1
00:00:00,890 --> 00:00:02,913
PROFESSOR: OK,
so, local picture.
2
00:00:10,140 --> 00:00:13,610
It's all about getting
insight into how
3
00:00:13,610 --> 00:00:15,330
the way function looks.
4
00:00:15,330 --> 00:00:18,360
That's what we'll need to get.
5
00:00:18,360 --> 00:00:20,500
These comments now
will be pretty useful.
6
00:00:20,500 --> 00:00:25,150
For this equation you
have one over psi,
7
00:00:25,150 --> 00:00:31,680
d second psi, d x
squared is minus 2 m
8
00:00:31,680 --> 00:00:37,980
over h squared, E minus v of x.
9
00:00:37,980 --> 00:00:41,660
Look how I wrote it, I
put the psi back here,
10
00:00:41,660 --> 00:00:44,440
and that's useful.
11
00:00:48,450 --> 00:00:52,560
Now, there's a whole
lot of discussion--
12
00:00:52,560 --> 00:00:56,090
many textbooks-- about
how the way function
13
00:00:56,090 --> 00:01:00,560
looks, and they say concave
or convex, but it depends.
14
00:01:00,560 --> 00:01:04,760
Let's try to make it very clear
how the wave function looks.
15
00:01:04,760 --> 00:01:08,060
For this we need two regions.
16
00:01:08,060 --> 00:01:20,990
So, the first case, A, is
when the energy minus v of x
17
00:01:20,990 --> 00:01:23,960
is less than 0.
18
00:01:23,960 --> 00:01:29,350
The energy is less than v of
x, that's a forbidden region--
19
00:01:29,350 --> 00:01:31,310
as you can see there--
20
00:01:31,310 --> 00:01:33,710
so it's a classically forbidden.
21
00:01:33,710 --> 00:01:35,710
Not quantum
mechanically forbidden,
22
00:01:35,710 --> 00:01:39,134
but classically forbidden.
23
00:01:48,040 --> 00:01:51,820
What is the main thing about
this classically forbidden
24
00:01:51,820 --> 00:01:56,140
region is that the right
hand side of this equation
25
00:01:56,140 --> 00:01:57,378
is positive.
26
00:02:04,220 --> 00:02:07,760
Now, this gives you
two possibilities.
27
00:02:07,760 --> 00:02:16,030
It may be that psi at some
point is positive, in which case
28
00:02:16,030 --> 00:02:21,040
the second psi must
also be positive,
29
00:02:21,040 --> 00:02:24,580
because psi and the
second psi appear here.
30
00:02:24,580 --> 00:02:27,580
If both are positive,
this is positive.
31
00:02:27,580 --> 00:02:33,090
Or, it may be case two,
that psi is negative,
32
00:02:33,090 --> 00:02:35,380
and the second psi--
33
00:02:35,380 --> 00:02:38,914
the x squared--
it's also negative.
34
00:02:42,050 --> 00:02:44,900
Well, how do we plot this?
35
00:02:44,900 --> 00:02:49,630
Well, you're at some
point x, and here it
36
00:02:49,630 --> 00:02:52,460
is, a positive
wave function seems
37
00:02:52,460 --> 00:02:56,030
to be one type of convexity,
another type of convexity
38
00:02:56,030 --> 00:02:59,090
for a negative, that's why
people get a little confused
39
00:02:59,090 --> 00:03:00,110
about this.
40
00:03:00,110 --> 00:03:03,890
There's a way to see in a way
that there's is no confusion.
41
00:03:03,890 --> 00:03:07,640
Look at this, it's positive,
second derivative positive.
42
00:03:07,640 --> 00:03:10,520
When you think of a second
derivative positive,
43
00:03:10,520 --> 00:03:14,030
I think personally of
a parabola going up.
44
00:03:14,030 --> 00:03:19,720
So, that's how it could look.
45
00:03:19,720 --> 00:03:23,510
The wave function is
positive, up, it's all real.
46
00:03:23,510 --> 00:03:26,680
We're using the thing we
proved at the beginning
47
00:03:26,680 --> 00:03:30,870
of this lecture: you can work
with real things, all real.
48
00:03:30,870 --> 00:03:37,280
So, the wave from here
is x, and here negative.
49
00:03:37,280 --> 00:03:41,230
And the negative
opening parabola,
50
00:03:41,230 --> 00:03:43,190
that's something they got.
51
00:03:43,190 --> 00:03:44,620
So nice.
52
00:03:44,620 --> 00:03:47,420
So, the wave
function at any point
53
00:03:47,420 --> 00:03:50,670
could look like this
if it's positive,
54
00:03:50,670 --> 00:03:54,220
or, it could look like
this if it's negative.
55
00:03:54,220 --> 00:03:57,820
So, it doesn't look like
both, it's not double value.
56
00:03:57,820 --> 00:04:00,290
So, either one or the other.
57
00:04:00,290 --> 00:04:04,630
But, this is easy
to say in words,
58
00:04:04,630 --> 00:04:11,770
it is a shape that is
convex towards the axis.
59
00:04:11,770 --> 00:04:15,340
From the axis it's convex
here and convex there.
60
00:04:15,340 --> 00:04:22,732
So, convex towards the axis.
61
00:04:25,690 --> 00:04:30,070
Now, there's another
possibility I want to just
62
00:04:30,070 --> 00:04:32,050
make sure you visualize this.
63
00:04:32,050 --> 00:04:38,280
Sometimes this looks funny--
doesn't mean actually the way
64
00:04:38,280 --> 00:04:41,080
function can look like that--
65
00:04:41,080 --> 00:04:46,390
but, it's funny because
of the following reason.
66
00:04:46,390 --> 00:04:49,700
It's funny because if you
imagine it going forever,
67
00:04:49,700 --> 00:04:51,310
it doesn't make
sense because you're
68
00:04:51,310 --> 00:04:54,200
in a classically the
forbidden region.
69
00:04:54,200 --> 00:04:56,320
And the way function's
becoming bigger and bigger
70
00:04:56,320 --> 00:04:57,700
is going to blow up.
71
00:04:57,700 --> 00:05:00,670
So, eventually
something has to happen.
72
00:05:00,670 --> 00:05:03,070
But, it can look like this.
73
00:05:03,070 --> 00:05:05,530
So, actually what happens
is that when you're
74
00:05:05,530 --> 00:05:08,290
going to minus infinity--
75
00:05:08,290 --> 00:05:11,080
here is x and we
use minus infinity--
76
00:05:11,080 --> 00:05:13,100
it can look like this.
77
00:05:13,100 --> 00:05:18,440
This is an example of this
piece that is asymptotic,
78
00:05:18,440 --> 00:05:21,820
and it's positive, and the
second derivative is positive.
79
00:05:21,820 --> 00:05:25,310
Or, negative and the second
derivative is negative.
80
00:05:25,310 --> 00:05:28,180
So that's a left asymptote.
81
00:05:28,180 --> 00:05:31,530
Or, you could have
a right asymptote,
82
00:05:31,530 --> 00:05:34,870
and it looks like this.
83
00:05:34,870 --> 00:05:38,350
Again, second derivative
positive, positive wave
84
00:05:38,350 --> 00:05:38,890
function.
85
00:05:38,890 --> 00:05:41,710
Second derivative negative,
negative wave function.
86
00:05:41,710 --> 00:05:45,500
So, you may find this at
the middle of the potential,
87
00:05:45,500 --> 00:05:48,250
but then eventually
something has to take over.
88
00:05:48,250 --> 00:05:52,720
Or, you may find this behavior,
or this behavior, at plus minus
89
00:05:52,720 --> 00:05:54,040
infinity.
90
00:05:54,040 --> 00:05:59,380
But, in any case you are
in a classically forbidden,
91
00:05:59,380 --> 00:06:02,559
you're convex towards the axis.
92
00:06:02,559 --> 00:06:04,100
That's the thing
you should remember.
93
00:06:08,718 --> 00:06:17,550
On the other hand, we can be on
the classically allowed region.
94
00:06:24,540 --> 00:06:26,357
So, let's think of that.
95
00:06:29,040 --> 00:06:32,380
Any questions about the
classically forbidden?
96
00:06:41,110 --> 00:06:49,590
Classically allowed, B. E
minus v of x greater than 0,
97
00:06:49,590 --> 00:06:53,725
classically allowed.
98
00:06:56,920 --> 00:07:02,180
On the right hand side of
the equation is negative.
99
00:07:02,180 --> 00:07:12,480
So, you can have, one,
a psi that is positive,
100
00:07:12,480 --> 00:07:18,760
and a second derivative
that is negative.
101
00:07:18,760 --> 00:07:27,400
Or, two, a psi that is negative,
and a second derivative
102
00:07:27,400 --> 00:07:31,330
that this positive.
103
00:07:31,330 --> 00:07:34,760
So, how does that look?
104
00:07:34,760 --> 00:07:38,260
Well, positive and second
derivative negative,
105
00:07:38,260 --> 00:07:42,070
I think of some wave function
as positive, and negative
106
00:07:42,070 --> 00:07:43,620
is parabolic like that.
107
00:07:48,500 --> 00:07:53,400
And then, negative and
second derivative positive,
108
00:07:53,400 --> 00:07:57,990
it's possible to have this.
109
00:07:57,990 --> 00:08:00,790
The wave function
there it's negative,
110
00:08:00,790 --> 00:08:04,020
but the second
derivative is positive.
111
00:08:04,020 --> 00:08:06,690
These things are not very good--
112
00:08:06,690 --> 00:08:10,440
they're not very
usable asymptotically,
113
00:08:10,440 --> 00:08:13,740
because eventually
if you are like this,
114
00:08:13,740 --> 00:08:15,330
you will cross these points.
115
00:08:15,330 --> 00:08:18,300
And then, if you're still
in the allowed region
116
00:08:18,300 --> 00:08:19,050
you have to shift.
117
00:08:19,050 --> 00:08:23,340
But, this is done nicely in a
sense if you put it together
118
00:08:23,340 --> 00:08:25,425
you can have this.
119
00:08:28,140 --> 00:08:30,180
Suppose all of this is
classically allowed.
120
00:08:30,180 --> 00:08:33,390
Then you can have the wave
function being positive,
121
00:08:33,390 --> 00:08:35,370
the second derivative
being negative,
122
00:08:35,370 --> 00:08:38,120
matching nicely
with the other half.
123
00:08:38,120 --> 00:08:41,299
The second derivative positive,
the wave function negative,
124
00:08:41,299 --> 00:08:43,380
and that's what the
psi function is.
125
00:08:46,160 --> 00:08:49,030
It just goes one after another.
126
00:08:49,030 --> 00:08:51,880
So, that's what
typically things look
127
00:08:51,880 --> 00:08:55,820
in the classically
allowed region.
128
00:08:55,820 --> 00:09:03,770
So, in this case, we say that
it's concave towards the axis.
129
00:09:15,910 --> 00:09:19,480
That's probably
worth remembering.
130
00:09:19,480 --> 00:09:25,270
So, one more case.
131
00:09:25,270 --> 00:09:37,210
The case C, when E is equal to E
minus v of x not is equal to 0.
132
00:09:37,210 --> 00:09:41,680
So, we have the negative,
the positive, 0.
133
00:09:41,680 --> 00:09:45,550
How about when you have
the situation where
134
00:09:45,550 --> 00:09:49,060
the potential at some point
is equal to the energy?
135
00:09:49,060 --> 00:09:52,150
Well, that's the
turning points there--
136
00:09:52,150 --> 00:09:54,890
those were our turning points.
137
00:09:54,890 --> 00:10:01,530
So, this is how x 0
is a turning point.
138
00:10:08,550 --> 00:10:12,440
And, something else happens,
see, the right hand side is 0.
139
00:10:16,380 --> 00:10:21,500
We have that one over
psi, the second psi,
140
00:10:21,500 --> 00:10:26,550
the x squared is equal to 0.
141
00:10:26,550 --> 00:10:34,150
And, if psi is different
from 0, then you
142
00:10:34,150 --> 00:10:41,400
have the second derivative
must be 0 at x not.
143
00:10:46,210 --> 00:10:50,450
And, the second derivative
being 0 is an inflection point.
144
00:11:00,590 --> 00:11:11,820
So, if you have a wave function
that has an inflection point,
145
00:11:11,820 --> 00:11:16,950
you have a sign that you've
reached a turning point.
146
00:11:16,950 --> 00:11:19,360
An inflection point
in a wave function
147
00:11:19,360 --> 00:11:24,190
could be anything like that.
148
00:11:24,190 --> 00:11:28,090
Second derivative
is positive here--
149
00:11:28,090 --> 00:11:32,430
I'm sorry-- is negative here,
second derivative is positive,
150
00:11:32,430 --> 00:11:34,036
this is an inflection point.
151
00:11:37,020 --> 00:11:40,870
It's a point where the
second derivative vanishes.
152
00:11:40,870 --> 00:11:43,600
So, that's an inflection point.
153
00:11:43,600 --> 00:11:48,240
And, it should be remarked
that from that differential
154
00:11:48,240 --> 00:11:53,940
equation, you also get that
the second psi, the x squared,
155
00:11:53,940 --> 00:12:00,670
is equal to E minus v times
psi, which is constant.
156
00:12:00,670 --> 00:12:04,240
And, therefore,
when psi vanishes,
157
00:12:04,240 --> 00:12:08,700
you also get inflection
points automatically
158
00:12:08,700 --> 00:12:10,790
because the second
derivative vanishes.
159
00:12:10,790 --> 00:12:19,360
So, inflection points
also at the nodes.
160
00:12:25,070 --> 00:12:27,580
Turning point is
an inflection point
161
00:12:27,580 --> 00:12:29,910
where you have this situation.
162
00:12:29,910 --> 00:12:37,060
Look here, you have negative
second derivative, positive
163
00:12:37,060 --> 00:12:39,490
second derivative, the
point where the wave
164
00:12:39,490 --> 00:12:44,550
function vanishes and joins them
is an inflection point as well.
165
00:12:44,550 --> 00:12:46,690
Is not the turning point--
166
00:12:46,690 --> 00:12:48,740
turning point are
more interesting--
167
00:12:48,740 --> 00:12:52,420
but inflection points
are more generic.