
have to be able to measure
position with some precision,

00:03:17.740 --> 00:03:22.000
with a precision a
little smaller than this.

00:03:22.000 --> 00:03:23.950
Otherwise if you
measure with precision

00:03:23.950 --> 00:03:26.410
10 times that, well
maybe it's to the left,

00:03:26.410 --> 00:03:28.010
maybe it's somewhere else.

00:03:28.010 --> 00:03:40.130
So you need to measure position
with delta x a little smaller

00:03:40.130 --> 00:03:44.390
than 1 over kappa, otherwise you
cannot really tell it's inside

00:03:44.390 --> 00:03:45.500
the forbidden region.

00:03:48.290 --> 00:03:58.930
But now the problem is that if
you do a position measurement,

00:03:58.930 --> 00:04:00.720
and you localize
the wave function,

00:04:00.720 --> 00:04:02.670
there is some
momentum uncertainty.

00:04:02.670 --> 00:04:05.110
The particle that
you're looking at,

00:04:05.110 --> 00:04:09.670
as opposed to the particle
to the left, has no momentum.

00:04:09.670 --> 00:04:11.800
It's a different kind
of wave function.

00:04:11.800 --> 00:04:13.850
There's no momentum
really associated

00:04:13.850 --> 00:04:16.149
or well-defined momentum to it.

00:04:16.149 --> 00:04:19.779
So because you make
a position, you're

00:04:19.779 --> 00:04:23.050
localizing x, whatever
wave function you have.

00:04:23.050 --> 00:04:30.090
You're going to have some
uncertainty, and some momentum

00:04:30.090 --> 00:04:37.910
that is going to be kind of
bigger than h bar over delta x.

00:04:37.910 --> 00:04:45.690
So a momentum that is bigger
than, or a little bigger,

00:04:45.690 --> 00:04:49.110
than h bar kappa.

00:04:49.110 --> 00:04:51.820
If delta x is less
than that inequality,

00:04:51.820 --> 00:04:53.310
it goes in the same direction.

00:04:53.310 --> 00:04:56.040
So there's going to
be an uncertainty P.

00:04:56.040 --> 00:05:02.160
And therefore, this particle
has now some kinetic energy

00:05:02.160 --> 00:05:04.360
due to this uncertain momentum.

00:05:04.360 --> 00:05:14.900
So uncertainty in the
kinetic energy is how much?

00:05:14.900 --> 00:05:22.430
It's P squared over 2m, where
P is this uncertain momentum.

00:05:22.430 --> 00:05:29.660
So this is equal to h bar
kappa squared over 2m,

00:05:29.660 --> 00:05:36.420
which is equal to v0 minus E.

00:05:36.420 --> 00:05:40.560
So actually, if you think
about it, here is v0.

00:05:40.560 --> 00:05:45.090
This difference is v0 minus
E. And you were going to say,

00:05:45.090 --> 00:05:48.690
oh, I found the particle, it
has negative kinetic energy.

00:05:48.690 --> 00:05:49.250
But no.

00:05:49.250 --> 00:05:51.960
The uncertainty principle
says, you found it localized?

00:05:51.960 --> 00:05:53.000
OK.

00:05:53.000 --> 00:05:55.020
Your kinetic energy,
I'm sorry, no.

00:05:55.020 --> 00:05:56.140
There's an uncertainty.

00:05:56.140 --> 00:05:57.310
How much?

00:05:57.310 --> 00:05:59.070
v0 minus E.

00:05:59.070 --> 00:06:03.180
So whatever you wanted to
prove, it has been disproved.

00:06:03.180 --> 00:06:04.050
You can't do it.

00:06:04.050 --> 00:06:11.080
The total energy,
total energy is now

00:06:11.080 --> 00:06:14.860
E plus the uncertainty
in the energy, which

00:06:14.860 --> 00:06:20.980
is E plus v0 minus
E. And it's therefore

00:06:20.980 --> 00:06:25.140
greater than or equal to v0.

00:06:25.140 --> 00:06:28.470
And no real contradiction.

00:06:28.470 --> 00:06:30.630
So the uncertainty
principle sort of

00:06:30.630 --> 00:06:34.980
conspires to prevent you
from finding a particle

00:06:34.980 --> 00:06:37.360
with negative kinetic energy.

00:06:37.360 --> 00:06:40.380
And if you do detect a particle
in the forbidden region,

00:06:40.380 --> 00:06:45.990
it will have total energy 0,
or total kinetic energy 0.

00:06:45.990 --> 00:06:47.580
It will be a normal particle.

00:06:47.580 --> 00:06:50.050
Nothing strange about it.