1
00:00:00,499 --> 00:00:03,030
PROFESSOR: So let's write
the Hamiltonian again
2
00:00:03,030 --> 00:00:07,490
in terms of v and v dagger.
3
00:00:07,490 --> 00:00:13,020
So for this equation, v
dagger v, from this equation,
4
00:00:13,020 --> 00:00:19,982
is equal to 2 h bar over
m omega, a dagger a.
5
00:00:22,530 --> 00:00:27,885
And immediately above
equation v dagger v's, this--
6
00:00:27,885 --> 00:00:34,420
we substitute into
the Hamiltonian.
7
00:00:34,420 --> 00:00:42,870
And Hamiltonian becomes the nice
object h bar omega, a dagger
8
00:00:42,870 --> 00:00:51,680
a plus 1/2 if you want.
9
00:00:56,580 --> 00:00:57,550
All right.
10
00:00:57,550 --> 00:01:00,870
We did this hard work
of factorization.
11
00:01:00,870 --> 00:01:03,390
We have to show what's good for.
12
00:01:03,390 --> 00:01:05,430
Well, in fact, we're
going to be able to solve
13
00:01:05,430 --> 00:01:08,280
the harmonic oscillator
without ever talking
14
00:01:08,280 --> 00:01:12,250
about differential-- almost
ever talking about differential
15
00:01:12,250 --> 00:01:12,750
equations.
16
00:01:12,750 --> 00:01:16,830
In fact, we will not talk about
the second order differential
17
00:01:16,830 --> 00:01:19,190
equation.
18
00:01:19,190 --> 00:01:21,292
Thanks to our
great work here, we
19
00:01:21,292 --> 00:01:24,640
will have to talk about a first
order differential equation,
20
00:01:24,640 --> 00:01:26,600
and a much simpler one.
21
00:01:26,600 --> 00:01:31,130
And only one, not for
and n 1, 2, 3, infitinty,
22
00:01:31,130 --> 00:01:33,480
infinity number of polynomials.
23
00:01:33,480 --> 00:01:36,060
It's a great simplification.
24
00:01:36,060 --> 00:01:39,660
Other Hamiltonians
admit factorization.
25
00:01:39,660 --> 00:01:44,070
In fact, there's whole books
of factorizable Hamiltonians,
26
00:01:44,070 --> 00:01:49,360
because those are the nicest
Hamiltonians to solve.
27
00:01:49,360 --> 00:01:50,920
Let's see why, though.
28
00:01:50,920 --> 00:01:54,620
We haven't said why yet.
29
00:01:54,620 --> 00:01:59,040
Here is the leading
thing that we can do.
30
00:01:59,040 --> 00:02:06,460
Remember we recalled 5
psi the integral dx phi
31
00:02:06,460 --> 00:02:11,314
star of x psi of x.
32
00:02:11,314 --> 00:02:15,090
This is just notation.
33
00:02:15,090 --> 00:02:21,200
So the expectation value,
calculate the expectation value
34
00:02:21,200 --> 00:02:25,050
of the Hamiltonian
in some state psi.
35
00:02:28,184 --> 00:02:33,100
Could be the general state sum.
36
00:02:33,100 --> 00:02:35,526
So what are you supposed to do?
37
00:02:35,526 --> 00:02:44,030
You're supposed to do psi h psi.
38
00:02:44,030 --> 00:02:47,670
This is normalized state,
the expectation value
39
00:02:47,670 --> 00:02:50,840
is the integral
of psi star h psi.
40
00:02:50,840 --> 00:02:52,700
That's what this is.
41
00:02:52,700 --> 00:02:56,300
But now, let's put
in this information.
42
00:03:02,992 --> 00:03:11,600
And the expectation value
of this would be psi h--
43
00:03:11,600 --> 00:03:16,530
or let me do it this way-- h
bar ome-- well, let's go slow.
44
00:03:16,530 --> 00:03:33,860
Psi h omega a dagger a a
psi plus h omega over 2 psi.
45
00:03:39,100 --> 00:03:43,910
So I just calculated
h on psi, and I
46
00:03:43,910 --> 00:03:47,980
wrote what it is-- h
omega this, this term.
47
00:03:47,980 --> 00:03:50,890
So this is two terms--
48
00:03:50,890 --> 00:04:04,236
h omega psi a dagger a psi
plus h omega over 2 psi psi.
49
00:04:10,910 --> 00:04:11,940
OK.
50
00:04:11,940 --> 00:04:15,660
So what did I gain
with the factorization?
51
00:04:15,660 --> 00:04:18,019
So far, it looks like nothing.
52
00:04:18,019 --> 00:04:21,420
But here we go--
53
00:04:21,420 --> 00:04:26,240
this term is equal to 1,
because the wave function
54
00:04:26,240 --> 00:04:27,800
is normalized.
55
00:04:27,800 --> 00:04:30,110
And here I can do one thing--
56
00:04:30,110 --> 00:04:37,210
I can remember my definition
of a Hermitian conjugate.
57
00:04:37,210 --> 00:04:40,880
I can move an operator and
put its Hermitian conjugate
58
00:04:40,880 --> 00:04:42,810
on the other side.
59
00:04:42,810 --> 00:04:48,596
So think of this
operator, a dagger--
60
00:04:48,596 --> 00:04:53,550
a dagger is acting on
this wave function.
61
00:04:53,550 --> 00:04:55,050
What is a dagger?
62
00:04:55,050 --> 00:04:56,030
It's this.
63
00:04:56,030 --> 00:04:59,190
And p is h bar over idex.
64
00:04:59,190 --> 00:05:02,030
So this-- you know how to act.
65
00:05:02,030 --> 00:05:06,770
But if a dagger is here, I
can put it on the first wave
66
00:05:06,770 --> 00:05:10,630
function, but I must put
the dagger of this operator,
67
00:05:10,630 --> 00:05:13,550
and the dagger of a dagger is a.
68
00:05:13,550 --> 00:05:26,165
So this is h omega, a psi
a psi plus h omega over 2.
69
00:05:31,270 --> 00:05:33,700
Now here comes the next thing.
70
00:05:33,700 --> 00:05:40,630
If this is an inner
product, any phi phi
71
00:05:40,630 --> 00:05:45,375
is greater or equal than 0,
because you would have phi star
72
00:05:45,375 --> 00:05:48,670
phi, and that's positive.
73
00:05:48,670 --> 00:05:52,050
So any of that [INAUDIBLE]
is greater or equal than 0.
74
00:05:52,050 --> 00:05:55,210
Note-- here you have
some function, but here
75
00:05:55,210 --> 00:05:57,100
the same function.
76
00:05:57,100 --> 00:05:58,660
It is this case.
77
00:05:58,660 --> 00:06:04,060
That thing is greater
or equal than 0.
78
00:06:04,060 --> 00:06:08,540
That is the great benefit of
the factorized Hamiltonian--
79
00:06:08,540 --> 00:06:14,455
if h has a v dagger v, you
can flip the v dagger here
80
00:06:14,455 --> 00:06:18,830
and it becomes v psi v
psi, and it's positive.
81
00:06:18,830 --> 00:06:21,920
And you've learned
something very important,
82
00:06:21,920 --> 00:06:25,210
and you can get
positive energies.
83
00:06:25,210 --> 00:06:29,090
In fact, from here,
since this is positive,
84
00:06:29,090 --> 00:06:34,540
this must be greater or
equal than h omega over 2.
85
00:06:34,540 --> 00:06:38,530
Because this is greater
than equal than 0.
86
00:06:38,530 --> 00:06:43,720
So the expectation value
of the Hamiltonian--
87
00:06:43,720 --> 00:06:49,130
if you would be thinking
now of energy eigenstates,
88
00:06:49,130 --> 00:06:52,750
the energy eigenvalue
is the expectation value
89
00:06:52,750 --> 00:06:55,240
of the Hamiltonian in
an energy eigenstate
90
00:06:55,240 --> 00:07:00,680
must be greater than
h bar omega over 2.
91
00:07:00,680 --> 00:07:05,710
And in some blackboard
that has been erased,
92
00:07:05,710 --> 00:07:10,820
we remember that the lowest
energy state had energy--
93
00:07:10,820 --> 00:07:12,250
there it is.
94
00:07:12,250 --> 00:07:18,010
The lowest energy state
has energy h omega over 2.
95
00:07:18,010 --> 00:07:23,860
So look at this, and
you say, OK, this
96
00:07:23,860 --> 00:07:29,590
shows that any eigenstate
must have energy
97
00:07:29,590 --> 00:07:33,060
greater than h omega over 2.
98
00:07:33,060 --> 00:07:39,520
But could there be one state
for which the energies exactly
99
00:07:39,520 --> 00:07:41,920
h omega over 2.
100
00:07:41,920 --> 00:07:46,010
Yes, if this inner product is 0.
101
00:07:46,010 --> 00:07:49,760
But for an inner product
of two things to be 0,
102
00:07:49,760 --> 00:07:52,270
each function must be 0.
103
00:07:52,270 --> 00:08:01,160
So from this, we conclude that
if there is a ground state,
104
00:08:01,160 --> 00:08:05,940
it's a state for which a phi--
105
00:08:05,940 --> 00:08:10,750
or a psi is equal to 0.
106
00:08:10,750 --> 00:08:14,880
So this is a very
nice conclusion.
107
00:08:14,880 --> 00:08:29,240
So if the lower
bound is realized,
108
00:08:29,240 --> 00:08:37,980
so that you get a state with
energy equal h bar over 2,
109
00:08:37,980 --> 00:08:46,733
then it must be true
that a psi is equal to 0.
110
00:08:46,733 --> 00:08:59,410
And a psi equal to 0 means x
plus ip over m omega on psi
111
00:08:59,410 --> 00:09:07,340
is equal to 0, or x plus
p is h bar over i ddx,
112
00:09:07,340 --> 00:09:16,050
so this is h bar over m omega
d/dx on psi of x is equal to 0.
113
00:09:16,050 --> 00:09:20,700
And that was the promised fact.
114
00:09:20,700 --> 00:09:25,940
We have turned the second
order differential equation
115
00:09:25,940 --> 00:09:29,070
into a first order
differential equation.
116
00:09:29,070 --> 00:09:32,630
Think of that magic that
has happened to do that.
117
00:09:32,630 --> 00:09:35,305
You had a second order
differential equation
118
00:09:35,305 --> 00:09:39,480
because the Hamiltonian
has x squared b squared.
119
00:09:39,480 --> 00:09:45,960
By factorizing, you go two first
order differential operators.
120
00:09:45,960 --> 00:09:50,420
And by Hermeticity, you
were led to the condition
121
00:09:50,420 --> 00:09:55,500
that the lowest energy
state had to be killed by a.
122
00:09:55,500 --> 00:09:59,760
That's why a is called
the annihilation operator.
123
00:09:59,760 --> 00:10:01,010
It should be killed.
124
00:10:01,010 --> 00:10:04,065
And now you have to solve
a first order differential
125
00:10:04,065 --> 00:10:08,640
equation, which is a game.
126
00:10:08,640 --> 00:10:12,570
An easy game compared with
a second order differential
127
00:10:12,570 --> 00:10:14,020
equation.
128
00:10:14,020 --> 00:10:16,110
So let's, of course, solve it.
129
00:10:16,110 --> 00:10:18,450
It doesn't take any time.
130
00:10:21,030 --> 00:10:24,270
Let's call this
the ground state.
131
00:10:24,270 --> 00:10:27,190
If it exists.
132
00:10:27,190 --> 00:10:35,850
And this gives you d psi 0 v x
is equal to minus m omega over
133
00:10:35,850 --> 00:10:38,665
h bar x psi .
134
00:10:38,665 --> 00:10:39,165
0.
135
00:10:46,410 --> 00:10:50,920
This can be degraded easily
or you can guess the answer.
136
00:10:50,920 --> 00:10:52,110
It's an exponential.
137
00:10:52,110 --> 00:10:55,230
Anything that differentiates
that you should extend
138
00:10:55,230 --> 00:10:57,300
the same function
as an exponential--
139
00:10:57,300 --> 00:11:05,310
e to the minus m omega
2 h squared x squared
140
00:11:05,310 --> 00:11:06,290
is the solution.
141
00:11:06,290 --> 00:11:11,670
Psi 0 of x is equal to
some number times that.
142
00:11:11,670 --> 00:11:16,620
This was-- the number is
the Hermit polynomials
143
00:11:16,620 --> 00:11:20,125
sub 0, and that exponential,
this exponential,
144
00:11:20,125 --> 00:11:23,430
we wrote a few blackboards ago.
145
00:11:23,430 --> 00:11:24,702
It's a good exponential.
146
00:11:24,702 --> 00:11:28,500
It's a perfect Gaussian.
147
00:11:28,500 --> 00:11:31,250
It's our ground state.
148
00:11:31,250 --> 00:11:34,110
And 0, if you want
to normalize it,
149
00:11:34,110 --> 00:11:40,970
m 0 is equal to m omega
over phi h bar to the 1/4.
150
00:11:45,650 --> 00:11:49,430
And that is the ground state.
151
00:11:49,430 --> 00:11:55,010
And it has energy,
h omega over 2.
152
00:11:55,010 --> 00:11:58,520
You could see what
the energy is by doing
153
00:11:58,520 --> 00:12:00,480
this very simple calculation.
154
00:12:00,480 --> 00:12:03,910
Look, get accustomed
to these things.
155
00:12:03,910 --> 00:12:06,690
H hat psi 0.
156
00:12:06,690 --> 00:12:07,690
What is h?
157
00:12:07,690 --> 00:12:17,285
Is h omega a dagger a
plus 1/2 acting on psi 0.
158
00:12:20,300 --> 00:12:25,120
The a acting on psi
0 already kills it.
159
00:12:25,120 --> 00:12:26,920
Because that's the
defining equation.
160
00:12:26,920 --> 00:12:28,600
Well that's 0.
161
00:12:28,600 --> 00:12:37,030
And you get 1/2 h bar omega,
confirming that you did
162
00:12:37,030 --> 00:12:44,480
get this thing to be correct.
163
00:12:44,480 --> 00:12:50,040
So this is only the
beginning of the story.
164
00:12:50,040 --> 00:12:52,690
We found the ground
state, and now we
165
00:12:52,690 --> 00:12:54,760
have to find the excited states.
166
00:12:54,760 --> 00:12:57,880
Let me say a couple
of words to set up
167
00:12:57,880 --> 00:13:00,012
this discussion for next time.
168
00:13:04,670 --> 00:13:08,640
The excited states appear
in a very nice way as well.
169
00:13:12,608 --> 00:13:23,612
So first a tiny bit
of language, of h bar.
170
00:13:23,612 --> 00:13:29,350
This equal h omega,
a dagger a is usually
171
00:13:29,350 --> 00:13:31,070
called the number operator.
172
00:13:31,070 --> 00:13:33,870
We'll explain more
on that next time.
173
00:13:33,870 --> 00:13:37,985
So n number operator
is a dagger a.
174
00:13:37,985 --> 00:13:42,620
It's a permission operator,
and it's pretty much
175
00:13:42,620 --> 00:13:43,572
the Hamiltonian.
176
00:13:43,572 --> 00:13:45,380
It's the number, it's called.
177
00:13:45,380 --> 00:13:49,250
Why is it called the number
is what we have to figure out.
178
00:13:49,250 --> 00:13:53,030
It is a counting operator--
it just looks at the state
179
00:13:53,030 --> 00:13:56,990
and counts things.
180
00:13:56,990 --> 00:13:59,270
So what does this give us?
181
00:13:59,270 --> 00:14:03,080
Well, we also know that
the number operator kills
182
00:14:03,080 --> 00:14:07,137
phi 0, because a kills psi 0.
183
00:14:10,480 --> 00:14:11,450
A kills it.
184
00:14:11,450 --> 00:14:13,890
So that's what we have.
185
00:14:13,890 --> 00:14:20,270
So we did say that
a was a destruction
186
00:14:20,270 --> 00:14:25,710
operator, annihilation
operator, because it annihilates
187
00:14:25,710 --> 00:14:28,280
the ground state.
188
00:14:28,280 --> 00:14:32,800
So if a annihilates
the ground state,
189
00:14:32,800 --> 00:14:36,290
a dagger cannot annihilate
the ground state.
190
00:14:36,290 --> 00:14:37,790
Why?
191
00:14:37,790 --> 00:14:44,130
Because a dagger with a
computator is equal to 1.
192
00:14:44,130 --> 00:14:45,070
Look at this.
193
00:14:45,070 --> 00:14:49,520
This is a a dagger
minus a dagger a.
194
00:14:49,520 --> 00:14:51,140
Act on the ground state.
195
00:14:54,804 --> 00:14:56,500
That's it.
196
00:14:56,500 --> 00:14:59,760
Now this term kills it.
197
00:14:59,760 --> 00:15:01,840
But this term
better not kill it,
198
00:15:01,840 --> 00:15:03,900
because it has to give
you back the ground
199
00:15:03,900 --> 00:15:05,230
state if this is true.
200
00:15:05,230 --> 00:15:06,950
And this is true.
201
00:15:06,950 --> 00:15:11,080
So a dagger doesn't
kill the ground state.
202
00:15:11,080 --> 00:15:16,000
Since it doesn't kill it, it's
called a creation operator.
203
00:15:16,000 --> 00:15:20,590
So you have this
state, but now there's
204
00:15:20,590 --> 00:15:25,710
also this state a dagger
acting on the vacuum.
205
00:15:25,710 --> 00:15:31,130
And there's a state a dagger
a dagger acting on the vacuum.
206
00:15:31,130 --> 00:15:33,480
And all those.
207
00:15:33,480 --> 00:15:35,950
And what we will
figure out next time
208
00:15:35,950 --> 00:15:38,780
is that, yes, this
is the ground state.
209
00:15:38,780 --> 00:15:40,890
And this is the
first excited state.
210
00:15:40,890 --> 00:15:43,280
And this is the
second excited state.
211
00:15:43,280 --> 00:15:45,150
And goes on forever.
212
00:15:45,150 --> 00:15:49,440
So we'll have a very compact
formula for the excited states
213
00:15:49,440 --> 00:15:51,310
of the harmonic oscillator.
214
00:15:51,310 --> 00:15:56,260
They're just creation
operators acting on the ground
215
00:15:56,260 --> 00:15:58,620
state or the [INAUDIBLE].