1
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Okay. So we finished last time.
2
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We're just about pushed through the calculation of what else squared is the differential operator,
3
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which we did, if you recall, by multiplying L the latter operators, L minus Snell plus together.
4
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And it was rather a tedious calculation, but at the end of the day, with luck,
5
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we ended up with this and we should recognise that this l squared is minus this combination of partial derivatives,
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vector theatre and PHI is the it's minus r squared times the angular part of of del
7
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squared the la plaza and when looked in when put into spherical polar coordinates,
8
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if you take if you take this thing and put it here,
9
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this minus sign cancels that minus sign and we get a one over all squared sine theta db peter sine teacher etc. which I hope you recognise as,
10
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as del squared. So, so you might ask yourself so physically what's happening.
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We have the kinetic energy operator i.e. so we got this is H sub k which means p squared over 2my,
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this is squared plus p y squared plus p z squared is also minus h squared over two times del squared.
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So in the position representation, this operator becomes this right because p is minus h bar times gradient.
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And classically. We have that l is equal to v tangential times the radius.
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So L squared is v tangential squared times radius squared l squared over our squared is equal to v tangential squared.
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So we have, we have that HK is equal to.
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Well sorry that's suggestive something isn't it, that this, this narrative squared can be written in terms of some radial derivative plus.
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So we could say that. Yeah, it was alright. We can say that HK is equal to some radial minus h bar squared over to m one overall squared D by d
19
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r squared D by the R and then we're going to have plus h bar squared L squared over all squared.
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I think just by substituting into there over to m sorry, over to him.
21
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And what's this going to be? We've defined the angular momentum operator l squared to be dimensionless,
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so putting an edge bar in front of it h bar L is the classical animal, right?
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So H bar operator is the analogue, I should say, is equal to it's the analogue of classical angular momentum, total angular momentum.
24
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So this that you have here has the dimensions of total incremental squared.
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It's the classically understood thing. So this term here is looks is looking awfully like v tangential squared.
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Sorry, I need a mass here. Right.
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The. So.
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The classical angular momentum is m v tangential r So the square is m squared v tangential squared r squared.
29
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Move the r squared down here. And this is the classical relationship that L squared overall squared is m squared V tangential squared.
30
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So this is looking like this in the back here is looking like half am v tangential squared.
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That's what this suggests to us, the quantum mechanical formula, which is correct.
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But it's suggesting to us that it's this sort of natural translation of classical physics is this and this is clearly the tangential kinetic energy.
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So this is the the k the tangential, the kinetic energy associated with tangential motion.
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Which suggests that this here should be the kinetic energy associated with radial motion.
35
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And that's what we want now to put on a firm intellectual footing.
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So we're going to show that this thing is minus H by squared over two M times PR squared, where PR is the radial momentum.
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So the question we want to address now is what is the radial momentum?
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Operator. We found the tangential operator.
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We found it in some sense, the tangential momentum operator in the sense Al.
40
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Now we want to find the radial one. So classically momentum is a vector and we can say that the radial momentum is simply our dot p overall number
41
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is the unit vector all dotted into P must surely be radial momentum momentum in the regular direction.
42
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But there's a problem with this from the perspective of quantum mechanics, because this operator doesn't compute with this operator.
43
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So it's well, what does that mean, this thing.
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Oh sorry in QM oh dot P overall is not emission me prove that to you but it's easier to prove that in general than in particular.
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Okay, so let me have two emission operators. A dagger is A and B, dagger is B, then let's look at a B dagger.
46
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If I multiply these together, what is I guess an operator ab is this operator mission?
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Find out that's b dagger. A dagger.
48
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Because the rule for taking her emission that joint is you reverse the order and dagger the individual bits, but B's B dagger and A's a dagger.
49
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So this is equal to be. So is this equal to AB?
50
00:07:01,070 --> 00:07:04,910
Well, clearly it is if and only if b in a commute.
51
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So. So this is only if woops a comma b equals nought in words.
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The product of two emission operators is itself emission only if those two operators commute if and only if those two operators commute.
53
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So this are dog P, which is shorthand of course, for x plus y,
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if y plus there p z would be emission could be emission only if X and commuted Y and p y commuted.
55
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Well, they don't. Therefore, this is not a mission. Therefore, this is not an observable.
56
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So it can't be what we're looking for. We're looking for something, the momentum in the radical direction, which is which is observable.
57
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All right. Well, there's a fix to this problem. There's a general fix, and we're going to use it.
58
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But if we do a half of AB plus B.A., this is a mission.
59
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Right. Because if you take the dagger of this. This one, we just proved the dagger of this one is that.
60
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And the dagger of that one is this. So this thing, the dagger of this bracket is itself.
61
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So when you've got two non-commissioned operators sorry, you've got two mission operators that don't commute and you want to make the product.
62
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The way to go is to is to take the average of them.
63
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You know, it's really naive thing to do. So let's do that. So let's let's so so we try.
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Let's have a look at the emission operator PR,
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which we're going to define to be a half of our dot p overall where it's
66
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important that that all this R here is in front of the PR plus P dot R overall.
67
00:09:07,980 --> 00:09:14,010
So this thing here will be emission and I'm going to show that it is what we require.
68
00:09:17,220 --> 00:09:22,620
So in the position representation so we we you can do this calculation in the abstract,
69
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not in the position representation, but it's easiest in position representation. So that's how we'll do it.
70
00:09:31,120 --> 00:09:36,250
So PR is equal to. So this P gets replaced by minus H by a grad, right?
71
00:09:36,820 --> 00:09:41,590
So this is going to be minus I h bar over to common factor.
72
00:09:41,890 --> 00:09:45,760
We're going to have our dot grad overall.
73
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But this of course is the scalar plus the divergence of our overall well.
74
00:10:00,270 --> 00:10:03,820
Now the issue is this. When what?
75
00:10:03,870 --> 00:10:09,180
This isn't quite the divergence of this. What this means is, is this is remember an operator.
76
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It's waiting for a wave function to come and stand in front, get operated on.
77
00:10:13,230 --> 00:10:19,110
Right. So this differential operator operates when everything to its right.
78
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It operates on this and it operates on these two. This here, this differential operator operates everything to its right, which is only the ABC.
79
00:10:31,480 --> 00:10:38,549
So we have to when we, when we expand this out,
80
00:10:38,550 --> 00:10:43,590
we're going to get three terms because we're going to get this thing operating on this that that is standing idly by,
81
00:10:43,860 --> 00:10:49,860
this thing operating on this with this and this standing idly by and this thing operating on this with these two standing idly by,
82
00:10:49,860 --> 00:11:02,049
which is the same as that. So this is going to be a minus I HPR bar dot grad overall, right?
83
00:11:02,050 --> 00:11:07,510
So I'm taking this one that I've got and the one that I'm promised at the end of all this reduction here.
84
00:11:08,800 --> 00:11:12,580
So that's where the two went to. So that's, that's those two.
85
00:11:12,580 --> 00:11:16,690
And now I've got these two bits minus my brackets.
86
00:11:17,170 --> 00:11:21,010
We're going to have over two. Sorry, that's this factor here.
87
00:11:21,460 --> 00:11:28,270
Then I'm going to have this thing operating on this, the divergence of RS three, and then I've got this thing operating on that.
88
00:11:28,510 --> 00:11:32,020
It's going to be all dotted into the gradient of one overall.
89
00:11:32,800 --> 00:11:38,670
So that's going to be minus oh dotted into the gradient of one overall.
90
00:11:38,680 --> 00:11:46,270
The gradient of one overall has to be a well, it is the vector R overall squared.
91
00:11:50,170 --> 00:11:55,120
This minus sign comes from the differentiating of the one overall. Right. Because I'm reminding you of prelims maths.
92
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Now the gradient of R itself is the is the vector are divided by all.
93
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This is dimensionless animal because that has dimensions of one of a length that has dimensions of length.
94
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So it's the vector is the unit vector R And that's what we've been using.
95
00:12:11,410 --> 00:12:14,980
Looks at this. Excuse me. So I made a mistake. This should be r cubed.
96
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Maybe we need to do this. Sorry. So what? What?
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Just to fill in here.
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So the gradient of one overall is equal by the ordinary rules of differentiation, minus one overall squared times the gradient of R,
99
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but the gradient of R I've just said is the vector are divided by R, hence the R cubed.
100
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These two dot together make an R squared which cancel most of those.
101
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So, so this minus sign, these two can be combined to a two overall.
102
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The twos go away and guess what we end up with is minus h bar grade overall plus one overall.
103
00:12:56,710 --> 00:13:07,810
Okay. So that's what this this stuff reduces to what we next want to know is so what is R dot gradient.
104
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Well, I want to know what this is in spherical polar coordinates.
105
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Well, the thing to do is just to write down let's write down r d by the R, it's easy to see that that is going to be x deep.
106
00:13:25,990 --> 00:13:34,110
Well, let's, let's do it o then we're going to have by the chain rule the X by the r d by the x plus d.
107
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Y. By the r. Debate Y plus d z by the R debate, you said that's just the chain rule.
108
00:13:49,230 --> 00:13:59,730
But but what is the x by the r x is equal to our sine theta cos phi so d x by the r is equal to
109
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x overall and for the same reason do y by the r is equal to Y over r and so on and so forth.
110
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So this is equal to x d by the x plus y, DPD, y plus z, dvt z because this is x overall,
111
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but this all cancels the R on the bottom y overall all cancels what's on the bottom and what's this?
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00:14:25,200 --> 00:14:30,930
This is a vector product of x, y, common z with nebula DVD, X, Y, or Z.
113
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In other words, this is this is the animal we're interested in our dog.
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Great. So I have now that PR is equal to minus I h bar r dot grad we've just agreed is deep idr.
115
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So this is go away and we have D by the R plus one overall.
116
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So we have an interesting result.
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We have the momentum associated with the radius is not simply d by d radius like the momentum associated with x is D Barry X.
118
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There's also this additional term in here. But just to convince you this really is the momentum associated with RADIUS.
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Let's for fun calculate r comma pr.
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So what is that? That is minus h bar of r deep 80 plus one which is r times pr minus D by the R of r minus one.
121
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Yeah, right. Yes.
122
00:16:06,450 --> 00:16:10,919
Sorry. I mean, we're getting zero on time. That's the trouble. I want to get minus.
123
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I want to get bar out of this. What the [INAUDIBLE] did I do wrong?
124
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DVD all PR minus DVD are plus one overall on working on.
125
00:16:25,440 --> 00:16:32,089
Ah. Yeah. Yes.
126
00:16:32,090 --> 00:16:35,270
Yes. Yes. Yes. Sorry. Sorry, sorry. Yes, yes. Yes. Yes. Yes. It's perfectly correct.
127
00:16:35,600 --> 00:16:41,060
Okay, so the ones go away. This is this is this this sort of thing is confusing right now, as I say.
128
00:16:41,360 --> 00:16:45,320
What does this mean? This means DVD are of everything to it's right.
129
00:16:45,530 --> 00:16:51,160
And there's a phantom wave function here waiting to be operated on. So this is the derivative of ah upside.
130
00:16:51,530 --> 00:16:59,950
When we take the derivative of the R, we get one times of psi and then the R stands idly by and we do the gradient of upside.
131
00:17:00,710 --> 00:17:06,980
The second term cancels on this because our times, the gradient of upside is occurring here with a plus sign,
132
00:17:07,250 --> 00:17:15,200
and there it will be occurring with a minus sign. So what we're left with is the D by the R times of PSI, the R, which makes one.
133
00:17:15,210 --> 00:17:27,890
So this is equal to plus because it's a minus sign coming here h bar so that it's these two operators satisfy the canonical commutation relations.
134
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Right. Canonical commutation relation.
135
00:17:35,880 --> 00:17:39,630
So so PR really is the momentum associated with.
136
00:17:39,630 --> 00:17:44,530
Ah. Okay. So what, what are we really trying to do here.
137
00:17:44,550 --> 00:17:52,800
We're trying to show that that one overall square or square video is essentially PR squared.
138
00:17:52,800 --> 00:18:06,420
So let's calculate PR squared. PR squared is going to be minus H bar squared because they'll be two, they'll be two minus bars and then it's D by t,
139
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r plus one overall bracket D by the R plus one overall, which is equal to minus H bar squared.
140
00:18:17,100 --> 00:18:24,389
Obviously this on this is D two by the R squared we will get this differential
141
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operator will differentiate that and would use a minus one over squared.
142
00:18:31,450 --> 00:18:38,350
We will otherwise get a one overall D by the R and also a one overall D by the R.
143
00:18:38,950 --> 00:18:44,350
So we'll get to of one overall D by the R.
144
00:18:49,430 --> 00:18:54,900
And sorry and I haven't finished. And I also get this thing on this thing is a plus one overall squared.
145
00:18:56,090 --> 00:19:02,060
So these two terms, these two terms cancel and we're left staring at this.
146
00:19:07,200 --> 00:19:10,270
Uh, I should have had two of these terms.
147
00:19:10,800 --> 00:19:15,960
I think I said I was going to get two terms because I have a one over audio and I have a one over DVD after this.
148
00:19:15,990 --> 00:19:24,300
Operator when this operator works on this, it reduces that, but also it works on the phantom wave function sitting over here without standing idly by.
149
00:19:24,600 --> 00:19:27,780
So we get two of these. I think I said that, but I didn't write it.
150
00:19:27,810 --> 00:19:38,460
I'm not sure. So we have a minus h bar squared D two by the R squared plus two overall D by the R,
151
00:19:38,910 --> 00:19:47,520
which can also be written as minus h bar squared over all squared d, buddy r of r squared d, buddy r.
152
00:19:47,670 --> 00:19:53,850
Right. Because if you differentiate out this product, you get R squared on R squared times D two by the R squared,
153
00:19:53,850 --> 00:20:00,210
which is this term and you also get a2r overall squared two overall times D video.
154
00:20:01,020 --> 00:20:05,840
So yeah, this, this term here, we've now shown the Del's,
155
00:20:06,120 --> 00:20:23,220
we've now shown that HK the momentum operator which is minus H squared over two M Del squared is also minus h bar squared over two M of sorry.
156
00:20:26,760 --> 00:20:31,990
Yeah. Of.
157
00:20:32,920 --> 00:20:36,760
Well, let's leave that outside. Let's take the ball square into the bracket.
158
00:20:37,990 --> 00:20:45,340
We're going to have a one over all square debris, all that stuff, which we've just shown is is PR squared.
159
00:20:47,110 --> 00:20:51,219
And then oops. But there was a minus sign.
160
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So that soaks up this minus sign, a PR squared. And then similarly there's plus H bar squared, L squared, overall squared.
161
00:21:02,980 --> 00:21:10,840
This is a very important formula that we will need when doing hydrogen and therefore fundamental to.
162
00:21:12,250 --> 00:21:19,210
So it's expressing your kinetic energy in terms of your radial kinetic energy and your tangential kinetic energy.
163
00:21:19,960 --> 00:21:25,150
And that's one of the reasons why the total orbital increment operator is important,
164
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because it encodes your sort of energy of going around and around.
165
00:21:31,940 --> 00:21:39,770
So with that we are now finished with we can massively finish with orbital and lumentum and we can get on to spin.
166
00:21:46,020 --> 00:21:55,169
And this is somewhat more interesting in the sense that it's quantum mechanics has more remarkable things to say and
167
00:21:55,170 --> 00:22:01,050
it's less tedious because all that stuff with as part of the French operator's DVD theatre and stuff is not much fun.
168
00:22:01,440 --> 00:22:13,830
It has to be set right. So we have identified two types of of generations of rotations, the angle, the total and element of operators.
169
00:22:21,060 --> 00:22:28,140
And they generate we introduce them in order to generate complete road to complete rotations.
170
00:22:33,950 --> 00:22:49,970
So you have alpha being e to the minus I alpha not j rotates system as on turntable.
171
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So it moves. It moves your system around the origin.
172
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If you put you you put your system.
173
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It's as if you put your system on a turntable centred at the origin, the axis of the origin, and you turn the turntable around.
174
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Your system moves through space and it rotates simultaneously.
175
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Whatever internal structure it has. But we also have shown that ally the orbital angle momentum.
176
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Moves system on circles.
177
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So it moves it around physically. It translates it around the origin, but it is not rotated at the same time.
178
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It leaves its orientation fixed. And we have some.
179
00:23:48,960 --> 00:23:52,190
Well, okay. So we found the commutation relations here.
180
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We found that we found that j i comma j j is equal to i some have a k excellent i j k j k.
181
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And we found that it was also true that l i l j was equal to i sum David k
182
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epsilon i j k l k They had the same commutation relations amongst themselves,
183
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these operators, which is why we could use the work we did demonstrating what the eigenvalues of these could be.
184
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Also down here, this this implied that and that that that j squared has a values.
185
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J j plus one for j is nothing.
186
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A half one three halves, etc.
187
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And. From these commutation relations, we infer those are possible values for the eigen values of these operators.
188
00:24:59,680 --> 00:25:05,169
But we also had the principle that if we wrote if we translated something completely around the origin,
189
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we proved that that that was the identity transformation.
190
00:25:10,720 --> 00:25:22,240
So we concluded that l l plus one had to be l equals l equals nought one to integers only allowed.
191
00:25:23,260 --> 00:25:33,870
In this case, what we now going to do is introduce s i is by definition j i minus l i.
192
00:25:33,880 --> 00:25:37,480
It's the difference between these two. What does that mean physically?
193
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It means that s II is going to be the generator of rotations of a thing about its own axis.
194
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So we're not going to be this rotated on a turntable.
195
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So it rotates it and moves it. This simply moves it around a circle.
196
00:25:56,380 --> 00:26:00,040
So this is going to only rotate it on its own axis.
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00:26:00,160 --> 00:26:19,430
It's not going to move it. It's only going to rotate it. That's what we expect to happen.
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But we'll have to be guided to some extent by the mathematics of all these what is going to happen.
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00:26:26,780 --> 00:26:27,410
Right. So what about.
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So what we want in having these new fangled operators, it's important to figure out what the computation relations are going to be.
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Now, see, comma SJ is going to be j i minus l i comma j i minus sorry j j minus lj.
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00:26:50,070 --> 00:26:55,220
All right, we're going to get this commuting with this.
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So we're going to get I some David K Excellent.
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00:27:01,400 --> 00:27:16,540
I j k this commuting with this will produce a j k this commuting with this will produce an elk this commuting with this.
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00:27:17,150 --> 00:27:21,260
Now, we didn't write that down, but I commuting with elk.
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This is a vector operator and therefore this thing when you when commutes with this always produces the the missing component of this vector.
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So this is going to be minus elk.
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And similarly, this thing on this thing is going to produce swap them over and you're going to.
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Well, there are several signs here that we could found down,
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but this thing is we're looking fundamentally at the same thing as the commentator of this on this we're looking at.
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00:28:02,840 --> 00:28:14,180
Looking at minus l i comma jj is equal to is equal to obviously jj l i is equal to
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00:28:14,210 --> 00:28:26,450
i epsilon j i k l k is equal to minus i epsilon summed over k This is a sum of a
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k of epsilon i j k l k So all right j i j i k And but I would like to have this in
214
00:28:42,020 --> 00:28:46,610
the order k So I swap those two over and introduced a minus sign to compensate.
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00:28:47,720 --> 00:28:56,540
And then at the end and then you can see this thing, including that minus sign is the same thing as this thing, including that minus sign.
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00:28:56,540 --> 00:29:03,800
So we have a minus another L Okay. So this is the justification for that last term there.
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00:29:04,610 --> 00:29:09,230
So what do we end up with at the end of the day? These three L K's collapsed into just one.
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00:29:09,290 --> 00:29:19,070
Okay, it's going to be j k minus L. In other words, this is going to be I someday have a k epsilon i j k as k.
219
00:29:20,150 --> 00:29:25,520
So these spin operators, so they say that they're going to be this we call them the spin operators.
220
00:29:25,760 --> 00:29:31,430
They have exactly the same commutation relations as the J. Therefore, we know what their eigenvalues are.
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00:29:32,900 --> 00:29:36,320
So this implies that the eigenvalues.
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00:29:38,780 --> 00:29:55,070
Of X squared, which is of course s which is x squared plus x y squared plus z squared, r s s plus one where s is equal to a half.
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00:29:55,730 --> 00:29:59,450
Sorry, nothing. A half one three halves.
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00:29:59,720 --> 00:30:06,860
Blah blah blah. Okay. Because these these results follow ideally from the fact of having the commutation relations.
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00:30:06,860 --> 00:30:15,470
Asi como se j is I excellent? I ask all the half integer values allowed.
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Answer you will have a half integer.
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When Jay does. Because.
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00:30:32,140 --> 00:30:36,010
L does not. All right.
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00:30:36,040 --> 00:30:51,819
Why is that? That's because Se Z is equal to Jay Z minus L.Z. and Jay Z comma S.Z. equals nothing, which is also the same as L.Z.,
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comma, Jay Z, etc. All these three operators commute with each other, so there's a complete set of mutual aid and states.
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00:31:11,680 --> 00:31:17,190
And if so, we can now see that if this is half integer, this is using half integer eigenvalues.
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So the eigenvalues of this are going to be a going to be the difference between the eigenvalues of this and the eigenvalues of this.
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00:31:24,430 --> 00:31:31,540
So if this has half integer eigenvalues, therefore this will have to have half integer eigenvalues because this one has integer eigenvalues.
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00:31:36,270 --> 00:31:41,429
So. So if J has half integer eigenvalues then does correspondingly.
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00:31:41,430 --> 00:31:45,120
If J doesn't s doesn't. It just takes along behind J.
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00:31:46,290 --> 00:31:49,290
And indeed, that's how we tend to think about it.
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00:31:49,290 --> 00:31:54,960
We tend to think that the integer amounts of angular momentum come from orbital motion,
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L.Z. and the half integer values, if present, come from S said.
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00:32:00,510 --> 00:32:05,520
And that's why J has half integer values. That's how we tend to think about it.
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00:32:12,220 --> 00:32:27,670
Okay. Now.
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00:32:27,940 --> 00:32:35,470
I think I've claimed a few times that spin is something to do with spin is something to do with the orientation of our system.
242
00:32:35,590 --> 00:32:40,960
And now it's time to to make good this claim that.
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00:32:44,340 --> 00:32:51,090
That the eigenvalues of the spin operator or your response to the spin operators encodes how a particle is oriented.
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00:32:51,960 --> 00:32:55,110
And this is a strange area of very quantum mechanical area.
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00:32:57,310 --> 00:33:14,580
Okay. So in general, the internal configuration of a system could be written that we could write if PSI is equal to the sum of s m upside.
246
00:33:16,680 --> 00:33:26,310
So we got a complete set of mutual we've got a complete set of mutual aid and states of S squared and s z.
247
00:33:26,380 --> 00:33:37,740
Right? So this is the so we're saying that s squared on s m is equal to S s plus one whoops one of s m and we're
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saying that s said on s m is equal to m s m and there should be a complete set of eigen states of this,
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of these mutual sets, of these operators.
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00:33:51,150 --> 00:33:59,130
So I should be able to expand any, any, any state as a linear combination of these of these states.
251
00:33:59,820 --> 00:34:02,070
And what are we going to have to sum over?
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00:34:02,490 --> 00:34:11,850
We're going to have to sum of S is equal to nought half and we're going to have to some of em is in modulus less than or equal to s.
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This. This should be a generally valid statement,
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but for the wave functions or the states of view of real occurring systems in the laboratory, the good news is you don't.
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00:34:28,480 --> 00:34:32,110
You only have to. There's only one value of s will occur in this sum.
256
00:34:32,110 --> 00:34:43,150
So. So this is this this would be generally the case. But, uh, for the systems in the lab of microscopic systems, this wouldn't be true.
257
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So if we were to embed a cricket ball, we would need to do a sum over all of these things up to ten to the 40 or something.
258
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But if we're dealing with an electron or an atom or whatever, we don't have to do some of all these things.
259
00:35:00,220 --> 00:35:03,580
We only only one value of s has non vanishing amplitude.
260
00:35:03,580 --> 00:35:07,870
So this is going to vanish. For real systems. Except when s takes one particular value.
261
00:35:08,410 --> 00:35:12,490
So for real. From microscopic. Microscopic.
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00:35:20,230 --> 00:35:29,810
Only one value of s occurs. Normal vanishing.
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00:35:36,760 --> 00:35:48,309
As any sign of this. We can say at the outset that the amplitude to find a value of s other than a value that's peculiar to the system is zero.
264
00:35:48,310 --> 00:35:52,750
So we just don't need to consider it. So we're able to so we're able to write that up.
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00:35:52,750 --> 00:36:03,010
PSI is equal to the sum from m equals minus s2s of s and.
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00:36:04,000 --> 00:36:10,580
CI said. That means.
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00:36:10,670 --> 00:36:18,320
What does that mean? We're encoding? Well, the state of the system is described by these numbers here.
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There will be two plus one complex numbers.
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00:36:22,760 --> 00:36:27,950
These these numbers here. And they're telling us they're encoding somehow the way the system is oriented.
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00:36:28,370 --> 00:36:40,960
So this is two plus one. Complex numbers, encoding, encoding orientation.
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00:36:40,970 --> 00:36:45,650
And what we want to do now is get some feel for how this encoding works.
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In simple cases, how would you encode the orientation of a macroscopic body?
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00:36:56,790 --> 00:37:04,530
Well, the traditional procedure is you use only two angles. You write down three angles which describe how you get some access in the body,
274
00:37:05,370 --> 00:37:13,439
you know which way it points with respect to the z-axis of some fixed fixed coordinates and then how it rotates around that.
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So you usually encode the orientation of a body in three Euler angles for a classical system,
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for quantum mechanical system, you encode it in a certain number of complex amplitudes.
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The amplitudes defined find it in various orientations. It's carrying the same information.
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00:37:29,760 --> 00:37:39,299
Right, in a very funny way. Okay, so let's let's consider the various cases.
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The case equals nought. In other words, if the only occurring thing here is nought, there's basically no information.
280
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There's only one, one amplitude.
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00:37:56,160 --> 00:37:59,390
Nought. Nought. State of your system.
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00:38:01,040 --> 00:38:11,330
And also, if you if you if you you rotate your system, you try and rotate system.
283
00:38:13,660 --> 00:38:26,010
With you, you alpha, which is of the minus I alpha dot s this is the so s is the generator of rotations of the system around its own axis.
284
00:38:26,020 --> 00:38:34,090
Right? So I've written down the unitary rotation that makes me a new system which is rotated around the Axis Alpha by the magnitude of alpha.
285
00:38:36,250 --> 00:38:41,170
Then if I use this you alpha on upside, what do I get?
286
00:38:41,470 --> 00:38:45,340
I get using that expansion. I get nothing.
287
00:38:45,700 --> 00:38:50,290
Nothing psi times you alpha or nothing.
288
00:38:51,130 --> 00:39:07,240
Nothing. Well let me replace this by e to the minus I alpha dot s when when this operator sees this cat, it thinks you know nothing doing zero, right?
289
00:39:07,480 --> 00:39:12,460
You simply get as all of these think, that that becomes nothing.
290
00:39:13,420 --> 00:39:17,740
So we have easier than nothing, which is one. So this return, this goes back to itself.
291
00:39:21,780 --> 00:39:23,090
Which is the same thing as upside.
292
00:39:23,760 --> 00:39:30,060
In other words, when Earth is one, you can't tell the difference between the system before you rotated and after you rotated it.
293
00:39:30,270 --> 00:39:35,070
It's like an absolutely immaculate and perfect sphere. If you rotate it, it stays the same.
294
00:39:39,910 --> 00:39:44,080
So a particle that that does this that has x equals nought.
295
00:39:44,080 --> 00:39:54,850
So no spin implies same after rotation.
296
00:39:58,960 --> 00:40:03,250
So if you like, for a classical animal, we would say this is strictly symmetric.
297
00:40:12,310 --> 00:40:19,630
And there's only one amplitude we need to bother with.
298
00:40:20,470 --> 00:40:30,940
So we say it's a scalar. So the simplest particles are spin zero particles.
299
00:40:31,450 --> 00:40:38,290
They there's just one amplitude, which is you say to say what the energy is or to say what the location is,
300
00:40:38,500 --> 00:40:41,709
because there's no issue of how is it oriented. It's a silly question.
301
00:40:41,710 --> 00:40:45,310
How is it oriented? It's it doesn't have an orientation. You can't tell it.
302
00:40:45,580 --> 00:40:51,310
If it doesn't know it doesn't have an orientation. That's the best thing to say. Nothing changes if you try and reorient it.
303
00:40:52,990 --> 00:40:56,380
And because of the one amplitude business, you say it's a scalar particle.
304
00:40:59,100 --> 00:41:02,639
Right. So that first equals. Next in the hierarchy of.
305
00:41:02,640 --> 00:41:08,430
And for an example, a pile is an elementary particle which is a scalar particle.
306
00:41:08,640 --> 00:41:15,110
Right. Let's do X equals a half. So unfortunately, there are not many particles.
307
00:41:15,120 --> 00:41:18,270
There's not much in physics that's a scalar particle or scalar field.
308
00:41:19,020 --> 00:41:25,250
This is a rare case. The the really important cases are X equals one and as equals a half and this equals one.
309
00:41:25,290 --> 00:41:30,120
We're all built out of X equals or half particles. Right. So this is electrons.
310
00:41:32,450 --> 00:41:36,100
Protons. Quarks.
311
00:41:38,270 --> 00:41:46,760
Therefore neutrons. Wide range of things as he made out of an odd number of quarks will be in as equals a half particle.
312
00:41:46,780 --> 00:41:55,510
What will prove that later on? Okay. So this is a really big class we have that the state of the system can now be written as.
313
00:41:58,990 --> 00:42:04,629
There's going to be a half a half of CI plus a half.
314
00:42:04,630 --> 00:42:09,340
Minus a half. I'm sorry. I need two times a half.
315
00:42:10,090 --> 00:42:17,680
Half plus upside times the half minus a half.
316
00:42:20,530 --> 00:42:30,460
So any stage of the system of this particle can be written as a linear combination of this state, which is to say that M is,
317
00:42:30,670 --> 00:42:36,610
you know, you're guaranteed to get the answer half if you measure a said and a linear and a linear combination of this.
318
00:42:36,610 --> 00:42:41,410
So we now have a non-trivial linear combination with two possibilities. This is a very cumbersome notation.
319
00:42:41,680 --> 00:42:53,409
So people don't use it. They either write that this is plus cy plus plus minus, abassi minus.
320
00:42:53,410 --> 00:42:56,800
That's a handy notation, right?
321
00:42:57,130 --> 00:43:02,110
Because no, there's really no point in writing down this half because we know the first half we know for certain will always be there.
322
00:43:03,250 --> 00:43:09,550
And instead of writing a half, we write plus shorthand for plus a half and minus shorthand for minus a half.
323
00:43:10,000 --> 00:43:16,390
Or since this is only a boring, complex number, we often write A plus plus, B minus.
324
00:43:17,860 --> 00:43:25,570
So any state is a linear combination of these two basis states where we're guaranteed to get plus a half or so,
325
00:43:25,610 --> 00:43:29,739
minus a half rest, and there are amplitudes. So this is the this is the amplitude.
326
00:43:29,740 --> 00:43:35,410
It's Mod Square gives you the probability. If you would measure the Z components of spin of this particle, you would get a half.
327
00:43:35,770 --> 00:43:38,889
And B is obviously the amplitude. What square of that would be?
328
00:43:38,890 --> 00:43:41,980
The probability that you got said it was minus a half.
329
00:43:59,210 --> 00:44:06,440
So let's do some stuff with. They would spend a half particles.
330
00:44:14,340 --> 00:44:17,400
So if I take an arbitrary spin operator.
331
00:44:18,720 --> 00:44:26,040
Well, so let that be the this is the the the spin component.
332
00:44:29,650 --> 00:44:38,809
Along the unit vector. And what is it?
333
00:44:38,810 --> 00:44:49,190
Mathematically, it's an dot sw. In other words, it's an x as x plus and y as y plus NZ said.
334
00:44:51,500 --> 00:44:56,570
Wouldn't this that I've just written down would apply for any spin, not just spin a half.
335
00:45:00,280 --> 00:45:09,459
And I would like to to do calculations like if I do an on ABC, what do I get?
336
00:45:09,460 --> 00:45:16,960
I get some new state fi say, all right, this is an operator, I usual state, I get a new state.
337
00:45:17,110 --> 00:45:21,090
I want to be able to to do calculations like this.
338
00:45:21,100 --> 00:45:24,040
We will we'll find this is crucial.
339
00:45:25,330 --> 00:45:31,030
Now, we know that this everything every one of these states can be written as a linear combination of plus and minus.
340
00:45:31,330 --> 00:45:34,569
So this can be written a C plus plus.
341
00:45:34,570 --> 00:45:41,690
D minus. His essay in.
342
00:45:42,350 --> 00:45:47,930
And this can be written as a linear combination of A-plus plus B minus.
343
00:45:52,030 --> 00:45:59,829
And the name of the game is given the numbers A and B, which characterise that to calculate the numbers C and D,
344
00:45:59,830 --> 00:46:04,810
which characterise that we need an apparatus that does that for us and that's easily obtained.
345
00:46:05,050 --> 00:46:09,460
What we do to find C of course is we draw through with plus.
346
00:46:12,550 --> 00:46:31,930
And then on the left we get C plus nothing. So we find that C is equal to plus s n plus times, a plus plus and minus times B and growing through.
347
00:46:31,930 --> 00:46:35,980
By minus we get an equation which tells us the value of D,
348
00:46:36,700 --> 00:46:50,109
which is equal to minus S and plus of a plus minus s, n, minus B, and there's a handy a way of writing this.
349
00:46:50,110 --> 00:47:07,690
We write this is C, D is equal to a matrix, but this is just a boring, complex number plus s and plus that's a boring complex number minus s n plus.
350
00:47:09,370 --> 00:47:16,240
Plus S, N minus, minus S and minus.
351
00:47:17,050 --> 00:47:23,440
And that's operating on the column vector a, b. So here we have a concrete apparatus.
352
00:47:23,830 --> 00:47:29,830
This is a matrix, a two by two matrix of complex numbers, working on the given complex numbers that characterise ASI,
353
00:47:30,160 --> 00:47:39,810
which gives us the two complex numbers that characterise FY. So for example, suppose we take.
354
00:47:41,780 --> 00:47:50,900
Just got time to do this. Suppose we take an is equal to is equal to nothing come and nothing come of one.
355
00:47:51,800 --> 00:47:54,860
Then this becomes s z. This becomes s.
356
00:47:54,860 --> 00:48:03,680
I mean, all these s ends become assets. And then said on plus is a half of plus.
357
00:48:03,980 --> 00:48:11,750
So this number evaluates to a half. Then see the is equal to a half.
358
00:48:13,410 --> 00:48:18,569
As I said, on minus is minus a half times minus, but a minus is orthogonal to this.
359
00:48:18,570 --> 00:48:24,150
So we get a nothing. Nothing. And as I said on this is minus, minus.
360
00:48:24,570 --> 00:48:28,620
So this one minus half of minus. So this thing evaluates to minus a half.
361
00:48:29,520 --> 00:48:43,379
So in this particular case, it evaluates to this which people write as a half of Sigma Z, A, B, a column B by Sigma.
362
00:48:43,380 --> 00:48:49,810
That is the Pauli matrix. One minus one.
363
00:48:49,820 --> 00:48:52,940
Nothing. Nothing. Apparently matrix.
364
00:49:00,360 --> 00:49:11,550
So these matrixes we really want are these matrices, but it's handy to take the half outside of the matrix and write them in terms of these.
365
00:49:11,820 --> 00:49:17,340
Well, this is and this is the first of them. And if we had time, which we haven't now, we'll have to do it tomorrow.
366
00:49:17,490 --> 00:49:24,450
We will derive what Sigma X's and Sigma. Why are the three apparently matrices that enable us then to write the matrix
367
00:49:24,450 --> 00:49:29,760
belonging to any one of these spin operators and then do calculations on these?
368
00:49:30,000 --> 00:49:31,470
So it's it's time to stop.
00:00:04,340 --> 00:00:08,690
Okay. So we finished last time.
2
00:00:12,050 --> 00:00:17,570
We're just about pushed through the calculation of what else squared is the differential operator,
3
00:00:17,570 --> 00:00:26,770
which we did, if you recall, by multiplying L the latter operators, L minus Snell plus together.
4
00:00:26,770 --> 00:00:30,739
And it was rather a tedious calculation, but at the end of the day, with luck,
5
00:00:30,740 --> 00:00:40,520
we ended up with this and we should recognise that this l squared is minus this combination of partial derivatives,
6
00:00:40,520 --> 00:00:49,940
vector theatre and PHI is the it's minus r squared times the angular part of of del
7
00:00:49,940 --> 00:00:54,670
squared the la plaza and when looked in when put into spherical polar coordinates,
8
00:00:54,800 --> 00:00:58,670
if you take if you take this thing and put it here,
9
00:01:00,260 --> 00:01:10,580
this minus sign cancels that minus sign and we get a one over all squared sine theta db peter sine teacher etc. which I hope you recognise as,
10
00:01:10,850 --> 00:01:19,160
as del squared. So, so you might ask yourself so physically what's happening.
11
00:01:19,550 --> 00:01:31,930
We have the kinetic energy operator i.e. so we got this is H sub k which means p squared over 2my,
12
00:01:31,940 --> 00:01:43,009
this is squared plus p y squared plus p z squared is also minus h squared over two times del squared.
13
00:01:43,010 --> 00:01:51,200
So in the position representation, this operator becomes this right because p is minus h bar times gradient.
14
00:01:54,500 --> 00:02:04,280
And classically. We have that l is equal to v tangential times the radius.
15
00:02:05,300 --> 00:02:15,230
So L squared is v tangential squared times radius squared l squared over our squared is equal to v tangential squared.
16
00:02:16,640 --> 00:02:25,190
So we have, we have that HK is equal to.
17
00:02:26,030 --> 00:02:36,310
Well sorry that's suggestive something isn't it, that this, this narrative squared can be written in terms of some radial derivative plus.
18
00:02:37,070 --> 00:02:50,899
So we could say that. Yeah, it was alright. We can say that HK is equal to some radial minus h bar squared over to m one overall squared D by d
19
00:02:50,900 --> 00:03:03,230
r squared D by the R and then we're going to have plus h bar squared L squared over all squared.
20
00:03:05,520 --> 00:03:10,590
I think just by substituting into there over to m sorry, over to him.
21
00:03:12,240 --> 00:03:21,270
And what's this going to be? We've defined the angular momentum operator l squared to be dimensionless,
22
00:03:21,750 --> 00:03:27,479
so putting an edge bar in front of it h bar L is the classical animal, right?
23
00:03:27,480 --> 00:03:35,970
So H bar operator is the analogue, I should say, is equal to it's the analogue of classical angular momentum, total angular momentum.
24
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So this that you have here has the dimensions of total incremental squared.
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It's the classically understood thing. So this term here is looks is looking awfully like v tangential squared.
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Sorry, I need a mass here. Right.
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The. So.
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The classical angular momentum is m v tangential r So the square is m squared v tangential squared r squared.
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Move the r squared down here. And this is the classical relationship that L squared overall squared is m squared V tangential squared.
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So this is looking like this in the back here is looking like half am v tangential squared.
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That's what this suggests to us, the quantum mechanical formula, which is correct.
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But it's suggesting to us that it's this sort of natural translation of classical physics is this and this is clearly the tangential kinetic energy.
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So this is the the k the tangential, the kinetic energy associated with tangential motion.
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Which suggests that this here should be the kinetic energy associated with radial motion.
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And that's what we want now to put on a firm intellectual footing.
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So we're going to show that this thing is minus H by squared over two M times PR squared, where PR is the radial momentum.
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So the question we want to address now is what is the radial momentum?
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Operator. We found the tangential operator.
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We found it in some sense, the tangential momentum operator in the sense Al.
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Now we want to find the radial one. So classically momentum is a vector and we can say that the radial momentum is simply our dot p overall number
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is the unit vector all dotted into P must surely be radial momentum momentum in the regular direction.
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But there's a problem with this from the perspective of quantum mechanics, because this operator doesn't compute with this operator.
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So it's well, what does that mean, this thing.
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Oh sorry in QM oh dot P overall is not emission me prove that to you but it's easier to prove that in general than in particular.
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Okay, so let me have two emission operators. A dagger is A and B, dagger is B, then let's look at a B dagger.
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If I multiply these together, what is I guess an operator ab is this operator mission?
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Find out that's b dagger. A dagger.
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Because the rule for taking her emission that joint is you reverse the order and dagger the individual bits, but B's B dagger and A's a dagger.
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So this is equal to be. So is this equal to AB?
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Well, clearly it is if and only if b in a commute.
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So. So this is only if woops a comma b equals nought in words.
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The product of two emission operators is itself emission only if those two operators commute if and only if those two operators commute.
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So this are dog P, which is shorthand of course, for x plus y,
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if y plus there p z would be emission could be emission only if X and commuted Y and p y commuted.
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Well, they don't. Therefore, this is not a mission. Therefore, this is not an observable.
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So it can't be what we're looking for. We're looking for something, the momentum in the radical direction, which is which is observable.
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All right. Well, there's a fix to this problem. There's a general fix, and we're going to use it.
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But if we do a half of AB plus B.A., this is a mission.
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Right. Because if you take the dagger of this. This one, we just proved the dagger of this one is that.
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And the dagger of that one is this. So this thing, the dagger of this bracket is itself.
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So when you've got two non-commissioned operators sorry, you've got two mission operators that don't commute and you want to make the product.
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The way to go is to is to take the average of them.
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You know, it's really naive thing to do. So let's do that. So let's let's so so we try.
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Let's have a look at the emission operator PR,
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which we're going to define to be a half of our dot p overall where it's
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important that that all this R here is in front of the PR plus P dot R overall.
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So this thing here will be emission and I'm going to show that it is what we require.
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00:09:17,220 --> 00:09:22,620
So in the position representation so we we you can do this calculation in the abstract,
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not in the position representation, but it's easiest in position representation. So that's how we'll do it.
70
00:09:31,120 --> 00:09:36,250
So PR is equal to. So this P gets replaced by minus H by a grad, right?
71
00:09:36,820 --> 00:09:41,590
So this is going to be minus I h bar over to common factor.
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We're going to have our dot grad overall.
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But this of course is the scalar plus the divergence of our overall well.
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Now the issue is this. When what?
75
00:10:03,870 --> 00:10:09,180
This isn't quite the divergence of this. What this means is, is this is remember an operator.
76
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It's waiting for a wave function to come and stand in front, get operated on.
77
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Right. So this differential operator operates when everything to its right.
78
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It operates on this and it operates on these two. This here, this differential operator operates everything to its right, which is only the ABC.
79
00:10:31,480 --> 00:10:38,549
So we have to when we, when we expand this out,
80
00:10:38,550 --> 00:10:43,590
we're going to get three terms because we're going to get this thing operating on this that that is standing idly by,
81
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this thing operating on this with this and this standing idly by and this thing operating on this with these two standing idly by,
82
00:10:49,860 --> 00:11:02,049
which is the same as that. So this is going to be a minus I HPR bar dot grad overall, right?
83
00:11:02,050 --> 00:11:07,510
So I'm taking this one that I've got and the one that I'm promised at the end of all this reduction here.
84
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So that's where the two went to. So that's, that's those two.
85
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And now I've got these two bits minus my brackets.
86
00:11:17,170 --> 00:11:21,010
We're going to have over two. Sorry, that's this factor here.
87
00:11:21,460 --> 00:11:28,270
Then I'm going to have this thing operating on this, the divergence of RS three, and then I've got this thing operating on that.
88
00:11:28,510 --> 00:11:32,020
It's going to be all dotted into the gradient of one overall.
89
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So that's going to be minus oh dotted into the gradient of one overall.
90
00:11:38,680 --> 00:11:46,270
The gradient of one overall has to be a well, it is the vector R overall squared.
91
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This minus sign comes from the differentiating of the one overall. Right. Because I'm reminding you of prelims maths.
92
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Now the gradient of R itself is the is the vector are divided by all.
93
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This is dimensionless animal because that has dimensions of one of a length that has dimensions of length.
94
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So it's the vector is the unit vector R And that's what we've been using.
95
00:12:11,410 --> 00:12:14,980
Looks at this. Excuse me. So I made a mistake. This should be r cubed.
96
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Maybe we need to do this. Sorry. So what? What?
97
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Just to fill in here.
98
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So the gradient of one overall is equal by the ordinary rules of differentiation, minus one overall squared times the gradient of R,
99
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but the gradient of R I've just said is the vector are divided by R, hence the R cubed.
100
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These two dot together make an R squared which cancel most of those.
101
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So, so this minus sign, these two can be combined to a two overall.
102
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The twos go away and guess what we end up with is minus h bar grade overall plus one overall.
103
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Okay. So that's what this this stuff reduces to what we next want to know is so what is R dot gradient.
104
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Well, I want to know what this is in spherical polar coordinates.
105
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Well, the thing to do is just to write down let's write down r d by the R, it's easy to see that that is going to be x deep.
106
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Well, let's, let's do it o then we're going to have by the chain rule the X by the r d by the x plus d.
107
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Y. By the r. Debate Y plus d z by the R debate, you said that's just the chain rule.
108
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But but what is the x by the r x is equal to our sine theta cos phi so d x by the r is equal to
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x overall and for the same reason do y by the r is equal to Y over r and so on and so forth.
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So this is equal to x d by the x plus y, DPD, y plus z, dvt z because this is x overall,
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but this all cancels the R on the bottom y overall all cancels what's on the bottom and what's this?
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This is a vector product of x, y, common z with nebula DVD, X, Y, or Z.
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In other words, this is this is the animal we're interested in our dog.
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Great. So I have now that PR is equal to minus I h bar r dot grad we've just agreed is deep idr.
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So this is go away and we have D by the R plus one overall.
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So we have an interesting result.
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We have the momentum associated with the radius is not simply d by d radius like the momentum associated with x is D Barry X.
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There's also this additional term in here. But just to convince you this really is the momentum associated with RADIUS.
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Let's for fun calculate r comma pr.
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So what is that? That is minus h bar of r deep 80 plus one which is r times pr minus D by the R of r minus one.
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Yeah, right. Yes.
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Sorry. I mean, we're getting zero on time. That's the trouble. I want to get minus.
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I want to get bar out of this. What the [INAUDIBLE] did I do wrong?
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DVD all PR minus DVD are plus one overall on working on.
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Ah. Yeah. Yes.
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Yes. Yes. Yes. Sorry. Sorry, sorry. Yes, yes. Yes. Yes. Yes. It's perfectly correct.
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Okay, so the ones go away. This is this is this this sort of thing is confusing right now, as I say.
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What does this mean? This means DVD are of everything to it's right.
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And there's a phantom wave function here waiting to be operated on. So this is the derivative of ah upside.
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When we take the derivative of the R, we get one times of psi and then the R stands idly by and we do the gradient of upside.
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The second term cancels on this because our times, the gradient of upside is occurring here with a plus sign,
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and there it will be occurring with a minus sign. So what we're left with is the D by the R times of PSI, the R, which makes one.
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So this is equal to plus because it's a minus sign coming here h bar so that it's these two operators satisfy the canonical commutation relations.
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Right. Canonical commutation relation.
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So so PR really is the momentum associated with.
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Ah. Okay. So what, what are we really trying to do here.
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We're trying to show that that one overall square or square video is essentially PR squared.
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So let's calculate PR squared. PR squared is going to be minus H bar squared because they'll be two, they'll be two minus bars and then it's D by t,
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r plus one overall bracket D by the R plus one overall, which is equal to minus H bar squared.
140
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Obviously this on this is D two by the R squared we will get this differential
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operator will differentiate that and would use a minus one over squared.
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We will otherwise get a one overall D by the R and also a one overall D by the R.
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So we'll get to of one overall D by the R.
144
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And sorry and I haven't finished. And I also get this thing on this thing is a plus one overall squared.
145
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So these two terms, these two terms cancel and we're left staring at this.
146
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Uh, I should have had two of these terms.
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I think I said I was going to get two terms because I have a one over audio and I have a one over DVD after this.
148
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Operator when this operator works on this, it reduces that, but also it works on the phantom wave function sitting over here without standing idly by.
149
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So we get two of these. I think I said that, but I didn't write it.
150
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I'm not sure. So we have a minus h bar squared D two by the R squared plus two overall D by the R,
151
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which can also be written as minus h bar squared over all squared d, buddy r of r squared d, buddy r.
152
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Right. Because if you differentiate out this product, you get R squared on R squared times D two by the R squared,
153
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which is this term and you also get a2r overall squared two overall times D video.
154
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So yeah, this, this term here, we've now shown the Del's,
155
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we've now shown that HK the momentum operator which is minus H squared over two M Del squared is also minus h bar squared over two M of sorry.
156
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Yeah. Of.
157
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Well, let's leave that outside. Let's take the ball square into the bracket.
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We're going to have a one over all square debris, all that stuff, which we've just shown is is PR squared.
159
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And then oops. But there was a minus sign.
160
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So that soaks up this minus sign, a PR squared. And then similarly there's plus H bar squared, L squared, overall squared.
161
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This is a very important formula that we will need when doing hydrogen and therefore fundamental to.
162
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So it's expressing your kinetic energy in terms of your radial kinetic energy and your tangential kinetic energy.
163
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And that's one of the reasons why the total orbital increment operator is important,
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because it encodes your sort of energy of going around and around.
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So with that we are now finished with we can massively finish with orbital and lumentum and we can get on to spin.
166
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And this is somewhat more interesting in the sense that it's quantum mechanics has more remarkable things to say and
167
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it's less tedious because all that stuff with as part of the French operator's DVD theatre and stuff is not much fun.
168
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It has to be set right. So we have identified two types of of generations of rotations, the angle, the total and element of operators.
169
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And they generate we introduce them in order to generate complete road to complete rotations.
170
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So you have alpha being e to the minus I alpha not j rotates system as on turntable.
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So it moves. It moves your system around the origin.
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If you put you you put your system.
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It's as if you put your system on a turntable centred at the origin, the axis of the origin, and you turn the turntable around.
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Your system moves through space and it rotates simultaneously.
175
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Whatever internal structure it has. But we also have shown that ally the orbital angle momentum.
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Moves system on circles.
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So it moves it around physically. It translates it around the origin, but it is not rotated at the same time.
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It leaves its orientation fixed. And we have some.
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Well, okay. So we found the commutation relations here.
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We found that we found that j i comma j j is equal to i some have a k excellent i j k j k.
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And we found that it was also true that l i l j was equal to i sum David k
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epsilon i j k l k They had the same commutation relations amongst themselves,
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these operators, which is why we could use the work we did demonstrating what the eigenvalues of these could be.
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Also down here, this this implied that and that that that j squared has a values.
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J j plus one for j is nothing.
186
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A half one three halves, etc.
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And. From these commutation relations, we infer those are possible values for the eigen values of these operators.
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But we also had the principle that if we wrote if we translated something completely around the origin,
189
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we proved that that that was the identity transformation.
190
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So we concluded that l l plus one had to be l equals l equals nought one to integers only allowed.
191
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In this case, what we now going to do is introduce s i is by definition j i minus l i.
192
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It's the difference between these two. What does that mean physically?
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It means that s II is going to be the generator of rotations of a thing about its own axis.
194
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So we're not going to be this rotated on a turntable.
195
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So it rotates it and moves it. This simply moves it around a circle.
196
00:25:56,380 --> 00:26:00,040
So this is going to only rotate it on its own axis.
197
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It's not going to move it. It's only going to rotate it. That's what we expect to happen.
198
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But we'll have to be guided to some extent by the mathematics of all these what is going to happen.
199
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Right. So what about.
200
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So what we want in having these new fangled operators, it's important to figure out what the computation relations are going to be.
201
00:26:34,040 --> 00:26:49,250
Now, see, comma SJ is going to be j i minus l i comma j i minus sorry j j minus lj.
202
00:26:50,070 --> 00:26:55,220
All right, we're going to get this commuting with this.
203
00:26:56,000 --> 00:27:01,370
So we're going to get I some David K Excellent.
204
00:27:01,400 --> 00:27:16,540
I j k this commuting with this will produce a j k this commuting with this will produce an elk this commuting with this.
205
00:27:17,150 --> 00:27:21,260
Now, we didn't write that down, but I commuting with elk.
206
00:27:22,760 --> 00:27:31,970
This is a vector operator and therefore this thing when you when commutes with this always produces the the missing component of this vector.
207
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So this is going to be minus elk.
208
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And similarly, this thing on this thing is going to produce swap them over and you're going to.
209
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Well, there are several signs here that we could found down,
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but this thing is we're looking fundamentally at the same thing as the commentator of this on this we're looking at.
211
00:28:02,840 --> 00:28:14,180
Looking at minus l i comma jj is equal to is equal to obviously jj l i is equal to
212
00:28:14,210 --> 00:28:26,450
i epsilon j i k l k is equal to minus i epsilon summed over k This is a sum of a
213
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k of epsilon i j k l k So all right j i j i k And but I would like to have this in
214
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the order k So I swap those two over and introduced a minus sign to compensate.
215
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And then at the end and then you can see this thing, including that minus sign is the same thing as this thing, including that minus sign.
216
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So we have a minus another L Okay. So this is the justification for that last term there.
217
00:29:04,610 --> 00:29:09,230
So what do we end up with at the end of the day? These three L K's collapsed into just one.
218
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Okay, it's going to be j k minus L. In other words, this is going to be I someday have a k epsilon i j k as k.
219
00:29:20,150 --> 00:29:25,520
So these spin operators, so they say that they're going to be this we call them the spin operators.
220
00:29:25,760 --> 00:29:31,430
They have exactly the same commutation relations as the J. Therefore, we know what their eigenvalues are.
221
00:29:32,900 --> 00:29:36,320
So this implies that the eigenvalues.
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00:29:38,780 --> 00:29:55,070
Of X squared, which is of course s which is x squared plus x y squared plus z squared, r s s plus one where s is equal to a half.
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00:29:55,730 --> 00:29:59,450
Sorry, nothing. A half one three halves.
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00:29:59,720 --> 00:30:06,860
Blah blah blah. Okay. Because these these results follow ideally from the fact of having the commutation relations.
225
00:30:06,860 --> 00:30:15,470
Asi como se j is I excellent? I ask all the half integer values allowed.
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00:30:15,740 --> 00:30:19,250
Answer you will have a half integer.
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00:30:24,520 --> 00:30:29,260
When Jay does. Because.
228
00:30:32,140 --> 00:30:36,010
L does not. All right.
229
00:30:36,040 --> 00:30:51,819
Why is that? That's because Se Z is equal to Jay Z minus L.Z. and Jay Z comma S.Z. equals nothing, which is also the same as L.Z.,
230
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comma, Jay Z, etc. All these three operators commute with each other, so there's a complete set of mutual aid and states.
231
00:31:11,680 --> 00:31:17,190
And if so, we can now see that if this is half integer, this is using half integer eigenvalues.
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00:31:17,200 --> 00:31:24,190
So the eigenvalues of this are going to be a going to be the difference between the eigenvalues of this and the eigenvalues of this.
233
00:31:24,430 --> 00:31:31,540
So if this has half integer eigenvalues, therefore this will have to have half integer eigenvalues because this one has integer eigenvalues.
234
00:31:36,270 --> 00:31:41,429
So. So if J has half integer eigenvalues then does correspondingly.
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00:31:41,430 --> 00:31:45,120
If J doesn't s doesn't. It just takes along behind J.
236
00:31:46,290 --> 00:31:49,290
And indeed, that's how we tend to think about it.
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00:31:49,290 --> 00:31:54,960
We tend to think that the integer amounts of angular momentum come from orbital motion,
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L.Z. and the half integer values, if present, come from S said.
239
00:32:00,510 --> 00:32:05,520
And that's why J has half integer values. That's how we tend to think about it.
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00:32:12,220 --> 00:32:27,670
Okay. Now.
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00:32:27,940 --> 00:32:35,470
I think I've claimed a few times that spin is something to do with spin is something to do with the orientation of our system.
242
00:32:35,590 --> 00:32:40,960
And now it's time to to make good this claim that.
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00:32:44,340 --> 00:32:51,090
That the eigenvalues of the spin operator or your response to the spin operators encodes how a particle is oriented.
244
00:32:51,960 --> 00:32:55,110
And this is a strange area of very quantum mechanical area.
245
00:32:57,310 --> 00:33:14,580
Okay. So in general, the internal configuration of a system could be written that we could write if PSI is equal to the sum of s m upside.
246
00:33:16,680 --> 00:33:26,310
So we got a complete set of mutual we've got a complete set of mutual aid and states of S squared and s z.
247
00:33:26,380 --> 00:33:37,740
Right? So this is the so we're saying that s squared on s m is equal to S s plus one whoops one of s m and we're
248
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saying that s said on s m is equal to m s m and there should be a complete set of eigen states of this,
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of these mutual sets, of these operators.
250
00:33:51,150 --> 00:33:59,130
So I should be able to expand any, any, any state as a linear combination of these of these states.
251
00:33:59,820 --> 00:34:02,070
And what are we going to have to sum over?
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00:34:02,490 --> 00:34:11,850
We're going to have to sum of S is equal to nought half and we're going to have to some of em is in modulus less than or equal to s.
253
00:34:14,590 --> 00:34:17,080
This. This should be a generally valid statement,
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but for the wave functions or the states of view of real occurring systems in the laboratory, the good news is you don't.
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You only have to. There's only one value of s will occur in this sum.
256
00:34:32,110 --> 00:34:43,150
So. So this is this this would be generally the case. But, uh, for the systems in the lab of microscopic systems, this wouldn't be true.
257
00:34:43,390 --> 00:34:52,090
So if we were to embed a cricket ball, we would need to do a sum over all of these things up to ten to the 40 or something.
258
00:34:54,130 --> 00:35:00,220
But if we're dealing with an electron or an atom or whatever, we don't have to do some of all these things.
259
00:35:00,220 --> 00:35:03,580
We only only one value of s has non vanishing amplitude.
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00:35:03,580 --> 00:35:07,870
So this is going to vanish. For real systems. Except when s takes one particular value.
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00:35:08,410 --> 00:35:12,490
So for real. From microscopic. Microscopic.
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Only one value of s occurs. Normal vanishing.
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As any sign of this. We can say at the outset that the amplitude to find a value of s other than a value that's peculiar to the system is zero.
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So we just don't need to consider it. So we're able to so we're able to write that up.
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PSI is equal to the sum from m equals minus s2s of s and.
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CI said. That means.
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What does that mean? We're encoding? Well, the state of the system is described by these numbers here.
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There will be two plus one complex numbers.
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These these numbers here. And they're telling us they're encoding somehow the way the system is oriented.
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So this is two plus one. Complex numbers, encoding, encoding orientation.
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And what we want to do now is get some feel for how this encoding works.
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In simple cases, how would you encode the orientation of a macroscopic body?
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Well, the traditional procedure is you use only two angles. You write down three angles which describe how you get some access in the body,
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you know which way it points with respect to the z-axis of some fixed fixed coordinates and then how it rotates around that.
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So you usually encode the orientation of a body in three Euler angles for a classical system,
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for quantum mechanical system, you encode it in a certain number of complex amplitudes.
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The amplitudes defined find it in various orientations. It's carrying the same information.
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Right, in a very funny way. Okay, so let's let's consider the various cases.
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The case equals nought. In other words, if the only occurring thing here is nought, there's basically no information.
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There's only one, one amplitude.
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Nought. Nought. State of your system.
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And also, if you if you if you you rotate your system, you try and rotate system.
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00:38:13,660 --> 00:38:26,010
With you, you alpha, which is of the minus I alpha dot s this is the so s is the generator of rotations of the system around its own axis.
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Right? So I've written down the unitary rotation that makes me a new system which is rotated around the Axis Alpha by the magnitude of alpha.
285
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Then if I use this you alpha on upside, what do I get?
286
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I get using that expansion. I get nothing.
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00:38:45,700 --> 00:38:50,290
Nothing psi times you alpha or nothing.
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00:38:51,130 --> 00:39:07,240
Nothing. Well let me replace this by e to the minus I alpha dot s when when this operator sees this cat, it thinks you know nothing doing zero, right?
289
00:39:07,480 --> 00:39:12,460
You simply get as all of these think, that that becomes nothing.
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So we have easier than nothing, which is one. So this return, this goes back to itself.
291
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Which is the same thing as upside.
292
00:39:23,760 --> 00:39:30,060
In other words, when Earth is one, you can't tell the difference between the system before you rotated and after you rotated it.
293
00:39:30,270 --> 00:39:35,070
It's like an absolutely immaculate and perfect sphere. If you rotate it, it stays the same.
294
00:39:39,910 --> 00:39:44,080
So a particle that that does this that has x equals nought.
295
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So no spin implies same after rotation.
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00:39:58,960 --> 00:40:03,250
So if you like, for a classical animal, we would say this is strictly symmetric.
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And there's only one amplitude we need to bother with.
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So we say it's a scalar. So the simplest particles are spin zero particles.
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They there's just one amplitude, which is you say to say what the energy is or to say what the location is,
300
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because there's no issue of how is it oriented. It's a silly question.
301
00:40:41,710 --> 00:40:45,310
How is it oriented? It's it doesn't have an orientation. You can't tell it.
302
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If it doesn't know it doesn't have an orientation. That's the best thing to say. Nothing changes if you try and reorient it.
303
00:40:52,990 --> 00:40:56,380
And because of the one amplitude business, you say it's a scalar particle.
304
00:40:59,100 --> 00:41:02,639
Right. So that first equals. Next in the hierarchy of.
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00:41:02,640 --> 00:41:08,430
And for an example, a pile is an elementary particle which is a scalar particle.
306
00:41:08,640 --> 00:41:15,110
Right. Let's do X equals a half. So unfortunately, there are not many particles.
307
00:41:15,120 --> 00:41:18,270
There's not much in physics that's a scalar particle or scalar field.
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00:41:19,020 --> 00:41:25,250
This is a rare case. The the really important cases are X equals one and as equals a half and this equals one.
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00:41:25,290 --> 00:41:30,120
We're all built out of X equals or half particles. Right. So this is electrons.
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00:41:32,450 --> 00:41:36,100
Protons. Quarks.
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00:41:38,270 --> 00:41:46,760
Therefore neutrons. Wide range of things as he made out of an odd number of quarks will be in as equals a half particle.
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00:41:46,780 --> 00:41:55,510
What will prove that later on? Okay. So this is a really big class we have that the state of the system can now be written as.
313
00:41:58,990 --> 00:42:04,629
There's going to be a half a half of CI plus a half.
314
00:42:04,630 --> 00:42:09,340
Minus a half. I'm sorry. I need two times a half.
315
00:42:10,090 --> 00:42:17,680
Half plus upside times the half minus a half.
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00:42:20,530 --> 00:42:30,460
So any stage of the system of this particle can be written as a linear combination of this state, which is to say that M is,
317
00:42:30,670 --> 00:42:36,610
you know, you're guaranteed to get the answer half if you measure a said and a linear and a linear combination of this.
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So we now have a non-trivial linear combination with two possibilities. This is a very cumbersome notation.
319
00:42:41,680 --> 00:42:53,409
So people don't use it. They either write that this is plus cy plus plus minus, abassi minus.
320
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That's a handy notation, right?
321
00:42:57,130 --> 00:43:02,110
Because no, there's really no point in writing down this half because we know the first half we know for certain will always be there.
322
00:43:03,250 --> 00:43:09,550
And instead of writing a half, we write plus shorthand for plus a half and minus shorthand for minus a half.
323
00:43:10,000 --> 00:43:16,390
Or since this is only a boring, complex number, we often write A plus plus, B minus.
324
00:43:17,860 --> 00:43:25,570
So any state is a linear combination of these two basis states where we're guaranteed to get plus a half or so,
325
00:43:25,610 --> 00:43:29,739
minus a half rest, and there are amplitudes. So this is the this is the amplitude.
326
00:43:29,740 --> 00:43:35,410
It's Mod Square gives you the probability. If you would measure the Z components of spin of this particle, you would get a half.
327
00:43:35,770 --> 00:43:38,889
And B is obviously the amplitude. What square of that would be?
328
00:43:38,890 --> 00:43:41,980
The probability that you got said it was minus a half.
329
00:43:59,210 --> 00:44:06,440
So let's do some stuff with. They would spend a half particles.
330
00:44:14,340 --> 00:44:17,400
So if I take an arbitrary spin operator.
331
00:44:18,720 --> 00:44:26,040
Well, so let that be the this is the the the spin component.
332
00:44:29,650 --> 00:44:38,809
Along the unit vector. And what is it?
333
00:44:38,810 --> 00:44:49,190
Mathematically, it's an dot sw. In other words, it's an x as x plus and y as y plus NZ said.
334
00:44:51,500 --> 00:44:56,570
Wouldn't this that I've just written down would apply for any spin, not just spin a half.
335
00:45:00,280 --> 00:45:09,459
And I would like to to do calculations like if I do an on ABC, what do I get?
336
00:45:09,460 --> 00:45:16,960
I get some new state fi say, all right, this is an operator, I usual state, I get a new state.
337
00:45:17,110 --> 00:45:21,090
I want to be able to to do calculations like this.
338
00:45:21,100 --> 00:45:24,040
We will we'll find this is crucial.
339
00:45:25,330 --> 00:45:31,030
Now, we know that this everything every one of these states can be written as a linear combination of plus and minus.
340
00:45:31,330 --> 00:45:34,569
So this can be written a C plus plus.
341
00:45:34,570 --> 00:45:41,690
D minus. His essay in.
342
00:45:42,350 --> 00:45:47,930
And this can be written as a linear combination of A-plus plus B minus.
343
00:45:52,030 --> 00:45:59,829
And the name of the game is given the numbers A and B, which characterise that to calculate the numbers C and D,
344
00:45:59,830 --> 00:46:04,810
which characterise that we need an apparatus that does that for us and that's easily obtained.
345
00:46:05,050 --> 00:46:09,460
What we do to find C of course is we draw through with plus.
346
00:46:12,550 --> 00:46:31,930
And then on the left we get C plus nothing. So we find that C is equal to plus s n plus times, a plus plus and minus times B and growing through.
347
00:46:31,930 --> 00:46:35,980
By minus we get an equation which tells us the value of D,
348
00:46:36,700 --> 00:46:50,109
which is equal to minus S and plus of a plus minus s, n, minus B, and there's a handy a way of writing this.
349
00:46:50,110 --> 00:47:07,690
We write this is C, D is equal to a matrix, but this is just a boring, complex number plus s and plus that's a boring complex number minus s n plus.
350
00:47:09,370 --> 00:47:16,240
Plus S, N minus, minus S and minus.
351
00:47:17,050 --> 00:47:23,440
And that's operating on the column vector a, b. So here we have a concrete apparatus.
352
00:47:23,830 --> 00:47:29,830
This is a matrix, a two by two matrix of complex numbers, working on the given complex numbers that characterise ASI,
353
00:47:30,160 --> 00:47:39,810
which gives us the two complex numbers that characterise FY. So for example, suppose we take.
354
00:47:41,780 --> 00:47:50,900
Just got time to do this. Suppose we take an is equal to is equal to nothing come and nothing come of one.
355
00:47:51,800 --> 00:47:54,860
Then this becomes s z. This becomes s.
356
00:47:54,860 --> 00:48:03,680
I mean, all these s ends become assets. And then said on plus is a half of plus.
357
00:48:03,980 --> 00:48:11,750
So this number evaluates to a half. Then see the is equal to a half.
358
00:48:13,410 --> 00:48:18,569
As I said, on minus is minus a half times minus, but a minus is orthogonal to this.
359
00:48:18,570 --> 00:48:24,150
So we get a nothing. Nothing. And as I said on this is minus, minus.
360
00:48:24,570 --> 00:48:28,620
So this one minus half of minus. So this thing evaluates to minus a half.
361
00:48:29,520 --> 00:48:43,379
So in this particular case, it evaluates to this which people write as a half of Sigma Z, A, B, a column B by Sigma.
362
00:48:43,380 --> 00:48:49,810
That is the Pauli matrix. One minus one.
363
00:48:49,820 --> 00:48:52,940
Nothing. Nothing. Apparently matrix.
364
00:49:00,360 --> 00:49:11,550
So these matrixes we really want are these matrices, but it's handy to take the half outside of the matrix and write them in terms of these.
365
00:49:11,820 --> 00:49:17,340
Well, this is and this is the first of them. And if we had time, which we haven't now, we'll have to do it tomorrow.
366
00:49:17,490 --> 00:49:24,450
We will derive what Sigma X's and Sigma. Why are the three apparently matrices that enable us then to write the matrix
367
00:49:24,450 --> 00:49:29,760
belonging to any one of these spin operators and then do calculations on these?
368
00:49:30,000 --> 00:49:31,470
So it's it's time to stop.
