1
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Okay. So now we've investigated the states are a well-defined momentum, which as you recall with these plane waves with a wave number.

2
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Pierpont Momentum upon HPR,

3
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we can go back to this problem of two split interference and ask ourselves why it is that quantum interference isn't observed

4
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in macroscopic objects and also assess what experimental setup will be required to see quantum interference with electrons.

5
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So we put some numbers into this experiment.

6
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So as you recall, we had some gun here that was symmetrically placed with respect to a couple of slits here in an obscuring screen screen,

7
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it fired out particles.

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Some of the particles got through the holes and they came to here and we said that the amplitude up here would be the sum of the quantum.

9
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The quantum amplitude to arrive at this point would be the sum of the amplitude to go via this route or to go for this route.

10
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These routes. Well, the the the distance from the gun to the two slits by set up is symmetrical is equal.

11
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So any difference in the amplitude that they come there is to do with the change in the amplitude when it goes along this route as

12
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against along this route so that D D plus b the distance from the upper slit to this place and D minus the distance to the lower slit.

13
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Then we can write a formula that D plus by Pythagoras theorem is going to be L squared plus x minus x squared square root,

14
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and correspondingly D minus is going to be the square root of l squared plus x plus s squared.

15
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And we make the reasonable conjecture.

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I mean, the right way to think about this is these particles.

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Let's imagine that we've got our gun here, has been tuned to emit particles with some well-defined energy.

18
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That means that as they go along here,

19
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the particles will have some reasonably well-defined momentum because they the energy will consist of that kinetic energy.

20
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So along here, we will have that, that the amplitude is going to be on the order of e to the i t upon h bar times x.

21
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So that's a reasonable model for what the how the amplitude varies with position.

22
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That's the new information that we bring to bear on this. So what will be the difference?

23
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So the probability to arrive at X, as you recalled, was equal to the amplitude to arrive by the top slot,

24
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plus the amplitude to arrive at the bottom slot mod squared.

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And the argument that this was a plus mod squared plus a minus mod squared plus twice the real part of a plus a minus.

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And these were about equal and well, these were the classical probabilities.

27
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So these were P plus plus, P minus.

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And then we have this quantum interference term, which we want to assess which.

29
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So so what is this quantum interference term?

30
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So the interference term. Is going to be.

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Mod plus mod minus neither.

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Not very interesting.

33
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And the crucial thing is that this one is going to be e to the eye p upon bar d plus and the other one's going to be e to the minus i pierpont bar.

34
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Sorry. Plus. Sorry.

35
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One of these needs to have a star in it, which means that one of these requires a minus sign.

36
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So if we're interested. Excuse me in the real in the real part of this and the real part so this thing

37
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can be written as cos Pierpont bar de plus de minus minus D plus etc. etc. etc.

38
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So this is looking like to the probability to go either way times through

39
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through each hole separately times the cosine of p over each bar d plus minus,

40
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d minus. Actually it's easier to do it the other way around it.

41
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Minus plus. So what is this difference of distances?

42
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This difference of distances from up there?

43
00:05:18,380 --> 00:05:28,550
Well, let's binomial expand this and because we can argue that el is going to be big experimentally compared to x, x will be mm.

44
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L will be a metre or so.

45
00:05:30,230 --> 00:05:44,750
So we can binomial expand this and say that this is l it's about half of l brackets one plus x minus x squared of a l squared plus a dot, dot, dot.

46
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And the other one's going to be about half L one plus x plus s.

47
00:05:51,370 --> 00:05:57,430
Squared over l squared plus dot, dot, dot. So when we tell you the difference of those two.

48
00:05:59,380 --> 00:06:11,380
So this is going to be two p plus minus this probability times the cosine of p upon bar of the

49
00:06:11,380 --> 00:06:21,280
difference is going to be x plus x squared over l minus x minus x squared over two l in fact.

50
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So when you take the difference of those two, you will be looking at two.

51
00:06:29,190 --> 00:06:42,110
X. S of l. Yes to excessive l is what that difference will be.

52
00:06:44,130 --> 00:06:48,410
So. So now let's put it now. We need to put in some numbers.

53
00:06:48,920 --> 00:06:53,930
All right. Suppose we take the energy, which is P squared over two.

54
00:06:54,560 --> 00:06:59,010
What do we want to do? We want to know. So what does this do?

55
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This gives us a probability of arrival as a function of X,

56
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which is going to consist of a of twice the sum of these two, which are going to be about equal.

57
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And so about twice this plus twice. This thing times this cosine.

58
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So it's going to be an oscillating commodity which will be doing this.

59
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And there's some characteristic distance between these, between these minima which so this will call this delta X say no, let's call it big x.

60
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That's what the so we call this big X the distance between the places where it's a

61
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minimum because the quantum interference term is cancelling the classical term.

62
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And this difference is what causes that.

63
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The argument is that cosine to become two pi so we can write the two pi is equal to p over four times two s of l times x.

64
00:08:05,660 --> 00:08:15,410
In other words, we have a formula for the distance between the minima, which is two pi to pi h bar, but h bar is h planck's constant over two pi.

65
00:08:15,410 --> 00:08:21,170
So that's going to be h over p h l sorry.

66
00:08:21,380 --> 00:08:24,620
Yes. Of S.

67
00:08:27,200 --> 00:08:31,259
Yeah. And we could also write this.

68
00:08:31,260 --> 00:08:34,620
I guess we we could also say that.

69
00:08:35,700 --> 00:08:40,830
Oh, no, it's not, Paula. So let's put some numbers in.

70
00:08:41,160 --> 00:08:44,520
Let's say that e the energy is.

71
00:08:44,670 --> 00:08:48,420
So in order to get a big value of X, we want to take a big value of L.

72
00:08:48,420 --> 00:08:57,090
Needless to say, we want to take a small value of P and of course a small value of S two on the bottom of.

73
00:08:57,090 --> 00:09:05,340
I've lost a two. Yeah, you're quite right. So, so let's take L to be a metre.

74
00:09:05,820 --> 00:09:12,510
Let's take S to be what got I think.

75
00:09:16,720 --> 00:09:20,100
What did I do. I think I think it's Yeah.

76
00:09:20,140 --> 00:09:24,100
A micron because you want to make it as small as you can.

77
00:09:24,640 --> 00:09:30,670
But if you make it much smaller than a micron, you'll find it difficult to make the hole using ordinary materials.

78
00:09:31,420 --> 00:09:37,270
And let's and we also in order to get a small value of P, we want to take a small value of the energy,

79
00:09:37,270 --> 00:09:45,280
but you can't take too small a value of the energy or your particles will be deflected by stray electromagnetic fields and stuff,

80
00:09:45,280 --> 00:09:48,280
and it'll be difficult to keep any kind of coherence.

81
00:09:48,670 --> 00:09:55,630
So let's take 100 ev say as a, as a sort of convenient low speed if you plug all this stuff into there.

82
00:09:56,200 --> 00:10:01,060
So that gives you what does it give you a 20th of a millimetre or something.

83
00:10:01,150 --> 00:10:09,580
I think it's is it 0.06 millimetres which is obviously perfectly, perfectly observable.

84
00:10:09,580 --> 00:10:20,530
Yeah. So this this such an interference experiment is, is, is possible but hard using electrons if you do the same thing with bullets.

85
00:10:24,320 --> 00:10:26,930
When we're not expecting anything to happen, what could we do?

86
00:10:27,170 --> 00:10:33,920
We would we would take the velocity for a gun is say 300 metres a second might be a bit faster these days.

87
00:10:33,920 --> 00:10:37,129
I'm not sure. But that's a classical that's, you know, faster than sound.

88
00:10:37,130 --> 00:10:39,470
So that's sort of a reasonable ballpark figure.

89
00:10:39,710 --> 00:10:47,120
Suppose we took L to be one kilometre a thousand yards, pretty reasonable shooting distance for a rifle.

90
00:10:48,560 --> 00:11:04,790
And if we took the mass to be ten grams, put it into the same formula and we discover that X is some ridiculous figure ten to the -29 metres.

91
00:11:04,910 --> 00:11:11,330
So it's obvious that you cannot observe this interference using anything like a bullet, any kind of macroscopic, any,

92
00:11:11,630 --> 00:11:18,320
any kind of macroscopic object, because it's going to be vastly bigger itself than the than the size of the interference pattern.

93
00:11:18,320 --> 00:11:22,850
Obviously, an absolutely basic requirement for this experiment to work is the physical size of

94
00:11:22,850 --> 00:11:25,959
your particle has to be smaller than the value of X that you derive out of this.

95
00:11:25,960 --> 00:11:37,700
So you haven't a hope of measuring this interference. So that's why classically we don't we are unaware of this interference term.

96
00:11:37,700 --> 00:11:42,950
But I would remind you that in the last lecture we recovered classical results,

97
00:11:42,950 --> 00:11:52,850
which which explain why cricket balls move as they do, why satellites and so on move as they do by interpreting.

98
00:11:53,090 --> 00:11:55,040
We calculated this.

99
00:11:55,730 --> 00:12:06,559
We obtained results which recovered classical physics by decomposing the amplitude to arrive into a sum of contributions from states of different,

100
00:12:06,560 --> 00:12:10,250
well-defined momentum. And these were all interfering with each other.

101
00:12:10,250 --> 00:12:13,670
And the classical physics came back as a result of quantum interference.

102
00:12:13,670 --> 00:12:18,379
So this quantum interference, on the one hand, is something which is very hard to observe with classical objects.

103
00:12:18,380 --> 00:12:24,410
On the other hand, our entire picture of the classical world, a classical world, is only recovered through quantum interference.

104
00:12:25,700 --> 00:12:32,120
It's not is it's not some esoteric corner of the subject, but it is hard to.

105
00:12:38,130 --> 00:12:41,510
It's hard to have it happened in a controlled way. Okay.

106
00:12:41,930 --> 00:12:50,100
So we. Yeah.

107
00:12:50,700 --> 00:12:57,479
We should just. So we've done, we've done the position representation in just one dimension.

108
00:12:57,480 --> 00:13:00,780
Everything has been a one dimensional motion motion long x.

109
00:13:01,110 --> 00:13:10,320
We obviously need to generalise the position representation to three dimensions because we live in a three dimensional world for whatever reason,

110
00:13:11,400 --> 00:13:14,940
and the generalisation is nice and trivial. We don't need to worry about it.

111
00:13:14,940 --> 00:13:24,820
We have we now have three position operators X, Y and Z, also known as X, Y.

112
00:13:24,840 --> 00:13:39,240
All right. And we have, of course, three momentum operators, three more operators, X, Y and Z, also known as P.

113
00:13:42,330 --> 00:13:46,890
And we have that every one of these operators commutes with the other one.

114
00:13:47,250 --> 00:13:50,760
So we have that exi comma sj.

115
00:13:53,320 --> 00:13:59,530
It's nothing. And every one of the momentum operators commutes P-I, comma, PGA.

116
00:14:02,070 --> 00:14:06,240
Equals nothing. So it is possible to simultaneously know your x, coordinate,

117
00:14:06,240 --> 00:14:10,620
your y coordinate and your z coordinate as a complete set of eigen function of can states.

118
00:14:12,270 --> 00:14:16,169
Of well-defined states. States. Where you know all those strings simultaneously.

119
00:14:16,170 --> 00:14:23,400
Or you can know all three components of momentum, but you can't know there's not a complete set of states for knowing and so on.

120
00:14:24,840 --> 00:14:30,240
And the only other interesting thing we have to have is XY commuted with PJ.

121
00:14:33,460 --> 00:14:41,950
He's h baa delta. So it is possible to know the exposition and the Y momentum, but it's not possible to know the exposition and the momentum.

122
00:14:42,820 --> 00:14:50,100
So most of these operators commute with well, each operator commutes with five of the of the sorry,

123
00:14:50,110 --> 00:14:56,370
four of the remaining five operators, but it does not commute with its own momentum.

124
00:14:56,380 --> 00:14:59,920
That's what each of these position operators. So that's the generalisation there.

125
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What else do we have to say? Well, we used to have a wave function of PSI being a function of scalar x.

126
00:15:07,900 --> 00:15:11,170
We now it's trivial the argument of the wave.

127
00:15:11,170 --> 00:15:16,360
We we can now label a complete set of states by X,

128
00:15:16,360 --> 00:15:21,520
Y and Z so we can write that there's a we have states of well-defined position

129
00:15:22,570 --> 00:15:27,820
which are labelled by a vector now vector position x because there are three.

130
00:15:27,970 --> 00:15:32,049
This is an eigen state of the x operator. It's an elegant state of the Y operator.

131
00:15:32,050 --> 00:15:41,290
And then I can say to the Z operator, so we need three eigenvalues written inside here to describe what this is.

132
00:15:41,680 --> 00:15:49,749
It is that's the mathematical level of the physical level. This is the state of being at the location position vector x correspondingly,

133
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our wave functions become functions of X, Y and Z because they become these complex numbers.

134
00:15:55,330 --> 00:16:01,890
Right? That's still a complex number, this complex number, but it's a function of X and Y and Z, the locations of the particles.

135
00:16:02,320 --> 00:16:06,370
Similarly, we have states of well-defined momentum up.

136
00:16:09,920 --> 00:16:13,720
We have states. Yeah. Yeah. You. You p.

137
00:16:14,420 --> 00:16:18,710
Of X, which is x. P.

138
00:16:20,960 --> 00:16:25,550
So now we have. Here we have p, p, p, z.

139
00:16:25,550 --> 00:16:33,110
Because we have a state of well-defined momentum which is labelled with all three components of momentum.

140
00:16:34,820 --> 00:16:39,220
So we have this function of a complex of three components, right?

141
00:16:39,320 --> 00:16:43,730
This complex function of three variables X, Y and Z labelled by the momentum.

142
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This is just an identical notation. Whereas in single, when we were doing this in one dimension, we found that this was e to the.

143
00:16:54,410 --> 00:16:58,160
P over h bar times. X not vectors.

144
00:16:58,190 --> 00:17:01,610
Now that's a vector. That's a vector.

145
00:17:01,880 --> 00:17:06,860
And whereas on the bottom we used to have h bar to the one half, now we have h bar two,

146
00:17:06,860 --> 00:17:14,480
the three halves reflecting the fact that there are there's an X component to this, y component to this and Z component to this.

147
00:17:14,960 --> 00:17:21,890
So. So this the way this wave function of a state of well-defined momentum has now become a plain wave.

148
00:17:22,220 --> 00:17:26,450
Whose wave whose wave surfaces are normal to the vector p.

149
00:17:28,210 --> 00:17:32,150
And that's and that's what it is. It's easy to check that that stuff works.

150
00:17:32,180 --> 00:17:40,100
It's a it's a very straightforward generalisation of what we did before. And I think that's all we have to say.

151
00:17:40,610 --> 00:17:45,310
Oh, no, not quite. We also want to say what the momentum operator P looks like.

152
00:17:45,320 --> 00:17:49,639
So previously we had that x. P xp.

153
00:17:49,640 --> 00:18:00,560
Sorry. Not right. Talking about. Yeah. X p ci was minus h, but it was introduced by this formula here.

154
00:18:00,830 --> 00:18:05,530
DPD x of x ci. All right.

155
00:18:05,710 --> 00:18:15,930
That was what we did in one dimension. That generalises in three dimensions, very straightforwardly to x p psi.

156
00:18:16,390 --> 00:18:20,050
So. So that's become a vector.

157
00:18:20,440 --> 00:18:25,389
That's become a vector because we have to write down what it is for P and p z.

158
00:18:25,390 --> 00:18:28,870
This is really going to be a shorthand for three formulae and it's going to be

159
00:18:28,870 --> 00:18:37,060
minus H bar gradient operator on the function of three variables functional space.

160
00:18:37,330 --> 00:18:43,670
This one here. All right. The wave function. So this is a vector reflecting the fact that that's a vector.

161
00:18:43,690 --> 00:18:49,490
This is just a label which appears on both sides of the equation. That's what this this formula generates.

162
00:18:49,630 --> 00:19:01,450
Generalise this to that formula. I don't think we need to be detained about that any longer before we leave the position representation.

163
00:19:01,690 --> 00:19:07,470
It's good to do. And a useful result.

164
00:19:09,520 --> 00:19:16,299
Which falls into our laps now because of what we've already done call the variable theorem, which is a a theorem.

165
00:19:16,300 --> 00:19:21,160
It's a result in classical physics, which you may not have met, I don't know, but you in a way, should have met.

166
00:19:21,730 --> 00:19:25,120
Did you cover the variable theorem in classical mechanics anyway?

167
00:19:26,480 --> 00:19:32,120
No. Anyway, so it's there's nothing quantum mechanical about the variable theorem, but it has a classical counterpart,

168
00:19:33,260 --> 00:19:37,730
but it's going to fall into our laps because we've got this powerful machinery.

169
00:19:38,120 --> 00:19:45,740
So do you recall if we are in a stationary state, that is to say, a state in which the result of measuring energy is certain then.

170
00:19:47,670 --> 00:19:52,170
All expectation values for such a state are constants.

171
00:19:52,260 --> 00:19:55,110
That's why we call it a stationary state. It's going nowhere.

172
00:19:55,800 --> 00:20:02,700
So every expectation value for a stationary state for a state of well-defined energy is independent of time.

173
00:20:06,500 --> 00:20:09,770
So we want to exploit that result. So for a stationary state.

174
00:20:11,410 --> 00:20:16,840
Is this just recalling what we already had? It was it was a consequence of our first theorem for a stationary state.

175
00:20:18,280 --> 00:20:35,730
We have that DVD time or even deep DVD time of E Q E equals nought for all for all operators.

176
00:20:35,740 --> 00:20:42,340
Q It doesn't matter what observable you stuff in there, as long as the observable doesn't is defined in a way that is independent of time.

177
00:20:42,340 --> 00:20:47,979
So it's something like position, momentum, angular momentum, whatever it has,

178
00:20:47,980 --> 00:20:53,200
it has a vanishing rate of change or with respect to time, it's a constant.

179
00:20:54,520 --> 00:21:03,910
So we now apply this result to CU is equal to x dot p.

180
00:21:09,930 --> 00:21:14,070
So then we have that nought is equal to.

181
00:21:18,300 --> 00:21:25,680
T. Sorry.

182
00:21:26,820 --> 00:21:38,300
Subtle. Some moment of doubt. Yeah.

183
00:21:39,380 --> 00:21:46,580
Yeah. So I want to apply this to export p e and let's divide.

184
00:21:49,440 --> 00:21:58,830
No stinking H-bomb. That by Ehrenfest theorem is x dot p comma h.

185
00:21:59,970 --> 00:22:10,940
Whoops. So Aaron, First Theorem tells us that this rate of change which vanishes is equal to this here and now.

186
00:22:10,940 --> 00:22:15,380
Let's take suppose we're dealing with a particle which has.

187
00:22:20,050 --> 00:22:25,480
Which has kinetic energy and potential energy. So we'll take the Hamiltonian to be of that form, which is.

188
00:22:28,280 --> 00:22:37,290
Pretty useful form. And stuff it in there and we're going to have that nought is equal to E!

189
00:22:40,980 --> 00:22:48,720
X dot p comma p squared over two m plus v close square brackets.

190
00:22:52,490 --> 00:22:55,220
So now we need to work out what this comitato is.

191
00:22:55,490 --> 00:23:00,920
And this is where a little bit of so this is where we get a bit of practice in using the three dimensional generalisation.

192
00:23:02,810 --> 00:23:09,300
We obviously have two things to work out. We've got a combination of X come up with P squared, so let's work that out.

193
00:23:09,320 --> 00:23:12,560
X come up with P squared.

194
00:23:14,510 --> 00:23:21,920
Now we write that in components. We write x x dot piece, x dot peters I say comma exactly comma squared.

195
00:23:23,360 --> 00:23:37,790
This x dot can be written as a sum over j equals 1 to 3 of x i p i sorry sj pj so that's just a way of writing that.

196
00:23:38,090 --> 00:23:43,480
And now I have a some peak. Well.

197
00:23:44,650 --> 00:23:47,860
P-square. Okay, so I'm something of a K as well.

198
00:23:50,590 --> 00:23:55,640
All right, p squared is p squared plus p y squared plus p said squared.

199
00:23:56,330 --> 00:23:59,510
Now we can work out this using our rules for a commentator.

200
00:23:59,510 --> 00:24:08,930
We had that rule that a comma B sorry, ap comma C was equal to a comma c.

201
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B plus a, B, C, P.

202
00:24:18,560 --> 00:24:22,520
We know that peak commutes with PGA that's been written down up there.

203
00:24:22,880 --> 00:24:27,020
So that commentator vanishes. That's this one here in some sense.

204
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Sorry, this this one here in some sense. And so what we're left with is so we have this double sum, we're going to have X Jay Peak, PJ.

205
00:24:43,490 --> 00:24:48,020
Sorry. That squared. Squared. Comma.

206
00:24:48,970 --> 00:24:52,090
No, no, no comma. So that's what we get. So.

207
00:24:52,300 --> 00:24:55,750
So this has to be commuted with that? That's what I've written down, I hope.

208
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And then there should in principle, be another term, this commuting with this.

209
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But that vanishes because commutes would peak for all for all jank.

210
00:25:04,030 --> 00:25:09,579
So we have to work out what this one is now and we can use the same rule.

211
00:25:09,580 --> 00:25:14,590
If we're being pedantic, we would say this is X on peak peak.

212
00:25:14,860 --> 00:25:23,830
So we would say that this is x j peak, peak plus peak.

213
00:25:24,850 --> 00:25:32,080
The commentator of x and peak. I'm using the same rule and that all has to be multiplied by p j.

214
00:25:33,910 --> 00:25:37,270
The same because. Because I'm not writing p squared is peak. Peak.

215
00:25:37,780 --> 00:25:41,650
But this is h bar. This is bar.

216
00:25:41,770 --> 00:25:53,860
So these two terms actually contribute the same thing. This becomes to h bar p p times, delta j k times, peak times.

217
00:25:53,870 --> 00:25:57,010
PJ And I'm sorry, I've lost track of the sum sign.

218
00:25:57,790 --> 00:26:02,840
Here we have a sum sine. With something over J and with something over K.

219
00:26:04,550 --> 00:26:11,750
Some have a J and you get nothing because of this delta j k unless j is equal to K, so this becomes PCP.

220
00:26:11,750 --> 00:26:16,430
K Some do have a k, but PCP k some of AK is the same thing as p squared.

221
00:26:16,700 --> 00:26:20,060
So this is 2ih bar p squared.

222
00:26:20,690 --> 00:26:24,590
That's what the commentator is of x dot p with p squared.

223
00:26:26,460 --> 00:26:35,220
Now let's let's right now, let's do the exact P commentator with V, which is itself a function of X, of course.

224
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These things ought to have hats, really. But one gets difficult to write down enough things.

225
00:26:44,260 --> 00:26:50,839
Well. What we want to do is is write this thoroughly in the position.

226
00:26:50,840 --> 00:27:00,050
Representation in the position representation x dot p is minus i h bar x dot gradient.

227
00:27:01,310 --> 00:27:09,800
Right? That's what this becomes in the position representation on v, which becomes a function of x just so this is in the position representation.

228
00:27:15,510 --> 00:27:24,060
So what does that mean. That means I ball minus age brackets x dot gradient.

229
00:27:24,930 --> 00:27:31,980
Uh, working on v minus the x dot gradient.

230
00:27:34,480 --> 00:27:40,450
And this is an operator statement. So it's waiting for you to put in the function of your choice of PSI on the right.

231
00:27:40,480 --> 00:27:43,600
Right. There's a virtual function there for it to work on.

232
00:27:43,630 --> 00:27:45,880
That's what that's the meaning of this vehicle gradient.

233
00:27:46,420 --> 00:27:53,020
And this exact gradient V doesn't mean extra gradient only V means of everything that is the right of it, including Europe PSI.

234
00:27:53,770 --> 00:28:03,820
So when you use this exact v on v times alone, you'll get a term and then you will still have to use the V on the up side.

235
00:28:04,060 --> 00:28:09,040
But the result of using the V on the up side will be killed by here and next v on up say.

236
00:28:09,400 --> 00:28:16,690
So what this is equal to is minus h bar x dot gradient of v.

237
00:28:17,980 --> 00:28:23,410
That's all it survives. This is the action of of the nebula, the gradient operator on the potential itself,

238
00:28:23,740 --> 00:28:31,960
the operation of the action of the gradient operator on the wave function that's virtually sitting here is cancelled by this contribution here.

239
00:28:34,810 --> 00:28:39,040
So we now have we we can put these results back into what we had up there.

240
00:28:39,370 --> 00:28:46,150
So what we had was nought is equal to yeah.

241
00:28:46,330 --> 00:28:57,550
Is equal to the sum of these of these commentators is equal to e x dot p comma p squared.

242
00:29:00,740 --> 00:29:07,760
E plus e x dot p v.

243
00:29:08,690 --> 00:29:13,290
That's just summarising where we stand. This we've discovered to be.

244
00:29:13,700 --> 00:29:18,439
This is to i h bar. This commentator turned out to be p squared.

245
00:29:18,440 --> 00:29:23,959
So it becomes the expectation. Oops, there should have been over two m on this, shouldn't it?

246
00:29:23,960 --> 00:29:29,690
Because it was the Hamiltonian p squared over to him. Yeah.

247
00:29:29,960 --> 00:29:35,060
This P came from the Hamiltonian where it was P squared over two m this v came from the Hamiltonian where it was just V.

248
00:29:35,840 --> 00:29:51,020
So we have over two M Uh, no, let's leave it alone of p squared over to m e plus we figured out that this one was a minus h bar.

249
00:30:12,320 --> 00:30:15,380
So we want to cancel what we.

250
00:30:15,830 --> 00:30:19,370
This is the expectation value of the kinetic energy. Clearly.

251
00:30:19,370 --> 00:30:22,969
Right. P squared over two m is the kinetic energy.

252
00:30:22,970 --> 00:30:24,590
That is the expectation value of it.

253
00:30:25,700 --> 00:30:34,730
So cancelling the ball, we can say that two times the expectation value of the kinetic energy is equal to this stuff.

254
00:30:44,520 --> 00:30:57,360
That's as far as we can go in general. But consider now very important cases have that V of X is proportional to model X to the alpha.

255
00:30:58,650 --> 00:31:05,160
So, for example, for a simple harmonic oscillator we're about to discuss, the potential energy goes like x squared alphas two.

256
00:31:05,460 --> 00:31:12,810
If we were dealing with a dealing with a Coulomb interaction, the potential and potential energy goes like one over radius.

257
00:31:13,200 --> 00:31:17,070
So it would be V of all is proportional to one overall.

258
00:31:18,660 --> 00:31:21,980
Well, this model X is R, so alpha would be minus one.

259
00:31:21,990 --> 00:31:25,680
So we can say that alpha equals two in simple harmonic motion.

260
00:31:26,250 --> 00:31:29,790
Alpha equals minus one is coulomb.

261
00:31:31,620 --> 00:31:37,260
There are, you know, you can think of other power laws which are relevant in this case.

262
00:31:37,770 --> 00:31:41,640
So then we ask ourselves, what is X dot?

263
00:31:43,800 --> 00:31:47,790
Gradient of. Evie.

264
00:31:49,560 --> 00:31:55,690
Well, that's going to be so sad. We'll say that this is equal to some constant A times, X to the alpha.

265
00:31:55,710 --> 00:32:13,020
What is this going to be? It's going to be alpha model X to the alpha minus one times X dot the gradient of model X and the gradient of Model X is.

266
00:32:15,920 --> 00:32:24,020
The gradient of Model X is. X the unit vector x, so it's the vector x over model x.

267
00:32:24,530 --> 00:32:37,460
So this is equal to. Sorry, this is a we have a X to the alpha minus one times x dot x over monarch's.

268
00:32:38,480 --> 00:32:41,870
So here this mod x is going to make this an alpha to the minus two.

269
00:32:42,050 --> 00:32:45,110
But from this x dot x we're going to get mod x squared.

270
00:32:45,440 --> 00:32:48,560
So this is going to be and I've lost sorry this was an alpha.

271
00:32:49,400 --> 00:32:53,060
That was also an A unfortunately. Yeah. Sorry. We need an A and an alpha.

272
00:32:54,590 --> 00:33:02,030
This is going to be alpha times a x to the alpha, which is alpha times V.

273
00:33:02,480 --> 00:33:12,200
So if V has a power lower dependence on distance from the origin, then x dot grad v is simply alpha times v.

274
00:33:14,270 --> 00:33:18,829
So when we put this result back into that formula, back into this statement,

275
00:33:18,830 --> 00:33:28,100
here we have that twice the k e expectation value is equal to alpha times the expectation value of the potential energy.

276
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So that's our Kepler formula. In the case of simple harmonic motion, alpha is two, and kinetic energy is equal potential energy.

277
00:33:37,300 --> 00:33:45,160
In the case of Coulomb interaction, where alpha is minus one, you have that the potential energy is minus twice the kinetic energy,

278
00:33:45,160 --> 00:33:55,629
which is to say that the particle has lost two units of energy, radiate in falling in from infinity into a bound orbit.

279
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It's lost two units of energy, one units being sent off to infinity and radiation or something, and one unit is used as kinetic energy of its orbit.

280
00:34:07,540 --> 00:34:27,440
So that's the this is a very theorem. So now we open a new chapter, as it were, by talking about harmonic motion.

281
00:34:39,860 --> 00:34:43,550
The harmonic oscillator is the single most important dynamical system in physics.

282
00:34:44,510 --> 00:34:48,070
Most of field theory, most of contents of quantum field theory,

283
00:34:48,080 --> 00:34:59,320
most of condensed matter physics is fiddling with more or less with harmonic oscillators, which which are which are decorated in some way.

284
00:34:59,330 --> 00:35:01,880
So the basic physics is that of the harmonic oscillator.

285
00:35:02,420 --> 00:35:09,980
And it's worth just taking a moment to understand why harmonic oscillators are all over the place, the universe, the physicists.

286
00:35:11,450 --> 00:35:16,910
A fundamental position of physicists. Well, physicists like to represent the universe is a collection of harmonic oscillators.

287
00:35:17,330 --> 00:35:21,800
And this is partly because physicists may be brighter than some other people, but they're still pretty stupid.

288
00:35:21,950 --> 00:35:28,129
We have a quite a small bag of tricks and harmonic oscillators is a trick that we we have.

289
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And it's an incredibly useful trick for this reason that if you plot force in some direction versus displacement from a point of equilibrium,

290
00:35:38,810 --> 00:35:44,690
you will get a curve which does something like this, the force.

291
00:35:46,170 --> 00:35:52,200
Vanishes at an equilibrium at the point of equilibrium of a system.

292
00:35:53,830 --> 00:35:59,260
The force on it obviously vanishes. So if you do a plot of force versus distance, you'll get a curve.

293
00:35:59,260 --> 00:36:04,720
Something like this passing through zero at the point of equilibrium, which I happen to put at the origin of X.

294
00:36:05,260 --> 00:36:08,380
But, you know, that's by construction, clearly.

295
00:36:08,590 --> 00:36:13,479
And the general idea is that most of the time you can destroy.

296
00:36:13,480 --> 00:36:21,210
That's meant to go through the through the origin. Most of the time, you can represent this to some, to a good approximation,

297
00:36:21,220 --> 00:36:29,010
you can say that F of X is about equal to x plus order of x squared or whatever,

298
00:36:29,020 --> 00:36:35,830
so to lowest order approximation, because f has to vanish to the point of equilibrium in the neighbourhood of the point of equilibrium.

299
00:36:35,980 --> 00:36:42,670
F is going to be proportional to x and if is if we neglect this, if this is small,

300
00:36:43,690 --> 00:36:47,420
then we have harmonic motion for displacements, these small displacements around here.

301
00:36:47,440 --> 00:36:54,040
So this is why harmonic oscillators are ubiquitous, a very credibly, an incredibly valuable model.

302
00:36:54,040 --> 00:36:59,470
We can apply. We can use to understand many, many systems because many systems for small displacements,

303
00:37:00,280 --> 00:37:03,820
almost all systems for small displacements look like a harmonic oscillator.

304
00:37:05,770 --> 00:37:08,800
Okay, so let's agree what the Hamiltonian of this thing should be.

305
00:37:09,040 --> 00:37:16,720
The Hamiltonian of our harmonic oscillator should be P squared over two M plus a half K X squared.

306
00:37:17,140 --> 00:37:21,190
Right. Because if the force is going like that, you integrated up, this becomes the potential energy.

307
00:37:21,490 --> 00:37:27,760
And this is of course the kinetic energy. We're familiar with that already. It's better, though, to write this in a different way,

308
00:37:27,940 --> 00:37:38,600
to anticipate results that are to come and to write this is p squared plus omega x squared over to M.

309
00:37:41,180 --> 00:37:51,050
So, uh, and of course omega squared is k over m so it's easy defining omega squared to be okay over him.

310
00:37:51,080 --> 00:37:54,290
It's easy to write this formula like that, and that's how I want to write it.

311
00:37:55,760 --> 00:37:59,060
So you want to reproduce this formula? Just think about dimensional analysis.

312
00:37:59,330 --> 00:38:03,350
We want to have piece going over to him because it's the kinetic energy. We're always saying that.

313
00:38:03,620 --> 00:38:10,880
And here I want something that's proportional to x squared and has the dimensions of momentum and obviously omega x has dimensions of speed.

314
00:38:11,060 --> 00:38:17,750
So M omega X's dimensions momentum. So that's why, you know, that enables you to recover that quickly from this.

315
00:38:18,170 --> 00:38:24,500
And that's the way to go for practical purposes. So that's our Hamiltonian and we're trying, of course, to solve.

316
00:38:25,490 --> 00:38:30,170
So these stationary states are the key to understanding dynamics because they have this

317
00:38:30,170 --> 00:38:35,810
trivial time evolution and by by decomposing any initial condition into with some of,

318
00:38:36,140 --> 00:38:39,290
of stationary states, into a linear superposition of stationary states,

319
00:38:39,530 --> 00:38:46,579
then evolving the stationary states, we find out how any arbitrary initial condition evolves in time.

320
00:38:46,580 --> 00:38:49,850
So that's why we want these stationary states. I've said that before and I'll say that again.

321
00:38:51,410 --> 00:38:55,610
So we want to find states of well-defined energy.

322
00:38:56,480 --> 00:39:00,710
This is the problem we want to solve. And this is a completely generic situation in physics.

323
00:39:01,160 --> 00:39:05,900
First of all, you think about your physical system on the grounds of physics, you write down the Hamiltonian.

324
00:39:06,110 --> 00:39:11,450
Then the next thing you do is you find the down stationary states, because once you've got those, you can do anything you want, pretty much.

325
00:39:14,390 --> 00:39:22,130
So that's what we're trying to solve. The the the way to do this is the proper way to, to find these states.

326
00:39:23,060 --> 00:39:29,450
So we need to find the energies that that are possible and we need to find the corresponding states.

327
00:39:29,450 --> 00:39:32,390
And the way to do this is to introduce some new operators.

328
00:39:32,660 --> 00:39:44,270
Let's introduce a which is m omega x plus i p over the square root of two m omega or h bar omega.

329
00:39:48,260 --> 00:39:51,340
Why do I write that down? Well, basically because I know where I'm going.

330
00:39:51,350 --> 00:40:00,110
But just to give you some sense of direction, the general idea here is that we want to factories, that that's the general idea.

331
00:40:00,110 --> 00:40:03,320
We want the factories, the Hamiltonian into.

332
00:40:03,650 --> 00:40:07,270
So it's a quadratic expression. It seems kind of reasonable to factories.

333
00:40:07,280 --> 00:40:13,690
It if these were if these weren't operators because these are operators, sorry.

334
00:40:13,880 --> 00:40:17,060
In future I'm not going to even attempt to put hats on operators.

335
00:40:17,060 --> 00:40:18,770
Right. These are operators.

336
00:40:18,770 --> 00:40:24,740
Despite the absence of hats, it's just too difficult to remember to put the hats on and takes too much time and grown ups never do.

337
00:40:25,400 --> 00:40:32,090
But these are operators now, if they but if they weren't operators this in its complex conjugate would factories that.

338
00:40:32,810 --> 00:40:33,680
So that's the drift.

339
00:40:33,680 --> 00:40:43,460
Okay let's write down it's it's a bit complex this is this of course is an operator and it's not an observable it's not a mission operator.

340
00:40:43,820 --> 00:40:52,219
It's what is its dagger? A dagger is not joint is this thing dagger, which is itself because X is a mission operation.

341
00:40:52,220 --> 00:41:00,020
It's own dagger. So it's M omega x plus this thing dagger p is its own dagger.

342
00:41:00,200 --> 00:41:05,359
But I has the the dagger of I the joint of eyes minus II.

343
00:41:05,360 --> 00:41:08,569
So this is minus I p over.

344
00:41:08,570 --> 00:41:12,710
Of course, this on the bottom is a real number. So it's own it's its own complex conjugate.

345
00:41:14,390 --> 00:41:18,440
So here we have two operators and the general idea is they're going to factories h or

346
00:41:18,440 --> 00:41:24,650
they almost do whatever because the plan and this is called in annihilation operator.

347
00:41:28,720 --> 00:41:33,310
And this is a correctional operator and. Well, the reason they have these names will emerge.

348
00:41:33,310 --> 00:41:36,310
But it is that if you use this on a state,

349
00:41:37,030 --> 00:41:43,780
this operator increases the excitation of our harmonic oscillator and this oscillator reduces the expectation of our harmonic oscillator.

350
00:41:44,410 --> 00:41:48,910
And since in quantum field theory, particles are excitations of the vacuum.

351
00:41:49,180 --> 00:41:53,440
This thing creates a particle because it creates an excitation, which is a particle,

352
00:41:53,620 --> 00:41:56,350
and this thing destroys a particle because it destroys the excitation.

353
00:42:01,810 --> 00:42:09,280
So what we next do is work out what a a dagger A is, because the idea was that this product would be more or less the Hamiltonian.

354
00:42:10,180 --> 00:42:23,560
So what exactly is it? Let's get this right. M omega x minus IP and omega x plus i p over to m bar omega.

355
00:42:24,670 --> 00:42:27,700
Now, when we write this out, we have the obvious terms.

356
00:42:27,700 --> 00:42:32,919
We have p squared and we have m omega x squared.

357
00:42:32,920 --> 00:42:46,510
So let's write those down. That's P squared plus m omega x squared all over two bar into m omega bar, whatever.

358
00:42:49,420 --> 00:43:01,820
And. And then we have some additional terms which would class that would cancel in classical algebra but don't now because we have an X,

359
00:43:02,150 --> 00:43:11,180
we have an M Omega X times IP and here we have an m omega x on the so we have an I minus IP times name and we are x.

360
00:43:11,570 --> 00:43:16,250
So the additional term is an m omega x.

361
00:43:17,500 --> 00:43:28,090
Am Omega I x comma p and it's a gain of a two and h bar omega.

362
00:43:31,010 --> 00:43:43,660
Right. So this is the Hamiltonian overreach for Omega and this is an H bar which and the I's make a minus one with this.

363
00:43:43,670 --> 00:43:48,910
So this is going to be minus a half and everything else will cancel.

364
00:43:51,120 --> 00:43:54,560
Right. Because we'll we've got an Omega here.

365
00:43:54,570 --> 00:44:01,649
We're going to get an H bar from there. So the rest cancels. So I should have I should have explained.

366
00:44:01,650 --> 00:44:07,170
Sorry, what is the factories, this and this on the bottom, this normalising factor on the bottom is put in.

367
00:44:07,170 --> 00:44:12,150
It's not really essential, but it's very convenient and it's put in in order to make this dimensionless.

368
00:44:14,100 --> 00:44:19,770
So just to check that that's true. H Bar has dimensions of position, times momentum.

369
00:44:20,730 --> 00:44:23,910
Right. So it has the dimensions of position, times momentum.

370
00:44:24,210 --> 00:44:33,570
So what we have here is m x sorry, m omega x which we've agreed has dimensions of momentum, times p which has dimensions of momentum.

371
00:44:33,570 --> 00:44:34,590
Then we take the square root.

372
00:44:34,590 --> 00:44:41,250
So this on the bottom has dimensions of the whole square root of momentum and therefore cancel the dimensions of what's on the top.

373
00:44:41,760 --> 00:44:48,600
So it's dimensionless. That's the purpose of the that's the purpose of the horrible square root.

374
00:44:51,030 --> 00:44:56,759
So we find that this product, which is dimensionless, is equal to the Hamiltonian divided by H Bar Omega,

375
00:44:56,760 --> 00:45:01,440
which has the dimensions of energy because H Bar also has the dimensions of energy,

376
00:45:01,440 --> 00:45:06,300
times, time, omega, of course, has dimensions of one over time it's a free is the frequency of the oscillator.

377
00:45:06,630 --> 00:45:11,670
So this is has dimensions of energy minus a half, which is obviously dimensionless.

378
00:45:12,000 --> 00:45:15,270
So we have indeed almost fact arised.

379
00:45:15,270 --> 00:45:24,720
We have the statement now that H can be written as H Bar Omega, which carries the dimensions time z dagger a plus a half.

380
00:45:25,800 --> 00:45:36,720
We've almost factorisation just this that there. The next thing we want to do is calculate the commentator a dagger, a comma, a yes.

381
00:45:36,720 --> 00:45:40,950
We just got time to do this, a dagger of these two operations.

382
00:45:41,910 --> 00:45:52,110
So of course we will have a one over two H Bar Omega as a factor on the bottom because each of these A's brings in its own square root.

383
00:45:52,830 --> 00:46:10,500
And then we will have the commentator of m omega x minus i p on m omega x plus i p now.

384
00:46:11,400 --> 00:46:16,170
We and we have this breaks down into four commentators in principal.

385
00:46:16,440 --> 00:46:23,639
There's the commentator of this with this and the commentator of this with this the commentator of this

386
00:46:23,640 --> 00:46:30,000
with this obviously vanishes because excuse with itself and the commentator of this with this is.

387
00:46:36,030 --> 00:46:44,579
So we're going to have an m omega AI times X comma p, that's the computation of this with this.

388
00:46:44,580 --> 00:46:50,520
And now we have to deal with these with these terms. This produces a non-negotiable commentator.

389
00:46:50,520 --> 00:46:57,270
With that, we're going to have minus M omega ai times p, comma x.

390
00:47:00,050 --> 00:47:02,960
And then we'll have the commentator repeat with self, which will vanish.

391
00:47:05,780 --> 00:47:13,070
If I swap those two over, then clearly I change the sign in front and then this becomes a plus x comma p.

392
00:47:13,250 --> 00:47:14,690
It becomes this thing all over.

393
00:47:14,960 --> 00:47:27,980
So that cancels this and this whole caboodle is going to equal i x comma p over h bar that because we're going to get a two.

394
00:47:27,980 --> 00:47:30,920
These two terms are going to add together to make us a two, which cancels with this.

395
00:47:31,220 --> 00:47:36,020
And the moment is clearly go x, come a p is itself equal to h bar.

396
00:47:36,200 --> 00:47:41,270
So the i's make a minus one, the bars cancel and this is equal to minus one.

397
00:47:41,930 --> 00:47:48,710
So these two operators have non vanishing comitato actually equal to minus one.

398
00:47:51,780 --> 00:47:56,190
Yeah. Well, we seem to still have time to to nail this this problem, I think.

399
00:47:56,580 --> 00:48:01,770
So let us suppose we have got a state of a stationary state.

400
00:48:07,590 --> 00:48:12,900
Let us now apply the operator a dagger to both sides of this equation.

401
00:48:13,080 --> 00:48:16,440
Right then, this is just an eigenvalue. It's only a number.

402
00:48:16,440 --> 00:48:19,470
So I can then write E a dagger.

403
00:48:20,310 --> 00:48:23,940
E is equal to a dagger.

404
00:48:24,660 --> 00:48:28,260
H e that's obvious.

405
00:48:29,820 --> 00:48:37,410
I would like to swap these over so I jolly well do. I say this is equal to AJ Dagger plus a dagger.

406
00:48:37,980 --> 00:48:50,660
Commentator H. So this commentator puts in what I'm supposed to have and takes away what I'm not supposed to have but have previously written down.

407
00:48:53,390 --> 00:49:00,620
But we know what age is in terms we have that age is equal to there it is age bar, omega eight Dagger.

408
00:49:00,740 --> 00:49:08,450
So let's use that. So this is H, a dagger plus commentator of dagger.

409
00:49:09,170 --> 00:49:14,810
And H turns out to be a dagger. A plus a half.

410
00:49:15,800 --> 00:49:19,520
Close brackets. H bar omega to carry the dimensions.

411
00:49:20,420 --> 00:49:24,590
Close that, close that and stick in our E that we first thought of.

412
00:49:27,130 --> 00:49:30,460
So all I have done is replace H by an expression we already derived.

413
00:49:32,080 --> 00:49:38,110
Yeah. Now I have to take the commentator of a dagger with this.

414
00:49:38,230 --> 00:49:44,740
And with this. The commentator of Dagen with a half clearly vanishes because a half is just a number, not an operator.

415
00:49:45,790 --> 00:49:49,089
The commentator of a dagger with itself then vanishes.

416
00:49:49,090 --> 00:49:54,610
So. So when we do the commentator with this product, there should in principle be two terms, but only one of them survives.

417
00:49:55,060 --> 00:50:00,540
And that term is. That term is this sticks.

418
00:50:01,390 --> 00:50:04,870
This stands idly by while the dagger works on that.

419
00:50:08,980 --> 00:50:12,310
And then I have an omega. Omega.

420
00:50:12,910 --> 00:50:16,240
Close brackets. Close brackets. He.

421
00:50:18,530 --> 00:50:22,640
But we just worked this thing out and found that it's minus one, right?

422
00:50:22,820 --> 00:50:26,840
A dagger turned out to be minus one. So this is equal to.

423
00:50:29,340 --> 00:50:40,110
H a dagger e minus H bar omega a dagger e.

424
00:50:41,070 --> 00:50:46,440
Just to remind you, remind us what we had on the left. What we had on the left was e a dagger.

425
00:50:46,450 --> 00:50:49,590
E is just a restatement of what's been at the top.

426
00:50:50,460 --> 00:51:02,880
So we take this a dagger E and we obviously join it on to that degree and we discover that h on a dagger e is equal to well, h working on this.

427
00:51:03,240 --> 00:51:15,210
The cap that you get by using a dagger on E is equal to E plus h bar omega of dagger working on E, what does this tell us?

428
00:51:15,630 --> 00:51:26,070
It tells us that we have out of a state which had energy e we have constructed a state by multiplying by a dagger which has energy e plus h by omega.

429
00:51:26,730 --> 00:51:42,180
So this means that a dagger E is equal to a constant normalising constant not discussed times e plus h bar omega, a new stationary state.

430
00:51:50,350 --> 00:51:55,360
This is an incredibly powerful result because it immediately follows that we have states,

431
00:51:55,480 --> 00:52:03,340
if we can find a state e we can immediately generate E plus h bar omega by using this a dagger based.

432
00:52:03,670 --> 00:52:11,890
And also if we use a dagger on this, it follows we're going to get E plus two H Bar Omega and we're going to get another.

433
00:52:12,040 --> 00:52:17,710
If we use a dagger on this, we're going to get E plus three H Bar Omega.

434
00:52:18,100 --> 00:52:27,040
So we're going to get a whole infinite series of states of ever increasing energy simply by applying a dagger again and again and again.

435
00:52:27,640 --> 00:52:34,540
So what remains is to find what e what number E is, and that we will do first thing tomorrow.