1
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Yes. Well, we had where we had arrived with the harmonic oscillator yesterday.

2
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Woods. We had established that the expectation value of x squared in the excited state was two and plus one l squared.

3
00:00:24,140 --> 00:00:26,930
Yes, of course. Obviously on dimensional grounds. Yes, that's correct.

4
00:00:27,500 --> 00:00:33,200
So and I said the next item on the agenda would be to connect that back to classical physics.

5
00:00:33,680 --> 00:00:37,820
Always a valuable exercise because it tells you something about quantum mechanics, it checks your results.

6
00:00:40,100 --> 00:00:48,650
So of course, quantum classical physics doesn't know anything about two N plus one and the quantum number, the excitation number.

7
00:00:48,980 --> 00:00:54,050
But it does know about the energy and we know that n plus a half bar Omega is the energy.

8
00:00:54,320 --> 00:01:02,719
So we can write this is two times the energy of a h bar omega and this l squared.

9
00:01:02,720 --> 00:01:06,140
Well, L was defined to be h bar.

10
00:01:06,140 --> 00:01:13,640
I think of a2m omega. I probably had better check that my memory as that is correct.

11
00:01:13,640 --> 00:01:20,030
Yes. So this L squared is an H bar over two omega.

12
00:01:20,510 --> 00:01:27,380
So therefore this is the energy various things cancel of omega squared.

13
00:01:27,440 --> 00:01:32,120
The twos cancel ups. We need an m that survives in the algebra.

14
00:01:32,120 --> 00:01:35,929
Is that correct? So let's ask ourselves, what do we expect?

15
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What do we expect? Classically, classically? We have that.

16
00:01:44,160 --> 00:01:48,090
What do we expect? We expect that the time average to write this down now.

17
00:01:48,180 --> 00:01:58,510
Right. So the time average. Of X squared, which we'll call x squared bar.

18
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Since x is a simple harmonic function of time, the same average should be a half.

19
00:02:04,750 --> 00:02:12,190
Sure. This. This, this thing should be a half of x max squared.

20
00:02:12,730 --> 00:02:15,970
Right. The average of cost squared is a half.

21
00:02:16,330 --> 00:02:29,799
So if we are writing that x is equal to x costs omega t, it follows that x squared bar is a half of x squared the maximum perturbation.

22
00:02:29,800 --> 00:02:37,450
And what's the energy? The energy classically is a half k x squared because what is that?

23
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Maximum extension has no kinetic energy. It only has potential energy.

24
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That's how much it has. Omega squared is root omega is root K over m.

25
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So this is a half. So so k it's some.

26
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Omega is the square root of k of rhythm for a harmonic oscillator, that being the spring constant.

27
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So if I want to get rid of K, I have to declare it to be omega squared.

28
00:03:07,590 --> 00:03:11,130
M squared, omega squared, x squared. Right.

29
00:03:11,490 --> 00:03:22,890
So that leads to the conclusion that I'm expecting that x squared is equal to two E over m squared omega squared,

30
00:03:23,610 --> 00:03:28,960
which is, but this thing is equal to two of x squared bar.

31
00:03:29,370 --> 00:03:35,580
So this leads to the conclusion that x squared bar is equal to E of m squared.

32
00:03:35,700 --> 00:03:39,059
Omega squared in perfect agreement with the quantum mechanical result.

33
00:03:39,060 --> 00:03:46,330
Oh, do we need to put this up, don't we? And I've got a drift by name, so let me get rid of that stupid screen.

34
00:03:46,350 --> 00:03:49,739
I can do one thing at a time. Two stupid lights and blinds.

35
00:03:49,740 --> 00:03:56,250
Uh, screens up right screen, left screen.

36
00:03:56,250 --> 00:03:59,580
Left screen. No. Which. So who. Who are we talking about? You or me?

37
00:04:03,970 --> 00:04:07,500
Okay. Right.

38
00:04:07,500 --> 00:04:10,559
So there was a complaint. What went wrong?

39
00:04:10,560 --> 00:04:13,950
What went wrong was the squares of m. Where did I. Where did I goof on that?

40
00:04:13,950 --> 00:04:18,090
That was because E was a half. Uh, no, no, that was correct.

41
00:04:19,710 --> 00:04:23,200
Same number of square. Yes.

42
00:04:23,230 --> 00:04:26,469
Because it was inside the square. Root, root sign. Yes, exactly.

43
00:04:26,470 --> 00:04:29,890
So this shouldn't have been there. So this shouldn't shouldn't have been there.

44
00:04:29,890 --> 00:04:39,070
And then everybody's happy. Thank you. Okay, so doing this check, right?

45
00:04:39,400 --> 00:04:51,930
Of what? What have we done? We have. We've we've checked the classical sorry, the quantum mechanical result agrees with the classical result.

46
00:04:51,970 --> 00:04:57,150
Now, actually, amazingly, we've been able to do this independent of end, right?

47
00:04:57,180 --> 00:05:03,030
In other words, classical physics or quantum mechanics has recovered classical physics for all end.

48
00:05:04,050 --> 00:05:12,570
But we we we believe that we have to recover from quantum mechanics, classical physics, only in the limits of logic.

49
00:05:12,570 --> 00:05:17,040
And because our classical experiments are all ones where we're moving macroscopic bodies around,

50
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where the expectation energy will be large in some natural units.

51
00:05:22,140 --> 00:05:32,010
So we the exercise that of QM goes to classical physics.

52
00:05:37,830 --> 00:05:41,969
For a large and large quantum number.

53
00:05:41,970 --> 00:05:45,900
Here's our first example of a quantum number, a relevant quantum number. This is the correspondence.

54
00:05:45,900 --> 00:05:49,050
Principle of correspondence.

55
00:05:53,370 --> 00:06:03,910
But. And this is a an early example in some senses, perhaps not a brilliant example because we get perfect agreement for all end right.

56
00:06:03,920 --> 00:06:07,230
But what we're always requiring is agreement for login.

57
00:06:07,460 --> 00:06:11,600
But we really must have agreement for login because classical physics is is,

58
00:06:11,710 --> 00:06:15,050
is about, you know, it's been validated by experiments conducted at login.

59
00:06:16,520 --> 00:06:19,850
So in the same spirit. So let's talk about now the dynamics of oscillators.

60
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So what? So far we have found these stationary states and I've said several times these stationary states are highly artificial.

61
00:06:37,540 --> 00:06:46,689
One way you can see their artificiality is that that anti this the the state with energy and plus a

62
00:06:46,690 --> 00:06:54,520
half h bar omega at the time t is equal to that state at time t equals zero times e to the minus o.

63
00:06:55,300 --> 00:07:05,680
Now this is e overreach bar party. But since e is n plus a half h bar omega, this is n plus a half omega t.

64
00:07:06,670 --> 00:07:12,469
So each and every one of these states has a phase which increments in time at a frequency and plus a half,

65
00:07:12,470 --> 00:07:18,879
eight and plus a half omega t, but the oscillator oscillates at a frequency omega.

66
00:07:18,880 --> 00:07:22,840
Right. So we have to explain how it is that the oscillator oscillates at a frequency omega.

67
00:07:23,170 --> 00:07:26,620
But none of the states has a evolves in time.

68
00:07:26,620 --> 00:07:35,859
None of these stationary states evolves in time with the frequency omega not one so that the amp and moreover the the

69
00:07:35,860 --> 00:07:41,979
oscillators that we're familiar with in the school laboratory masses on weights and stuff will have values of n,

70
00:07:41,980 --> 00:07:49,120
which are like ten to the ten to the 28 or 34 or that kind of simply ginormous values of.

71
00:07:49,120 --> 00:07:56,950
And so the frequency here will be stupendous and nothing and nothing in the laboratory is happening at that frequency.

72
00:07:57,280 --> 00:08:00,910
So this is totally this is totally fantasyland. We have to get back to reality.

73
00:08:02,290 --> 00:08:06,699
We get back to reality by concentrating on expectation values, because it's our connection to classical physics,

74
00:08:06,700 --> 00:08:14,230
which is what we call is what we are pleased to call reality. So let's calculate the expectation value of X.

75
00:08:14,860 --> 00:08:25,630
If we do it for N, we know we're going to get a constant right, because when we take the complex conjugate of this complex number,

76
00:08:25,840 --> 00:08:29,470
it will multiply together with a complex number over here and make one.

77
00:08:29,650 --> 00:08:34,330
But we already know this. We already know that a stationary state has no time evolution whatsoever.

78
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So to get time evolution, what we need to do is, is say let's at the state of our system, we have to consider we have something that moves.

79
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We have to consider a system which does not have well-defined energy, which means that it's wave function,

80
00:08:53,590 --> 00:08:58,809
it's its state vector wave function, whatever is a linear combination of states, a well-defined energy.

81
00:08:58,810 --> 00:09:02,320
And let's suppose let's take a simple example. Let's suppose there are just two.

82
00:09:06,070 --> 00:09:11,820
Two states present. No, sorry.

83
00:09:11,910 --> 00:09:15,809
The proposal is. Let me take this. Let's do a song. Let's do it in all generality.

84
00:09:15,810 --> 00:09:21,600
So we get to write this as a and E to the minus I.

85
00:09:28,980 --> 00:09:33,150
So this is totally safe. Any state could be written like this.

86
00:09:33,210 --> 00:09:36,300
This is absurd. The state of my system at time times.

87
00:09:36,450 --> 00:09:40,460
It's a linear combination of states, of well-defined energy. There's no question I can do.

88
00:09:40,510 --> 00:09:45,570
That's the general initial condition. Let's work out the expectation value of X.

89
00:09:46,200 --> 00:10:01,710
So what is it? It's going to be the sum and star e to the eye and plus a half omega t times and times x times m.

90
00:10:02,430 --> 00:10:06,870
Times a m. Not start.

91
00:10:07,200 --> 00:10:14,270
It is a minus. I m plus a half omega t.

92
00:10:14,580 --> 00:10:20,100
So we can clean this this stuff up to.

93
00:10:20,190 --> 00:10:27,540
This is a sum of and an m of course it's going to be the sum and a in star.

94
00:10:27,570 --> 00:10:33,340
A m e to the. When we put these two together,

95
00:10:33,340 --> 00:10:48,729
we're going to have an E to the eye and minus M omega T times and x m and yesterday we already saw what the stylish way is to handle this expectation.

96
00:10:48,730 --> 00:10:56,170
Value here is to take advantage of an expression that we showed that the operator x can be written as L Times,

97
00:10:56,620 --> 00:11:03,699
a plus a dagger where l is the thing we were discussing earlier on.

98
00:11:03,700 --> 00:11:11,860
It's a square root of h bar over two and omega characteristic length.

99
00:11:13,750 --> 00:11:21,070
So and we also saw what happened when we took an expectation value where we were doing a slightly harder problem yesterday.

100
00:11:21,610 --> 00:11:35,680
So this is going to be very straightforward now because it's going to be L and a m plus and a dagger.

101
00:11:39,220 --> 00:11:48,160
All right. And this remember a on M produces M plus one in an amount.

102
00:11:48,250 --> 00:11:59,080
The square root of emphasise one. So this is going to be l root M plus one and m plus the square root.

103
00:12:00,040 --> 00:12:04,540
Excuse me. This A produces m minus one looks at minus one zero.

104
00:12:04,540 --> 00:12:11,760
I'm not concentrating at all right. But it's the square root of the bigger number, the biggest number that occurs.

105
00:12:11,770 --> 00:12:15,639
So this is a squared of m sorry. A on m produces m minus one.

106
00:12:15,640 --> 00:12:23,020
How much? The square root of the biggest integer that's involved. That's the square root of M this is going to produce M plus one.

107
00:12:25,900 --> 00:12:33,580
And with this normalisation, which is the square root of the biggest number involved, which is M plus one, so and M plus one.

108
00:12:35,850 --> 00:12:42,100
What we want is we have a here are some of N and M, let's do the sum of M first.

109
00:12:42,170 --> 00:12:49,570
Right. Bearing in mind that that thing is the sum of a delta n and minus one and a delta and plus one.

110
00:12:50,290 --> 00:12:55,540
So we're going to have the for our oscillator, the expectation value of X is going to be.

111
00:12:57,640 --> 00:13:06,970
Okay, let's take this first one. When we do this first one sum over n, we're going to have that this is a and star a how much?

112
00:13:08,800 --> 00:13:14,380
In order for this to be to be not zero, m has to be one bigger than N.

113
00:13:14,890 --> 00:13:18,190
So this is going to be the square root of n plus one.

114
00:13:19,300 --> 00:13:25,390
And everywhere where I everywhere where I see an m, I'm going to have to write an N plus one.

115
00:13:31,470 --> 00:13:37,150
E to the eye. And now in this case, we've agreed that M is one bigger than ends.

116
00:13:37,150 --> 00:13:47,010
This is going to be easier than minus omega T, so that's what we're going to get from this term and then try from this term when that goes in there.

117
00:13:47,280 --> 00:13:52,560
And now we have to put this in there and now M is going to be one to get a zero contribution.

118
00:13:53,190 --> 00:13:59,370
M is going to N plus one is going to have to be N and so M is going to be n minus one.

119
00:13:59,640 --> 00:14:07,110
So we're going to get an A and store and minus one times the square root.

120
00:14:09,360 --> 00:14:13,760
Of MN. E.

121
00:14:14,790 --> 00:14:19,319
And now in this case, M is going to be smaller than N is going to be easy.

122
00:14:19,320 --> 00:14:28,840
Plus, I mean, Kitty. And I've lost my l somewhere along the line that may be reinstated.

123
00:14:29,100 --> 00:14:34,680
Right. There's this L here. I hope we have everything. Slightly scared that we haven't.

124
00:14:36,270 --> 00:14:41,370
Let me just check them. Now seems to be okay.

125
00:14:41,910 --> 00:14:50,190
So. And we're still swimming.

126
00:14:50,700 --> 00:14:54,840
I've lost some sign we are still swimming over end.

127
00:15:04,510 --> 00:15:07,600
Why don't we declare that in this?

128
00:15:07,690 --> 00:15:12,390
These are two separate sums. And in this sum I can introduce a new notation.

129
00:15:12,400 --> 00:15:23,290
I can say that. And primed is equal to and is equal to and primed minus one.

130
00:15:25,440 --> 00:15:35,880
In this sum here. And then some of end primed, and then I can relabel the end, primed end.

131
00:15:36,150 --> 00:15:40,860
And this term becomes the same as that term becomes the complex conjugate of that term.

132
00:15:40,870 --> 00:15:56,489
So when I do this, I'm going to have a sum now of rn primed l a primed primed a and primed minus one a and that start that stored

133
00:15:56,490 --> 00:16:05,670
and primed times the square root of and primed e to the minus I make a t and the other sum is still over.

134
00:16:05,670 --> 00:16:08,790
And and that's a star.

135
00:16:09,900 --> 00:16:22,590
And A and minus one e to the I make a t, but these n and primed are the same other.

136
00:16:22,710 --> 00:16:28,260
I mean, they're just dummy indices we just summing over them. So this sum is in fact the complex conjugative that some.

137
00:16:29,570 --> 00:16:35,240
These two things are complex conjugates. So what we we.

138
00:16:36,980 --> 00:16:53,130
If we. If we write a and a and minus one is is equal to say X and E to the I phi which we can do.

139
00:16:53,250 --> 00:16:57,620
This is real. And this, of course, is real, too.

140
00:16:58,130 --> 00:17:01,370
If I'm writing a complex number. Sorry, that needs a star in it.

141
00:17:03,260 --> 00:17:11,750
Then I'm going to be taking x e to the minus i omega t minus ify.

142
00:17:11,810 --> 00:17:15,050
Plus this stuff is the omega t plus sci fi.

143
00:17:15,980 --> 00:17:30,460
And we're going to be able to combine the two exponentials and discover that x is equal is equal to L times the sum of x and costs omega t plus five.

144
00:17:47,760 --> 00:17:52,440
Sorry. We need to fly in on that. Excuse me. We need to find him. So this obviously needs an end.

145
00:17:52,440 --> 00:17:58,200
And that is an end because each of these complex numbers has, as its own matter, has its own fate.

146
00:17:58,250 --> 00:18:09,350
So what do we discovered? We discovered that, lo and behold, the position, the expectation, value the position does oscillate with periodically.

147
00:18:09,360 --> 00:18:17,790
This is now this. We have indeed sinusoidal.

148
00:18:20,920 --> 00:18:27,530
Oscillation. But period.

149
00:18:29,870 --> 00:18:36,950
Two Pi of omega. We have, in fact, recovered classical physics to the classical motion.

150
00:18:37,490 --> 00:18:45,620
So the motion at this frequency occurs because of interference, quantum interference between states.

151
00:18:48,110 --> 00:18:53,540
So these these these terms, we've got quantum interference between states of different energy.

152
00:18:55,580 --> 00:18:59,710
Right? It was. Why do we have states of different energy involved?

153
00:18:59,720 --> 00:19:02,820
It was because X that.

154
00:19:04,730 --> 00:19:11,270
Oh. It it was because.

155
00:19:12,950 --> 00:19:20,899
So when we went to the expectation value of X, we've got this huge long sum which involved cross terms between states of it it inclu,

156
00:19:20,900 --> 00:19:31,840
it included the term and x and that was also involved in here in which the same the state of a given energy was was present on on both sides of X.

157
00:19:32,420 --> 00:19:38,840
But that made no contribution to the sum because x is a sum can be written as a sum of these letter operators,

158
00:19:39,110 --> 00:19:44,690
of these annihilation creation operators. And if you put the same state on either side, you get nothing.

159
00:19:44,990 --> 00:19:50,330
You only get something if the states on either side differ in energy by one unit of excitation.

160
00:19:51,230 --> 00:19:58,880
So as a result, all arose from interference between states which differ in energy by one by one excitation.

161
00:20:00,080 --> 00:20:01,489
And it's a peculiar.

162
00:20:01,490 --> 00:20:13,280
So that's a very general phenomenon and a peculiar feature of of of this problem is that those differences in energy are all the same.

163
00:20:13,580 --> 00:20:18,640
They're all H Bar Omega and and the frequency.

164
00:20:18,680 --> 00:20:24,709
Well, we'll see this in a moment. The frequency of these oscillations. So all these terms have the same sinusoidal.

165
00:20:24,710 --> 00:20:29,090
We have an infinite number of contributions still, but they all have the same sinusoidal behaviour.

166
00:20:30,680 --> 00:20:38,749
So we've recovered the important feature of a harmonic oscillator that the, that the period is independent of the amplitude of the excitation,

167
00:20:38,750 --> 00:20:47,000
the amplitude of the excitation is controlled by which of these ends are significantly large.

168
00:20:47,060 --> 00:20:52,670
Right. Because and it's the amplitude to have energy and plus a half h bar omega.

169
00:20:52,670 --> 00:20:56,080
So a highly excited oscillator has uh,

170
00:20:56,510 --> 00:21:04,309
has the non-zero values of N are all clustered around a large value of N and a very gently

171
00:21:04,310 --> 00:21:12,379
excited oscillator has the A ends around around zero or small values of and being fairly large.

172
00:21:12,380 --> 00:21:20,959
And therefore this sum will be and this sum will be dominated by whatever region has the large values of a but the result

173
00:21:20,960 --> 00:21:31,430
we've got is that there's harmonic motion at frequency to a frequency omega regardless of which terms in this sum dominating.

174
00:21:32,650 --> 00:21:37,780
And that's this property that that the period does not depend on the amplitude.

175
00:21:39,510 --> 00:21:43,290
So let's let's be more realistic and invest.

176
00:21:43,410 --> 00:21:50,400
See how much more of classical physics we can we can get out of this by talking about an and harmonic and harmonic oscillator.

177
00:21:58,570 --> 00:22:07,660
So I, I introduced harmonic oscillators by saying that they, they're widespread because if you have a point of equilibrium,

178
00:22:07,930 --> 00:22:13,060
if you plot against displacement from from point of equilibrium, you plot the force.

179
00:22:13,690 --> 00:22:21,579
You have some curve that looks like this and should pass through zero and should pass

180
00:22:21,580 --> 00:22:25,690
through zero at the point of equilibrium by definition of a point of equilibrium,

181
00:22:26,560 --> 00:22:33,760
but that if you displace yourself from either side of the point of equilibrium, if it's stable the force slopes like this, it's positive.

182
00:22:35,130 --> 00:22:37,870
That's right. Really should be negative, shouldn't it, actually? And I come to think of it.

183
00:22:38,290 --> 00:22:42,820
Sorry, I should draw the the graph this way around sun in order to get a stable force.

184
00:22:43,210 --> 00:22:48,760
So if I displace myself positively in X, the force becomes negative and pushes me back.

185
00:22:48,760 --> 00:22:52,910
If I displace myself negatively, the force becomes positively and pushes me back.

186
00:22:52,930 --> 00:23:02,770
So that's that's a stable equilibrium. And if a harmonic oscillator arises, if we replace if we approximate the curve,

187
00:23:03,700 --> 00:23:09,340
the curve of this force versus distance by the straight line, that's tangent to it at that point.

188
00:23:09,910 --> 00:23:18,940
So basically what we're doing is so any, any force versus distance curve could be expanded as some kind of a Taylor series.

189
00:23:19,930 --> 00:23:26,530
And if we just take the first non-trivial term in that Taylor series, we have a harmonic oscillator.

190
00:23:26,530 --> 00:23:32,140
If we take subsequent terms, we will have a not harmonic oscillator and harmonic oscillator.

191
00:23:34,020 --> 00:23:39,200
And typically. And typically the force versus.

192
00:23:39,470 --> 00:23:44,480
So for a harmonic oscillator, the force versus distance is a is a straight line that goes all the way to infinity,

193
00:23:44,750 --> 00:23:50,330
which means that in order to pull your your spring apart, you have to do infinite work.

194
00:23:51,200 --> 00:23:56,030
Because to get X to go to infinity, you have to overcome a force that goes to infinity.

195
00:23:56,480 --> 00:24:04,840
So infinite work is required to pull this thing apart. But all real oscillators, all macroscopic ones, certainly you can just just break them.

196
00:24:04,880 --> 00:24:09,560
So only a finite amount of energy is required to push X off to infinity.

197
00:24:09,980 --> 00:24:15,950
And that's reflected in the fact that typically the force versus distance curve slopes over like this,

198
00:24:17,000 --> 00:24:32,510
so that if we if we plot the potential V versus X in the harmonic case, we have a parabola that looks like this and disappears off to infinity.

199
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So this is the harmonic oscillator. But in a real oscillator, the force, the potential curve starts from some finite value is infinity.

200
00:24:47,340 --> 00:24:53,220
Sorry. And I should. I need to draw it so it becomes tangent to this and then disappears off like this.

201
00:24:53,400 --> 00:24:56,190
So this is a this is a more realistic curve.

202
00:24:58,920 --> 00:25:09,030
And the mark oscillator is is a good model if the parabola is tangent to the the realistic curve over a decent range.

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That's the main idea. So what we should do is investigate.

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00:25:14,470 --> 00:25:18,370
Let's. Let's to see what quantum mechanics has to say about more realistic oscillators.

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00:25:19,270 --> 00:25:31,719
Let us take. So this is just an example. Supposing we take V of X is minus some constant A squared plus x squared.

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And I suppose we need an x squared on top to get the dimension straight.

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So supposing we take that to be our potential curve, then we can no longer.

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We can no longer we now sit down.

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00:25:52,910 --> 00:25:59,569
We have a perfectly well-defined Hamiltonian p squared over to him, plus this v of x,

210
00:25:59,570 --> 00:26:03,200
but we can no longer solve this analytically any more than we can actually

211
00:26:03,200 --> 00:26:07,680
analytically integrate the equations of motion classically in this potential.

212
00:26:07,700 --> 00:26:14,680
So in either case, you you can't do it. But it's pretty straightforward to solve this problem.

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00:26:15,790 --> 00:26:20,260
Eight G equals E numerically.

214
00:26:24,020 --> 00:26:36,680
We do it in the position representation we browse through by x and have that x squared over to m e plus x.

215
00:26:37,760 --> 00:26:49,760
The E is equal to e x e, which turns into this by the rules we've already discussed.

216
00:26:49,760 --> 00:26:57,320
This turns into an ordinary differential equation minus h squared over to m d to you

217
00:26:58,160 --> 00:27:06,140
by the x squared plus v of x u is equal to EU where you of course is equal to x.

218
00:27:08,060 --> 00:27:18,950
Okay, so this is an ordinary differential equation, second order, etc. and it's linear and it is, it's pretty straightforward to solve numerically.

219
00:27:19,250 --> 00:27:21,290
If you look in the book as a footnote, that explains how to do that.

220
00:27:21,770 --> 00:27:28,490
Now, I should have I'm afraid I meant to bring my laptop with the with the official figures.

221
00:27:28,730 --> 00:27:31,610
But when you do this. So, so.

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00:27:33,590 --> 00:27:41,900
By discrete izing this differential equation, we turn it into a into a, an exercise in linear algebra, which your computer solves.

223
00:27:42,320 --> 00:27:52,990
So you, you write this basically as a matrix and uh, on, on you, which becomes a column vector.

224
00:27:53,000 --> 00:27:58,820
The value that you takes of the different of the different positions in X is equal to E.

225
00:27:59,060 --> 00:28:02,870
You. So you turn it into a matrix equation. And computers are very good at matrix equations.

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00:28:02,870 --> 00:28:09,440
When you do that, you discover what the values of are, and you can also discover what these what these way functions look like.

227
00:28:09,650 --> 00:28:21,740
And the crucial thing is that you find that if you plot the possible energies, you you get a distribution that looks like this.

228
00:28:21,740 --> 00:28:26,060
It starts off looking like an equally spaced ladder for the harmonic oscillator.

229
00:28:26,270 --> 00:28:30,330
There are steps, each one of which is separated by a bar omega for an alpha.

230
00:28:30,380 --> 00:28:37,340
So we start off like that with the spacing given by the harmonic oscillator that the tangent to the bottom of the curve.

231
00:28:37,580 --> 00:28:44,750
But as we go up, the spacing gets gets less and less and less and less and less and less.

232
00:28:45,380 --> 00:28:49,940
And what essentially the the algebra is doing is giving you an infinite number

233
00:28:49,940 --> 00:28:58,460
of allowed energies or already in a finite range because this is v nought.

234
00:29:00,840 --> 00:29:09,809
So so that potential allows that potential as X goes to infinity, goes to a finite value, not mine, where it goes to zero.

235
00:29:09,810 --> 00:29:13,920
So this is zero and I guess this is minus v nought.

236
00:29:15,500 --> 00:29:17,780
So that the lowest energy is somewhere down here.

237
00:29:18,110 --> 00:29:26,690
So with only a finite range in energy, you pack in an infinite number of allowed energies with a harmonic oscillator.

238
00:29:28,460 --> 00:29:35,180
You have to you pack in an infinite number of these things, but in an infinite energy range, because this ladder goes on forever.

239
00:29:35,180 --> 00:29:39,889
Right up to the heavens. Okay.

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00:29:39,890 --> 00:29:43,100
So that's the this is this is a very generic behaviour.

241
00:29:45,130 --> 00:29:48,220
That we will encounter again in real systems.

242
00:29:49,500 --> 00:29:54,659
Now. What's the physical consequence of that? Suppose we have.

243
00:29:54,660 --> 00:30:08,370
So now let's let's say our initial condition is this that it consists simply of two terms and of n plus a n plus one of n plus one.

244
00:30:09,480 --> 00:30:13,140
And so the time evolution is going to be.

245
00:30:20,400 --> 00:30:34,300
And this one's going in a self some space and plus one E to the minus e and plus one T on h bar and plus one.

246
00:30:39,410 --> 00:30:45,920
So that's that's not a completely general initial condition now because I'm assuming that there are only two non vanishing amplitudes.

247
00:30:45,920 --> 00:30:49,650
So my, my state, uh.

248
00:30:51,370 --> 00:30:55,540
It happens to be such. There are only two possible values of the energy that I can measure.

249
00:30:55,750 --> 00:31:02,050
There's an amplitude. And to measure the energy and and there's an aptitude a and plus one to measure the next highest energy.

250
00:31:03,050 --> 00:31:05,260
Okay. So this is this is kind of a special case.

251
00:31:06,160 --> 00:31:22,390
If we now work out what the expectation value of X is for this special case, we find that it is a and star e to the i e in t h bar.

252
00:31:23,380 --> 00:31:34,990
And this is all very similar to the other case and plus one star e i e in plus 1th bar.

253
00:31:38,090 --> 00:31:44,630
X. And then the same stuff on and then.

254
00:31:45,620 --> 00:31:50,360
And he's the minus I ian to your natural.

255
00:32:07,580 --> 00:32:17,280
Now, when we multiply this stuff out, we will get. We generally get only two terms.

256
00:32:17,290 --> 00:32:22,930
We'll have this on this and this on this.

257
00:32:23,350 --> 00:32:37,360
The reason for that is that we will show later on that for that potential X and so for the harmonic oscillator, this is true.

258
00:32:38,350 --> 00:32:42,370
But it's not only true for a harmonic oscillator that this thing vanishes here.

259
00:32:42,640 --> 00:32:50,020
It's going to be true for any potential. Well, which is symmetrical around x equals nought.

260
00:32:50,260 --> 00:33:03,600
So this follows from symmetry. A V of x that v of x is an even function of x so long as vivek's the potential is an even function.

261
00:33:03,600 --> 00:33:06,929
The same behaviour at minus x is plus x. This will.

262
00:33:06,930 --> 00:33:12,060
This will vanish. We will show this as we as we go along. We haven't shown it yet, but that that will be true.

263
00:33:12,720 --> 00:33:17,530
So given that that so. Quite generally.

264
00:33:20,600 --> 00:33:26,140
My expectation value here is going to be a star.

265
00:33:26,810 --> 00:33:42,020
So it's going to be this on this e to the I D and minus E and plus one t on h bar times the matrix element and X and plus one.

266
00:33:44,080 --> 00:33:51,100
Plus excuse me and I'm needing here some a and still a and plus one.

267
00:33:51,100 --> 00:33:55,209
Right. So that's that. On that.

268
00:33:55,210 --> 00:34:16,600
And then we will have this on this a and and plus one star e to the minus I e and minus C and plus one T on both times end plus one and x and.

269
00:34:19,780 --> 00:34:25,240
Hope I've done that right. So what do we have now?

270
00:34:25,260 --> 00:34:28,649
We have a gain that this term is the complex.

271
00:34:28,650 --> 00:34:32,160
Well, we have this term is a complex conjugate of this term.

272
00:34:34,160 --> 00:34:38,000
So we're looking at a sinusoidal function. Plus it's complex conjugate.

273
00:34:38,270 --> 00:34:38,749
Therefore,

274
00:34:38,750 --> 00:34:58,790
we're looking at something which is x and could be written as x and cos e and cos e and minus C and plus one T on h bar plus a possible phase factor.

275
00:34:59,660 --> 00:35:09,290
Right. Where just to be concrete x and is the model is the modulus of and and plus one.

276
00:35:09,620 --> 00:35:16,640
So and AA plus one is a complex number. I have its modulus sticking out here and I stuck its phase into that.

277
00:35:20,620 --> 00:35:25,630
So what do we what do we observe? Again, we have harmonic motion, sinusoidal motion.

278
00:35:26,350 --> 00:35:36,790
But look now at the period of this sinusoidal motion, the period of the frequency of this sinusoidal motion now depends on PN because it's the again,

279
00:35:36,790 --> 00:35:40,480
it's the difference of two energies of adjacent energies, which counts.

280
00:35:41,470 --> 00:35:48,430
And as we increase n according to my bad sketch up there, the difference between adjacent energies gets smaller.

281
00:35:49,600 --> 00:35:55,610
So the frequency. Becomes.

282
00:36:00,010 --> 00:36:02,800
Smaller with increasing in.

283
00:36:09,060 --> 00:36:17,940
So we're recovering a classical fact, which is that if you have if you have an uncommon oscillator of a typical type and you make bigger,

284
00:36:18,000 --> 00:36:22,470
you kick it to a bigger oscillations, it it's period will slow down.

285
00:36:23,010 --> 00:36:29,100
And that's true of an ordinary pendulum. An ordinary pendulum has its highest frequency.

286
00:36:29,100 --> 00:36:34,620
If you you have it go to and fro with a small amplitude the clock makers make their pendulums go to.

287
00:36:34,980 --> 00:36:39,240
But the period goes if you as you increase the amplitude of the oscillations.

288
00:36:39,240 --> 00:36:43,830
So this is this is, as it were, high, high omega for a pendulum.

289
00:36:45,450 --> 00:36:52,210
As you boost the period to the point at which it's about to go overtop dead centre, you know,

290
00:36:52,230 --> 00:36:58,500
if you have if you break it swings it goes like this and then like this the period goes to infinity formally.

291
00:36:58,830 --> 00:37:00,030
Well, it does go to infinity.

292
00:37:00,030 --> 00:37:06,510
It's hard to do experimentally as you increase the amplitude to the point at which it would go or just keep keep on going.

293
00:37:06,780 --> 00:37:16,079
It had enough energy to go right through top to centre. So so this slowing of the period with increasing amplitude is, is manifested in a,

294
00:37:16,080 --> 00:37:19,950
in a standard pendulum and we see how it emerges from the structure.

295
00:37:19,950 --> 00:37:28,620
So here we're learning something important, the way in which the dynamics is encoded in the spacing of the energy levels of the of the stationary.

296
00:37:28,620 --> 00:37:35,580
The energies of the stationary states. This is these are just simple examples of what's totally generic.

297
00:37:36,960 --> 00:37:47,790
Okay. Something else that we can learn about this on harmonic oscillator is that so more generally, this was a simple example.

298
00:37:47,790 --> 00:37:51,209
I said let's consider in order to get something to move.

299
00:37:51,210 --> 00:37:54,480
I considered a state that was under it had undefined energy.

300
00:37:54,750 --> 00:37:55,709
To keep it simple,

301
00:37:55,710 --> 00:38:07,950
I took just two non vanishing amplitudes the only two 3 to 5 non vanishing amplitudes in the expansion of the state and stationary states.

302
00:38:08,550 --> 00:38:13,770
Realistically, we would have if you take an ordinary pendulum like that and you and you give it a jog,

303
00:38:13,980 --> 00:38:20,130
you will the energy will be uncertain by zillions of values of Omega.

304
00:38:20,310 --> 00:38:25,590
And many, many, many, many, many of these coefficients will be non will be non vanishing.

305
00:38:25,590 --> 00:38:35,680
So more generally. We're going to have the upside is equal to we will have many terms and let's just write down a few

306
00:38:35,680 --> 00:38:48,160
of these terms and minus one e to the minus i e n minus one t on each bar plus a and e to the I in,

307
00:38:49,000 --> 00:38:57,489
etc. W So there will be many, many of these coefficients, but if we know pretty much what the energy of the oscillator is,

308
00:38:57,490 --> 00:39:03,280
right, we've, we've lifted our bulb up to 30 degrees or something and let it go.

309
00:39:03,280 --> 00:39:08,139
The energy is not completely undetermined. And what that means is that the many of these will be non-zero,

310
00:39:08,140 --> 00:39:20,530
but they will all be clustered around some particular value of N so that if you look at the at the value of one of these amplitudes,

311
00:39:20,740 --> 00:39:28,300
the modulus of it as a function of n, you'll find that you'll get a pattern sort of like this somehow.

312
00:39:28,630 --> 00:39:36,190
Right? There'll be an N at which the at which the amplitudes peak and they'll be small values here because we're pretty

313
00:39:36,190 --> 00:39:40,810
certain the energy isn't that small and small values here because we're pretty certain the energy isn't that large.

314
00:39:41,260 --> 00:39:43,450
So that's the the generic situation.

315
00:39:44,920 --> 00:39:52,569
And when we come in of calculate the expectation value of X, what we now have is the same sort of thing up as up there.

316
00:39:52,570 --> 00:40:00,940
But it's it's somewhat more complicated. We're going to have an and still a n minus one.

317
00:40:01,240 --> 00:40:19,060
Uh, some of the things that we had before e to the I n minus E and minus one T on h bar times, some matrix element and x and minus one.

318
00:40:19,930 --> 00:40:23,850
And then we will have to do this the other way around and.

319
00:40:29,680 --> 00:40:34,839
And then I need to put in a minus sign. Right. Then I will have the next one.

320
00:40:34,840 --> 00:40:38,350
I will have because an x and will vanish.

321
00:40:38,350 --> 00:40:40,360
The next one I will buy the cemetery property.

322
00:40:40,630 --> 00:40:58,930
I will have a and plus one store and e to the minus I e and minus E and plus one t on each bar each four times and x and plus one.

323
00:40:59,470 --> 00:41:07,330
And then I will have not plus equals, but plus a and plus three store.

324
00:41:07,510 --> 00:41:23,180
And this will be the next term. Each of the i e in minus e and plus three t over h bar and X and plus three plus two.

325
00:41:23,230 --> 00:41:30,580
Not at all. Right. This is this is a specimen of a disgusting expression which would give us the expectation value of X to the.

326
00:41:36,010 --> 00:41:40,900
This combination of terms we've already seen, this is nothing really new.

327
00:41:41,770 --> 00:41:44,500
The harmonic oscillator had just this kind of thing.

328
00:41:44,740 --> 00:41:52,000
In the case of a harmonic oscillator, this energy difference was exactly minus this energy difference,

329
00:41:52,360 --> 00:41:55,600
making this exponential in this extra amount complex conjugates.

330
00:41:56,440 --> 00:42:04,690
They will not now be. Exactly. This will not be exactly this, because this is the difference between an and N minus one.

331
00:42:04,900 --> 00:42:11,950
And this is the. So one step in the latter. And this is the the size of the step one above it, in the ladder, which we slightly smaller.

332
00:42:13,900 --> 00:42:18,440
And I think you have your problem. Oh, thank you.

333
00:42:18,680 --> 00:42:21,760
I have. That's right. Because I changed my mind how I was going to do this.

334
00:42:21,770 --> 00:42:28,850
And I thank you very much. So this should be in my plus one and that should be in and this should be plus three.

335
00:42:30,650 --> 00:42:33,740
Okay. It's good to know that.

336
00:42:34,680 --> 00:42:38,330
There's understanding in the room. So.

337
00:42:41,670 --> 00:42:46,800
So that's one thing that's going to happen because this is a more realistic oscillator as these frequencies will be changing.

338
00:42:47,190 --> 00:42:56,610
And this crucially, this frequency here will be present where it wasn't present in the harmonic oscillator case.

339
00:42:56,610 --> 00:42:59,760
In the harmonic oscillator case, this number here vanished.

340
00:42:59,850 --> 00:43:05,280
In principle, this this would have been here. But this matrix element vanishes for the harmonic oscillator.

341
00:43:10,940 --> 00:43:14,150
But it's not generally going to vanish. It's going to stick around.

342
00:43:14,570 --> 00:43:22,550
Now that has an important consequence because this frequency so e n plus three

343
00:43:22,910 --> 00:43:34,910
minus e n is going to be on the order of three times E and minus E and minus one.

344
00:43:37,620 --> 00:43:43,040
Right. Because it's the difference. It's it's three steps on the ladder and that's only one step on the ladder.

345
00:43:43,050 --> 00:43:49,560
And if we think of of that, the size of the steps in the ladder becoming small only gradually as we go up the ladder,

346
00:43:49,560 --> 00:43:53,160
which will be is, is, is a good picture to use.

347
00:43:53,730 --> 00:44:02,190
Then then this term is going to be essentially three times the frequency associated with the other two, which we can regard is about about the same.

348
00:44:03,000 --> 00:44:05,909
So when we, when we assemble all this stuff,

349
00:44:05,910 --> 00:44:18,810
we're going to find that x the expectation value of X looks like some number times costs of e n minus the N plus one.

350
00:44:18,990 --> 00:44:22,920
Now let's, let's, let's declare this to be three omega n, right?

351
00:44:22,920 --> 00:44:28,379
So we defining omega n to be this quantity here and we're going to find that we

352
00:44:28,380 --> 00:44:36,540
have a cos some sum term like cos Omega and T and we're going to have some other

353
00:44:36,540 --> 00:44:43,079
term with some other coefficient times cos three Omega and T and we're going to

354
00:44:43,080 --> 00:44:49,350
have some other term with x five some other coefficient times cos five omega n t.

355
00:44:54,230 --> 00:44:58,410
So the, this number will depend on, uh,

356
00:44:59,540 --> 00:45:12,560
will contain products of stuff like a and a and plus three star and a and plus three A and plus five star, etc.

357
00:45:12,590 --> 00:45:18,120
Right. And this will contain things like a and a and, uh.

358
00:45:22,430 --> 00:45:26,230
Plus. Plus five.

359
00:45:31,110 --> 00:45:35,600
But we will have these other frequencies present. And this is what leads to.

360
00:45:35,610 --> 00:45:39,840
So this. This series implies periodic motion.

361
00:45:41,740 --> 00:45:53,620
But on harmonic motion. So in a musical instrument.

362
00:45:56,100 --> 00:46:00,360
You know, the the. The.

363
00:46:01,430 --> 00:46:07,610
The note. The motion of the of the string in a piano or the vibrations in an organ tube or flute tube

364
00:46:07,610 --> 00:46:15,290
or whatever has its it has a it has a well-defined frequency which that sets its pitch.

365
00:46:15,560 --> 00:46:23,690
But the but the particular tone of the instrument is determined by the characteristic numbers of of higher harmonics,

366
00:46:23,690 --> 00:46:27,670
which are present because it's an and harmonic oscillation. Uh.

367
00:46:29,440 --> 00:46:41,180
Typically. But there's there's more that we can do here, which is connecting to classical physics,

368
00:46:41,600 --> 00:46:50,000
which is to make the point that if we arrange this stuff, right, this expectation value of X, we take out the leading term.

369
00:46:50,270 --> 00:47:00,140
We say that this is E to the minus omega and T, and then we're going to have some some.

370
00:47:04,390 --> 00:47:08,060
Well. No. Let's, let's.

371
00:47:08,060 --> 00:47:09,200
Let's try it. It's better.

372
00:47:09,720 --> 00:47:15,510
It gets very complicated using about the expectation, but it makes it's easier if we think about website itself as a function of time.

373
00:47:16,450 --> 00:47:25,280
If we look at upside cells as a function of time, we can take an E to the minus I, e and T on each bar out.

374
00:47:25,290 --> 00:47:46,440
And we can say that this is dot, dot, dot a and and plus e to the i e in plus one e and plus one t on each bar.

375
00:47:48,780 --> 00:47:55,630
And plus one times a and sorry.

376
00:47:55,820 --> 00:48:03,639
Right. This needs to be. Yeah.

377
00:48:03,640 --> 00:48:10,750
That's you need a minus sign there or I need a T on all that. So I've taken the common factor out so this one doesn't have any exponential.

378
00:48:10,760 --> 00:48:16,870
This one should have its proper exponential minus the thing that I've taken out the next one should have.

379
00:48:16,870 --> 00:48:18,310
And this should be a and plus one.

380
00:48:18,760 --> 00:48:33,970
This should be plus and plus to e to the minus i e n plus two minus e n t over h for and plus two and so on and so forth.

381
00:48:36,730 --> 00:48:43,300
So you have to a lowest order approximation. These differences here are all multiples of a common frequency.

382
00:48:43,600 --> 00:48:58,090
And after a, after a period. So when, uh, e and t over h bar is equal to two pi, these things.

383
00:48:58,330 --> 00:49:03,280
Uh, sorry. And.

384
00:49:04,410 --> 00:49:16,640
Plus one minus C in. After after the time it takes for this to come around to to pay, this will come round two for pi almost, and so on and so forth.

385
00:49:16,940 --> 00:49:18,470
And so the wave function will look.

386
00:49:19,040 --> 00:49:26,089
All this sum will be the same as it was at t equals nought, because all of these exponentials would have come round to one.

387
00:49:26,090 --> 00:49:33,049
Again, that's in the case. That's the that's in the case that these things are all multiples of the same frequency.

388
00:49:33,050 --> 00:49:36,050
But as we've seen, they're not quite multiples of the same frequency.

389
00:49:36,470 --> 00:49:39,920
This is slightly smaller than twice this.

390
00:49:40,340 --> 00:49:50,630
And so in the time it takes for this one to come around to PI, this one has come round isn't quite round to pi and even more so further down the line.

391
00:49:50,810 --> 00:49:57,650
And therefore the wave function isn't quite back to where it was when a t equals nought.

392
00:49:58,220 --> 00:50:05,900
And as we as as this, as time goes on and we count more periods so this becomes two pi end,

393
00:50:06,470 --> 00:50:10,730
these discrepancies become more and more and more important in these terms down here.

394
00:50:10,910 --> 00:50:20,150
When this one is come round to two to end pi, this one will be significantly off to end pi and this one even more so.

395
00:50:20,780 --> 00:50:24,380
And that means that the wave function is not returning to its original value.

396
00:50:24,860 --> 00:50:27,680
And we're have we're looking at motion which is not periodic.

397
00:50:28,310 --> 00:50:34,610
And whereas initially because we released our particle from some particular point in the potential, well,

398
00:50:34,970 --> 00:50:43,190
these wave functions all constructively interfered here at a particular value of X after a certain number of basic periods,

399
00:50:43,430 --> 00:50:49,460
the interference, the constructive interference here and the destructive interference everywhere else will become less and less exact.

400
00:50:50,330 --> 00:50:56,900
And the disruption and the distribution of our particle will become more and more vague until after a very long period.

401
00:50:57,500 --> 00:51:01,430
The phases of these will be essentially random, and we'll have no knowledge of where it is.

402
00:51:01,910 --> 00:51:05,629
And this is precisely mirrored in classical physics. In classical physics,

403
00:51:05,630 --> 00:51:15,740
the small uncertainty in energy that was associated with having more than one and in this series was associated with a small uncertainty in period.

404
00:51:17,450 --> 00:51:22,189
If the periods, if the if the energy was very high, the period would be very long.

405
00:51:22,190 --> 00:51:28,009
And after a long time, the particle would have gone around and around a million times and a million and a bit times.

406
00:51:28,010 --> 00:51:28,790
And here it would be.

407
00:51:29,450 --> 00:51:39,379
But if it was slightly different, slightly lower energy would have a slightly higher frequency and it would have done it 1,000,001 oscillations,

408
00:51:39,380 --> 00:51:46,760
and it would be over here so that you can see that the small uncertainty in energy is going to lead off for a long time through

409
00:51:46,760 --> 00:51:52,820
the small uncertainty in period to a large uncertainty and phase and a total scrambling of our prediction of where it is.

410
00:51:53,690 --> 00:52:01,220
So again, quantum mechanics is returning in a rather complicated way and through quantum interference, a result that we're very familiar with.

411
00:52:01,850 --> 00:52:04,850
If we think about the classical situation, it's time to stop.