1
00:00:00,000 --> 00:00:12,580
PROFESSOR: Would solving this
equation for some potential,
2
00:00:12,580 --> 00:00:20,054
and since h is Hermitian,
we found the results
3
00:00:20,054 --> 00:00:22,230
that we mentioned last time.
4
00:00:22,230 --> 00:00:27,950
That is the
eigenfunctions of h are
5
00:00:27,950 --> 00:00:33,970
going to form an
orthonormal set of functions
6
00:00:33,970 --> 00:00:35,300
that span the space.
7
00:00:35,300 --> 00:00:38,690
You can expand
anything on there.
8
00:00:38,690 --> 00:00:43,460
This is what we proved for
a general condition operator
9
00:00:43,460 --> 00:00:44,540
to some degree.
10
00:00:44,540 --> 00:01:03,082
So the eigenfunctions
form an orthonormal set
11
00:01:03,082 --> 00:01:09,960
that spans the space.
12
00:01:09,960 --> 00:01:14,110
So you're going to define
that psi 1 with an E1
13
00:01:14,110 --> 00:01:22,762
and psi 2 with an E2,
and then this continues.
14
00:01:22,762 --> 00:01:33,140
And this is called the
spectrum of the theory
15
00:01:33,140 --> 00:01:38,970
because energy eigenstates are
considered the gold standard.
16
00:01:38,970 --> 00:01:42,088
If you want to find
solving a theory
17
00:01:42,088 --> 00:01:45,984
means finding the
energy eigenstates.
18
00:01:45,984 --> 00:01:50,120
Because if you find the energy
eigenstates, you can solve,
19
00:01:50,120 --> 00:01:52,960
you can write any wave
function of superposition
20
00:01:52,960 --> 00:01:57,334
of energetic states and
then just let them evolve.
21
00:01:57,334 --> 00:02:01,230
And the energetic
states involve easily
22
00:02:01,230 --> 00:02:04,863
because they are just
stationary states.
23
00:02:04,863 --> 00:02:09,199
So the spectrum of the
theory is the collection
24
00:02:09,199 --> 00:02:13,980
of numbers that are the
allowed energies and of course,
25
00:02:13,980 --> 00:02:17,295
the associated eigenfunctions.
26
00:02:17,295 --> 00:02:22,280
So the energies may be many,
maybe discrete, maybe it
27
00:02:22,280 --> 00:02:25,550
has a little bit of
continuous partners,
28
00:02:25,550 --> 00:02:27,560
all kind of varieties.
29
00:02:27,560 --> 00:02:32,340
But your task is to find
those for any problem.
30
00:02:32,340 --> 00:02:37,100
So the equation that
we're trying to solve
31
00:02:37,100 --> 00:02:39,570
is now re-written.
32
00:02:39,570 --> 00:02:42,265
We're going to try to solve it.
33
00:02:45,199 --> 00:02:49,754
So let's look at it.
34
00:02:53,163 --> 00:02:57,624
It's a second order
differential equation
35
00:02:57,624 --> 00:03:00,660
with a potential in general.
36
00:03:00,660 --> 00:03:04,040
So we had an example there.
37
00:03:04,040 --> 00:03:04,786
It's there.
38
00:03:04,786 --> 00:03:06,450
It's boxed.
39
00:03:06,450 --> 00:03:10,270
So we'll write it
slightly different,
40
00:03:10,270 --> 00:03:13,800
remove the potential
to the right-hand side
41
00:03:13,800 --> 00:03:16,890
and get rid of the
constants here.
42
00:03:16,890 --> 00:03:46,540
So d x squared is equal
to 2m over h squared.
43
00:04:24,110 --> 00:04:26,100
So this is the equation
we have to solve.
44
00:04:33,650 --> 00:04:35,760
So whenever you
have a problem, you
45
00:04:35,760 --> 00:04:39,963
may encounter a
potential, v of x.
46
00:04:39,963 --> 00:04:43,440
And the question is how
bad this potential can be.
47
00:04:43,440 --> 00:04:53,650
Well, the potential may be nice
and simple, or it may be nice
48
00:04:53,650 --> 00:04:57,620
but then has some jumps.
49
00:04:57,620 --> 00:05:02,950
It may have infinite
jumps, like a potential is
50
00:05:02,950 --> 00:05:12,040
a complete barrier, or it
may have delta functions.
51
00:05:15,456 --> 00:05:19,165
all these are v of
x equal possibles.
52
00:05:23,940 --> 00:05:26,270
All of them.
53
00:05:26,270 --> 00:05:28,370
Many things can happen
with a potential.
54
00:05:28,370 --> 00:05:31,830
In fact, the potential
can be as strange
55
00:05:31,830 --> 00:05:35,610
as you're one, depending on
what problems you want to solve.
56
00:05:35,610 --> 00:05:39,930
So it's your choice.
57
00:05:39,930 --> 00:05:46,700
Now, we're going to accept, in
fact, all of those potentials
58
00:05:46,700 --> 00:05:48,510
for our analysis.
59
00:05:48,510 --> 00:05:50,240
May be nice and smooth.
60
00:05:50,240 --> 00:05:52,560
There may have discontinuities.
61
00:05:52,560 --> 00:05:55,466
It may have infinite
discontinuities,
62
00:05:55,466 --> 00:05:58,630
and worse things
like delta function.
63
00:05:58,630 --> 00:06:04,420
But worse things than
that we will ignore,
64
00:06:04,420 --> 00:06:08,640
and there are worse
things than that.
65
00:06:08,640 --> 00:06:14,710
Maybe a potential discontinues
at every point, or maybe
66
00:06:14,710 --> 00:06:18,060
a potential has delta
functions and derivatives
67
00:06:18,060 --> 00:06:20,760
of delta functions.
68
00:06:20,760 --> 00:06:26,490
Or potentials that blow up
and do all kinds of things.
69
00:06:26,490 --> 00:06:30,600
And I'm not saying you
should never consider that.
70
00:06:30,600 --> 00:06:35,120
I'm saying that we don't know
of any very useful case where
71
00:06:35,120 --> 00:06:37,500
you get anything
interesting with that.
72
00:06:37,500 --> 00:06:41,810
But a conceivable a particular
time a singular potential one
73
00:06:41,810 --> 00:06:44,530
day could be used.
74
00:06:44,530 --> 00:06:46,912
So we'll look at
these potentials
75
00:06:46,912 --> 00:07:00,390
and try to understand how to
set up boundary conditions.
76
00:07:00,390 --> 00:07:05,796
And we're going to worry
about basically psi
77
00:07:05,796 --> 00:07:08,820
and how does it behave.
78
00:07:08,820 --> 00:07:15,996
And my first claim is that
psi of x has to be continuous.
79
00:07:23,870 --> 00:07:26,385
So psi of x cannot jump.
80
00:07:26,385 --> 00:07:32,090
The wave function move
along but cannot jump.
81
00:07:32,090 --> 00:07:35,960
And the reason is a
differential equation.
82
00:07:35,960 --> 00:07:42,915
Look, if psi of x
was not continuous,
83
00:07:42,915 --> 00:07:47,520
if psi of x was
like this, and just
84
00:07:47,520 --> 00:07:53,290
had a discontinuity,
psi of x equal to x,
85
00:07:53,290 --> 00:07:58,466
psi prime of x would
contain a delta function
86
00:07:58,466 --> 00:08:00,831
and this is continuity.
87
00:08:00,831 --> 00:08:04,620
The derivative is infinite.
88
00:08:04,620 --> 00:08:08,440
And psi double prime of
x, the second derivative,
89
00:08:08,440 --> 00:08:13,350
would have a derivative of a
delta function which is worse
90
00:08:13,350 --> 00:08:16,570
because a delta
function, we think of it
91
00:08:16,570 --> 00:08:20,064
as a spike that is becoming
thinner and higher,
92
00:08:20,064 --> 00:08:24,740
but the derivative of the delta
function first goes to infinity
93
00:08:24,740 --> 00:08:30,710
and then goes to minus infinity
and then comes back up.
94
00:08:30,710 --> 00:08:35,669
It's much worse in many ways.
95
00:08:35,669 --> 00:08:39,679
And look, if you have
this differential equation
96
00:08:39,679 --> 00:08:45,300
and psi is not continuous,
well, the right-hand side
97
00:08:45,300 --> 00:08:46,920
is not continuous.
98
00:08:49,626 --> 00:08:51,790
Or if you have a
delta function, then
99
00:08:51,790 --> 00:08:55,810
something not continuous,
but left-hand side,
100
00:08:55,810 --> 00:08:59,630
we've had a derivative of a
delta function that is nowhere
101
00:08:59,630 --> 00:09:02,000
on the right-hand side.
102
00:09:02,000 --> 00:09:05,510
On the right-hand side,
the worst that could exist
103
00:09:05,510 --> 00:09:09,540
is a delta function in v of x.
104
00:09:09,540 --> 00:09:12,950
But the derivative of a
delta function doesn't exist.
105
00:09:12,950 --> 00:09:18,477
So you cannot afford to have
a psi that is discontinuous.
106
00:09:18,477 --> 00:09:22,880
Psi has to be continuous.
107
00:09:22,880 --> 00:09:24,680
There's other ways
to argue this.
108
00:09:24,680 --> 00:09:27,090
You might put them
in your notes,
109
00:09:27,090 --> 00:09:30,488
but I'll leave it like that.
110
00:09:35,420 --> 00:09:38,020
Now how about the next case?
111
00:09:38,020 --> 00:09:40,560
I will say the
following happens too.
112
00:09:40,560 --> 00:09:53,510
Sine prime of x is
continuous unless v of x
113
00:09:53,510 --> 00:09:55,188
has a delta function.
114
00:10:00,424 --> 00:10:05,950
You see, potentials of
delta functions are nice,
115
00:10:05,950 --> 00:10:07,769
they are interesting.
116
00:10:07,769 --> 00:10:09,290
We will consider that.
117
00:10:09,290 --> 00:10:12,190
Delta functions potentials
can be attractive
118
00:10:12,190 --> 00:10:15,830
potentials, repulsive
potentials of [INAUDIBLE].
119
00:10:15,830 --> 00:10:22,430
So I claim now that psi prime
of x has to also be continuous.
120
00:10:22,430 --> 00:10:25,170
Why are we worrying
about psi and psi
121
00:10:25,170 --> 00:10:28,835
prime is because you need two
conditions whenever you're
122
00:10:28,835 --> 00:10:33,230
going to solve this differential
equation at an interface,
123
00:10:33,230 --> 00:10:35,550
you will need to know
psi is continuous
124
00:10:35,550 --> 00:10:37,550
and psi prime is
continuous because
125
00:10:37,550 --> 00:10:41,500
of second-order
differential equations.
126
00:10:41,500 --> 00:10:48,250
So suppose psi
prime is continuous.
127
00:10:48,250 --> 00:10:52,030
Then there is no problem.
128
00:10:52,030 --> 00:10:55,856
If psi prime is continuous,
the worse that can happen
129
00:10:55,856 --> 00:10:59,200
is that the second
derivative is discontinuous.
130
00:10:59,200 --> 00:11:01,900
And the second derivative
is discontinuous
131
00:11:01,900 --> 00:11:07,330
could happen with a potential
of this discontinuous,
132
00:11:07,330 --> 00:11:12,644
so one problem if psi
prime is continuous.
133
00:11:12,644 --> 00:11:18,100
But psi prime can fail to be
continuous if the potential has
134
00:11:18,100 --> 00:11:19,585
a delta function.
135
00:11:19,585 --> 00:11:21,266
And let's see that.
136
00:11:21,266 --> 00:11:32,760
If psi prime is
discontinuous, then
137
00:11:32,760 --> 00:11:36,724
psi double prime is proportional
to a delta function.
138
00:11:42,158 --> 00:11:47,250
If psi prime is
discontinuous, double prime
139
00:11:47,250 --> 00:11:50,194
is proportional to
a delta function.
140
00:11:50,194 --> 00:11:54,302
But here psi just
takes some value--
141
00:11:54,302 --> 00:11:55,852
there's nothing
strange about it--
142
00:11:55,852 --> 00:12:01,420
in order to have delta function,
which is psi double prime.
143
00:12:01,420 --> 00:12:05,230
To be equal to the
right-hand side, v of x
144
00:12:05,230 --> 00:12:07,216
must have a delta function.
145
00:12:10,618 --> 00:12:19,239
And v will have
a delta function.
146
00:12:19,239 --> 00:12:23,002
So it will be a somewhat
similar potential,
147
00:12:23,002 --> 00:12:30,790
but we're going to look at
them in about a week from now.
148
00:12:30,790 --> 00:12:36,444
But this will be our
guidance to solve problems.
149
00:12:36,444 --> 00:12:38,980
The continuity of
the wave function
150
00:12:38,980 --> 00:12:41,310
and the continuity of the
derivative of the wave
151
00:12:41,310 --> 00:12:41,810
function.
152
00:12:41,810 --> 00:12:48,060
And for this slightly
more complicated problems
153
00:12:48,060 --> 00:12:51,690
in which the potential
has a delta function,
154
00:12:51,690 --> 00:12:56,070
then you will have a
discontinuity in psi prime,
155
00:12:56,070 --> 00:12:59,130
and it will be calculable,
and it's manageable,
156
00:12:59,130 --> 00:13:00,810
and it's all very nice.
157
00:13:00,810 --> 00:13:02,695
Now, we do it a
little complicated,
158
00:13:02,695 --> 00:13:06,480
and everything is
mixed up, but you will
159
00:13:06,480 --> 00:13:09,550
see that it's quite doable.