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So where we go to yesterday was that we had reduced the horrible partial differential equation in six variables,

2
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which is posed by the time independent Schrodinger equation for a hydrogen like ion two to this equation here.

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Sorry. I'm being premature. To this.

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So the internal motion had been separated from the translational motion of the atom as a whole,

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and we had exploited we by using a change of variables, the central mass coordinates and the separation variable.

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And we had used the work that we did with the angular momentum operator earlier on to show that we could express the,

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the internal Hamiltonian in in in this format in this format here.

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So this was called this was called. Ah, I think that's right for the internal measurement.

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Right. And then I pointed out that this Hamiltonian computed with the total with the angular momentum operators

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because the only mention of the Angles feature and Phi sits inside this total angular momentum operator.

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And therefore we can seek therefore we can seek stationary status.

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There's going to be a complete set of stationary states, which is simultaneous eigen functions of E and L, right?

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So we will also have the statement that L squared on E and L is equal to l, l plus one, e and L.

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And then when we're using these stationary states, the action of this this operator can be replaced by this eigenvalue.

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And now we do end up with what I wrote originally, which is that we're going to have a Hamiltonian which has a subscript L,

16
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meaning it's the one that's valid for four stationary states which have total incrementing quantum number L which is going to

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be PR squared over two mu plus l l plus one h bar squared over two mu squared minus z squared over four pi epsilon nought.

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Oh. And this is fundamentally a Hamiltonian analogous.

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This is a Hamiltonian analogous to that for the harmonic oscillator in the

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sense that we have now only one surviving coordinate of our six coordinates.

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We're down to one. That's this here, the modulus of the separation vector and it's conjugate momentum.

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PR So we've made an enormous simplification, made tremendous progress,

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and we're going to knock this into submission using the same approach as we did with the harmonic oscillator.

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We're going to define what will turn out to be a letter operator ale, which is going to be defined to be dimensionless.

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It's going to be a zero of a root two.

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I better write this down from notes rather than my memory, because these factors do matter.

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I p r over each bar minus l plus one over a plus z of l plus 1a0 where a zero is the

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following not very helpful item for pi epsilon nought h bar squared over mu e squared.

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So this is an object. I'll make it convincing. Well, let's make it convincing now.

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What is this object? This is the. This is the so-called ball radius.

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So it was introduced by Niels Bohr before there was proper quantum mechanics using what turns out to be a fallacious model of hydrogen, the poor atom.

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And the way to see what is dimensions are are actually to to write, to bring this up to this side and say,

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look, this is going to be squared over four pi epsilon nought a nought on this side.

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And what's on the top, not the bottom. So therefore, I have to say this is equal to h bar over a nought squared of amu.

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So out of this equation by putting that E on the top and this for perhaps in the nose on the bottom,

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dividing through by a nought squared to get this on the bottom so that this is something we understand to be this is obviously electrostatic energy.

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And what's this that we see on the right side? Well, H Bach is momentum, right?

38
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So this is more or less P squared over mass. This is the reduced mass recall.

39
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This is the reduced mass, more or less the mass of an electron.

40
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So what we have on the right hand side is p squared.

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When it is P squared for k is equal to one over a nought.

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So if you have if you have a wave which has a wavelength which is comparable to one over the this scale radius here sorry.

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A wavelength comparable to this scale radius here. Not worrying about two PI's at the moment.

44
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Then what you have on the right here is.

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So this side is two times kinetic energy associated with a particle which has a wavelength which is on the order of zero.

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And what we have on the left side here is the electric energy. So the scale that's being said this, this, this dimension of this dimensional quantity,

47
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the ball radius is the natural scale at which the kinetic energy associated with the uncertainty principle,

48
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the zero point in it,

49
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what with the harmonic oscillator we would have called the zero point motion is is on the is on the order of the electrostatic energy.

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That's dimensionally where this number comes from. That number comes from so with.

51
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The thing to notice now is that ail is dimensionless.

52
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Why is it dimensionless? Well, this cancels the dimensions of this.

53
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Obviously this in. This cancels this. This is all dimensionless.

54
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This factor is zero. And here we're looking at again, this is PR divided by with this put on the bottom h for over a zero.

55
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Therefore H Bach. Therefore something with the dimensions of P. So this thing here, this operator here is dimensionless.

56
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Same as the letter operations in the harmonic oscillator with dimensionless.

57
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Okay. So what do we do with that? What we do is we calculate what a.

58
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And this thing, of course, carries this subscript little L because little L the orbital quantum number is appearing in in its definition.

59
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And what we do is we now calculate l dagger a l.

60
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Right. So what does this give us?

61
00:07:31,860 --> 00:07:39,749
We have all this in a zero squared of two out front because we're going to have two zeros and root twos and then we're going to have PR.

62
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We showed was a commission what we we engineered that it was a commission operator.

63
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So when I take the commission now joint of what I have up there, I get a minus IPR.

64
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The minus sign is from the I over H bar and then the rest of course is the same because it's it's commission, it's just boring numbers plus z.

65
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Well, actually, this is an operator, strictly speaking, but we're in the position representation,

66
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so it looks like a number and that has to be multiplied onto IPR over each bar.

67
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So no minus sign because this is Al I'm writing down now minus L plus one overall plus Z overall plus 1a0.

68
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So we have to multiply this stuff out and the way we do it is we regard this in the back here as one factor and that in the front is another factor.

69
00:08:32,310 --> 00:08:43,590
So this is looking like the product of a number minus minus some number, a number plus some number.

70
00:08:43,590 --> 00:08:50,460
Right? So the usual pattern, just so this is mirroring very closely what we did with the harmonic oscillator.

71
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So we get a zero squared over two.

72
00:08:55,890 --> 00:09:05,670
These two obviously multiply together and we get PR squared over each bar squared and then these I have to multiply together.

73
00:09:05,670 --> 00:09:10,200
So what do we get? We get the square basically of this number, this of this number here.

74
00:09:11,100 --> 00:09:28,440
So we have plus l plus one overall squared plus Z over L plus 1a0 squared minus the cross term here which is going to be 12.

75
00:09:28,440 --> 00:09:35,639
There'll be two minus twice twos and those are going to cancel and we will have a zero or so.

76
00:09:35,640 --> 00:09:39,300
That's, that's the, the two easy parts. Right, because.

77
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It's the front things squared, plus the back thing squared.

78
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And now we have to think about the cross terms which would vanish because this is a plus B into A minus B spiritually,

79
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these cross terms would vanish if we weren't dealing with operators,

80
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and they fail to vanish only because we are dealing with operators so that there's a failure of computation.

81
00:10:04,880 --> 00:10:08,630
Otherwise the PR well, okay, so PR commutes with this.

82
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So for the cross terms, we do we don't get anything from this thing on this,

83
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but we do get something from this thing on this, namely we get the relevant commentator.

84
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So the extra term that arises because we're working with operators is going to be.

85
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I. Overreach bar.

86
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That's that. So. So this minus sign and this minus sign cancel.

87
00:10:34,760 --> 00:10:43,400
We have that L plus one and then I have a p r comma one overall plays big bracket.

88
00:10:43,430 --> 00:10:46,579
That's what I'm going to get from this term on this term.

89
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Not cancelling on this term, this term on this term.

90
00:10:54,650 --> 00:11:05,990
Okay. So that's going.

91
00:11:06,020 --> 00:11:10,570
So what? So what is this going to come to? Well, let's rearrange things.

92
00:11:11,380 --> 00:11:24,870
He's very squared of two PR squared over each bar squared plus L plus one squared of R squared minus.

93
00:11:24,880 --> 00:11:29,450
I'm going to put this one down next to Z of a zero.

94
00:11:30,790 --> 00:11:36,309
Then this term plus z squared over L plus one squared.

95
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A zero squared. When I have to do this.

96
00:11:41,130 --> 00:11:49,730
What I remember is something we handled very last term that when I had to do it arose when I was doing that,

97
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when we were doing the comitato of P and V, the potential function of X that turned out to be the right DV by the X Times.

98
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The commentator of P and X. This is exactly the situation we have here because this is the operator canonically conjugate to r.

99
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So what we have here is the derivative of one overall, which is so minus i l plus one l plus one over h bar two times minus one over

100
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all squared because that's the derivative of one over all times PR commercial.

101
00:12:30,360 --> 00:12:33,870
But this is a piece of a canonical commutation relation.

102
00:12:34,170 --> 00:12:39,990
This is equal to minus I. All right. We show that oh come a PR is equal to H bar.

103
00:12:40,320 --> 00:12:45,750
So in this order it's minus H bar. So we have rather a load of minus signs.

104
00:12:45,750 --> 00:12:54,630
Let me see. I think we have one, two, three minus signs and another minus sign coming from here.

105
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So I think in total we have a plus sign. So this stuff here, I believe comes to L plus one over each bar R squared.

106
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Sorry, the bar get the bar cancels this edge bar on top.

107
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That's on the bottom. So we get an L plus one over squared and this can now be combined with this.

108
00:13:21,410 --> 00:13:26,270
Except I've got the wrong blinking sign. Let me just double check that.

109
00:13:27,800 --> 00:13:32,860
Yep. I'm looking for minus. Is it mine as well?

110
00:13:32,980 --> 00:13:36,400
I want it to be mine. So let's declare it to be minus. And I'm sure it is minus.

111
00:13:36,400 --> 00:13:43,570
I'm sure it is minus. Let's not spend time chasing down some wretched sign because now what we're going to do is combine this.

112
00:13:43,570 --> 00:13:52,120
So let's this is the side calculation here. We're going to have L plus one overall squared brackets, L plus one from up here.

113
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That's the old plus one squared minus one times this stuff.

114
00:13:56,410 --> 00:14:01,000
So you can see we're going to fit this and this is going to give me an L, l plus one, which is exactly what I want.

115
00:14:01,450 --> 00:14:13,540
So this is going to be a nought squared over to p r squared over each bar squared plus l l plus one.

116
00:14:15,760 --> 00:14:24,580
Overall squared minus two Z over 800.

117
00:14:26,680 --> 00:14:30,999
Close brackets, and then I'm going to take this and join in on that.

118
00:14:31,000 --> 00:14:38,770
So we get plus garbage in the garbage term is z squared over two L plus one squared.

119
00:14:39,460 --> 00:14:46,600
This should all be dimensionless. I think it probably is dimensionless on the grounds that it's the product of two dimensionless operators.

120
00:14:46,720 --> 00:14:53,860
Now what is the point of this ridiculous exercise? The point is that we should see the original hamiltonians peaking out here.

121
00:14:54,280 --> 00:14:57,610
We should basically have our original Hamiltonian plus garbage.

122
00:14:57,940 --> 00:15:12,280
So the. In order to get our original Hamiltonian, we need to have a new under here and a mew under here.

123
00:15:12,290 --> 00:15:14,990
So why don't we multiply by a meal on the top? On the bottom?

124
00:15:15,500 --> 00:15:25,640
So this is a nought squared mew and we want to take this bar outside and then that won't be under there as we want.

125
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We can allow that to to wander inside and then this bracket becomes hl this stuff here becomes HL.

126
00:15:35,640 --> 00:15:44,660
And we still got unwanted garbage in the back. But that's exactly how it worked with the harmonic oscillator.

127
00:15:44,660 --> 00:15:50,000
Remember, a dagger A was equal to H.

128
00:15:50,000 --> 00:15:53,360
The Hamiltonian overreached by Omega minus a half.

129
00:15:53,630 --> 00:15:55,820
There was garbage in the back, which in that case was a half.

130
00:15:57,380 --> 00:16:02,930
So this is obviously some constant with the dimensions of energy, and I better make sure that I've done that right.

131
00:16:04,070 --> 00:16:07,210
It's the instruction B squared. Yes, of course it should, because it was the bar I took out from there.

132
00:16:07,220 --> 00:16:11,240
Otherwise it is correct. And that's that's just just the business.

133
00:16:11,810 --> 00:16:16,130
So we've expressed h let's write down what we let's write this down in the other way.

134
00:16:16,340 --> 00:16:24,740
We've said that H is equal to H bar squared over a nought squared mu which must provide the dimensions of energy,

135
00:16:26,360 --> 00:16:34,160
a dagger l a l minus z squared over two l plus one squared.

136
00:16:34,820 --> 00:16:40,880
Another way of writing. We've expressed h basically as dimensionless constant times as of a dagger.

137
00:16:41,060 --> 00:16:45,770
A dagger. So this is the harmonic oscillator trick and it's all just looking a little bit, little bit messier.

138
00:16:45,920 --> 00:16:49,549
But this is only a boring number, right? It's just I mean, what's the difference in a half?

139
00:16:49,550 --> 00:16:53,629
And this it's just a number. They're both. They're both. They're both just numbers.

140
00:16:53,630 --> 00:16:56,830
Rational numbers. Okay. So what do we do next? What do we do?

141
00:16:56,840 --> 00:17:03,560
In the harmonic oscillator, we calculated a a dagger comma, a, we found the commentator.

142
00:17:03,920 --> 00:17:06,830
That was the next thing that we did. And that's what we do just right now.

143
00:17:07,640 --> 00:17:13,700
Make sure I'm don't know it's more convenient to work it out the other way around later use.

144
00:17:13,700 --> 00:17:22,489
All right. So what is that? We have to write this horrible thing down again so we can have an in or squared of a to open over in a square bracket

145
00:17:22,490 --> 00:17:32,810
because we're talking comitatus now write down I pr on each bar because I'm writing down a now which is just fair.

146
00:17:33,740 --> 00:17:37,340
I need to write down minus L plus one over R.

147
00:17:37,730 --> 00:17:41,450
I don't need to write down the boring constant in the back,

148
00:17:41,450 --> 00:17:48,860
the z of real plus 1a0 that will commute with everything in sight and that will make no contribution to the commentator.

149
00:17:49,460 --> 00:17:58,840
Then I have to write down the corresponding parts of a dagger, which is minus EPR over bar, minus L plus one overall.

150
00:17:58,850 --> 00:18:04,700
And now I can rest easy. So this is what the commentator that I have to do obviously.

151
00:18:05,120 --> 00:18:10,850
So PR commutes with itself nothing doing their PR on the other hand does not compute with this.

152
00:18:11,210 --> 00:18:18,230
So we are going to get a zero squared over two i.

153
00:18:19,880 --> 00:18:34,440
And I l plus one that's that l plus one over h bar that's that age bar p r comma one overall commentator I get, I get that from that list.

154
00:18:34,600 --> 00:18:39,170
Live in hope that I do. What did I do with the minus sign? I put it in the bin.

155
00:18:39,560 --> 00:18:48,240
I shouldn't have done right. There should be a leading minus sign. Then I have the same term actually,

156
00:18:48,240 --> 00:18:56,580
because here I am going to get plus but everything plus plus this except I'm going to have a one overall comma PR, which of course is minus this.

157
00:18:56,790 --> 00:19:00,150
So I'm going to get this all over again. So why don't you just rub out the two and then it's right.

158
00:19:02,480 --> 00:19:05,630
What's this? Commentator? We've already discussed that problem.

159
00:19:06,470 --> 00:19:11,629
It's going to be the rate of change. It's going to be D by the R of this times.

160
00:19:11,630 --> 00:19:20,810
The commentator. So this is equal to minus a zero squared i l plus one over h bar.

161
00:19:21,380 --> 00:19:27,530
And then I'm going to have a minus one over all squared for the derivative of this times.

162
00:19:27,530 --> 00:19:33,650
PR Commodore. Commentator Because that's how these things work.

163
00:19:33,860 --> 00:19:36,980
But this once again is minus h bar.

164
00:19:38,030 --> 00:19:39,919
So the two minus is here.

165
00:19:39,920 --> 00:19:49,550
Cancel the two i's, make another minus sign which kills this minus sign all being well, this is equal to a zero squared l plus one.

166
00:19:51,170 --> 00:19:56,480
The poles cancel all squared checks sine.

167
00:20:01,160 --> 00:20:05,550
Yeah. Uh, should have a.

168
00:20:07,880 --> 00:20:12,390
No, that's correct. That's correct. Good. All right. So we want to express this.

169
00:20:12,410 --> 00:20:16,520
Remember the commentator? So in the harmonic oscillator case, what was this commentator?

170
00:20:16,820 --> 00:20:24,400
This commentator was actually one. So the bad news is, it's not looking very promising at this point.

171
00:20:24,410 --> 00:20:28,420
You think, oh, no, it's not good, because I'll come to some damn function of Oh,

172
00:20:28,430 --> 00:20:33,620
but we've seen that damn function of while somewhere before, up in the Hamiltonian.

173
00:20:33,620 --> 00:20:36,080
Basically, I've lost that. I've lost Hamiltonian. There it is.

174
00:20:36,470 --> 00:20:42,410
So we've got l l plus one plus squared of A to B to mu r squared is appearing in the Hamiltonian.

175
00:20:42,710 --> 00:20:49,640
So supposing I would take h l plus one and from it I would take h l.

176
00:20:50,210 --> 00:20:58,250
Then everything in the hamiltonians would cancel except that middle term which has the right form, in namely it contains a 1 to 4 squared.

177
00:20:58,250 --> 00:21:17,569
And what would we have. We have h squared over two mu I think R squared brackets L plus one l plus two minus l l plus one to do that.

178
00:21:17,570 --> 00:21:23,380
Right. Given hope. Okay. So obviously there's going to be a factor of L plus one common.

179
00:21:23,570 --> 00:21:27,290
And then we're looking at the difference between L plus two and L, in other words,

180
00:21:27,290 --> 00:21:37,820
to the two is going to cancel this and this is going to equal l plus one h squared over over mu r squared.

181
00:21:38,390 --> 00:21:42,200
So we can express with this little side calculation,

182
00:21:42,200 --> 00:21:49,099
we can go back up the board and express this as an appropriate multiple of the differences in the Hamiltonian.

183
00:21:49,100 --> 00:21:51,700
So it's going to be a nought squared. That's the say nought squared.

184
00:21:53,780 --> 00:22:02,690
Then we will want to multiply by mu divide by h bar squared and then we'll be able to say This is h l plus one minus eight gel.

185
00:22:06,140 --> 00:22:13,280
Check that we haven't got any horrible factors in this. Okay.

186
00:22:18,510 --> 00:22:21,150
All right. What was the next thing we did in the harmonic oscillator?

187
00:22:22,320 --> 00:22:30,040
Having got the commentator of the A and the Dagger and having expressed H is a product of A and a dagger.

188
00:22:30,060 --> 00:22:35,520
The next thing we did was calculate the commentator. Use these results to calculate the commentator of A with H.

189
00:22:35,880 --> 00:22:42,560
So that's what we do now. I will do it here so that we can see our results.

190
00:22:42,570 --> 00:22:51,570
So I want to calculate the commentator a l comma h l and I can do that by expressing this.

191
00:22:51,570 --> 00:22:55,950
HL HL is basically a product of the A's.

192
00:22:55,980 --> 00:23:01,050
Now I've only got to locate the wretched product. It's at the top there, isn't it?

193
00:23:01,100 --> 00:23:08,100
Uh oh. So hard from this position to see what you need to see this product, etcetera, etcetera, etcetera.

194
00:23:08,110 --> 00:23:14,730
There's a statement. This is the statement I'm looking for. I want that statement. All right, so this h l can be traded in for that product.

195
00:23:15,210 --> 00:23:21,960
So we have h bar squared over a0 squared mu times.

196
00:23:21,960 --> 00:23:31,620
The commentator of a l and a l i l dagger round and I can rest easy.

197
00:23:31,620 --> 00:23:37,600
There are no need to put this stuff inside a commentator because it be with everything inside so it can't contribute to the commentator.

198
00:23:38,250 --> 00:23:47,100
So this is the commentator I have to evaluate and this is easy peasy because this is a commentator, a product, some A with B and C,

199
00:23:47,370 --> 00:23:54,090
which in principle is the commentator of A with B, C standing idly by, and then the commentator of A with C, B standing idly by.

200
00:23:54,480 --> 00:23:59,160
But A, of course, computes with itself. So there's only one term which is an L.

201
00:24:01,370 --> 00:24:10,610
Al mutated with Al Dagger so this is equal to h bar squared over a zero squared mu al comma al dagger.

202
00:24:12,870 --> 00:24:21,030
And we just worked that one out. And the answer was and the answer was, it was here.

203
00:24:21,870 --> 00:24:27,970
Sorry, I did something wrong. And. Yes, it's very important.

204
00:24:27,990 --> 00:24:31,810
I need an extra factor, Al, and thank you very much. In the back that stands out.

205
00:24:31,830 --> 00:24:36,540
This is standing idly by while Al works on his companion. All right, so.

206
00:24:40,380 --> 00:24:47,160
So I now need to plug this in for this commentator. And you can see that all these all these factors are going to cancel.

207
00:24:47,160 --> 00:24:49,890
This factor is going to cancel on this factor.

208
00:24:50,520 --> 00:25:00,390
And so we're simply going to get h l plus one minus h l times the al I've been very helpfully told to include.

209
00:25:01,470 --> 00:25:05,879
So let me just. Right. So we're now we're now in wonderful shape.

210
00:25:05,880 --> 00:25:10,650
We so we've completed all three steps of the simple oscillator calculation.

211
00:25:11,190 --> 00:25:28,800
And now we just need to go for the point of the exercise which was which is so we're given that we always were given that HL on L is equal to e of L,

212
00:25:28,920 --> 00:25:33,870
right. What we want to do is make ourselves a new.

213
00:25:33,900 --> 00:25:40,110
So we've got one stationary state. We want to make ourselves a new stationary state by multiplying by al obviously both sides of the equation.

214
00:25:40,560 --> 00:25:42,959
So let me write down the right side of the equation first.

215
00:25:42,960 --> 00:25:54,180
This implies that E which is a boring number times h l being l is equal to a l l of e an l usual business.

216
00:25:54,540 --> 00:26:00,599
Swap these over h l a l which I'm not entitled to.

217
00:26:00,600 --> 00:26:04,290
Plus the commentator for that restores order.

218
00:26:08,900 --> 00:26:09,290
On E!

219
00:26:09,350 --> 00:26:23,750
Now, we just worked that out and found that it was the difference of two ages times ail so this becomes h l plus one minus h l with an al in back.

220
00:26:24,290 --> 00:26:29,000
Guess what? We have an el al here with a minus sign and hll here with a plus sign.

221
00:26:29,330 --> 00:26:33,920
So the whole thing is equal to h l plus one ail e and l.

222
00:26:36,490 --> 00:26:47,290
So we have achieved what we wanted to achieve. That is to say, we have shown that this state.

223
00:26:49,240 --> 00:26:55,930
Is all is an iron state, not of HL, but of l plus one and for the same energy.

224
00:26:56,410 --> 00:27:02,620
So the map is looking a little different now from harmonic oscillator, but nonetheless we have a very powerful result.

225
00:27:02,920 --> 00:27:08,170
We have generated ourselves from a state which identity and angular momentum quantum number.

226
00:27:08,170 --> 00:27:23,500
L We've made ourselves a state. Which can only be characterised as e l plus one.

227
00:27:24,730 --> 00:27:28,990
We've made up of a state of the same energy but more angular momentum.

228
00:27:30,870 --> 00:27:34,109
So what have we done? This operator. This is. This is.

229
00:27:34,110 --> 00:27:40,550
This is some normalising constant. Right? We had just this kind of thing in the case of the harmonic oscillator.

230
00:27:46,290 --> 00:27:49,569
So. Physically.

231
00:27:49,570 --> 00:27:53,760
What have we done? Well, classically.

232
00:27:55,300 --> 00:28:02,980
What have we done? We've taken an orbit that might look like this, and we've turned it into an orbit.

233
00:28:03,580 --> 00:28:08,330
I'm trying to make an orbit that has about the same Semi-major axis and is rounder.

234
00:28:08,350 --> 00:28:14,290
You know about Kepler orbits. So we weave with the same supply of energy.

235
00:28:14,300 --> 00:28:17,780
We have increased the angular momentum. So we've made the orbit less eccentric.

236
00:28:29,740 --> 00:28:33,580
So in the civil law case, what did we do? We made a service in orbit with less energy.

237
00:28:34,690 --> 00:28:44,410
And then we argued that the. The energy, we were able to show that the energy could never be negative.

238
00:28:44,980 --> 00:28:49,420
So we said to ourselves, So there must be some.

239
00:28:52,170 --> 00:28:58,920
So so given given this state now we could apply a L plus one,

240
00:28:59,670 --> 00:29:05,400
a sub l plus one to this and make ourselves e comma l plus two with even more angular momentum.

241
00:29:05,790 --> 00:29:09,420
So like an isolated case, we said we can make ourselves an orbit with even less energy.

242
00:29:10,620 --> 00:29:17,470
Is this possible with a given supply of energy, with a bound orbit two to have more and more more angling momentum?

243
00:29:17,490 --> 00:29:22,379
No, it's not. At some point, you've got the maximum angular momentum you can have for that given energy,

244
00:29:22,380 --> 00:29:27,870
which in classical physics is what we would call a circular orbit. We've completely destroyed the the radial motion.

245
00:29:27,870 --> 00:29:31,440
So what we've been doing here is we've been shifting kinetic energy.

246
00:29:32,310 --> 00:29:37,410
We've shifted k e from.

247
00:29:39,580 --> 00:29:50,080
PR squared over two mu 2ll squared h bar squared over two mu squared.

248
00:29:50,080 --> 00:29:53,110
Right. This was the tangential kinetic energy. This was the regular kinetic energy.

249
00:29:53,110 --> 00:29:59,820
We've shifted energy from here to here when we've got no energy left in the arrow as little as the theory, you know,

250
00:29:59,830 --> 00:30:06,250
as quantum mechanics allows, which won't be zero but will be some amount, then we won't be able to shift anymore.

251
00:30:06,250 --> 00:30:10,060
So there must be a maximum angular momentum for a given energy.

252
00:30:21,230 --> 00:30:26,700
We'll call this. Curly L All right.

253
00:30:26,710 --> 00:30:31,420
This is the maximum angular momentum and it is a function of energy.

254
00:30:31,420 --> 00:30:35,620
But we won't write. We won't write that it's a function, as you will. But we're going to find out what function of energy it is.

255
00:30:37,750 --> 00:30:44,920
So what does that mean? That means if we take the if we take the what are we going to call this?

256
00:30:44,920 --> 00:30:50,860
The the secularisation operator a l belonging to this maximum angular momentum.

257
00:30:51,280 --> 00:30:56,560
And we use it on the states which has the maximum angular momentum for the given energy.

258
00:30:56,800 --> 00:31:04,510
What do we get? Nothing. That's the only way we can be prevented from getting a state which has even more

259
00:31:04,510 --> 00:31:08,770
angular momentum for the same energy is if this operator simply kills this state.

260
00:31:09,190 --> 00:31:17,500
So we've used this argument twice before, once in the harmonic isolated case, and also in the case of the angular momentum operators.

261
00:31:18,430 --> 00:31:21,460
What do we mean by nothing? What we mean is the mod square of this is nothing.

262
00:31:22,420 --> 00:31:36,729
What does that map to? That maps to e maximum angular momentum, a l dagger, a l e curly thingy is nought.

263
00:31:36,730 --> 00:31:40,690
Where have we seen a dagger a before? I think we must have seen it in the Hamiltonian.

264
00:31:41,020 --> 00:31:44,530
We need to replace that by the Hamiltonian times.

265
00:31:44,950 --> 00:31:48,190
Some horrible factors. Yeah.

266
00:31:49,240 --> 00:31:56,290
All right. So. Well, that we already have it here. So a dagger a comes right down to this line here.

267
00:31:56,560 --> 00:31:59,920
So this line here can replace the dagger in here.

268
00:32:00,430 --> 00:32:16,660
So we get to have that e curly l onto hainaut squared mu over h bar squared uh h h curly l plus

269
00:32:17,320 --> 00:32:33,250
z squared over to curly l plus one squared close a bracket e curly l ain't much but eight.

270
00:32:33,250 --> 00:32:41,020
But this thing, this is an eigen function of this operator with eigenvalue e this is a boring number.

271
00:32:41,020 --> 00:32:44,740
So it stands by was this bangs into that and makes a one.

272
00:32:45,160 --> 00:32:48,580
This gives me e times this and this is left over.

273
00:32:48,580 --> 00:33:04,930
It bangs into this and makes a one. So this implies that h0 squared mu over h bar squared e plus z squared over two l plus one squared is nothing.

274
00:33:04,930 --> 00:33:10,400
Or perhaps I should write this as equals minus. So what have we done?

275
00:33:10,410 --> 00:33:15,750
We've got a relationship between the energy and the maximum allowed angular momentum.

276
00:33:30,450 --> 00:33:31,320
More than that,

277
00:33:31,560 --> 00:33:38,640
we know that these angular momentum quantum numbers because this is orbital increment and we're talking about we prove that those had to be integers.

278
00:33:40,350 --> 00:33:47,010
So in being defined to be is an integer.

279
00:33:47,880 --> 00:33:51,330
Integer? What integer?

280
00:33:52,050 --> 00:33:55,740
We know that Curly L is allowed to be nothing. One, two, three, four.

281
00:33:56,040 --> 00:34:02,250
So n is equal to the numbers it's allowed to be is 1 to 3, four, blah, blah.

282
00:34:02,280 --> 00:34:05,460
Nothing not included in the list because of that plus one.

283
00:34:06,960 --> 00:34:21,090
So we have shown that e the energy has to be of the form minus z squared h bar squared over a nought move.

284
00:34:21,090 --> 00:34:26,680
I done that right. Not squared mu one over.

285
00:34:26,700 --> 00:34:35,359
Now we need to hear one over n squared. So we have found the possible energies of a hydrogen atom.

286
00:34:35,360 --> 00:34:42,470
Well, in fact, for a hydrogen like Ion, because Z remembers this integer which controls the number of charge units on the nucleus.

287
00:34:42,800 --> 00:34:46,720
We have found this with the possible values, right?

288
00:34:46,730 --> 00:34:50,209
It's given by this constant, which we know what it is. We'll simplify it in the moment.

289
00:34:50,210 --> 00:34:54,590
We know what it is. Times one over n squared where n is one, two, three, four.

290
00:34:55,610 --> 00:35:07,579
So this gives the energy levels. We write this as minus z squared.

291
00:35:07,580 --> 00:35:11,780
Times are over and squared where.

292
00:35:11,990 --> 00:35:16,340
Oh. Curly.

293
00:35:16,340 --> 00:35:27,170
Ah is whatever you see it to be which is page bar squared over two a nought squared MMU which is not very intuitive.

294
00:35:27,440 --> 00:35:30,900
The way to make this intuitive is to take those zeros.

295
00:35:31,130 --> 00:35:35,520
There are two of them and turn one of them back into its h bars and things.

296
00:35:35,540 --> 00:35:39,229
Now where did we define a zero? For heaven's sake, it was right over here somewhere.

297
00:35:39,230 --> 00:35:44,940
Right? There it is. So take. So one of those two I'm going to replace by that garbage there.

298
00:35:44,960 --> 00:35:48,930
All right. So this is going to become on the bottom.

299
00:35:48,930 --> 00:35:52,880
We're going to have an eight pi epsilon nought a zero.

300
00:35:53,480 --> 00:35:58,070
All right. That's the four pi epsilon nought. The edge bar squared will cancel top and bottom.

301
00:35:58,070 --> 00:36:04,910
So that goes away. The mu in the E squared. Well the mu will go away with this and the E squared will sit on the top.

302
00:36:06,370 --> 00:36:16,600
So the red book is is what is squared over for pi ips and not a zero would be the potential energy of two charges of charge,

303
00:36:16,840 --> 00:36:20,170
you know, two electric charges that was separated by a zero.

304
00:36:20,950 --> 00:36:30,140
So this is half of the potential energy. A separation of a zero.

305
00:36:33,720 --> 00:36:40,840
And. So this is the fundamental energy scale of atoms.

306
00:36:42,540 --> 00:36:50,940
And what does it equal to 13.6 electron volts plus, you know, 13.6 to 3 significant figures.

307
00:36:53,670 --> 00:36:56,490
So the energy range of which we work, you know,

308
00:36:56,690 --> 00:37:06,060
the the battery that you stick into your you stick your camera or something has 1.5 volts basically because of that 13.6 EVs,

309
00:37:06,360 --> 00:37:11,670
it's all of all of condensed matter. Physics is a mere reflection of that number.

310
00:37:12,430 --> 00:37:19,260
We're all you know, that's why we live at one TV, not a one movie or 1 million every year or whatever.

311
00:37:22,680 --> 00:37:31,420
So what do we need to do next? Yeah.

312
00:37:31,450 --> 00:37:37,440
Jargon. This is called the principle.

313
00:37:38,520 --> 00:37:44,310
Principle. Al Quantum number.

314
00:37:49,910 --> 00:37:51,820
So in these hydrogen like ions,

315
00:37:51,830 --> 00:38:05,450
we've discovered that there are a whole series of different distinct states which have the same energy and different tangle of momenta.

316
00:38:06,110 --> 00:38:13,480
So let's talk a bit about degeneracy. Okay.

317
00:38:13,490 --> 00:38:23,950
So if a principal quantum number and equals one, we have that L which is equal to n minus one is equal to zero.

318
00:38:23,950 --> 00:38:30,240
And what is the largest angular momentum you can have is nothing. And on the ground, state of hydrogen.

319
00:38:30,540 --> 00:38:36,300
There's one electron. It sits in the state with the lowest energy, which is going to be associated with any equals one.

320
00:38:36,480 --> 00:38:40,710
And it has no angular momentum. It's on a totally radial orbit in classical physics.

321
00:38:40,860 --> 00:38:44,280
Right. Not going round and round at all. It just goes in and out.

322
00:38:44,280 --> 00:38:48,100
In and out. I mean, in classical physics.

323
00:38:48,700 --> 00:38:52,330
Quantum mechanics. But it doesn't have any angular momentum.

324
00:38:53,140 --> 00:38:56,680
So that's a surprising result for any equals.

325
00:38:56,680 --> 00:39:02,230
Two. L is the maximum angular momentum is equal to one.

326
00:39:04,540 --> 00:39:10,150
That means that l can be nought if you like, and l can be one.

327
00:39:10,160 --> 00:39:14,230
Right? This is the maximum angular momentum. So there's a slightly funny thing going on here.

328
00:39:14,440 --> 00:39:18,550
And was introduced as one plus the maximum angular momentum.

329
00:39:19,150 --> 00:39:22,780
But now I'm saying it's better to what we one standardly thinks about it.

330
00:39:22,840 --> 00:39:25,870
One thinks about what's the value of N from it.

331
00:39:26,050 --> 00:39:31,240
One one takes as N minus one the maximum angular momentum.

332
00:39:32,140 --> 00:39:36,190
So that's right. And in this sense, we have one state.

333
00:39:36,910 --> 00:39:38,020
It'll be two states.

334
00:39:39,400 --> 00:39:47,950
Well, this one here, basically we have this is for spin this party where it'll turn out to be two states when we include the spin of the electron.

335
00:39:48,220 --> 00:39:55,300
But remember, we were doing the growth strategy, which means we said we were going to forget about the spin of the electron here.

336
00:39:55,540 --> 00:40:00,309
We would have one state and here we would have three states, right?

337
00:40:00,310 --> 00:40:08,410
Because four equals one where we got total an element of one, which means we've got three possible orientations of it.

338
00:40:08,410 --> 00:40:12,880
M can be one nothing or minus one. So we have three quantum states here, one here.

339
00:40:12,890 --> 00:40:19,210
So we've got four states all with the same energy for an equal to one, four equals one.

340
00:40:19,390 --> 00:40:26,380
And so it goes down the line. So the number of states is is increasing rapidly.

341
00:40:29,430 --> 00:40:35,129
Because there'll be five states for any equals three. The maximum incremental will be two for two units of Anglo mentum.

342
00:40:35,130 --> 00:40:39,990
You've got five possible orientations and then you've still got three of these and one of those.

343
00:40:39,990 --> 00:40:43,740
So that's nine states, etc.

344
00:40:43,890 --> 00:40:47,910
So, so the structure that we've derived is extremely degenerate.

345
00:40:49,500 --> 00:40:51,600
What does this have to say about experiments?

346
00:41:02,620 --> 00:41:13,120
So stick some hydrogen atoms in a in a vessel and pass a electric current through and get the electrons knocked out of their out of their comfortable,

347
00:41:13,960 --> 00:41:20,620
out of their comfort zone. And you will get photons coming out at discrete frequencies.

348
00:41:22,240 --> 00:41:31,240
New is going to be the difference in the energies over H, which is going to be z squared.

349
00:41:31,900 --> 00:41:40,990
The rip constant over H of regional h bar one over NW prime squared minus one over and squared.

350
00:41:41,020 --> 00:41:46,420
This is for. And goes to end primes.

351
00:41:46,690 --> 00:41:54,040
So if you were in one of these higher states, for example, and equals to you will have less.

352
00:41:54,040 --> 00:41:59,200
Your energy would be a smaller negative number. Right? You'll have one of a one of a two squared of a quarter here.

353
00:41:59,620 --> 00:42:07,959
And this will be if you could then fall down to the state and primed equals one, in which case this will be one.

354
00:42:07,960 --> 00:42:13,300
So this will be this bracket will be, say, three quarters and you will get three quarters of this number coming out.

355
00:42:14,650 --> 00:42:21,160
So that gives you that gives you some frequency. And what we have is a series.

356
00:42:23,030 --> 00:42:27,770
Of lines of the way we think about this is that we have a series of lines each.

357
00:42:30,250 --> 00:42:36,280
Four fixed. And primed, i.e. bottom level.

358
00:42:40,280 --> 00:42:48,710
So if we fix and primed at one, we can have transitions from MN is 2 to 1 or and is 3 to 1 or and is 4 to 1.

359
00:42:48,920 --> 00:42:52,820
And these are the successive lines of the lineman series.

360
00:42:53,060 --> 00:42:58,879
So here we have the. So here, here, here is the energy of an equals one.

361
00:42:58,880 --> 00:43:09,020
His n equals two has an equal three. And Lyman Alpha is the name used for the for the spectral line associated with an electron tumbling from equals

362
00:43:09,020 --> 00:43:16,249
two down to when equals one and Lyman beta is associated with from an equals three down to and equals one,

363
00:43:16,250 --> 00:43:24,950
which is further to fall. So it emits more energy. So the line appears at higher frequency.

364
00:43:24,950 --> 00:43:28,870
Longer shorter wavelength. So so the Lyman series this.

365
00:43:32,730 --> 00:43:37,150
Is fat and primed equals one, if any equals two.

366
00:43:37,170 --> 00:43:41,940
We're looking at Lyman Alpha. That's what it's conventionally called, if any equals three.

367
00:43:42,270 --> 00:43:47,729
It's Lyman Beta. And this has, I think is 112 nanometres.

368
00:43:47,730 --> 00:43:57,170
Is that right? 121. Sorry. And as you go down to end equals infinity.

369
00:43:57,180 --> 00:44:04,800
In other words, if you fall all the way from not being bound into the bottom of the atom,

370
00:44:05,670 --> 00:44:12,140
then this is the Lyman at the beginning of the Lyman continuum. And that's what.

371
00:44:12,290 --> 00:44:15,349
What is it? 92. Nanometres.

372
00:44:15,350 --> 00:44:19,110
Roughly speaking, I've got a more accurate number here. 91.2.

373
00:44:23,160 --> 00:44:27,990
So these lines are all in the. These are all vacuum ultraviolet lines.

374
00:44:31,010 --> 00:44:40,790
They all carry. So this one is carrying 13.6 EV of energy and these are carrying this is carrying three quarters of 13.6 KV of energy.

375
00:44:41,390 --> 00:44:45,560
So they're carrying enough energy to kick electrons out of the air molecules.

376
00:44:46,400 --> 00:44:51,720
So, so the so these photons are heavily are absorbed by all kinds of things.

377
00:44:51,740 --> 00:44:59,630
They're very these. These are very easily absorbed photons because they carry enough energy to lift electrons out of most atoms.

378
00:45:01,280 --> 00:45:05,900
And then we have the next is the bottom of Ceres, which is where the whole story started.

379
00:45:09,060 --> 00:45:13,250
Which is so end prime just two and equals three.

380
00:45:13,260 --> 00:45:21,989
If you go from 3 to 2, that's called Boma Alpha, but it's written as H Alpha because that stands for Hydrogen Alpha.

381
00:45:21,990 --> 00:45:32,100
So Palmer was a was a Swiss schoolteacher who empirically fitted the formula we've we've derived.

382
00:45:32,100 --> 00:45:33,030
So I've lost it. There it is.

383
00:45:33,030 --> 00:45:42,749
He fitted that formula empirically to to measured frequencies of of lines that he identified as being the Balmer series lines.

384
00:45:42,750 --> 00:45:47,580
Well, a series of lines equal to a hydrogen series. So this this is called H Alpha.

385
00:45:48,060 --> 00:45:53,610
And it's a it's a pink photon. It's 600 and something nanometres six, five, six.

386
00:45:58,310 --> 00:46:05,480
So it's pink light. So many astronomical objects are pink because they are shining in alignment.

387
00:46:05,840 --> 00:46:11,360
Sorry. In Alpha. In Palma, Alpha Beta.

388
00:46:13,100 --> 00:46:26,630
If you. Then you go to passion. That's four in prime, just three and obviously and can be four or five, etc., etc., etc.

389
00:46:28,010 --> 00:46:31,010
So these start off as pink and they get bluer.

390
00:46:31,040 --> 00:46:37,250
So as you go down this list, the wavelengths get shorter as you go to infinity with Series Limited.

391
00:46:37,280 --> 00:46:45,860
I did write it down here. 364.

392
00:46:49,690 --> 00:46:54,100
Six nanometres. So they. They go from pink light right through the optical spec.

393
00:46:54,220 --> 00:46:58,630
The rest of the optical spectrum to the to the ultraviolet region.

394
00:46:59,650 --> 00:47:05,770
And the Passion series starts at 1875, I think.

395
00:47:07,080 --> 00:47:10,320
So any calls for you were looking at 18?

396
00:47:10,350 --> 00:47:15,840
Yep. So that's already these are sort of more or less optical.

397
00:47:18,500 --> 00:47:24,710
By now we're in the near-infrared, etc., etc., etc.

398
00:47:26,120 --> 00:47:33,089
So. That's pretty much the right place to stop.

399
00:47:33,090 --> 00:47:39,510
I think what we should do. Uh, there's just one other thing I would point out is that so you can apply these

400
00:47:39,510 --> 00:47:45,930
formulae to the inner electrons of to the innermost electrons of other atoms,

401
00:47:47,010 --> 00:47:54,430
like atoms that have more than one electron. You can't apply them to the outer electrons with any useful way because we've done all this right.

402
00:47:54,430 --> 00:47:58,800
Remember, with with no other electrons present, we got one nucleus and one electron.

403
00:47:59,100 --> 00:48:05,820
But there's one very important thing to take home, which is that this energy scale goes like Z squared.

404
00:48:06,390 --> 00:48:15,650
So the energies, the characteristic energies of the innermost electrons are going up like Z squared.

405
00:48:15,650 --> 00:48:20,430
And by the time you get uranium, which has 92 units of charge, the Z is 92.

406
00:48:20,820 --> 00:48:24,330
You're almost a factor of of ten to the four.

407
00:48:24,330 --> 00:48:32,070
You're almost a factor of of 10,000 higher in energy, which means that these electrons are moving essentially relativistic.

408
00:48:32,080 --> 00:48:35,880
LI So that's just the thing to bear in mind.

409
00:48:35,910 --> 00:48:38,610
Okay. And we'll look at the wave functions that go with this lot tomorrow.