1
00:00:00,499 --> 00:00:02,150
PROFESSOR: The simplest
quantum system.

2
00:00:13,810 --> 00:00:18,190
In order to decide what could
be the simplest quantum system

3
00:00:18,190 --> 00:00:20,320
you could say a
particle in a box.

4
00:00:20,320 --> 00:00:24,640
It's very simple, but in a
sense it's not all that simple.

5
00:00:24,640 --> 00:00:29,410
It has infinitely many states.

6
00:00:29,410 --> 00:00:34,650
All these functions on an
interval, and then the energy

7
00:00:34,650 --> 00:00:40,390
is where infinitely many of
them, so not that simple.

8
00:00:40,390 --> 00:00:44,420
OK, infinite bound says
something with one bound.

9
00:00:44,420 --> 00:00:48,880
OK, a delta function potential
just one bound state,

10
00:00:48,880 --> 00:00:52,550
but it has infinitely
many scattering states.

11
00:00:52,550 --> 00:00:55,300
It's still complicated.

12
00:00:55,300 --> 00:00:59,710
What could be simpler?

13
00:00:59,710 --> 00:01:02,530
Suppose you have the
Schrodinger equation.

14
00:01:07,270 --> 00:01:10,110
H psi.

15
00:01:10,110 --> 00:01:14,910
And we work in general we
know that this thing has

16
00:01:14,910 --> 00:01:18,730
energy eigenstates, and probably
we should focus on them.

17
00:01:18,730 --> 00:01:23,930
So Psi equal E to the
minus I, Et over H bar.

18
00:01:23,930 --> 00:01:29,360
Little psi, and then
have H Psi equal E psi.

19
00:01:32,420 --> 00:01:37,040
That is quantum mechanics,
and you could say,

20
00:01:37,040 --> 00:01:40,660
well it's up to me to decide
what the Hamiltonian is.

21
00:01:40,660 --> 00:01:46,330
If I want to invent the simplest
quantum mechanical system.

22
00:01:46,330 --> 00:01:48,100
On the other hand,
there are some things

23
00:01:48,100 --> 00:01:51,230
that should be true.

24
00:01:51,230 --> 00:01:55,360
These are complex
numbers, energies,

25
00:01:55,360 --> 00:02:02,850
H must be an operator
that has units of energy.

26
00:02:02,850 --> 00:02:06,910
And we also saw that if
we want probabilities

27
00:02:06,910 --> 00:02:09,699
that are going to be
associated with PSI

28
00:02:09,699 --> 00:02:16,790
squared to be conserved
we need H to be Hermitian.

29
00:02:16,790 --> 00:02:20,540
There should be some
notion of inner product.

30
00:02:20,540 --> 00:02:24,560
Some sort of operation
that gives us

31
00:02:24,560 --> 00:02:30,305
numbers we used to defy
PSI that gives a number.

32
00:02:33,000 --> 00:02:36,720
To complex numbers
in general, and

33
00:02:36,720 --> 00:02:39,855
has the property of somewhat
conjugates this thing.

34
00:02:39,855 --> 00:02:43,590
It has this, and integrates,
but maybe if you're

35
00:02:43,590 --> 00:02:47,820
doing the simplest quantum
mechanical system in the world

36
00:02:47,820 --> 00:02:50,850
it will be simpler
than an integral.

37
00:02:50,850 --> 00:02:53,710
Integrals are complicated.

38
00:02:53,710 --> 00:02:56,740
But anyway we have
something like that,

39
00:02:56,740 --> 00:03:04,450
and we want H to be Hermitian.

40
00:03:04,450 --> 00:03:13,460
Let me write this in
for any operator A,

41
00:03:13,460 --> 00:03:20,370
this is equal to a dagger Psi.

42
00:03:20,370 --> 00:03:22,560
And that's a
Hermitian conjugate.

43
00:03:22,560 --> 00:03:27,990
That's a general definition,
and we want H to be Hermitian.

44
00:03:27,990 --> 00:03:37,630
H dagger equal H.
OK, in some sense

45
00:03:37,630 --> 00:03:41,810
you could say that's
quantum mechanics for you.

46
00:03:41,810 --> 00:03:48,580
It's a Schrodinger equation, a
Hamiltonian, an inner product,

47
00:03:48,580 --> 00:03:52,210
a notion of Hermitian
operators, and then you're

48
00:03:52,210 --> 00:03:53,890
supposed to solve it.

49
00:03:53,890 --> 00:03:58,270
And what we've done is solve
this for a whole semester,

50
00:03:58,270 --> 00:04:04,550
and try to understand
some physics out of it.

51
00:04:04,550 --> 00:04:09,380
But we started with the notion
that something simple would

52
00:04:09,380 --> 00:04:12,020
be a particle living
in one dimension,

53
00:04:12,020 --> 00:04:14,870
and that's a very
reasonable thought.

54
00:04:14,870 --> 00:04:18,500
Motivated from
classical mechanics

55
00:04:18,500 --> 00:04:23,060
that surely we have
particles that move,

56
00:04:23,060 --> 00:04:26,990
and moving in three dimensions
is more complicated.

57
00:04:26,990 --> 00:04:29,540
We waited towards the
end of the semester

58
00:04:29,540 --> 00:04:33,110
to do three dimensions,
but moving in one dimension

59
00:04:33,110 --> 00:04:37,940
is already kind of
interesting, and complicated.

60
00:04:37,940 --> 00:04:46,310
We had Psi of X that represented
the fact that the particle

61
00:04:46,310 --> 00:04:47,630
could be anywhere here.

62
00:04:55,840 --> 00:04:58,240
How can I simplify this?

63
00:04:58,240 --> 00:05:01,660
The key to simplifying
this is maybe

64
00:05:01,660 --> 00:05:06,520
not to be too attached to
the physics for a while,

65
00:05:06,520 --> 00:05:12,490
and try to visualize what could
you describe that was simpler.

66
00:05:12,490 --> 00:05:20,650
Suppose the particle could only
live at two points X1, and X2.

67
00:05:20,650 --> 00:05:25,960
The particle can
be here, or here.

68
00:05:25,960 --> 00:05:30,750
Now we've re-aligned
down to just two points.

69
00:05:30,750 --> 00:05:34,700
It can only be this
point, or that point.

70
00:05:34,700 --> 00:05:38,260
And you say, that's
very in physical.

71
00:05:38,260 --> 00:05:41,920
But let's wait a second,
and think of this.

72
00:05:41,920 --> 00:05:43,480
What does that mean?

73
00:05:43,480 --> 00:05:47,060
We used to have Psi effects
that could be anywhere,

74
00:05:47,060 --> 00:05:48,435
and we wrote it as a function.

75
00:05:51,790 --> 00:05:55,320
If I think of this
the simplest thing OK,

76
00:05:55,320 --> 00:05:58,560
the simplest thing is a
particle is just at one point.

77
00:05:58,560 --> 00:06:00,010
There is only one point.

78
00:06:00,010 --> 00:06:04,300
The whole world for the particle
is one point, and it's there.

79
00:06:04,300 --> 00:06:07,900
But that probably is
not too interesting

80
00:06:07,900 --> 00:06:09,250
because the particle is there.

81
00:06:09,250 --> 00:06:12,210
The probability defined
there is always one,

82
00:06:12,210 --> 00:06:15,040
and what can you do with It?

83
00:06:15,040 --> 00:06:18,820
But if you have two points
there's room for funny things

84
00:06:18,820 --> 00:06:20,440
to happen.

85
00:06:20,440 --> 00:06:24,250
We'll assume that the
particle can be in two points.

86
00:06:24,250 --> 00:06:34,360
From F of this Psi effects will
go to a new Psi effects that

87
00:06:34,360 --> 00:06:37,450
has two pieces of information.

88
00:06:37,450 --> 00:06:44,990
The value of PSI at x1,
and the value of Psi at X2.

89
00:06:44,990 --> 00:06:48,530
And those are two
numbers alpha, and beta.

90
00:06:53,530 --> 00:07:03,490
Alpha squared would be the
probability to be at the X1.

91
00:07:03,490 --> 00:07:07,750
Beta squared would be the
probability to be at X2.

92
00:07:14,480 --> 00:07:20,490
And this may remind you
already of something we're

93
00:07:20,490 --> 00:07:22,250
doing with interferometers.

94
00:07:22,250 --> 00:07:25,980
In which the photon could be in
the upper branch, or the lower

95
00:07:25,980 --> 00:07:28,290
branch, and you
have two numbers.

96
00:07:28,290 --> 00:07:30,960
This is somewhat
analogous except that

97
00:07:30,960 --> 00:07:34,920
the interferometer
you could eventually

98
00:07:34,920 --> 00:07:39,570
put more beam splitters, and
maybe later three branches,

99
00:07:39,570 --> 00:07:41,820
or four branches,
or things like that.

100
00:07:41,820 --> 00:07:48,120
Here I want to consider
two things, particle there.

101
00:07:48,120 --> 00:07:53,730
One thing that this could be
strictly that, but now let's

102
00:07:53,730 --> 00:07:57,340
relax our assumptions.

103
00:07:57,340 --> 00:07:59,790
It could also mean
for example, if you

104
00:07:59,790 --> 00:08:03,150
have a box, and a partition.

105
00:08:03,150 --> 00:08:05,250
And there's the left
side of the box,

106
00:08:05,250 --> 00:08:07,470
and the right side of the box.

107
00:08:07,470 --> 00:08:11,280
And the molecule can
either be on the left side,

108
00:08:11,280 --> 00:08:14,410
or on the right side.

109
00:08:14,410 --> 00:08:19,270
That's a fairly
physical question.

110
00:08:19,270 --> 00:08:27,490
Here you could be
probability the amplitude

111
00:08:27,490 --> 00:08:31,390
to be on the left, or
amplitude to be on the right.

112
00:08:31,390 --> 00:08:34,150
Two component vector
just like that.

113
00:08:34,150 --> 00:08:38,280
One would be the amplitude to
be in either one, and maybe

114
00:08:38,280 --> 00:08:40,720
that amplitude changes in time.

115
00:08:40,720 --> 00:08:47,390
Or it could be that
you have a particle,

116
00:08:47,390 --> 00:08:51,440
and suddenly you discovered that
yeah, the particle is at rest.

117
00:08:51,440 --> 00:08:53,010
It's not moving.

118
00:08:53,010 --> 00:08:54,620
It's not doing anything.

119
00:08:54,620 --> 00:08:57,920
It's one single
point, not two points.

120
00:08:57,920 --> 00:09:03,320
But actually this particle has
maybe something called spin,

121
00:09:03,320 --> 00:09:09,790
and the spin can be up,
or the spin can be down.

122
00:09:09,790 --> 00:09:11,620
We it could invent something.

123
00:09:11,620 --> 00:09:16,320
We could call it
spin, or a particle

124
00:09:16,320 --> 00:09:18,760
could be in this state,
or in that state.

125
00:09:18,760 --> 00:09:21,390
And if that's possible
for a particle

126
00:09:21,390 --> 00:09:28,670
you could have here the
amplitude for up spin,

127
00:09:28,670 --> 00:09:32,720
and the amplitude for down spin.

128
00:09:35,850 --> 00:09:37,350
And those would be
the two numbers.

129
00:09:39,960 --> 00:09:43,020
It's lots of possibilities
in the sense this

130
00:09:43,020 --> 00:09:48,600
is a classic problem waiting
for a physical application

131
00:09:48,600 --> 00:09:50,070
in quantum mechanics.

132
00:09:50,070 --> 00:09:53,520
Let's push it a little more.

133
00:10:03,450 --> 00:10:10,660
Now how would we
do inner products?

134
00:10:10,660 --> 00:10:16,630
We decided OK, you need
to do inner products.

135
00:10:16,630 --> 00:10:20,750
And what was the inner
product of two functions

136
00:10:20,750 --> 00:10:30,715
phi and psi was the integral,
the X of phi star of X1 times--

137
00:10:30,715 --> 00:10:35,440
phi star of X times psi of X.

138
00:10:35,440 --> 00:10:37,870
And what you're
really doing is taking

139
00:10:37,870 --> 00:10:42,430
the values of the first
wave function at one point.

140
00:10:42,430 --> 00:10:46,840
Complex conjugating it, take
the value of the second wave

141
00:10:46,840 --> 00:10:50,380
function at the same point
complex conjugating it.

142
00:10:50,380 --> 00:11:00,710
If you would have two vectors
like this alpha, and beta

143
00:11:00,710 --> 00:11:02,920
the first wave function.

144
00:11:02,920 --> 00:11:07,100
Alpha one, beta one, and
the second wave function.

145
00:11:07,100 --> 00:11:10,430
Alpha two, beta two.

146
00:11:10,430 --> 00:11:17,220
The inner products
psi 1, psi 2 should

147
00:11:17,220 --> 00:11:19,740
be the analog of
this thing which

148
00:11:19,740 --> 00:11:23,190
is multiply things
at the same point.

149
00:11:23,190 --> 00:11:27,430
You should do the
alpha one star.

150
00:11:27,430 --> 00:11:35,900
That's alpha two plus beta
one star times beta two.

151
00:11:35,900 --> 00:11:38,150
That would be the
nice way to do this.

152
00:11:42,510 --> 00:11:49,460
You could think of this as
having transposed this alpha

153
00:11:49,460 --> 00:11:53,140
one, and complex conjugated it.

154
00:11:53,140 --> 00:11:58,930
Beta one, and then the matrix
product with alpha one,

155
00:11:58,930 --> 00:12:01,720
beta one.

156
00:12:01,720 --> 00:12:05,610
You transpose complex
conjugate the first,

157
00:12:05,610 --> 00:12:11,050
and you do that with the second.

158
00:12:11,050 --> 00:12:14,740
When you study a little more
quantum mechanics in 805

159
00:12:14,740 --> 00:12:17,540
you will explore this
analogy even more

160
00:12:17,540 --> 00:12:23,180
in that you will think of a wave
function as a column vector,

161
00:12:23,180 --> 00:12:31,280
infinite one. psi at zero, psi
at epsilon, psi at two epsilon,

162
00:12:31,280 --> 00:12:34,780
psi at minus epsilon.

163
00:12:34,780 --> 00:12:40,720
So you've sliced the x-axis and
conserved an infinite vector.

164
00:12:40,720 --> 00:12:42,520
And that's all wave function.

165
00:12:42,520 --> 00:12:48,890
It's not so
unnatural to do this,

166
00:12:48,890 --> 00:12:51,560
and this will be
our inner product.

167
00:12:54,980 --> 00:12:58,410
How about H be in Hermitian.

168
00:13:02,500 --> 00:13:09,010
That just means for
matrices that H transpose

169
00:13:09,010 --> 00:13:16,860
complex conjugate that dagger,
Hermitian, is equal to H.

170
00:13:16,860 --> 00:13:21,720
And you may have seen that
that's what dagger means.

171
00:13:21,720 --> 00:13:25,300
You transpose a
complex conjugate.

172
00:13:25,300 --> 00:13:29,590
If you haven't seen it you
could prove it now using

173
00:13:29,590 --> 00:13:33,520
this rule for the inner product
because the inner product

174
00:13:33,520 --> 00:13:40,990
will tell you how to construct
the dagger of any operator.

175
00:13:40,990 --> 00:13:44,200
And you will find that indeed
the dagger what it does

176
00:13:44,200 --> 00:13:47,290
is transposes, and
complex conjugates it.

177
00:13:47,290 --> 00:13:51,790
And it sort of comes because
the inner product transposes,

178
00:13:51,790 --> 00:13:55,140
and complex conjugates
the first object.