WEBVTT
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Prof: So I got an
interesting email this weekend
00:07.450 --> 00:12.940
from two of your buddies,
Jerry Wang and Emma Alexander.
00:12.940 --> 00:20.830
The title of the email is this.
00:20.830 --> 00:26.150
It says Casaplanka,
so I already know these guys
00:26.147 --> 00:28.407
are up to no good.
00:28.410 --> 00:29.830
And the question is the
following.
00:29.830 --> 00:35.620
They said, "You've written
a wave function ψ(x) and
00:35.624 --> 00:37.004
A(p).
00:37.000 --> 00:41.080
There seems to be no reference
of time, so where is time in
00:41.077 --> 00:44.097
this," and they couldn't
stop there.
00:44.100 --> 00:47.070
That would be a good place to
stop but they went on to say,
00:47.070 --> 00:50.640
"Are you saying the ψ
is just a ψ
00:50.637 --> 00:52.537
as time goes by?"
00:52.540 --> 00:55.110
So that's the kind of stuff
that appeals to me,
00:55.110 --> 00:58.060
so I don't care if you don't
learn any quantum mechanics,
00:58.060 --> 01:02.590
but if you can do this kind of
stuff I'm not worried about you.
01:02.590 --> 01:05.800
But now I have to give a
serious answer to that serious
01:05.796 --> 01:08.186
question,
which I sort of mentioned
01:08.186 --> 01:12.136
before, which is that everything
I've done so far is for one
01:12.144 --> 01:13.424
instant in time.
01:13.420 --> 01:15.340
I hope I made it very clear.
01:15.340 --> 01:17.000
So here's the analogy.
01:17.000 --> 01:19.230
If someone comes to you and
says, "Tell me all about
01:19.232 --> 01:20.112
Newtonian mechanics.
01:20.110 --> 01:20.840
How does it work?
01:20.840 --> 01:22.360
What's the scheme like?"
01:22.360 --> 01:26.850
You say, "At any instant
someone has to give you the x
01:26.849 --> 01:31.489
and p of that particle at that
time," and that's all you
01:31.494 --> 01:32.814
need to know.
01:32.810 --> 01:34.850
That's all you can ask and
that's all you need to know.
01:34.849 --> 01:38.699
Everything about that one
particle is completely given by
01:38.699 --> 01:39.729
this x and p.
01:39.730 --> 01:43.060
That is called a state.
01:43.060 --> 01:45.790
That's the complete description
of any system is called at
01:45.789 --> 01:47.989
state,
and a state in classical
01:47.992 --> 01:52.432
mechanics for one particle in
one dimension is just a pair of
01:52.427 --> 01:53.237
numbers.
01:53.239 --> 01:57.609
It's a state because given that
I can predict the future for
01:57.614 --> 01:58.064
you.
01:58.060 --> 02:01.600
I can predict the future
because if I knew the initial
02:01.596 --> 02:03.996
momentum, namely initial
velocity.
02:04.000 --> 02:07.290
I know how fast the guy is
moving, and I know where it is
02:07.292 --> 02:08.412
at the initial x.
02:08.408 --> 02:12.368
A tiny amount of time later
it'll be at x vt,
02:12.370 --> 02:14.350
but t is very small.
02:14.348 --> 02:18.448
Then since I know the
acceleration from Newton's Laws,
02:18.449 --> 02:20.439
which is rate of change of
velocity,
02:20.438 --> 02:24.308
if I knew the initial velocity
or initial momentum I can get
02:24.311 --> 02:26.281
the momentum a little later.
02:26.280 --> 02:28.840
Then in that manner I can inch
forward in time.
02:28.840 --> 02:32.900
So you need Newton's Laws,
and Newton's Laws tell you,
02:32.902 --> 02:36.052
if you like,
rate of change of momentum is
02:36.045 --> 02:37.115
the force.
02:37.120 --> 02:43.240
So if you like you can write
this mdx/dt squared is the
02:43.241 --> 02:44.151
force.
02:44.150 --> 02:48.810
So this part is called dynamics
and this part is called
02:48.807 --> 02:49.927
kinematics.
02:49.930 --> 02:52.330
So dynamics is how do things
change with time?
02:52.330 --> 02:56.350
Kinematics is what do you need
to know about the particle?
02:56.348 --> 02:59.728
In the quantum version of this
if you say,
02:59.729 --> 03:03.709
"What constitutes complete
knowledge of the
03:03.712 --> 03:07.362
particle?,"
the answer is this function
03:07.358 --> 03:08.458
ψ(x).
03:08.460 --> 03:11.680
If you knew this function you
know all that there is to know
03:11.675 --> 03:12.815
about the particle.
03:12.819 --> 03:14.479
We're not saying it's real.
03:14.479 --> 03:18.329
It can be complex.
03:18.330 --> 03:21.050
Then you can ask,
"If you knew the ψ
03:21.052 --> 03:24.212
at one time what's it going to
be later on?"
03:24.210 --> 03:26.170
That's the analog of this one.
03:26.169 --> 03:27.989
So I have not come to that yet.
03:27.990 --> 03:30.970
So this is a pretty long story.
03:30.970 --> 03:36.380
It takes so long just to tell
you what to do at one time.
03:36.378 --> 03:40.478
Then I'm going to tell you how
to go from now to later.
03:40.479 --> 03:44.259
In Newtonian mechanics it's
just a one-word answer,
03:44.263 --> 03:46.763
x and p now is the whole story.
03:46.758 --> 03:50.728
So anyway, since we've got four
lectures including today I will,
03:50.729 --> 03:54.569
of course, tell you before it's
all over what is the analog of
03:54.572 --> 03:55.142
this.
03:55.139 --> 03:57.949
Namely I'll tell you a formula
for dψ/dt.
03:57.949 --> 04:00.879
How does ψ change with time?
04:00.879 --> 04:02.659
So let's summarize what we know.
04:02.658 --> 04:06.518
I want to do this every class
so that it gets into your
04:06.516 --> 04:07.156
system.
04:07.158 --> 04:10.598
If it is true that ψ(x)
tells me everything I need to
04:10.599 --> 04:13.119
know let's ask questions of this
ψ.
04:13.120 --> 04:15.940
We can say, "Where is the
particle?"
04:15.939 --> 04:18.109
So there is some ψ.
04:18.110 --> 04:20.670
You're told then the
probability that you will find
04:20.666 --> 04:22.556
it somewhere is the absolute
value.
04:22.560 --> 04:28.290
Probability density is
proportional to ψ^(2).
04:28.290 --> 04:29.810
So it doesn't tell you where
the particle is.
04:29.810 --> 04:33.390
It gives you the odds,
and the odds are this.
04:33.389 --> 04:35.519
Then you can say,
"Okay, x is just one of
04:35.519 --> 04:37.129
the variables of interest to me.
04:37.129 --> 04:38.819
How about momentum?
04:38.819 --> 04:41.739
If I measure momentum what
answers will I get,
04:41.738 --> 04:45.368
and what are the odds for the
different answers?"
04:45.370 --> 04:48.870
That question has a longer
answer and it goes as follows.
04:48.870 --> 04:53.670
Take the ψ
that's given to you,
04:53.670 --> 04:58.420
and write is as a sum with some
coefficients of functions that I
04:58.415 --> 05:02.915
call ψ_p(x),
but I will tell you what they
05:02.920 --> 05:03.320
are.
05:03.319 --> 05:11.499
They are e^(ipx/ℏ) by square
root of L.
05:11.500 --> 05:13.540
Then I said,
"If you manage to write
05:13.536 --> 05:15.876
the ψ that's given to you in
this form,
05:15.879 --> 05:18.699
summing over all the allowed
values of p,
05:18.699 --> 05:24.469
then the probability that you
will get one of those allowed
05:24.466 --> 05:28.936
values is equal to the absolute
square of that
05:28.942 --> 05:31.132
coefficient."
05:31.129 --> 05:35.039
And then final question can be,
"How do I know what
05:35.038 --> 05:38.378
A(p) is given some
function ψ?,"
05:38.377 --> 05:42.567
and the answer is A(p)
is the integral of the complex
05:42.571 --> 05:45.131
conjugate of this ψ(x) dx.
05:45.129 --> 05:48.049
In our case it was a ring of
length L.
05:48.050 --> 05:50.740
Then it's 0 to L,
otherwise it's whatever your
05:50.735 --> 05:51.765
allowed region is.
05:51.769 --> 05:52.599
Yep?
05:52.600 --> 05:54.960
Student: Can you
explain ________ one more time
05:54.964 --> 05:55.364
what...
05:55.360 --> 05:56.810
Prof: What this guy
means?
05:56.810 --> 05:58.440
Student: Yeah.
05:58.440 --> 06:00.940
Prof: So let me give an
example.
06:00.939 --> 06:07.179
Suppose ψ
of p was cosine
06:07.182 --> 06:11.402
4Πx over L?
06:11.399 --> 06:20.539
Then that looks like
e^(4Πix/L) e^(-4Πix/L)
06:20.535 --> 06:23.385
divided by 2.
06:23.389 --> 06:30.879
Let me put the 2 here,
put the 2 here.
06:30.879 --> 06:33.489
Then my ψ
is not quite in the standard
06:33.488 --> 06:36.668
form because I don't have the
square root of Ls.
06:36.670 --> 06:42.640
So let me put a square root of
L here, a square root of L here
06:42.637 --> 06:45.277
and also put one outside.
06:45.279 --> 06:53.849
So my ψ looks like a square
root of L.
06:53.850 --> 06:55.070
Let me see.
06:55.069 --> 06:57.009
Oh, I know what the problem is.
06:57.009 --> 07:02.849
This ψ is not normalized
because the rule is if you
07:02.846 --> 07:04.856
don't--
first let's say it's not
07:04.857 --> 07:07.027
normalized,
so don't worry about any of
07:07.026 --> 07:07.916
these numbers.
07:07.920 --> 07:13.840
Can you see that it's made up
of two such functions with equal
07:13.836 --> 07:14.706
weight?
07:14.709 --> 07:17.559
That means if you measure the
momentum you'll only get the
07:17.560 --> 07:20.260
answer corresponding to what's
here and what's here.
07:20.259 --> 07:26.569
If you compare it to
e^(ipx/ℏ) you can see the p
07:26.567 --> 07:33.387
that goes here is really
4Πâ/L or -4Πâ/L.
07:33.389 --> 07:41.889
So a particle in this state can
have only two values of momentum
07:41.887 --> 07:46.637
if you measure it,
and the probability for the two
07:46.639 --> 07:50.539
you can see by symmetry must be
50/50 for each because they come
07:50.538 --> 07:51.838
with equal weight.
07:51.839 --> 07:54.029
But if you don't want to do
that you can,
07:54.029 --> 07:56.989
if you like, like this ψ,
normalize it,
07:56.985 --> 07:59.575
and then we square it,
integrate it and put a number
07:59.583 --> 08:02.023
in front so it comes out to have
length 1.
08:02.019 --> 08:05.209
Then the coefficients here
directly when squared will give
08:05.209 --> 08:06.329
you probabilities.
08:06.329 --> 08:06.979
Yes?
08:06.980 --> 08:07.620
Oh, a lot of questions.
08:07.620 --> 08:08.830
Yeah?
08:08.829 --> 08:13.409
Student: So
A_p has to be a real
08:13.413 --> 08:14.913
number, right?
08:14.910 --> 08:15.440
Prof: No.
08:15.439 --> 08:18.949
A_p does not have to
be a real number just like
08:18.949 --> 08:22.709
ψ(x) does not have to be a
real number A_p--
08:22.709 --> 08:26.259
because I'm taking the absolute
value of A^(2).
08:26.259 --> 08:29.589
So even if it's a complex
number the absolute value of A
08:29.586 --> 08:32.426
will always come out to be real
and positive.
08:32.429 --> 08:37.579
Is that your question?
08:37.580 --> 08:40.120
See A_p is not a
probability.
08:40.120 --> 08:43.380
The absolute value squared of
A_p is a probability
08:43.383 --> 08:45.433
which, of course,
must be positive.
08:45.428 --> 08:49.038
But A does not have to be
positive, does not have to be
08:49.041 --> 08:49.511
real.
08:49.509 --> 08:50.799
Yeah?
08:50.798 --> 08:59.508
Student: You said
A_p like the integral
08:59.508 --> 09:08.218
if you're multiplying ψ*
times ψ would that be in
09:08.216 --> 09:10.856
real numbers?
09:10.860 --> 09:12.350
Prof: You should be very
careful.
09:12.350 --> 09:12.930
Student: Really?
09:12.928 --> 09:17.178
Prof: A particular ψ
times that same ψ* is a
09:17.182 --> 09:18.872
real positive number.
09:18.870 --> 09:20.530
This is not the same function.
09:20.528 --> 09:23.418
I think I remember now from
previous years that students
09:23.424 --> 09:24.904
will get confused on this.
09:24.899 --> 09:29.849
This ψ(x) is the function
that was given to me of which it
09:29.845 --> 09:32.095
was asked,
"If I measure momentum
09:32.102 --> 09:33.442
what answer will I get?"
09:33.440 --> 09:36.500
This ψ_p is not
the function that was given to
09:36.500 --> 09:36.760
me.
09:36.759 --> 09:39.979
This is another function that
describes a particle whose
09:39.981 --> 09:40.921
momentum is p.
09:40.918 --> 09:44.778
So you've got to multiply that
function with the given function
09:44.777 --> 09:46.517
to get these coefficients.
09:46.519 --> 09:48.519
So you can change your mind and
say, "Oh,
09:48.515 --> 09:50.065
I'm not interested in that
ψ.
09:50.070 --> 09:52.110
I want to know the answer to
that ψ."
09:52.110 --> 09:53.850
Well, you take that ψ
and put it here.
09:53.850 --> 09:56.290
These guys will not change.
09:56.289 --> 09:57.259
Yeah?
09:57.259 --> 09:58.649
Student: I'm sort of
confused.
09:58.649 --> 10:01.849
Right next to what you were
explaining you have written
10:01.854 --> 10:03.164
ψ of p.
10:03.159 --> 10:04.039
Prof: Oh, I'm sorry.
10:04.039 --> 10:05.049
No, no, no.
10:05.048 --> 10:06.528
You have every right to be
confused.
10:06.529 --> 10:07.459
I did not mean that.
10:07.460 --> 10:09.830
This is ψ of x.
10:09.830 --> 10:12.420
Student: And is that
just ψ of x or that
10:12.423 --> 10:13.593
a ψ_p also?
10:13.590 --> 10:14.710
Prof: Here?
10:14.710 --> 10:16.380
Student: That's just
the ψ of x...
10:16.379 --> 10:17.289
Prof: Yes,
this is ψ
10:17.293 --> 10:17.783
of x.
10:17.779 --> 10:18.739
Student: Okay.
10:18.740 --> 10:20.660
Prof: This
ψ(x)--well,
10:20.658 --> 10:23.598
if I'm more disciplined,
I can do that for you.
10:23.600 --> 10:24.860
Maybe I should do it one more
time.
10:24.860 --> 10:26.350
It doesn't hurt.
10:26.350 --> 10:34.130
So let's so let's say ψ(x)
is cosine 4Πx/L.
10:34.129 --> 10:38.299
Now first job is to rescale the
ψ so that if you square it
10:38.304 --> 10:40.294
and integrate it you get 1.
10:40.288 --> 10:45.128
I know the answer to that is
square root of 2/L.
10:45.129 --> 10:45.799
You can check that.
10:45.798 --> 10:49.138
If you take this guy and you
square it and integrate it it'll
10:49.135 --> 10:51.745
work because the square of this
will be 2/L,
10:51.750 --> 10:54.330
and the average value of cosign
squared is half.
10:54.330 --> 10:56.510
And we integrate it from 0 to L
it will cancel.
10:56.509 --> 10:57.799
You'll get 1.
10:57.798 --> 11:04.248
If you do this particular
function then all you have to do
11:04.248 --> 11:10.308
is write this in terms of--
first write this one,
11:10.307 --> 11:16.787
4Πix/L divided by 2
e^(-4Πix/L) divided by 2
11:16.792 --> 11:23.412
because exponential is cosine
θ is e iθ
11:23.408 --> 11:27.508
e to the -iθ
over 2.
11:27.509 --> 11:35.989
Now I'm going to write this as
1/√2 times e^(4Πix/L)
11:35.990 --> 11:42.350
divided by square root of L
e^(-4ΠIx/L).
11:42.350 --> 11:45.880
Divided by square root of L.
11:45.879 --> 11:49.159
I'm just manipulating the
function that was given to me.
11:49.159 --> 11:51.019
Why do I like this form?
11:51.019 --> 11:56.229
Because this is of the form I
wanted, A of p times ψ(p)
11:56.229 --> 12:00.629
where ψ(p) I this and that's
also ψ(p).
12:00.629 --> 12:05.119
It's a normalized function
associated with momentum p.
12:05.120 --> 12:11.490
By comparing this you can see
that A is equal to 1 over square
12:11.488 --> 12:17.348
root of 2 for p = 4Πâ/L,
and A's also equal to
12:17.346 --> 12:21.176
1/√2 for p = -4Πâ/L.
12:21.179 --> 12:23.329
How do I know what p is?
12:23.330 --> 12:27.740
States of definite momentum
look like e^(ipx/ℏ).
12:27.740 --> 12:34.570
So whatever multiplies ix/ℏ
is the momentum that you can see
12:34.567 --> 12:36.357
it's 4Πâ.
12:36.360 --> 12:37.030
You see what I'm saying?
12:37.029 --> 12:41.199
If you want to put an ℏ down
here and put it on the top and
12:41.200 --> 12:42.730
make the comparison.
12:42.730 --> 12:46.840
So these are the only two
values for momentum.
12:46.840 --> 12:51.910
So this is a state which is
simultaneously in the state of
12:51.909 --> 12:55.379
momentum 4Πâ/L,
and -4Πâ/L.
12:55.379 --> 12:57.189
It has no definite momentum yet.
12:57.190 --> 13:00.950
If you measure it then you can
get this answer with probability
13:00.951 --> 13:03.801
1 over root 2 squared,
which is 1 half,
13:03.798 --> 13:08.618
or you can get this answer 1
over root 2 squared which is 1
13:08.620 --> 13:12.360
half and no probability for
anything else.
13:12.360 --> 13:14.220
You cannot get any other value.
13:14.220 --> 13:16.280
See, normally when you take a
ψ(x),
13:16.279 --> 13:21.249
a generic ψ(x) that I draw,
it typically has some non-zero
13:21.248 --> 13:24.068
value in all of space,
so you can find it anywhere.
13:24.070 --> 13:27.440
But this guy has only two
values of p in the sum and only
13:27.442 --> 13:28.952
those will be possible.
13:28.950 --> 13:30.240
Yeah?
13:30.240 --> 13:33.130
Student: ________
you're drawing ℏ over there
13:33.130 --> 13:33.720
________.
13:33.720 --> 13:34.670
Prof: This guy here?
13:34.669 --> 13:35.269
Student: Yeah.
13:35.269 --> 13:37.129
Prof: Yes,
I borrowed an ℏ,
13:37.129 --> 13:40.369
put it on top and bottom so you
can compare the expression.
13:40.370 --> 13:44.330
I'm just saying compare,
if you want without these ℏs,
13:44.330 --> 13:48.650
compare this expression to that
and you can see p is equal to
13:48.650 --> 13:49.300
this.
13:49.298 --> 13:52.698
Now this problem was simple in
the sense that you didn't have
13:52.703 --> 13:55.763
to do that integral,
but last time before end of
13:55.761 --> 13:58.231
class I took the following
function,
13:58.230 --> 14:05.540
ψ(x) =
e^(−α|x|).
14:05.538 --> 14:11.368
On a ring the function looked
like this, falling very rapidly
14:11.369 --> 14:16.809
and dying within a width roughly
Δx = 1/α.
14:16.809 --> 14:21.189
Beyond that it is gone.
14:21.190 --> 14:25.440
Then if you square that and
normalize it in the circle you
14:25.441 --> 14:28.501
find this is the correct
normalization.
14:28.500 --> 14:30.830
In other words,
if you took this ψ
14:30.833 --> 14:33.803
and you square and integrate it
you'll get 1.
14:33.798 --> 14:37.558
If you're very,
very careful it is not strictly
14:37.563 --> 14:42.153
1 because this function the x
goes from −L/2 to L/2
14:42.147 --> 14:46.807
whereas in the integral I took
it to go from minus to plus
14:46.812 --> 14:48.042
infinity.
14:48.038 --> 14:50.628
That's because this function is
falling so rapidly.
14:50.629 --> 14:53.649
The function is falling so
rapidly you don't care if you
14:53.649 --> 14:55.789
cut if off at L/2 or go to
infinity,
14:55.788 --> 14:57.458
so I did that to simplify the
math,
14:57.460 --> 14:59.650
but the idea is the same.
14:59.649 --> 15:05.649
Then if you want to know what's
the probability to get some
15:05.645 --> 15:11.635
number p you've got to take A of
p a square root of α
15:11.642 --> 15:17.952
times e^(ipx/â) square root
of L times e^(−α|x|)
15:17.948 --> 15:18.878
dx.
15:18.879 --> 15:20.299
You understand that?
15:20.298 --> 15:24.678
This is the ψ
with the root α
15:24.677 --> 15:25.597
in it.
15:25.600 --> 15:27.170
That's my ψ.
15:27.169 --> 15:35.999
That's my ψ_p*(x).
15:36.000 --> 15:38.740
If you come to me with a new
function tomorrow,
15:38.743 --> 15:41.133
I don't know,
maybe a function that looks
15:41.129 --> 15:44.349
like this, a constant in some
region and 0 beyond.
15:44.350 --> 15:47.050
We'll put that function here
and do these integrals.
15:47.048 --> 15:49.568
You'll get a different set of A
of p's.
15:49.570 --> 15:53.790
So for every function ψ(x)
that's a set of A_p's
15:53.789 --> 15:56.419
that you find by doing the
integral.
15:56.419 --> 15:57.639
Yeah?
15:57.639 --> 16:03.419
Student: So if you
measure the momentum and you get
16:03.418 --> 16:08.298
one of those answers so it
becomes that answer?
16:08.298 --> 16:08.938
Prof: Right.
16:08.937 --> 16:09.907
So I'm coming to that point.
16:09.908 --> 16:13.178
So the second thing is another
postulate.
16:13.178 --> 16:15.678
In the end I'll give you the
list of postulates.
16:15.678 --> 16:20.198
And the postulate says that if
you measure the momentum you'll
16:20.198 --> 16:24.788
get only those values for which
the corresponding A_p
16:24.792 --> 16:27.632
is not 0,
and the ones for which it is
16:27.625 --> 16:30.825
not 0 the probability for
getting the number is the
16:30.831 --> 16:33.911
absolute value squared of that
A_p.
16:33.908 --> 16:38.868
And right after the measurement
the wave function will change
16:38.865 --> 16:43.325
from this sum over all kinds of
momenta to the one term
16:43.325 --> 16:46.955
corresponding to the one answer
you got.
16:46.960 --> 16:49.450
You understand that?
16:49.450 --> 16:52.470
So in this simple example there
are only two values for
16:52.471 --> 16:53.871
p you can get.
16:53.870 --> 16:55.840
No other values are possible.
16:55.840 --> 16:57.110
They're all theoretically
allowed.
16:57.110 --> 16:59.380
For example,
6Πâ/L is a perfectly
16:59.380 --> 17:02.820
allowed momentum in that ring
because it corresponds to a
17:02.817 --> 17:05.717
periodic function,
but that's not contained in
17:05.723 --> 17:07.443
this particular wave function.
17:07.440 --> 17:12.140
This wave function is built out
of only two of them,
17:12.141 --> 17:15.461
namely 4Πâ/L and
-4Πâ/L.
17:15.460 --> 17:20.370
Once you measure it suppose you
got -4ℏ/L.
17:20.368 --> 17:23.508
The whole wave function,
which I had here in the simple
17:23.510 --> 17:25.490
example, has got only two terms.
17:25.490 --> 17:29.940
This part will simply
disappear, and the function
17:29.944 --> 17:33.384
after the measurement will be
this.
17:33.380 --> 17:36.070
The rough logic is that if I
measure momentum,
17:36.068 --> 17:38.978
and I got an answer,
that's got to mean something in
17:38.978 --> 17:42.398
any real sense that that is the
momentum of the particle,
17:42.400 --> 17:46.110
therefore it has to be true if
I immediately remeasure it.
17:46.108 --> 17:49.328
Immediately remeasure momentum,
and I want to get exactly the
17:49.326 --> 17:51.726
same answer,
that means the function after
17:51.731 --> 17:55.051
the first measurement should
contain only one momentum in its
17:55.046 --> 17:59.656
expansion,
so it'll reduce to that one
17:59.656 --> 18:00.506
term.
18:00.509 --> 18:01.719
Yep?
18:01.720 --> 18:04.010
Student: Is there a
difference between ψ(x) and
18:04.006 --> 18:06.446
negative ψ(x) because it
seems like everything's just...
18:06.450 --> 18:07.570
Prof: That is correct.
18:07.568 --> 18:13.488
I think I also explained the
other day that in quantum theory
18:13.490 --> 18:19.210
ψ(x) and say 92i
times ψ(x) are treated as
18:19.213 --> 18:25.043
physically equivalent because
they contain the same relative
18:25.036 --> 18:27.006
probabilities.
18:27.009 --> 18:30.659
Of course if ψ(x) had been
properly normalized to 1 this
18:30.655 --> 18:32.815
guy would not be normalized to
1.
18:32.818 --> 18:35.608
It's as if you live in a world
where all you care about is the
18:35.606 --> 18:37.846
direction of the vector,
but not the length of the
18:37.846 --> 18:38.346
vector.
18:38.349 --> 18:39.929
We don't live in such a world.
18:39.930 --> 18:41.470
If you say, "Where is Stop
& Shop,"
18:41.474 --> 18:42.864
you say,
"Well, go in that
18:42.863 --> 18:45.073
direction,"
and you're not told how far,
18:45.068 --> 18:46.978
well, you're missing some
information.
18:46.980 --> 18:51.320
Imagine a world where all you
need to know is which way to go.
18:51.318 --> 18:53.298
In fact, maybe if you go to
some town and ask people where
18:53.304 --> 18:55.144
is something they'll say,
"Go in that
18:55.141 --> 18:57.511
direction,"
that's useful information.
18:57.509 --> 18:59.979
It's as if the direction is the
only thing that matters,
18:59.983 --> 19:01.383
not the length of the vector.
19:01.380 --> 19:05.080
The analogous thing here is if
you multiply the function by any
19:05.076 --> 19:07.576
number you don't change the
information.
19:07.578 --> 19:10.438
They're all considered as the
same state.
19:10.440 --> 19:13.290
And I told you from this huge
family of functions all
19:13.291 --> 19:16.311
describing the same state I will
pick that function,
19:16.308 --> 19:23.638
that member whose square
integral happens to be 1.
19:23.640 --> 19:24.660
Yep?
19:24.660 --> 19:27.050
Student: So after the
wave function breaks down how
19:27.050 --> 19:29.650
long do you have to wait for the
wave function to rematerialize,
19:29.648 --> 19:30.348
or does it...
19:30.348 --> 19:31.298
Prof: Oh,
it's not break down.
19:31.298 --> 19:42.448
It's another wave function
because e^(ipx/ℏ),
19:42.450 --> 19:46.040
which is ψ(p),
this guy is as much a wave
19:46.040 --> 19:49.960
function as any other ψ(x)
you write down.
19:49.960 --> 19:51.920
So these are not different
creatures.
19:51.920 --> 19:53.250
It's like saying the following.
19:53.250 --> 19:54.980
I think the analogy may be
helpful to you.
19:54.980 --> 19:56.320
It may not be.
19:56.318 --> 19:59.338
ψ is like some vector in
three dimensions,
19:59.337 --> 20:02.757
and you know there are these
three unit vectors,
20:02.759 --> 20:06.799
i, j and
k, and you can write the
20:06.795 --> 20:09.835
vector as some
V_x times
20:09.839 --> 20:13.109
i V_y
times j
20:13.114 --> 20:16.164
V_z times
k where
20:16.160 --> 20:20.040
V_y =
V⋅j
20:20.042 --> 20:21.492
etcetera.
20:21.490 --> 20:24.890
Think of ψ
as a vector and each of these
20:24.894 --> 20:29.174
directions is corresponding to a
possible momentum,
20:29.170 --> 20:31.080
and you want to expand that
vector in terms of these
20:31.076 --> 20:31.446
vectors.
20:31.450 --> 20:34.650
This is ψ_p for
p = p_1.
20:34.650 --> 20:38.300
This is ψ_p for
p = p_2 and so
20:38.296 --> 20:38.596
on.
20:38.598 --> 20:41.258
So it's how much of the vector
is in each direction that
20:41.255 --> 20:43.565
determines the likelihood you'll
get that answer,
20:43.573 --> 20:45.073
that answer or that answer.
20:45.068 --> 20:47.878
But right after the
measurement, if you've caught it
20:47.882 --> 20:50.152
in this direction,
the entire vector,
20:50.148 --> 20:53.478
only the component of the
vector in the direction where
20:53.478 --> 20:55.388
you got your answer remains.
20:55.390 --> 20:57.260
The rest of it gets chopped out.
20:57.259 --> 20:59.989
It's really like Polaroid
glasses.
20:59.990 --> 21:03.440
If you've got Polaroid glasses
light can come in polarized this
21:03.443 --> 21:06.683
way or polarized that way,
but once it goes through the
21:06.681 --> 21:10.271
glass it's polarized only in the
one direction corresponding to
21:10.271 --> 21:12.011
the way the polaroid works.
21:12.009 --> 21:15.259
If it filters light with the e
field going this way the light
21:15.255 --> 21:17.685
on the other side will have only
up and up.
21:17.690 --> 21:20.620
So measurement is like a
filtering process.
21:20.618 --> 21:26.068
It filters out of the sum over
many terms the one term which
21:26.067 --> 21:29.757
corresponds to the one answer
you got.
21:29.759 --> 21:33.379
So I don't mind telling this to
you any number of times,
21:33.381 --> 21:36.741
but that is the way that
quantum mechanics works.
21:36.740 --> 21:41.170
Similarly, if you're chosen to
measure position starting with
21:41.166 --> 21:44.196
this ψ,
and you found the guy here,
21:44.199 --> 21:46.819
once you found it--
see, before you found it the
21:46.820 --> 21:47.840
odds may vary like this.
21:47.838 --> 21:50.618
Once you found it it's there,
and if you remeasure it
21:50.622 --> 21:53.032
infinitesimally later it'd
better be there.
21:53.029 --> 21:57.199
So the only function which has
the property is the big spike at
21:57.203 --> 21:58.823
wherever you found it.
21:58.818 --> 22:02.438
That's called a collapsible of
the wave function.
22:02.440 --> 22:05.580
The real difference is the
following.
22:05.578 --> 22:08.078
I gave you an example of
somebody tracking me to see
22:08.082 --> 22:08.722
where I am.
22:08.720 --> 22:13.480
I drew a probability graph,
and the probability graph may
22:13.476 --> 22:14.916
look like this.
22:14.920 --> 22:17.030
And if you caught me here,
if someone says,
22:17.025 --> 22:19.075
"Where will I find him
next?"
22:19.078 --> 22:22.948
The answer is right there
because the probability has
22:22.948 --> 22:27.188
collapsed from being anywhere to
where we just saw him.
22:27.190 --> 22:31.920
The only difference between me
and the quantum particle is if
22:31.921 --> 22:34.761
you got me here I really was
here.
22:34.759 --> 22:37.769
You couldn't have gotten me
anywhere else at that right now.
22:37.769 --> 22:40.889
Can anybody find me anywhere
else except right now here?
22:40.890 --> 22:42.340
You cannot.
22:42.338 --> 22:44.968
That's because my location's
being constantly measured.
22:44.970 --> 22:48.240
Photons are bouncing off to see
where I am, so my measurement is
22:48.238 --> 22:49.378
constantly measured.
22:49.380 --> 22:52.600
A quantum particle with a
function like this,
22:52.603 --> 22:56.123
which is then found here,
was not really here.
22:56.118 --> 22:58.778
It was in this state of limbo
which has no analog in which it
22:58.775 --> 22:59.965
could have been anywhere.
22:59.970 --> 23:02.530
It's the act of measuring it
that nailed it to there,
23:02.528 --> 23:05.378
but then right after that if
you measure it we expect it to
23:05.384 --> 23:06.324
be still there.
23:06.318 --> 23:07.948
Now the time over which you can
do that is
23:07.946 --> 23:08.936
infinitesimal.
23:08.940 --> 23:11.970
If you wait long enough the
wave function will change;
23:11.970 --> 23:14.700
I will tell you the laws for
the dynamics that will tell you
23:14.701 --> 23:17.611
how it will change,
but we all believe that if a
23:17.605 --> 23:21.445
measurement is immediately
repeated for that same variable
23:21.445 --> 23:23.665
we should get the same answer.
23:23.670 --> 23:27.700
Need not be,
it could have been even worse,
23:27.695 --> 23:30.855
but at least that much is true.
23:30.858 --> 23:34.228
So it's not going to get any
more familiar.
23:34.230 --> 23:37.680
It's a strange thing,
but hopefully you will know
23:37.680 --> 23:39.190
what the rules are.
23:39.190 --> 23:42.610
When I said no one understands
quantum mechanics what I meant
23:42.612 --> 23:45.182
was, of course,
by now you know the recipe.
23:45.180 --> 23:47.520
It doesn't mean you like it,
or it doesn't mean it looks
23:47.515 --> 23:48.785
like anything in daily life.
23:48.788 --> 23:51.698
Things in daily life they have
a location before you measure
23:51.703 --> 23:54.573
it, while you measure it and
right after you measure it.
23:54.568 --> 23:57.388
They're always in one place
doing one thing.
23:57.390 --> 24:01.720
Only in the quantum world they
can be in many places at the
24:01.722 --> 24:02.622
same time.
24:02.618 --> 24:05.068
But you should be very careful
if this is a wave function for
24:05.067 --> 24:05.637
an electron.
24:05.640 --> 24:07.530
The charge of the electron is
not spread out.
24:07.529 --> 24:09.019
I told you that.
24:09.019 --> 24:12.649
An electron is one little guy
you'll only catch in one place.
24:12.650 --> 24:15.730
It's the wave function that's
spread out.
24:15.730 --> 24:19.040
Anyway, I think what'll happen
is you will have to do a lot of
24:19.038 --> 24:20.848
problems,
and you will have to talk to a
24:20.849 --> 24:23.189
lot of people,
and you have to read a lot of
24:23.192 --> 24:23.612
stuff.
24:23.608 --> 24:26.458
You can teach quantum mechanics
for a whole semester,
24:26.455 --> 24:28.695
sometimes I taught it for a
whole year.
24:28.700 --> 24:30.650
There's more and more stuff.
24:30.650 --> 24:34.230
I want you at least to know how
it works.
24:34.230 --> 24:37.340
In real life when you go
forward I don't think you need
24:37.335 --> 24:38.135
all of this.
24:38.140 --> 24:39.960
If you go into physics,
or something,
24:39.958 --> 24:41.878
or chemistry,
they'll teach you quantum
24:41.877 --> 24:43.947
mechanics again,
but I wanted people doing
24:43.949 --> 24:46.629
something else;
they are the people I want to
24:46.634 --> 24:47.704
send a message to.
24:47.700 --> 24:49.220
Is here's a part of the world.
24:49.220 --> 24:51.960
If you ever hear the word
quantum, where does the word
24:51.964 --> 24:53.264
quantization come from?
24:53.259 --> 24:55.259
What's the funny business in
the microscopic world?
24:55.259 --> 24:58.009
I want you to have a felling
for how that works.
24:58.009 --> 25:00.219
And I claim that the
mathematics you need is not very
25:00.221 --> 25:00.521
much.
25:00.519 --> 25:05.289
You should know how to do
integrals, and you should know
25:05.288 --> 25:07.888
what e^(iθ) is.
25:07.890 --> 25:11.810
So the postulates right now are
the state has given me a
25:11.805 --> 25:13.865
function ψ
of x.
25:13.868 --> 25:17.458
State of momentum,
definite momentum looks like
25:17.455 --> 25:20.565
this,
and if you want the odds for
25:20.565 --> 25:25.975
any particular momentum expand
it in this fashion which is the
25:25.978 --> 25:31.568
same as A of p
ψ_p of x,
25:31.568 --> 25:33.508
but A of p,
again, is that integral.
25:33.509 --> 25:35.669
I don't want to write it over.
25:35.670 --> 25:37.980
Then if you measure p
you'll get one of the answers
25:37.980 --> 25:40.410
with the probability given by
the square of the corresponding
25:40.413 --> 25:43.253
number,
and right after the measurement
25:43.250 --> 25:45.070
the function collapses.
25:45.068 --> 25:48.178
So it always collapses to
whatever variable you measure.
25:48.180 --> 25:49.960
Yes, another question?
25:49.960 --> 25:51.960
Student: It looks like
the opposite.
25:51.960 --> 25:55.650
If you collapse it and you
measure x then what
25:55.648 --> 25:57.208
happens to p?
25:57.210 --> 25:57.730
Prof: Very good.
25:57.730 --> 26:00.590
His question was,
"If you collapse it to
26:00.588 --> 26:03.318
x what happens to
p?"
26:03.319 --> 26:03.939
I'm glad you asked.
26:03.940 --> 26:06.040
I mean, that is the real
problem.
26:06.039 --> 26:07.109
You had the same question?
26:07.108 --> 26:07.598
Student: No.
26:07.596 --> 26:09.076
I was going to say well don't
you not know about p?
26:09.079 --> 26:09.749
Prof: That is correct.
26:09.750 --> 26:13.430
So I will repeat what she said,
and I'll repeat what you said.
26:13.430 --> 26:16.760
I want to have this discussion
with you guys because it's very
26:16.758 --> 26:19.758
important because everyone's
thinking the same thing.
26:19.759 --> 26:23.019
If I first take a generic
ψ, so I want everyone to
26:23.021 --> 26:26.901
know what the answer's going to
be to any of these questions.
26:26.900 --> 26:29.410
So I take this ψ
somebody prepared for me.
26:29.410 --> 26:32.140
Let's not worry about how that
person knew this is ψ,
26:32.138 --> 26:32.868
you're given.
26:32.868 --> 26:35.888
The electron is in this stage
ψ.
26:35.890 --> 26:39.130
If you measure x,
and you got x equal to
26:39.132 --> 26:42.682
this point the function,
of course, becomes a big spike.
26:42.680 --> 26:45.660
This spike in principle should
be infinitesimally thin,
26:45.661 --> 26:46.711
but I don't care.
26:46.710 --> 26:49.970
Let this be the width of a
proton.
26:49.970 --> 26:52.690
If after that you say,
"What momentum will I
26:52.693 --> 26:55.423
get," well,
you know what you have to do.
26:55.420 --> 26:59.270
You've got to write spike
function equal to sum over these
26:59.267 --> 27:00.547
functions, right?
27:00.548 --> 27:02.928
You've got to take the integral
of the spike with this
27:02.925 --> 27:04.535
exponential and do that
integral,
27:04.538 --> 27:06.788
and you'll get a bunch of
numbers A of p for
27:06.785 --> 27:07.765
all values of p.
27:07.769 --> 27:11.969
And then if you measure
momentum, and you got the one
27:11.965 --> 27:15.915
corresponding to p =
6Πâ/L,
27:15.920 --> 27:19.710
the state which contains many,
many things will reduce to the
27:19.709 --> 27:23.309
one term corresponding to
p = 6Πâ/L.
27:23.308 --> 27:26.818
If you plot that function it
will be some oscillatory
27:26.817 --> 27:27.557
function.
27:27.558 --> 27:31.268
The real part and imaginary
part will both oscillate with
27:31.266 --> 27:33.646
some wavelength given by
p.
27:33.650 --> 27:34.830
Be very careful.
27:34.828 --> 27:36.958
This is not the absolute value
of ψ.
27:36.960 --> 27:38.860
It's the real or imaginary part.
27:38.859 --> 27:40.449
They're both sines and cosines.
27:40.450 --> 27:44.680
Absolute value will be flat.
27:44.680 --> 27:48.060
So it'll go from a particle of
known location to a particle
27:48.058 --> 27:51.088
whose probability's completely
flat on the circle.
27:51.089 --> 27:52.219
You understand?
27:52.220 --> 27:57.210
The wave function can look like
6Πi/L square root of
27:57.209 --> 28:00.399
L right after the
measurement.
28:00.400 --> 28:02.650
Let's call is ψ_6.
28:02.650 --> 28:05.770
The absolute value of ψ
is a constant,
28:05.773 --> 28:08.823
but the real and imaginary
parts of ψ
28:08.821 --> 28:11.491
oscillate with some wavelength.
28:11.490 --> 28:15.080
So right after the measurement
of momentum you don't know where
28:15.080 --> 28:17.740
the guy is, and you say,
"Let me find this
28:17.744 --> 28:18.734
fellow."
28:18.730 --> 28:21.870
You catch it somewhere then
it's a spike at that point,
28:21.865 --> 28:24.705
but then you have no guarantee
on the momentum.
28:24.710 --> 28:27.750
So you can never produce for me
something of perfectly well
28:27.748 --> 28:30.838
defined position and momentum
because once you squeeze it in
28:30.838 --> 28:33.038
x it gets broad it in
p.
28:33.038 --> 28:36.028
Once you squeeze it in p
it gets broad in x.
28:36.029 --> 28:39.179
This is really mathematical
property of Fourier analysis,
28:39.180 --> 28:41.590
that functions which are very
narrow in x when you do
28:41.589 --> 28:46.719
the Fourier expansion have many,
many wavelengths in them.
28:46.720 --> 28:50.490
And likewise,
a function with a very well
28:50.493 --> 28:54.363
defined wavelength,
because it's a complex
28:54.364 --> 28:58.804
exponential, has a magnitude
which is flat.
28:58.798 --> 29:03.408
So now I'm going to ask the
following question.
29:03.410 --> 29:07.320
So when I did ψ(x)
the probability for x was
29:07.320 --> 29:09.310
very easy, squared the ψ.
29:09.308 --> 29:11.048
When I said,
"Okay, I want to look at
29:11.048 --> 29:12.658
momentum,"
the answer was long and
29:12.659 --> 29:14.709
complicated,
namely, take these exponential
29:14.708 --> 29:15.958
functions,
write the ψ
29:15.963 --> 29:19.213
in terms of those,
find the coefficient, etcetera.
29:19.210 --> 29:23.360
Now I can say I want some other
variable I'm interested in.
29:23.358 --> 29:26.928
I want to know what happens if
I measure energy.
29:26.930 --> 29:30.220
Energy is a very,
very important variable.
29:30.220 --> 29:33.910
It's very, very important
because it turns out that if a
29:33.906 --> 29:37.456
particle starts out in a state
of definite energy,
29:37.460 --> 29:42.510
I will show that to you later,
it remains in that state.
29:42.509 --> 29:45.409
That's the only state that will
remain the way it is.
29:45.410 --> 29:48.440
If you start in a state of
definite momentum two seconds
29:48.438 --> 29:51.738
later it can have a different
momentum or it can be a mixture
29:51.742 --> 29:53.122
of different momenta.
29:53.118 --> 29:55.758
But if it starts in the state
of definite energy it will
29:55.756 --> 29:56.616
remain that way.
29:56.619 --> 29:57.419
That's not obvious.
29:57.420 --> 29:58.930
I'm going to prove that to you
later.
29:58.930 --> 29:59.990
That's why it's very important.
29:59.990 --> 30:03.470
So most atoms are in a state of
definite energy and they can
30:03.465 --> 30:06.235
stay that way forever,
but once in a while when
30:06.241 --> 30:09.501
they're tickled by something
they will either absorb light or
30:09.497 --> 30:10.797
they will emit light.
30:10.799 --> 30:12.379
So we draw a picture like this.
30:12.380 --> 30:15.540
We will see that the allowed
energies of the systems are some
30:15.535 --> 30:16.425
special values.
30:16.430 --> 30:18.580
Not every value's allowed.
30:18.578 --> 30:22.758
And this can be called
n=1, n=2,
30:22.761 --> 30:24.901
n=3, etcetera.
30:24.900 --> 30:26.890
And an atom,
for example,
30:26.885 --> 30:31.185
can sometimes jump from doing
that to doing that,
30:31.190 --> 30:35.200
and in that process it will
emit an energy which is
30:35.203 --> 30:39.383
E(n=3) −
E(n=1).
30:39.380 --> 30:43.740
That difference of energy will
come in the form of a photon and
30:43.736 --> 30:47.176
the energy of the photon is
âω,
30:47.180 --> 30:50.660
or if you like,
2Πâf where f
30:50.659 --> 30:53.889
is what you and I call
frequency.
30:53.890 --> 30:56.330
And from the frequency you can
find the wavelength.
30:56.328 --> 31:00.628
The wavelength is just the
velocity of light divided by
31:00.625 --> 31:01.575
frequency.
31:01.578 --> 31:04.868
So an atom will have only
certain allowed energies and
31:04.866 --> 31:08.706
when it jumps form one allowed
energy to another allowed energy
31:08.711 --> 31:11.381
it will emit a photon whose
frequency--
31:11.380 --> 31:14.430
in fact, you should probably
call this frequency
31:14.431 --> 31:18.331
f_31 meaning
what I get when I jump from the
31:18.326 --> 31:19.816
level 3 to level 1.
31:19.818 --> 31:23.528
Similarly, if you shine light
on this atom it won't take any
31:23.534 --> 31:24.294
frequency.
31:24.288 --> 31:27.588
It'll only take those
frequencies that connect it from
31:27.590 --> 31:30.520
one allowed energy to another
allowed energy.
31:30.519 --> 31:32.979
That's the fingerprint of the
atom.
31:32.980 --> 31:36.950
Both emission and absorption
betray the atom.
31:36.950 --> 31:39.520
That's how we know what atoms
there are in this star,
31:39.521 --> 31:41.651
or that star,
or what the composition is.
31:41.650 --> 31:45.610
No one's gone to any of these
stars, but we know because of
31:45.614 --> 31:48.764
the light they emit,
and it's all controlled by
31:48.758 --> 31:49.508
energy.
31:49.509 --> 31:53.219
So the question I'm going to
ask is here's the function
31:53.221 --> 31:54.461
ψ(x).
31:54.460 --> 32:00.340
Someone gave it to me in some
context, and I say if I measure
32:00.343 --> 32:05.053
the energy of this particle what
are the answers,
32:05.049 --> 32:07.599
and what are the odds?
32:07.598 --> 32:10.158
So how do you think that will
play out?
32:10.160 --> 32:11.980
You have to make a guess.
32:11.980 --> 32:14.770
Suppose you're inventing
quantum mechanics and someone
32:14.767 --> 32:17.767
says, "What do you think is
going to be the deal with
32:17.765 --> 32:18.655
energy?"
32:18.660 --> 32:23.760
You know what the scenario
might look like?
32:23.759 --> 32:24.719
You can take a guess.
32:24.720 --> 32:27.730
I mean, as I told you many
times I don't expect you to
32:27.732 --> 32:31.202
invent quantum mechanics on the
fly, but you should be able to
32:31.201 --> 32:31.771
guess.
32:31.769 --> 32:36.469
What form do you think the
answer will take?
32:36.470 --> 32:38.000
You want to guess?
32:38.000 --> 32:40.350
Student: ________
they'll be ________?
32:40.349 --> 32:42.919
Prof: Okay,
but if I want to know what
32:42.923 --> 32:47.863
energies I can get and with what
odds in analogy with momentum
32:47.858 --> 32:50.528
what do you think will happen?
32:50.529 --> 32:51.399
Yep?
32:51.400 --> 32:54.450
Student: Would you use
the formula p^(2)/2m?
32:54.450 --> 32:56.650
Prof: He said use
p^(2)/2m.
32:56.650 --> 32:57.820
That's a good answer.
32:57.818 --> 33:03.108
His answer was we know that the
energy is equal to 1 half m v
33:03.108 --> 33:06.988
squared which I can write as
p^(2)/2m,
33:06.988 --> 33:07.868
right?
33:07.868 --> 33:10.978
So you're saying if I measure
the momentum and I got a certain
33:10.984 --> 33:12.524
answer, well,
the energy's that
33:12.518 --> 33:13.538
p^(2)/2m.
33:13.538 --> 33:18.388
That's actually correct except
the energy of a particle is not
33:18.394 --> 33:21.024
always just the kinetic energy.
33:21.019 --> 33:23.429
For a free particle this is the
kinetic energy.
33:23.430 --> 33:28.020
For a particle moving in a
potential you know that you have
33:28.018 --> 33:29.678
to add V(x).
33:29.680 --> 33:34.380
That is the total energy.
33:34.380 --> 33:37.840
Now that's when we have a
problem.
33:37.838 --> 33:40.418
In classical mechanics once I
measure the x and
33:40.424 --> 33:43.554
p of the particle I don't
have to make another measurement
33:43.547 --> 33:44.227
of energy.
33:44.230 --> 33:45.370
Do you understand that?
33:45.368 --> 33:49.208
I just plug the values I got
into this formula.
33:49.210 --> 33:51.330
For example,
particle connected to a spring
33:53.246 --> 33:55.766
k is the force constant
of the spring,
33:55.769 --> 33:58.899
and the kinetic energy's always
p^(2)/2m.
33:58.900 --> 34:00.980
Say if I measured p and
I got some number,
34:00.980 --> 34:03.450
and measured x I got
some number, I can put that in
34:03.451 --> 34:04.971
the formula and get the energy.
34:04.970 --> 34:07.950
You don't have to do another
energy measurement.
34:07.950 --> 34:09.940
And you don't have to do an
angular momentum measurement
34:09.943 --> 34:10.273
either.
34:10.268 --> 34:12.578
In higher dimensions angular
momentum is
34:12.579 --> 34:15.709
r x p,
and if you already measured the
34:15.708 --> 34:19.198
position and you measured the
momentum just take the cross
34:19.204 --> 34:19.944
product.
34:19.940 --> 34:23.230
So in classical mechanics you
only need to measure x
34:23.228 --> 34:24.078
and p.
34:24.079 --> 34:27.349
In quantum mechanics he made a
pretty good guess that if you
34:27.349 --> 34:29.789
measured p,
the p^(2)/2m is the
34:29.789 --> 34:30.399
energy.
34:30.400 --> 34:34.470
That is true if the particle is
not in a potential.
34:34.469 --> 34:38.639
But if the particle is in a
potential can you tell me how to
34:38.641 --> 34:41.401
compute p^(2)/2m
V(x)?
34:41.400 --> 34:44.710
You realize you cannot really
compute it because if you knew
34:44.713 --> 34:47.693
the p exactly you have no
idea where it is,
34:47.690 --> 34:50.770
and if you knew the x exactly
you don't know what the momentum
34:50.766 --> 34:51.016
is.
34:51.018 --> 34:53.888
Maybe you know a little bit of
both, but still,
34:53.887 --> 34:56.317
how are you going to find the
energy?
34:56.320 --> 35:01.670
So the answer is you have to do
a separate energy measurement.
35:01.670 --> 35:03.550
You cannot infer that from x
and p,
35:03.547 --> 35:05.967
because first of all you cannot
even get a pair of x and
35:05.974 --> 35:07.114
p at a given time.
35:07.110 --> 35:08.830
I hope I convinced you.
35:08.829 --> 35:10.749
You measure this guy you screw
up that guy.
35:10.750 --> 35:12.970
Measure that one you mess up
this one.
35:12.969 --> 35:15.369
You can never get a state of
well defined x and
35:15.365 --> 35:16.175
p anyway.
35:16.179 --> 35:20.799
So the way to find energy is to
do a whole other calculation.
35:20.800 --> 35:24.230
So I will tell you what the
answer is, and hopefully you
35:24.226 --> 35:28.206
will realize it's not completely
different from the recipe we had
35:28.213 --> 35:28.903
before.
35:28.900 --> 35:31.910
I'm going to give you a rule
for the functions which
35:31.911 --> 35:35.281
correspond to a state in which
the particle has a definite
35:35.277 --> 35:36.457
energy E.
35:36.460 --> 35:41.520
Let's not worry about how you
get it, some function.
35:41.518 --> 35:45.958
You find all those functions,
or you're given all those
35:45.961 --> 35:50.571
functions, then can you imagine
what will happen next?
35:50.570 --> 35:55.750
If I give you all those
functions what do you think the
35:55.748 --> 35:57.858
rule is going to be?
35:57.860 --> 35:58.660
Yep?
35:58.659 --> 36:01.779
Student: You separate
it into a sum of all the
36:01.775 --> 36:03.475
energies with coefficients.
36:03.480 --> 36:04.370
Prof: Very good.
36:04.369 --> 36:05.459
Let me repeat what she said.
36:05.460 --> 36:08.420
I hope at least some of you
were thinking about the same
36:08.423 --> 36:08.913
answer.
36:08.909 --> 36:12.179
Her answer is take the function
ψ,
36:12.179 --> 36:14.539
write it in terms of these
functions ψ
36:14.538 --> 36:18.098
E of x with some
coefficient A_E
36:18.103 --> 36:21.213
summing this over the allowed
values of E,
36:21.210 --> 36:24.600
whatever they may be.
36:24.599 --> 36:27.639
And now that you said that what
do you think
36:27.644 --> 36:31.474
A_E is going to
be in a given case?
36:31.469 --> 36:32.549
Would you like to continue?
36:32.550 --> 36:37.230
Student: It's going to
be the integral or the wave
36:37.230 --> 36:39.530
function ________________.
36:39.530 --> 36:50.910
Prof: That's right.
36:50.909 --> 36:52.429
That is correct.
36:52.429 --> 36:54.809
The recipe's almost complete
except you don't know what these
36:54.813 --> 36:57.163
functions are,
but if you knew these functions
36:57.164 --> 36:59.274
you have to write the given
function,
36:59.268 --> 37:01.618
given wave function,
as a sum of these functions
37:01.619 --> 37:03.319
with some suitable coefficients.
37:03.320 --> 37:05.760
Coefficients are found by the
same rule,
37:05.760 --> 37:09.710
and then the probability that
you'll find an energy E
37:09.708 --> 37:11.138
is,
again,
37:11.141 --> 37:20.941
A_E^(2). And
everything else will be also
37:20.943 --> 37:22.323
true.
37:22.320 --> 37:25.140
Once you measure energy you've
got energy corresponding to
37:25.143 --> 37:27.473
E_1,
or E_2 or
37:27.471 --> 37:28.711
E_3.
37:28.710 --> 37:30.330
Let's say you've got
E_3,
37:30.331 --> 37:31.401
the third possible value.
37:31.400 --> 37:35.510
The whole wave function will
collapse from being a sum over
37:35.510 --> 37:39.550
many things to just this one
guy, E_3.
37:39.550 --> 37:40.530
The collapse is the same.
37:40.530 --> 37:42.530
The probability rule is the
same.
37:42.530 --> 37:46.170
The only thing you don't know
is who are these functions ψ
37:46.172 --> 37:47.012
of E?
37:47.010 --> 37:48.260
You understand?
37:48.260 --> 37:51.280
So, again, the analogy's the
following.
37:51.280 --> 37:55.470
There is a vector that we call
ψ.
37:55.469 --> 37:57.169
Sometimes you want to write it
in terms of
37:57.170 --> 37:59.810
i, j,
and k.
37:59.809 --> 38:00.799
They are like the A(p).
38:00.800 --> 38:04.140
Sometimes you may pick three
other mutually perpendicular
38:04.137 --> 38:06.877
vectors, i',
j' and k'.
38:06.880 --> 38:09.270
And if you know those
coefficients you'll get the
38:09.268 --> 38:11.158
probability for some other
variable.
38:11.159 --> 38:14.679
So you're expanding the same
function over and over in many
38:14.677 --> 38:18.617
possible ways depending on what
variable is of interest to you.
38:18.619 --> 38:21.809
If it's momentum you expand it
in terms of exponential ipx of
38:21.809 --> 38:22.129
ℏ.
38:22.130 --> 38:28.150
If it's energy you expand it in
terms of these functions.
38:28.150 --> 38:31.980
So the question is what is the
recipe going to be for these
38:31.981 --> 38:34.891
functions ψ_E of
x.
38:34.889 --> 38:38.219
By what means do I find them?
38:38.219 --> 38:40.679
Now you're getting more and
more and more recipes every day,
38:40.684 --> 38:42.234
but it's going to stop pretty
soon.
38:42.230 --> 38:44.500
This is about the last of the
recipes.
38:44.500 --> 38:48.800
Even this recipe I'll tell you
how to get from a master recipe,
38:48.795 --> 38:52.815
so it's not that many recipes,
but I have to reveal that to
38:52.815 --> 38:54.335
you one at a time.
38:54.340 --> 38:57.260
Now you can say,
"Okay, what function do
38:57.255 --> 39:01.295
you want me to use for every
energy, what is this function?
39:01.300 --> 39:07.720
After all, when it was momentum
you came right out and gave this
39:07.722 --> 39:08.642
answer.
39:08.639 --> 39:12.469
Why don't you do the same thing
here," and that's a problem
39:12.465 --> 39:12.885
here.
39:12.889 --> 39:15.629
The problem is the energy of a
particle depends on what
39:15.628 --> 39:18.668
potential it is in because it's
got a kinetic and a potential
39:18.672 --> 39:21.022
part,
so I cannot give you a
39:21.016 --> 39:25.186
universal answer for
ψ_E(x).
39:25.190 --> 39:28.900
I will have to first ask you;
"Tell me the potential the
39:28.903 --> 39:30.323
particle is in."
39:30.320 --> 39:33.620
Once I know the potential I
will give you the recipe.
39:33.619 --> 39:37.149
Imagine the potential has been
given to me, for example,
39:40.356 --> 39:41.956
what's called a well.
39:41.960 --> 39:45.260
You make a hole in the ground
or you build a little barrier.
39:45.260 --> 39:48.310
That's a possible potential.
39:48.309 --> 39:50.759
You can have a potential that
looks like harmonic isolator.
39:50.760 --> 39:53.060
That's a possible potential.
39:53.059 --> 39:57.239
You can have an electron and a
hydrogen atom -1/r.
39:57.239 --> 39:58.649
That's a possible potential.
39:58.650 --> 40:01.200
There are many-potentials,
and the answer's going to vary
40:01.195 --> 40:01.965
on the problem.
40:01.969 --> 40:05.949
There's no universal answer for
the energy functions.
40:05.949 --> 40:08.789
You tell me what the electron's
doing, what field it is in,
40:08.791 --> 40:10.311
what field of force it is in.
40:10.309 --> 40:12.779
Then for each field of force,
or for each potential,
40:12.782 --> 40:14.192
there's a different answer.
40:14.190 --> 40:16.860
And here is the master formula.
40:16.860 --> 40:20.120
This is the great
Schr�dinger equation.
40:20.119 --> 40:22.229
So the answer looks like this.
40:22.230 --> 40:26.360
The function ψ_E
obeys the following equation,
40:26.360 --> 40:35.250
-â^(2)/2m times
the second derivative of ψ
40:35.251 --> 40:41.671
V(x) times
ψ(x) is equal to
40:41.666 --> 40:46.766
E times ψ(x).
40:46.769 --> 40:47.849
Do not worry.
40:47.849 --> 40:49.929
I will see you through this
equation.
40:49.929 --> 40:52.169
Everything you need to know I
will tell you,
40:52.166 --> 40:55.126
but you should not be afraid of
what the equation says.
40:55.130 --> 40:58.200
It says those functions that
are allowed corresponding to
40:58.199 --> 41:01.379
definite energy will have the
property that if you took the
41:01.378 --> 41:04.198
second derivative,
multiplied it by this number,
41:04.199 --> 41:07.209
add it to that V(x)
times ψ_E you'll
41:07.210 --> 41:08.340
get some function.
41:08.340 --> 41:14.350
That function should be some
number times the very same
41:14.346 --> 41:15.566
function.
41:15.570 --> 41:18.550
If you can find me those
functions then you will find out
41:18.552 --> 41:21.162
when I will show you
mathematically that there are
41:21.161 --> 41:24.231
many solutions to the equation,
but they don't occur for every
41:24.231 --> 41:24.541
energy.
41:24.539 --> 41:27.759
Only some energies are allowed,
and the energies are usually
41:27.757 --> 41:29.717
labeled by some integer
n,
41:29.719 --> 41:31.619
and for every n,
1,2, 3,4,
41:31.619 --> 41:33.909
you'll get a bunch of energies,
ψ_E1,
41:33.905 --> 41:35.685
ψ _E2,
ψ_E3,
41:35.688 --> 41:37.608
3, and you write those
functions down.
41:37.610 --> 41:39.530
Then you can do everything
thing I said.
41:39.530 --> 41:42.590
But the only thing is here you
have to do some hard work.
41:42.590 --> 41:46.240
Whereas for momentum I gave you
that and for a state of definite
41:46.244 --> 41:49.324
positions x = x_0
I told you spike at
41:49.318 --> 41:50.768
x_0.
41:50.769 --> 41:53.799
You didn't have to do much work.
41:53.800 --> 42:02.690
For this you have to solve an
equation before you an even
42:02.693 --> 42:08.733
start, but we'll see how to do
that.
42:08.730 --> 42:15.600
So the first problem I want to
solve is the problem where there
42:15.596 --> 42:17.586
is no potential.
42:17.590 --> 42:21.900
That is called a free particle.
42:21.900 --> 42:24.860
A free particle is one for
which V is 0.
42:24.860 --> 42:31.750
And let me imagine it's living
on this line,
42:31.753 --> 42:39.453
the circle of length
2ΠR = L.
42:39.449 --> 42:41.989
Oh, by the way,
I should mention something
42:41.985 --> 42:42.415
else.
42:42.420 --> 42:46.050
In terms of all the postulates
you notice I never mentioned the
42:46.052 --> 42:49.482
uncertainty principle today,
ΔxΔp should
42:49.480 --> 42:50.930
be bigger than ℏ.
42:50.929 --> 42:54.129
I didn't mention it as a
postulate because once you tell
42:54.130 --> 42:57.210
me ψ is given by wave
function and that states are
42:57.213 --> 43:00.533
definite momentum have definite
wavelength it follows from
43:00.530 --> 43:03.910
mathematics that you cannot have
a function of well defined
43:03.905 --> 43:07.335
periodicity and wavelength also
localized in space.
43:07.340 --> 43:09.240
It's a mathematical consequence.
43:09.239 --> 43:12.979
Similarly, who told me that I
can expand every given ψ
43:12.980 --> 43:15.080
as a sum over these functions?
43:15.079 --> 43:18.499
There's a very general
mathematical theorem that tells
43:18.501 --> 43:22.051
you in what situations you can
actually expand any given
43:22.054 --> 43:24.964
function in terms of a set of
functions,
43:24.960 --> 43:27.300
namely are they like unit
vector i,
43:27.295 --> 43:28.715
j and k.?
43:28.719 --> 43:31.229
Suppose I had only i and
j and I don't have
43:31.226 --> 43:31.716
k.
43:31.719 --> 43:35.009
I cannot expand ever vector in
3D using i and j.
43:35.010 --> 43:37.720
So you've got to make sure
you've got enough basis
43:37.724 --> 43:40.944
functions and the theory tells
you that if you find all the
43:40.936 --> 43:43.926
solutions to that equation
together they can expand any
43:43.929 --> 43:44.759
function.
43:44.760 --> 43:47.810
Similarly the rule for
expansion is also not arbitrary.
43:47.809 --> 43:48.679
It all comes from that.
43:48.679 --> 43:50.899
It's a very,
very beautiful theory of the
43:50.896 --> 43:53.056
mathematics behind quantum
mechanics.
43:53.059 --> 43:56.529
If you learn linear algebra one
day, or if you've already
43:56.527 --> 43:58.877
learned it, it's all linear
algebra.
43:58.880 --> 44:03.010
So every great discovery in
physics is accompanied by some
44:03.007 --> 44:05.177
mathematical stuff you need.
44:05.179 --> 44:07.729
Like all of Newtonian mechanics
requires calculus.
44:07.730 --> 44:10.390
Without calculus you cannot do
Newtonian mechanics.
44:10.389 --> 44:14.759
Maxwell's theory for
electromagnetism requires vector
44:14.764 --> 44:15.694
calculus.
44:15.690 --> 44:18.710
Einstein's non-relativistic
theory doesn't require anything;
44:18.710 --> 44:21.470
it's algebra,
but the general theory requires
44:21.467 --> 44:25.227
what's called tensor calculus,
and quantum mechanics requires
44:25.228 --> 44:26.418
linear algebra.
44:26.420 --> 44:28.440
And string theory we don't know
what it requires.
44:28.440 --> 44:30.980
People are still discovering
new mathematics.
44:30.980 --> 44:34.440
But it's very true that very
often new mathematics is needed
44:34.440 --> 44:36.610
to express the new laws of
physics.
44:36.610 --> 44:39.560
And if you don't know the laws
you may find out you're not able
44:39.556 --> 44:40.456
to write it down.
44:40.460 --> 44:41.740
If you didn't know what a
gradient was,
44:41.739 --> 44:43.809
or if you did not know what a
curl is,
44:43.809 --> 44:47.879
and so on, then you cannot
write down the laws of
44:47.876 --> 44:50.246
electricity and magnetism.
44:50.250 --> 44:53.200
So we have to solve this
equation,
44:53.199 --> 44:55.539
and I'm going to solve it for
the easiest problem in the
44:55.536 --> 44:57.886
world,
a particle moving on a ring of
44:57.889 --> 45:00.649
length L with no
potential energy.
45:00.650 --> 45:02.680
So what does that equation look
like?
45:02.679 --> 45:12.769
It says -â;/2m
d^(2)ψ/dx^(2) no
45:12.773 --> 45:24.283
potential = E times
ψ_E(x).
45:24.280 --> 45:26.850
So let me rearrange the
equation so it looks like this,
45:26.849 --> 45:35.049
d^(2)ψ/dx^(2)
k^(2)ψ
45:35.052 --> 45:44.112
= 0 where k^(2) is
defined to be 2mE/ℏ^(2)
45:44.108 --> 45:48.038
= k^(2).
45:48.039 --> 45:51.709
All I've done is just taken
everything to one side and
45:51.711 --> 45:55.321
multiplied everything by
2m/ℏ^(2) and called
45:55.315 --> 45:57.875
that combination as
k^(2).
45:57.880 --> 45:59.460
So who is this number k?
45:59.460 --> 46:02.280
Let's see.
46:02.280 --> 46:04.780
The energy is
ℏk^(2)/2m,
46:04.784 --> 46:08.064
but we also know energy's
p^(2)/2m.
46:08.059 --> 46:14.419
So this number k will
turn out to be just momentum
46:14.422 --> 46:17.152
divided by ℏ.
46:17.150 --> 46:19.330
Well, momentum has not entered
the picture, but we will see.
46:19.329 --> 46:22.349
Let's solve this equation now.
46:22.349 --> 46:29.029
I say the solution to this
equation is ψ(x) = to
46:29.034 --> 46:34.804
any number A times
e^(ikx) any number
46:34.795 --> 46:38.825
B times e^(-ikx).
46:38.829 --> 46:40.899
Let's see if that is true.
46:40.900 --> 46:42.320
Take two derivatives of ψ.
46:42.320 --> 46:43.850
What do you get?
46:43.849 --> 46:47.539
You understand every time I
take a derivative you pull down
46:47.543 --> 46:48.503
an ik?
46:48.500 --> 46:51.590
If you pull it twice you'll
pull an ik squared which
46:51.594 --> 46:54.254
is -k^(2)ψ,
and the same thing will happen
46:54.251 --> 46:56.631
to this term that it'll pull
down a minus ik,
46:56.630 --> 46:59.240
but if you do it twice you'll
again get −k^(2).
46:59.239 --> 47:02.859
So both of them will have the
property that the second
47:02.864 --> 47:05.264
derivative of ψ
will be equal to
47:05.257 --> 47:08.817
-k^(2)ψ,
and that is the equation you
47:08.820 --> 47:09.850
want to solve.
47:09.849 --> 47:12.859
And A and B is
whatever you like.
47:12.860 --> 47:17.660
A and B are not
fixed by the equation because
47:17.657 --> 47:22.537
for any choice of A and
any choice of B this'll
47:22.538 --> 47:23.268
work.
47:23.269 --> 47:26.349
So let me write it as follows.
47:26.349 --> 47:28.629
Ae to the i.
47:28.630 --> 47:35.510
Let me write k,
k was a shorthand for
47:35.510 --> 47:43.770
square root of 2mE/ℏ^(2)
x B to the -i
47:43.766 --> 47:49.266
square root of 2mE/ℏ^(2)
x.
47:49.268 --> 47:52.178
I'm trying to show you that
these are really functions of
47:52.175 --> 47:54.455
definite energy
ψ_E and here is
47:54.458 --> 47:55.858
how the energy appears.
47:55.860 --> 47:58.480
So what does it look like to
you?
47:58.480 --> 48:03.280
Have you seen these functions
before?
48:03.280 --> 48:03.810
Yes?
48:03.809 --> 48:04.899
What does it look like?
48:04.900 --> 48:08.200
Student: One of the
spring wave theories?
48:08.199 --> 48:08.869
Prof: Pardon me?
48:08.869 --> 48:10.239
Student: One of the
spring wave theories?
48:10.239 --> 48:12.919
Prof: The equation is
like the spring equation.
48:12.920 --> 48:21.440
That is absolutely correct,
but what does this function
48:21.436 --> 48:23.326
look like?
48:23.329 --> 48:26.699
That function look like
something to you?
48:26.699 --> 48:27.509
Yep?
48:27.510 --> 48:29.600
Student: Isn't that a
cosine function?
48:29.599 --> 48:32.759
Prof: It's a cosine only
if A is equal to B.
48:32.760 --> 48:33.950
Forget the sines and cosines.
48:33.949 --> 48:35.319
They are your old flames.
48:35.320 --> 48:37.260
What's their new quantum flame?
48:37.260 --> 48:42.900
What's the function that means
a lot more in quantum mechanics
48:42.898 --> 48:45.208
than sines and cosines?
48:45.210 --> 48:50.610
No?
48:50.610 --> 48:55.450
I think I told you long back
that e to the i
48:55.454 --> 49:00.054
times dog x over ℏ
is a state where the
49:00.045 --> 49:02.505
momentum is equal to dog.
49:02.510 --> 49:05.090
In other words,
you can put anything you want
49:05.092 --> 49:06.152
in the exponent.
49:06.150 --> 49:10.110
If a function looks like that,
that fellow there is the
49:10.112 --> 49:10.922
momentum.
49:10.920 --> 49:14.530
So this is a state of momentum,
of definite momentum.
49:14.530 --> 49:21.000
In fact, look at this,
e to the i square
49:20.996 --> 49:27.706
root of 2mE/ℏ x
B times e to the
49:27.708 --> 49:29.538
minus that.
49:29.539 --> 49:31.499
Do you understand?
49:31.500 --> 49:34.580
This must be the momentum.
49:34.579 --> 49:37.379
It is the momentum because
e^(ipx/ℏ) is a state of
49:37.376 --> 49:40.146
momentum p,
but the momentum here can
49:40.150 --> 49:45.130
either have one value,
square root of 2mE,
49:45.134 --> 49:52.994
or it can have another value
minus square root of 2mE.
49:52.989 --> 49:58.849
So what do you think the
particle is doing in these
49:58.851 --> 50:00.261
solutions?
50:00.260 --> 50:01.230
Yep?
50:01.230 --> 50:03.470
Student: It's jumping
back and forth _______________
50:03.472 --> 50:04.312
one of those states.
50:04.309 --> 50:08.559
Prof: But in any one of
them what is it doing here?
50:08.559 --> 50:10.569
What is the sign of the
momentum here?
50:10.570 --> 50:12.010
Student: Oh, positive.
50:12.010 --> 50:14.700
Prof: Positive,
and here it's got negative
50:14.702 --> 50:15.322
momentum.
50:15.320 --> 50:16.990
And how much momentum does it
have?
50:16.989 --> 50:19.829
The momentum it has,
if you look at any of these
50:19.826 --> 50:23.596
things,
is that p^(2)/2m
50:23.601 --> 50:29.941
= E is what is satisfied
by the p that you have
50:29.943 --> 50:30.793
here.
50:30.789 --> 50:33.019
In other words,
what I'm telling you is in
50:33.019 --> 50:35.639
quantum theory,
in classical theory if I said,
50:35.635 --> 50:38.195
"I've got a particle of
energy E,
50:38.199 --> 50:40.499
what is its momentum?"
50:40.500 --> 50:42.040
You would say, "Well,
E is
50:42.041 --> 50:45.911
p^(2)/2m,
therefore p is equal to
50:45.905 --> 50:48.995
plus or minus square root of
2mE,"
50:49.003 --> 50:52.863
because in one dimension when I
give you the energy,
50:52.860 --> 50:55.220
the Kinetic energy if you like,
the particle has to have a
50:55.215 --> 50:57.895
definite speed,
but it can be to the left or it
50:57.896 --> 50:59.206
can be to the right.
50:59.210 --> 51:00.670
And the momentum is not
arbitrary.
51:00.670 --> 51:03.890
If the energy is E the
momentum has to satisfy the
51:03.891 --> 51:06.541
condition p^(2)/2m
= E.
51:06.539 --> 51:09.889
That's also exactly what's
happening in the quantum theory.
51:09.889 --> 51:14.149
The state of definite energy is
a sum over two possible things,
51:14.150 --> 51:17.930
one where the momentum is the
positive value for root of
51:17.929 --> 51:20.809
2mE;
other is a negative value,
51:20.807 --> 51:24.577
and these are the two allowed
values even in classical
51:24.577 --> 51:27.917
mechanics for a particle of
definite energy.
51:27.920 --> 51:30.610
But what's novel in quantum
mechanics,
51:30.610 --> 51:34.310
whereas in classical mechanics
if it's got energy E it
51:34.309 --> 51:37.329
can only be going clockwise or
anti-clockwise,
51:37.329 --> 51:40.929
but this fellow can be doing
both because it's not in a state
51:40.929 --> 51:42.909
of clockwise or anti-clockwise.
51:42.909 --> 51:46.379
In fact, the probability for
clockwise is proportional to
51:46.380 --> 51:47.310
A^(2).
51:47.309 --> 51:49.929
Probability for anti-clockwise
is proportional to B^(2).
51:49.929 --> 51:53.019
I've not normalized it,
but the relative odds are
51:53.021 --> 51:57.011
simply proportional to A^(2)
and B^(2). So that's
51:57.014 --> 52:00.884
what is bizarre about quantum
mechanics that the particle has
52:00.880 --> 52:03.070
indefinite sign of momentum.
52:03.070 --> 52:03.860
Yes?
52:03.860 --> 52:07.050
Student: Do we have
problem with m because
52:07.047 --> 52:09.607
m is too small to
________ properly?
52:09.610 --> 52:10.410
Prof: Which one?
52:10.409 --> 52:13.529
Student: With the mass
is there going to be some kind
52:13.534 --> 52:14.944
of issue if that's small?
52:14.940 --> 52:15.760
Prof: What about the
m?
52:15.760 --> 52:16.770
I'm sorry.
52:16.768 --> 52:18.718
Student: m
stands for the mass,
52:18.722 --> 52:19.072
right?
52:19.070 --> 52:20.570
Prof: Yeah,
m is the mass of the
52:20.567 --> 52:21.397
particle, that's right.
52:21.400 --> 52:23.950
Student: But is it too
small in some way for our
52:23.952 --> 52:26.462
equations to work because we had
that issue in previous
52:26.460 --> 52:27.110
equations?
52:27.110 --> 52:28.690
Prof: When did we have
the issue?
52:28.690 --> 52:31.780
m is whatever the mass
of the particle is.
52:31.780 --> 52:32.540
It can be small.
52:32.539 --> 52:35.549
It can be large.
52:35.550 --> 52:36.340
I didn't understand.
52:36.340 --> 52:37.400
Student:
>
52:37.400 --> 52:38.330
Prof: No,
no, go ahead.
52:38.333 --> 52:38.883
I want to know.
52:38.880 --> 52:45.980
Student:
>
52:45.980 --> 52:48.760
Prof: There are no
restrictions on the correctness
52:48.762 --> 52:49.262
of this.
52:49.260 --> 52:53.500
If the particle weighs a
kilogram then you will find
52:53.500 --> 52:56.410
that--well, we're coming to
this.
52:56.409 --> 52:59.689
So you find that these are the
allowed values of p.
52:59.690 --> 53:02.160
There are two values,
but p itself is not
53:02.155 --> 53:04.615
arbitrary, p itself is
not continuous.
53:04.619 --> 53:06.459
Maybe that's what you meant.
53:06.460 --> 53:09.580
There's a restriction on the
allowed values of p,
53:09.583 --> 53:13.223
and therefore a restriction on
the allowed values of E.
53:13.219 --> 53:16.829
All I'm telling you now is that
if you want to solve that
53:16.829 --> 53:20.629
equation it is obviously made up
of sines and cosines as you
53:20.632 --> 53:24.182
recognize from the oscillator,
or in the quantum world it's
53:24.184 --> 53:26.684
more natural to write them in
terms of exponentials,
53:26.679 --> 53:30.059
e^(ikx) and
e^(-ikx) where k is not
53:30.059 --> 53:33.179
independent of e,
k satisfies this
53:33.181 --> 53:36.001
condition, and if you call
ℏk as p,
53:36.000 --> 53:37.560
p satisfies this
condition.
53:37.559 --> 53:41.189
This is just a classical
relation between energy and
53:41.186 --> 53:41.966
momentum.
53:41.969 --> 53:46.399
The other difference is that
not every value of momentum is
53:46.396 --> 53:47.156
allowed.
53:47.159 --> 53:50.819
Not every value of momentum is
allowed for the same reason as
53:50.822 --> 53:53.512
when I did particles of definite
momentum.
53:53.510 --> 53:58.070
In other words,
if the particle is living on a
53:58.070 --> 54:03.640
circle and the state of energy
E it's given by
54:03.643 --> 54:08.513
Ae^(ipx/â)
Be^(−ipx/â) where
54:08.507 --> 54:15.397
p is related to E
by E = p^(2)/2m.
54:15.400 --> 54:18.880
We have the requirement that
when you go around a circle
54:18.880 --> 54:21.730
you've got to come back to where
you start.
54:21.730 --> 54:23.530
And that condition,
if you remember,
54:23.530 --> 54:28.340
says that p times
L over ℏ has to
54:28.342 --> 54:32.762
be a multiple of some integer,
or that the allowed values of
54:32.760 --> 54:35.190
p are labeled by some
index n,
54:35.190 --> 54:38.350
which is 2Πâ/L
times n.
54:38.349 --> 54:41.489
I don't want to use m
because m stands for
54:41.485 --> 54:44.045
particle mass,
n is the integer now.
54:44.050 --> 54:46.730
In other words,
when we studied the state of
54:46.728 --> 54:49.468
definite momentum,
namely the first function,
54:49.467 --> 54:52.767
we realized even then that
p is quantized.
54:52.768 --> 54:56.638
Because of the single valued
condition p is quantized.
54:56.639 --> 55:00.849
And if energy functions are
made up of such functions they
55:00.849 --> 55:03.139
also have to be singlevalued.
55:03.139 --> 55:08.949
That means the p here or
the -p here both have to
55:08.954 --> 55:12.524
satisfy the condition given by
this.
55:12.518 --> 55:18.528
Therefore, the allowed energies
are also labeled by an integer
55:18.534 --> 55:24.254
n and they are really
p n squared over
55:24.253 --> 55:29.483
2m where p
n is 2Πâ n
55:29.478 --> 55:31.548
over L .
55:31.550 --> 55:32.620
You see that?
55:32.619 --> 55:37.299
This is the quantization of
energy.
55:37.300 --> 55:41.780
So a particle in a ring has
only these allowed values of
55:41.777 --> 55:42.507
energy.
55:42.510 --> 55:48.160
So in a way this problem is
somewhat easy because it's a
55:48.164 --> 55:49.814
free particle.
55:49.809 --> 55:53.859
Once you've understood the
particles in terms of allowed
55:53.860 --> 55:58.570
momenta it turns out the allowed
momentum states are also allowed
55:58.574 --> 55:59.904
energy states.
55:59.900 --> 56:03.150
The allowed momentum states,
you remember,
56:03.152 --> 56:06.882
they either look this or they
look like this.
56:06.880 --> 56:10.390
And I can pick A and
B to be arbitrary,
56:10.389 --> 56:13.379
so one choice is to pick
A equal to
56:13.382 --> 56:17.252
1/√L and just take
e^(ipx/â).
56:17.250 --> 56:22.260
Other is to pick B equal
to 1/√L and pick
56:22.262 --> 56:27.752
e^(−ipx/â),
but you can also mix them up.
56:27.750 --> 56:29.620
You don't have to mix them.
56:29.619 --> 56:33.929
If you don't mix them up you
have a particle here which has a
56:33.934 --> 56:37.534
well-defined energy and a
well-defined momentum.
56:37.530 --> 56:39.760
This guy also has a
well-defined energy and a
56:39.762 --> 56:40.982
well-defined momentum.
56:40.980 --> 56:43.750
This guy only has a
well-defined energy,
56:43.748 --> 56:47.788
but not a well-defined momentum
because there's a two-fold
56:47.793 --> 56:49.643
ambiguity in momentum.
56:49.639 --> 56:50.329
You understand?
56:50.329 --> 56:52.319
Even in classical mechanics
it's true.
56:52.320 --> 56:54.830
If I give you the momentum you
can find the energy.
56:54.829 --> 56:57.789
If I give you the energy you
cannot find the momentum because
56:57.786 --> 57:00.836
there are two square roots you
can take because p^(2)/2m
57:00.842 --> 57:02.802
is E,
p is plus or minus
57:02.795 --> 57:03.995
square root of 2mE.
57:04.000 --> 57:08.350
It's the same uncertainty even
in classical mechanics.
57:08.349 --> 57:11.209
So what happens in quantum
theory is if you pick any
57:11.211 --> 57:14.521
particle of definite momentum on
the ring it'll already have
57:14.523 --> 57:17.723
definite energy which is simply
that momentum squared over
57:17.722 --> 57:18.622
2m.
57:18.619 --> 57:20.799
You don't need to find a new
function.
57:20.800 --> 57:24.340
What is novel is that since the
energy depends only on
57:24.342 --> 57:27.882
p^(2) you can take a
function with one value of
57:27.884 --> 57:30.174
p,
and you can take a function
57:30.172 --> 57:32.732
with the opposite value of
p and add them,
57:32.730 --> 57:36.790
there will still be a state of
definite energy because whether
57:36.789 --> 57:40.579
it's doing this or whether it's
doing that the energy will
57:40.583 --> 57:44.783
always be p^(2)/2m
and the minus signs drop out of
57:44.777 --> 57:45.507
that.
57:45.510 --> 57:50.280
So what is novel here is what's
called degeneracy.
57:50.280 --> 57:53.720
Degeneracy is the name,
but there's more than one
57:53.724 --> 57:57.964
solution for a given value of
the variable you're interested
57:57.956 --> 57:58.456
in.
57:58.460 --> 58:01.110
You saw the energy looks like
this.
58:01.110 --> 58:06.550
Therefore, you'll find there's
a state E_0
58:06.552 --> 58:09.932
which is 0 squared over
2m.
58:09.929 --> 58:15.249
Then there are two states
E_1.
58:15.250 --> 58:19.470
One has got momentum p
going clockwise and one has
58:19.474 --> 58:21.894
momentum going anti-clockwise.
58:21.889 --> 58:27.779
So they look like
(1/√L)
58:27.784 --> 58:31.554
e^(2Πix/L).
58:31.550 --> 58:33.370
That's this guy.
58:33.369 --> 58:37.659
Then you can have
(1/√L)
58:37.655 --> 58:43.245
e^(-2Πix/L) which is
the other guy.
58:43.250 --> 58:47.510
So at every-energy,
allowed energy except 0,
58:47.507 --> 58:50.277
there'll be two solutions.
58:50.280 --> 58:54.050
There are two quantum states
with the same energy.
58:54.050 --> 58:56.830
When you studied hydrogen atom
in high school maybe you
58:56.826 --> 58:59.136
remember there are these shells
with 2, and 4,
58:59.139 --> 59:00.579
and 8, and 10 and so on.
59:00.579 --> 59:03.559
They are called degeneracies
where the energy is not enough
59:03.559 --> 59:05.099
to tell you what it's doing.
59:05.099 --> 59:07.439
There at every-energy it can
have a different angular
59:07.443 --> 59:07.943
momentum.
59:07.940 --> 59:11.950
Here at every-energy it can
have two different momenta,
59:11.954 --> 59:14.264
clockwise and anti-clockwise.
59:14.260 --> 59:15.010
Yep?
59:15.010 --> 59:19.060
Student: Going back can
you explain how you got the
59:19.063 --> 59:22.073
________ of p L over ℏ
________?
59:22.070 --> 59:22.670
Prof: Here?
59:22.670 --> 59:24.450
Student: To the left.
59:24.449 --> 59:25.939
Prof: Here?
59:25.940 --> 59:26.990
You mean this one?
59:26.989 --> 59:28.329
Student: I don't where
that came from.
59:28.329 --> 59:30.969
Prof: It came just like
in the momentum problem.
59:30.969 --> 59:34.529
If you've got a function like
this you have a right to demand
59:34.525 --> 59:37.775
that if you go a distance
L around the circle you
59:37.784 --> 59:39.744
come back to where you start.
59:39.739 --> 59:44.489
So if you take any x here and
add to it an L it should not
59:44.485 --> 59:46.145
make a difference.
59:46.150 --> 59:49.900
And what you're adding is
pL/ℏ and that better be
59:49.900 --> 59:51.350
a multiple of 2Π.
59:51.349 --> 59:53.929
It's the same single valuedness
condition.
59:53.929 --> 59:56.989
So the momentum problem pretty
much does this problem for you,
59:56.985 --> 59:59.485
all the singlevalued stuff we
dealt with before.
59:59.489 --> 1:00:03.819
What is novel here is that
demanding energy have one value
1:00:03.817 --> 1:00:07.307
fixes the momentum to be one of
two values,
1:00:07.309 --> 1:00:10.639
and that double valuedness is
the same as in classical theory
1:00:10.644 --> 1:00:13.704
that a particle of energy
E can have two possible
1:00:13.702 --> 1:00:16.652
momenta plus or minus square
root of 2mE,
1:00:16.650 --> 1:00:21.580
and the quantum theory,
then, is the state of E
1:00:21.581 --> 1:00:26.421
is a sum of one value of
p and the other value
1:00:26.418 --> 1:00:30.698
of p with any coefficient
you like.
1:00:30.699 --> 1:00:35.079
So if this atom makes a jump
from,
1:00:35.079 --> 1:00:37.399
or this system makes a jump
from somewhere there to
1:00:37.400 --> 1:00:40.370
somewhere there you can find the
frequency of the photons it will
1:00:40.373 --> 1:00:42.513
emit because here are my allowed
energies.
1:00:42.510 --> 1:00:45.620
E_n is this.
1:00:45.619 --> 1:00:51.419
Suppose it jumps from
n=4 to n=3.
1:00:51.420 --> 1:00:56.310
The energy that's liberated
that goes to the photon is
1:00:56.306 --> 1:01:01.286
âω will be
4Π^(2) â^(2) over
1:01:01.286 --> 1:01:05.986
2mL squared times 4
squared - 3 squared.
1:01:05.989 --> 1:01:08.519
I'm just using this formula
with n=4,
1:01:08.518 --> 1:01:10.988
and n=3,
and take the difference.
1:01:10.989 --> 1:01:13.029
It's 16 - 9 which 7.
1:01:13.030 --> 1:01:16.590
You plug all that in you can
solve for the ω,
1:01:16.590 --> 1:01:20.600
or if you like frequency you
can write it as 2Πâf
1:01:20.599 --> 1:01:22.939
and you can find the
frequencies.
1:01:22.940 --> 1:01:24.250
This is actually true.
1:01:24.250 --> 1:01:27.490
If you've got a charged
particle moving in a ring and
1:01:27.492 --> 1:01:31.362
you want to excite it from one
state to a higher state you will
1:01:31.358 --> 1:01:34.848
have to give it only one of
these frequencies so that the
1:01:34.851 --> 1:01:37.591
frequency you give it,
âω,
1:01:37.585 --> 1:01:40.155
must match the energy
difference of the particle.
1:01:40.159 --> 1:01:42.669
You understand?
1:01:42.670 --> 1:01:44.770
So I've drawn these levels here.
1:01:44.768 --> 1:01:48.308
If you want the electrons,
say, in a metallic ring going
1:01:48.309 --> 1:01:50.819
in the lowest possible energy
state,
1:01:50.820 --> 1:01:53.180
if you want to jump,
if you want to crank it up to
1:01:53.175 --> 1:01:56.055
the next level you've got to
have photons at that energy,
1:01:56.059 --> 1:01:57.569
or that energy.
1:01:57.570 --> 1:01:59.680
Well, that happens to be the
same as this,
1:01:59.681 --> 1:02:02.311
but these are the only
frequencies it'll absorb from
1:02:02.306 --> 1:02:02.716
you.
1:02:02.719 --> 1:02:08.079
And when it cools down it'll
emit back those frequencies,
1:02:08.081 --> 1:02:11.531
and that's something you can
test.
1:02:11.530 --> 1:02:18.160
By the way, do you know why
there's only one state at
1:02:18.164 --> 1:02:22.634
E_0 and not
two?
1:02:22.630 --> 1:02:23.470
Yep?
1:02:23.469 --> 1:02:26.329
Student: Because 0
equals -0.
1:02:26.329 --> 1:02:27.599
Prof: That's right.
1:02:27.599 --> 1:02:30.929
The solution plus or minus
2m square root of
1:02:30.932 --> 1:02:34.802
2mE has only one-answer
when E is 0 because 0
1:02:34.800 --> 1:02:37.600
momentum and -0 momentum are the
same.
1:02:37.599 --> 1:02:41.609
Otherwise, any finite positive
momentum has a partner which is
1:02:41.605 --> 1:02:43.045
minus that momentum.
1:02:43.050 --> 1:02:46.320
There's a very interesting
piece of work being done
1:02:46.318 --> 1:02:50.178
experimentally at Yale which is
the claim that if you took a
1:02:50.177 --> 1:02:53.767
metallic ring in a magnetic
field it will have a current
1:02:53.773 --> 1:02:57.453
going one way or the other way,
unbalanced current,
1:02:57.445 --> 1:02:59.565
and it's not driven by a
battery.
1:02:59.570 --> 1:03:02.580
It's not driven by anything.
1:03:02.579 --> 1:03:05.789
Normally if you took an
ordinary ring the lowest energy
1:03:05.793 --> 1:03:09.373
state will be a field of zero
current while you go one way or
1:03:09.365 --> 1:03:11.445
the other,
but if you put it in the
1:03:11.454 --> 1:03:14.544
magnetic field one can show it
likes to go one way or the other
1:03:14.541 --> 1:03:14.891
way.
1:03:14.889 --> 1:03:17.879
And now measurements are being
done at Yale where you can
1:03:17.882 --> 1:03:20.932
actually measure the tiny
current due to one electron,
1:03:20.929 --> 1:03:24.929
or one net electron going one
way or the other.
1:03:24.929 --> 1:03:28.309
So this L is either a
mathematical convenience if
1:03:28.311 --> 1:03:31.821
you're talking about free space
and you want to be able to
1:03:31.817 --> 1:03:35.077
normalize your wave functions,
or it really is the
1:03:35.079 --> 1:03:36.949
circumference of a real system.
1:03:36.949 --> 1:03:41.129
Now that we can probe nano
systems very well we can vary
1:03:41.132 --> 1:03:45.472
the L and we can find out
all the energy levels.
1:03:45.469 --> 1:03:51.719
All right, so now I'm going to
do the one problem which is
1:03:51.715 --> 1:03:56.425
really a very standard
pedagogical exercise,
1:03:56.427 --> 1:04:01.027
and that's called a particle in
a box.
1:04:01.030 --> 1:04:05.410
I remember this example the
first time I remember seeing
1:04:05.405 --> 1:04:08.825
quantization,
which is more interesting than
1:04:08.826 --> 1:04:09.936
on a ring.
1:04:09.940 --> 1:04:11.670
A box is the following.
1:04:11.670 --> 1:04:16.860
If you dig a hole in the ground
and you are standing somewhere
1:04:16.855 --> 1:04:21.865
here you realize you're kind of
trapped unless you can scale
1:04:21.869 --> 1:04:23.059
this wall.
1:04:23.059 --> 1:04:25.879
Now, you can call that as a
ground level and think of it as
1:04:25.882 --> 1:04:29.102
a hole in the ground,
or you can think of this as the
1:04:29.096 --> 1:04:32.526
ground level and that's the
height of your barrier.
1:04:32.530 --> 1:04:38.420
So imagine a particle living in
a barrier that looks like this.
1:04:38.420 --> 1:04:40.530
This is the potential energy.
1:04:40.530 --> 1:04:41.750
This is like the height above
the ground,
1:04:41.750 --> 1:04:43.400
if you like,
and it has a height
1:04:43.400 --> 1:04:47.110
V_0 t here,
and is 0 here,
1:04:47.106 --> 1:04:55.186
and it goes from some
−L/2 to L/2.
1:04:55.190 --> 1:05:00.030
If the barrier
V_0 goes to
1:05:00.027 --> 1:05:03.367
infinity it's called a box.
1:05:03.369 --> 1:05:07.009
So a barrier that goes to
infinity, I'll just show you the
1:05:07.007 --> 1:05:08.857
part that you can see here.
1:05:08.860 --> 1:05:12.530
Here V is infinity,
and here V is infinity
1:05:12.527 --> 1:05:14.327
and inside V is 0.
1:05:14.329 --> 1:05:16.719
That's a particle in a box.
1:05:16.719 --> 1:05:20.429
So if the particle goes hits
against the wall it cannot,
1:05:20.429 --> 1:05:24.409
no matter how fast it's moving,
go over the top because it's
1:05:24.409 --> 1:05:25.759
infinitely high.
1:05:25.760 --> 1:05:27.670
And that's not realistic.
1:05:27.670 --> 1:05:32.360
Every barrier is finite,
but just to teach you the
1:05:32.360 --> 1:05:36.860
principles we always pick the
simple example.
1:05:36.860 --> 1:05:39.450
So now I want to solve this
problem.
1:05:39.449 --> 1:05:42.969
What are the allowed wave
functions, ψ,
1:05:42.972 --> 1:05:45.912
for a particle in this
potential?
1:05:45.909 --> 1:05:51.149
So let's go back to the same
equation which says
1:05:51.152 --> 1:05:57.402
-â^(2)/2m
d^(2)ψ/dx^(2)
1:05:57.400 --> 1:06:00.970
= E - V ψ.
1:06:00.969 --> 1:06:05.779
What I've done is I've taken
V to the other side.
1:06:05.780 --> 1:06:09.510
Now the potential is a constant
in the three regions.
1:06:09.510 --> 1:06:12.040
This is region I where the
potential is 0.
1:06:12.039 --> 1:06:14.639
This is region II where the
potential is infinity.
1:06:14.639 --> 1:06:19.229
This is region III where the
potential is infinity.
1:06:19.230 --> 1:06:25.590
I'm going to first look at the
region here, region III.
1:06:25.590 --> 1:06:30.880
What's the solution going to
look like in region III is what
1:06:30.882 --> 1:06:32.052
I'm asking.
1:06:32.050 --> 1:06:34.700
So in this region the energy's
something I don't know what the
1:06:34.697 --> 1:06:35.607
allowed values are.
1:06:35.610 --> 1:06:39.100
Whatever the value is V is an
extremely large number.
1:06:39.099 --> 1:06:40.629
Can you see that?
1:06:40.630 --> 1:06:43.470
In this region don't let the
barrier be infinity.
1:06:43.469 --> 1:06:47.179
Imagine it's one zillion,
a very high barrier.
1:06:47.179 --> 1:06:48.749
Then what's the solution?
1:06:48.750 --> 1:06:59.750
The solution will look like
d^(2)ψ/dx^(2)
1:06:59.753 --> 1:07:09.243
is = 2m/â^(2) times
V - E ψ.
1:07:09.239 --> 1:07:11.029
Then the solution to that is
very easy,
1:07:11.030 --> 1:07:20.200
ψ = Ae^(κx)
Be^(-κx) where
1:07:20.195 --> 1:07:28.315
κ is equal to all of this,
square root of
1:07:28.318 --> 1:07:34.528
2m/ℏ^(2) times V -
E.
1:07:34.530 --> 1:07:39.920
I'm saying this is what I'm
calling κ^(2).
1:07:39.920 --> 1:07:43.460
Look, I'm asking you give me a
function whose second derivative
1:07:43.461 --> 1:07:45.521
is some number times the
function.
1:07:45.518 --> 1:07:49.228
Well, that's obviously
exponential and it's the real
1:07:49.231 --> 1:07:52.941
exponential because it's a
positive number times the
1:07:52.943 --> 1:07:53.893
function.
1:07:53.889 --> 1:07:56.559
Now A and B are
free parameters,
1:07:56.559 --> 1:07:58.289
whatever they are it'll solve
the equation,
1:07:58.289 --> 1:08:01.749
but we don't want a function
that's growing exponentially
1:08:01.751 --> 1:08:05.521
when you go to infinity because
that means that particle would
1:08:05.521 --> 1:08:09.231
rather be at infinity than in
your box or near your box.
1:08:09.230 --> 1:08:13.630
So for that mathematical,
for that physical reason we
1:08:13.625 --> 1:08:15.395
junk this function.
1:08:15.400 --> 1:08:19.070
You pick it on physical grounds
as not having a part growing
1:08:19.074 --> 1:08:21.634
exponentially when you go to
infinity,
1:08:21.630 --> 1:08:24.760
but you do admit a part that's
falling exponentially when you
1:08:24.761 --> 1:08:25.651
go to infinity.
1:08:25.649 --> 1:08:26.949
That's okay.
1:08:26.949 --> 1:08:29.119
But how fast is it falling?
1:08:29.118 --> 1:08:32.368
It's going like B e to
the minus some blah,
1:08:32.365 --> 1:08:35.475
blah, blah times square root of
V - E.
1:08:35.479 --> 1:08:37.839
That's all I want you to look
at.
1:08:37.840 --> 1:08:39.800
Forget all the ℏ's and
m's.
1:08:39.800 --> 1:08:42.400
Make V larger,
and larger, and larger,
1:08:42.395 --> 1:08:45.045
and just tell me what you think
it will do.
1:08:45.050 --> 1:08:47.480
If V is larger and
larger it's like this
1:08:47.484 --> 1:08:50.984
e^(-αx) I wrote for
you where α's very large.
1:08:50.979 --> 1:08:54.169
So this function will fall
faster, and faster,
1:08:54.171 --> 1:08:58.571
and faster, and in the limit in
which V goes to infinity
1:08:58.570 --> 1:08:59.920
it will vanish.
1:08:59.920 --> 1:09:07.260
That basically means the
particle cannot be found outside
1:09:07.259 --> 1:09:08.569
the box.
1:09:08.569 --> 1:09:13.409
So your function ψ
is 0 here and 0 here.
1:09:13.408 --> 1:09:17.358
Because if you made the barrier
height finite you will find it's
1:09:17.360 --> 1:09:19.870
falling exponentially on either
side,
1:09:19.868 --> 1:09:22.098
but the exponential becomes
narrower and narrower as the
1:09:22.099 --> 1:09:23.599
barrier becomes higher and
higher,
1:09:23.600 --> 1:09:27.980
and in the limit in which the
wall is infinitely tall there is
1:09:27.978 --> 1:09:29.268
nothing outside.
1:09:29.270 --> 1:09:35.910
The wave function is non-0 only
inside.
1:09:35.909 --> 1:09:37.779
So what's the solution inside?
1:09:37.779 --> 1:09:39.109
Let me call this 0.
1:09:39.109 --> 1:09:41.029
Let me call this L.
1:09:41.029 --> 1:09:43.949
Inside the box there is no
potential,
1:09:43.948 --> 1:09:50.838
so this equation is
d^(2)ψ/dx^(2)
1:09:50.840 --> 1:09:55.250
(2mE/â^(2))ψ
= 0.
1:09:55.250 --> 1:09:59.580
It's like a free particle in
the box, but it cannot leave the
1:09:59.577 --> 1:10:00.007
box.
1:10:00.010 --> 1:10:04.210
This is what I called k,
remember?
1:10:04.210 --> 1:10:07.690
So now I'm going to purposely
write the solution in terms of
1:10:07.694 --> 1:10:09.234
trigonometric functions.
1:10:09.229 --> 1:10:11.439
You'll see in a minute why.
1:10:11.439 --> 1:10:16.409
So I'm going to write this
Ae^(ikx)
1:10:16.413 --> 1:10:18.723
Be^(-ikx).
1:10:18.720 --> 1:10:21.520
You can see if I take any of
these solutions it's going to
1:10:21.520 --> 1:10:25.270
satisfy the equation,
but k better be related to E in
1:10:25.265 --> 1:10:29.265
this form,
2mE/ℏ^(2) is equal to
1:10:29.273 --> 1:10:30.683
k^(2).
1:10:30.680 --> 1:10:35.680
Sorry that's k^(2).
1:10:35.680 --> 1:10:39.020
Because two derivatives of this
will give me
1:10:39.015 --> 1:10:43.665
−k^(2), and if you
put that particular value for
1:10:43.671 --> 1:10:47.631
−k^(2) here these
two will cancel.
1:10:47.630 --> 1:10:53.140
But I'm somehow going for--I'm
sorry, I'm going to not write it
1:10:53.143 --> 1:10:57.593
this way, but write it as
C cosine kx
1:10:57.590 --> 1:11:00.080
D sine kx.
1:11:00.078 --> 1:11:02.578
Do you realize I can always go
back and forth between
1:11:02.582 --> 1:11:05.282
exponentials and trigonometric
functions because one is a
1:11:05.277 --> 1:11:07.007
linear combination of the other?
1:11:07.010 --> 1:11:09.490
If you want,
write this as cos i sine
1:11:09.494 --> 1:11:12.794
and cos - i sine and
rearrange the coefficients.
1:11:12.788 --> 1:11:14.428
It'll look like something,
something cosine.
1:11:14.430 --> 1:11:16.530
I want to call that a C,
and something,
1:11:16.529 --> 1:11:18.819
something sine which I want to
call this D,
1:11:18.815 --> 1:11:20.725
D and C may be
complex.
1:11:20.729 --> 1:11:23.859
I'm not saying anything,
but you can write a solution
1:11:23.860 --> 1:11:27.650
either in terms of the sines and
cosines or E to the plus
1:11:27.654 --> 1:11:29.044
or minus something.
1:11:29.039 --> 1:11:31.409
So here's the function ψ.
1:11:31.408 --> 1:11:35.788
It looks like I have an answer
for every energy I want,
1:11:35.788 --> 1:11:39.878
because pick any energy you
like, find the corresponding
1:11:39.877 --> 1:11:44.547
k,
you put it here and you're done.
1:11:44.550 --> 1:11:49.720
But that's not allowed because
we have an extra condition which
1:11:49.715 --> 1:11:54.625
is the ψ was identically 0
here, ψ was identically 0
1:11:54.630 --> 1:11:55.380
here.
1:11:55.380 --> 1:11:57.410
We're going to demand that at
the two ends,
1:11:57.408 --> 1:11:59.218
it can do whatever it wants in
the middle,
1:11:59.220 --> 1:12:02.480
it must vanish at the two ends
for the continuity of ψ
1:12:02.476 --> 1:12:05.046
because if ψ
had two values you're getting
1:12:05.047 --> 1:12:08.017
two different probabilities for
the same point,
1:12:08.020 --> 1:12:09.600
so that's not allowed.
1:12:09.600 --> 1:12:14.000
So ψ must have--must match
at the two ends.
1:12:14.000 --> 1:12:16.480
But look at this function.
1:12:16.479 --> 1:12:20.919
It's got to vanish at the left
end at x = 0 and it's got to
1:12:20.918 --> 1:12:22.908
vanish at the right end.
1:12:22.908 --> 1:12:28.578
At x = 0 you can see
ψ of 0 is simply C
1:12:28.578 --> 1:12:31.008
because sine vanishes.
1:12:31.010 --> 1:12:35.650
Cosine is 1 and C has to
be 0 then because this guy has
1:12:35.654 --> 1:12:39.084
no business being non-zero on
the left end.
1:12:39.079 --> 1:12:42.519
That's good.
1:12:42.520 --> 1:12:45.540
This guy vanishes at the left
end so I allow it,
1:12:45.537 --> 1:12:47.527
but I have another condition.
1:12:47.529 --> 1:12:51.749
It should also vanish at the
right end.
1:12:51.750 --> 1:12:56.700
If it should vanish at the
right end I demand that sine
1:12:56.704 --> 1:13:01.844
kL should be 0 because
sine kL vanishing is
1:13:01.844 --> 1:13:02.674
fine.
1:13:02.670 --> 1:13:05.460
Then it'll vanish at both ends.
1:13:05.460 --> 1:13:14.430
But sine kL = 0 means kL
is a multiple of Π.
1:13:14.430 --> 1:13:20.120
That means k is
nΠ/L.
1:13:20.118 --> 1:13:25.368
This means the allowed values
of k in the problem are very
1:13:25.367 --> 1:13:30.607
special so here is n = 1,
ψ looks like
1:13:30.614 --> 1:13:37.194
sin(Πx/L),
and equal to 2 looks like
1:13:37.190 --> 1:13:42.850
sin(2Πx/L),
and so on with some numbers in
1:13:42.849 --> 1:13:45.379
front which I have not chosen
yet.
1:13:45.380 --> 1:13:47.270
If you plot them they look like
this.
1:13:47.270 --> 1:13:48.480
That's one guy.
1:13:48.479 --> 1:13:50.939
At a higher energy I've got
that guy.
1:13:50.939 --> 1:13:55.529
Then I've got that and so on
because these are exactly like
1:13:55.529 --> 1:13:59.329
waves on a string,
a violin string clamped at two
1:13:59.327 --> 1:14:00.037
ends.
1:14:00.038 --> 1:14:03.348
In fact, this wave equation's
identical to the wave equation
1:14:03.353 --> 1:14:04.143
on a string.
1:14:04.140 --> 1:14:06.870
The only requirement is that
the string is clamped at two
1:14:06.868 --> 1:14:07.208
ends.
1:14:07.210 --> 1:14:09.210
Here the ψ
is clamped at the two ends
1:14:09.212 --> 1:14:12.222
because it's got to vanish on
either side outside the box.
1:14:12.220 --> 1:14:15.180
So the allowed wavelengths are
the same except here the
1:14:15.180 --> 1:14:18.360
wavelength is connected to the
momentum that's connected to
1:14:18.359 --> 1:14:21.269
energy and E,
you remember,
1:14:21.273 --> 1:14:24.953
is
ℏ^(2)k^(2)/2m
1:14:24.953 --> 1:14:29.943
then becomes
ℏ^(2)/2m times
1:14:29.940 --> 1:14:33.740
k will be
n^(2)Π
1:14:33.740 --> 1:14:36.710
^(2)/L^(2).
1:14:36.710 --> 1:14:40.280
So this particle in a box can
have only these particular
1:14:40.275 --> 1:14:40.985
energies.
1:14:40.988 --> 1:14:42.238
So let me write it for you
nicely.
1:14:42.238 --> 1:14:48.788
The allowed energies are
â^(2) Π^(2)/2
1:14:48.786 --> 1:14:55.196
mL squared times an
integer n squared,
1:14:55.198 --> 1:14:58.538
and the corresponding wave
functions look like this.
1:14:58.538 --> 1:15:01.728
They're waves in which you've
got half an oscillation,
1:15:01.726 --> 1:15:04.486
or 2 half oscillations and 3
half oscillations,
1:15:04.493 --> 1:15:07.143
but you've got to start and
finish at 0.
1:15:07.140 --> 1:15:08.760
So it's the quantization.
1:15:08.760 --> 1:15:11.300
This is why when Schr�dinger
came up with this equation
1:15:11.302 --> 1:15:14.162
everybody embraced it right away
because you suddenly understood
1:15:14.162 --> 1:15:15.392
why energy's quantized.
1:15:15.390 --> 1:15:18.510
You're trying to fit some
number of waves into an interval
1:15:18.506 --> 1:15:21.676
and only some multiple of half
wavelengths are allowed,
1:15:21.680 --> 1:15:23.640
but wavelength translates into
momentum.
1:15:23.640 --> 1:15:25.170
That translates into energy.
1:15:25.170 --> 1:15:29.050
Suddenly you understand the
quantization of energy.
1:15:29.050 --> 1:15:30.950
So it's got 1 state
n = 1,
1:15:30.948 --> 1:15:32.798
and only one state
n = 2,
1:15:32.801 --> 1:15:34.471
one state n =
3.
1:15:34.470 --> 1:15:38.860
I'm going to come back to this
next time, but you should think
1:15:38.863 --> 1:15:39.803
about this.
1:15:39.800 --> 1:15:45.000