1
00:00:01,890 --> 00:00:16,740
Know. So the big idea introduced yesterday was that of a quantum amplitude,
2
00:00:16,740 --> 00:00:23,010
a complex number whose mode square gives you the probability for the outcome of some experiment, some measurement.
3
00:00:24,030 --> 00:00:35,099
And we have used the concept of a complete set of amplitude quantum amplitudes, so that if you knew these many all the amplitudes in a complete set,
4
00:00:35,100 --> 00:00:43,890
then you could calculate the amplitudes for any experiment that you might you might conceive any measurement that you might make.
5
00:00:44,820 --> 00:00:49,170
And I made the point that quantum mechanics is all about going from the amplitudes in some
6
00:00:49,170 --> 00:00:54,240
complete set to calculating these other amplitudes for the outcomes of other experiments.
7
00:00:56,780 --> 00:01:07,790
So this is there is a there is a very powerful analogy here between so a knowledge of the state,
8
00:01:07,790 --> 00:01:17,240
a dynamical state of our system is encapsulated in the values taken by these complete sets of amplitudes, some series, some set of complex numbers.
9
00:01:18,770 --> 00:01:26,630
And there's a very good analogy here between the way that we identify points in space and the coordinates of vectors.
10
00:01:27,620 --> 00:01:33,620
So we can use many different coordinate systems, many numbers to identify one in the same point in space.
11
00:01:33,620 --> 00:01:42,349
So the points in space are primitive notion in the sense the three numbers we use to identify them depend on well preference.
12
00:01:42,350 --> 00:01:45,559
And you might use their medical order systems.
13
00:01:45,560 --> 00:01:52,549
We might use many different Cartesian quarters systems, we might use public borders we have and the corners we use to identify a given
14
00:01:52,550 --> 00:01:57,620
point depend on the problem we're trying to solve may be most efficient use.
15
00:01:57,620 --> 00:02:02,160
Furthermore, this may be most efficient to use a particular Cartesian coordinate system, whatever.
16
00:02:02,600 --> 00:02:07,160
So we want and we find it very useful to have the concept of a position vector.
17
00:02:14,240 --> 00:02:21,860
Oh what we understand to be taken away from ANZ you said the three numbers.
18
00:02:22,460 --> 00:02:26,420
But it's more than a set of three numbers. It's really an equivalence class itself.
19
00:02:26,420 --> 00:02:31,000
It's freedom. It's because every different court and system, we have a different set of three numbers.
20
00:02:31,010 --> 00:02:40,010
On the same point we introduced the concept of a can of CI.
21
00:02:40,010 --> 00:02:52,100
So this symbol effectively characterises the symbol stands for the state of our system, the dynamic,
22
00:02:52,100 --> 00:02:59,090
the state of our system and you can think of it in symbolically stands for a one another.
23
00:02:59,210 --> 00:03:10,280
We have a new one, so it stands for a 1.334.1.1.
24
00:03:10,640 --> 00:03:16,430
Right. So we don't know how many quantum amplitudes we need in order to characterise our system.
25
00:03:16,700 --> 00:03:26,480
So it just goes on to. But the power of the notation is, is the power that we get from position vectors.
26
00:03:26,630 --> 00:03:32,030
Instead of writing all this, if we write all this stuff down, then we are committing ourselves to a particular coordinates,
27
00:03:32,030 --> 00:03:36,200
to a particular coordinate system, if you like, to a complete particular set of complete amplitudes.
28
00:03:36,890 --> 00:03:41,060
Whereas what we really want to do is focus on the dynamical state of our system.
29
00:03:41,300 --> 00:03:49,700
This is a dynamic side of our system. We we might even we might find it convenient to use the amplitudes to find the different possible energies.
30
00:03:49,850 --> 00:03:53,929
We might find it convenient to use instead the amplitudes with the different,
31
00:03:53,930 --> 00:03:58,250
different possible measurements of the momentum or the position or whatever.
32
00:03:58,340 --> 00:04:09,920
We leave that flexible by using by using, excuse me, that we have by using this symbol, said Kett.
33
00:04:10,850 --> 00:04:13,940
And of course, that is sorry. That is the back end of brackets.
34
00:04:13,940 --> 00:04:20,490
We will have browse in a moment. Okay.
35
00:04:20,670 --> 00:04:25,260
Now we know what it is. We can if we have got two cats.
36
00:04:27,450 --> 00:04:31,620
Supposing this stands for. These are not another dynamical state of the system.
37
00:04:31,860 --> 00:04:35,400
And let it be defined. Let it be in some particular system.
38
00:04:35,610 --> 00:04:45,570
Let it be these numbers. B1, B2, B3, etc. then because we know what it is to add amplitudes, indeed, we know we're under orders to add amplitudes.
39
00:04:45,780 --> 00:04:53,070
When something can happen by two different routes, it makes sense to define the object.
40
00:04:54,000 --> 00:05:06,690
We know what this object is. It is A1 plus B1 comma, a two plus B2, two comma and so on.
41
00:05:07,260 --> 00:05:15,030
So if you add two cats that that says the dynamical state of the system, which is described by the amplitude,
42
00:05:15,180 --> 00:05:18,900
the first amplitude being the sum of the amplitudes from the individual bits,
43
00:05:19,110 --> 00:05:23,850
the second amplitude being the sum of the amplitudes, the second amplitudes for the individual bits and so on.
44
00:05:23,880 --> 00:05:27,420
Right. So just as you add two vectors, if you add two vectors,
45
00:05:28,620 --> 00:05:35,310
you add the X components and you add the way components and you add the Z components to make a new set of three numbers.
46
00:05:35,520 --> 00:05:43,079
That's what we do with Cat. So we know to add cats now and we also know what it is to multiply cats.
47
00:05:43,080 --> 00:05:50,280
We can define a new cat by primed being, which we write like this Alpha PSI, which is just some complex number.
48
00:05:54,480 --> 00:06:01,920
We define this to be the cat alpha, a one comma, alpha a two comma, and so on.
49
00:06:02,130 --> 00:06:08,010
In other words, if you multiply a cat by some complex number alpha, what you mean is the dynamical state of the system that you would have,
50
00:06:08,490 --> 00:06:15,690
which has amplitudes, alpha times, the original alpha amplitude in every slot so we know how to add.
51
00:06:15,690 --> 00:06:19,050
These things are not to multiply these things by complex numbers.
52
00:06:19,350 --> 00:06:24,960
It follows that cats form a vector space.
53
00:06:28,970 --> 00:06:35,240
So you I guess you've been you've encountered this idea with in Professor Ashley's lectures, right?
54
00:06:35,240 --> 00:06:38,450
That the elements of a vector space for a mathematician,
55
00:06:38,450 --> 00:06:45,770
they are nothing but objects which you can add and objects you can multiply by numbers are the real numbers or complex numbers at your discretion.
56
00:06:46,580 --> 00:06:50,810
So that can form part of a vector space. We'll call this vector space Big V.
57
00:06:53,920 --> 00:06:57,100
You from those lectures, I hope know that what you get.
58
00:06:58,750 --> 00:07:01,870
No, let's let's let's let's get those lectures.
59
00:07:01,870 --> 00:07:06,910
I hope you've met the idea of a basis. A set of basis.
60
00:07:06,950 --> 00:07:11,110
Cats. What is a set of basis? Cats. It's set of objects.
61
00:07:11,640 --> 00:07:20,050
I well, you like this, which is such that any cat can be written as a linear combination.
62
00:07:25,170 --> 00:07:30,180
Whatever you need. It's a set of cat such that any cat does.
63
00:07:30,360 --> 00:07:35,100
For example, the dynamical state of our system can be written as a linear combination of these cats.
64
00:07:35,160 --> 00:07:40,720
Right. Then we have the idea of an adjoined space.
65
00:07:40,750 --> 00:07:53,110
I hope I'm just reminding you of stuff that you've already met. So if we consider the linea, we are going to be very interested in the linea complex.
66
00:07:53,110 --> 00:07:58,820
Valued. Complex valued.
67
00:08:01,530 --> 00:08:09,130
Functions on cats. Mathema a mathematician would say.
68
00:08:09,540 --> 00:08:25,200
On V functions on the elements of V. So as you might imagine, traditionally you would you would you would say, okay, f of SCI is a complex number.
69
00:08:28,130 --> 00:08:34,340
The complex number in question is going to be the amplitude. The reason why we care about these functions is because they're going to these complex
70
00:08:34,340 --> 00:08:39,020
numbers are going to be the all important amplitudes for something to happen,
71
00:08:39,230 --> 00:08:40,700
for something to be measured. Right.
72
00:08:41,450 --> 00:08:47,359
And that's you know, we completely focus the whole all this mathematical power is only there to help us to calculate these amplitudes,
73
00:08:47,360 --> 00:08:53,500
because if we can calculate amplitudes, we can take the mod square and we then have a prediction for what some experiment is going to.
74
00:08:54,920 --> 00:08:59,780
A probabilistic prediction for what some experiment is going to is going to yield.
75
00:09:00,710 --> 00:09:04,610
Okay. So so we're interested in these complex valued functions.
76
00:09:04,820 --> 00:09:08,000
I'm just I'm just saying that they're going to turn out to be the amplitudes.
77
00:09:08,000 --> 00:09:16,430
I'm not establishing that at this point. And the thing is, we don't actually use this notation, and the temptation we use is this.
78
00:09:17,000 --> 00:09:24,140
But these mean the same thing, a bracket opening, sort of angular bracket opening this way F of CI.
79
00:09:24,380 --> 00:09:30,980
This thing here means the function F evaluated on its side means that it is a complex number.
80
00:09:31,220 --> 00:09:34,130
It is going to be interpreted as an amplitude for something to happen.
81
00:09:34,970 --> 00:09:46,880
And this gives us the idea of saying that f which so this thing is a function, a linear, complex valued function is called the bra and the bra f.
82
00:09:50,210 --> 00:09:58,100
So we've got cats which define dynamical states of our system and we've got bras which are
83
00:09:58,100 --> 00:10:03,650
functions on the dynamical states of the system which extract the all important amplitudes,
84
00:10:04,460 --> 00:10:12,620
the cats form of vector space because it's a vector space, it must have bases like that up there.
85
00:10:13,790 --> 00:10:20,810
And the bras also form a vector space, as I hope you've discovered in in Professor Isserlis lectures.
86
00:10:23,540 --> 00:10:28,680
So the brass. Form the Adjoint space.
87
00:10:36,510 --> 00:10:41,220
Often called V primed. Why do they form a vector space?
88
00:10:41,250 --> 00:10:51,840
Because I know what it is to add to bras. If I give an if you give me a bra F and a bra G, I can form a new bra.
89
00:10:51,900 --> 00:10:55,800
Let's call it H for originality. Right.
90
00:10:56,070 --> 00:11:01,320
What do you want? In order to. In order to give meaning to this, I need to know what H does.
91
00:11:01,620 --> 00:11:04,800
What H does to any state website.
92
00:11:04,830 --> 00:11:09,660
I want to know. Function is defined by the value it takes on any on any possible argument.
93
00:11:09,990 --> 00:11:22,260
So I need to know what age of CI is, what number that is, and I define it to be efficaci plus skip CI,
94
00:11:23,220 --> 00:11:27,299
which of course is a perfectly well defined expression because this is a complex number.
95
00:11:27,300 --> 00:11:33,180
This is a complex number and we all know how to add complex numbers. So this is the definition of the function of the of the branch.
96
00:11:33,180 --> 00:11:40,830
So I know what it is to add two functions. And of course, I know what it is also to multiply a function by some constant thing.
97
00:11:41,040 --> 00:11:46,680
So I define the g primed meaning alpha g.
98
00:11:49,670 --> 00:12:01,229
By the rule. G primed of upside is alpha g of upside case of gain.
99
00:12:01,230 --> 00:12:06,480
This is perfectly well-defined because. That's just a complex number.
100
00:12:06,480 --> 00:12:12,510
And so this multiplication is well defined. So now I know what g primed what value it takes in every CI.
101
00:12:13,140 --> 00:12:19,410
So this is so this is the point that this is, this is the basic principle that establishes that the functions,
102
00:12:19,410 --> 00:12:24,210
the linear, complex valued functions on a vector space form, a vector space, the adjoined space.
103
00:12:24,540 --> 00:12:28,290
And we're going to be working extensively with both the cats and the pros.
104
00:12:30,820 --> 00:12:40,630
The only other thing that we need to remind ourselves is that the dimension of the adjoined space is equal to the dimension of the space itself.
105
00:12:42,480 --> 00:12:47,800
And so if we. And how do we how do we define this?
106
00:12:47,820 --> 00:13:01,500
We have a chorus where we prove so. So if we're given a basis of case I for each one of these, we define a a bra and we do it as follows.
107
00:13:01,500 --> 00:13:20,590
We say that. The bra j is the object is the function on the on the parts such that this complex number j i is equal to delta i j so in other words,
108
00:13:20,590 --> 00:13:30,970
it's nothing if if j the label j is not equal to the label I and it's one if the label label J is equal to the label, the label I write.
109
00:13:31,600 --> 00:13:36,160
So, so this do this this equation defines.
110
00:13:38,060 --> 00:13:54,470
J. The for all j the funk so that we're saying that that for example to the function to belonging to the second cat in our basis is defined.
111
00:13:54,800 --> 00:14:04,010
This is a function and it's defined such that two on two is one and two on anything else equals nought.
112
00:14:06,800 --> 00:14:12,709
So that is a perfectly good rule which defines the values that the function j takes in every element of the basis.
113
00:14:12,710 --> 00:14:15,320
And again from Professor S this lectures.
114
00:14:15,320 --> 00:14:22,190
I hope you are aware and can show that if you know what a function takes in every element of the basis, a linear function takes in every element.
115
00:14:22,190 --> 00:14:29,580
The basis you know what it takes in every cat whatsoever. So there's one final thing that we want to do in this abstract area.
116
00:14:29,850 --> 00:14:35,460
We want to say, supposing Abassi is equal to the sum I.
117
00:14:38,460 --> 00:14:48,890
Of of. So we take a state of our system and we have is a linear combination of the basis states then we define a function.
118
00:14:48,900 --> 00:14:54,180
This is the funny part, right? So so far I hope I think everything's been I hope everything's been fairly straightforward.
119
00:14:54,510 --> 00:14:57,299
But now I'm saying associated with the state of our system.
120
00:14:57,300 --> 00:15:08,190
I want to find a function on states and the function in question is defined by this rule that it's a complex conjugate.
121
00:15:11,930 --> 00:15:21,380
Times I. The bra i. So given that my state of my system is a certain linear combination of the basis states,
122
00:15:21,860 --> 00:15:29,689
I'm saying that the function associated with that state of the system is a certain linear combination of the functions,
123
00:15:29,690 --> 00:15:34,280
these functions which are associated with the basis states. Why do we do that?
124
00:15:35,450 --> 00:15:41,280
One reason we do that is in order that we can evaluate this important number of science.
125
00:15:41,360 --> 00:15:44,780
I. So let's have a look at that number.
126
00:15:45,080 --> 00:15:55,070
That is the sum. I write this out as a sum, a I star I sum of I.
127
00:15:55,160 --> 00:16:01,610
And then I have to write the this one out as a sum, a j of j.
128
00:16:02,750 --> 00:16:07,790
So I'm summing over J. These are just dummy labels, right? So I'm entitled to call one J and one I.
129
00:16:08,180 --> 00:16:13,040
So it's a sum over J is one to how many we need and i's one to have a many we need.
130
00:16:15,230 --> 00:16:18,490
This is a this is a linear function, right?
131
00:16:18,500 --> 00:16:21,560
We're evaluating this linear function on this dirty great sum.
132
00:16:22,040 --> 00:16:26,570
But because it's a linear function, the dirty great sum can be taken out side.
133
00:16:26,810 --> 00:16:35,090
So I can write this is the sum of I and now J being one two, whatever it is of a I star,
134
00:16:35,570 --> 00:16:50,550
a j of i j and there I've used the linearity of the function I and now I use the fact that this is by definition of this function delta right?
135
00:16:50,570 --> 00:16:53,959
J So it is nothing and less I equals.
136
00:16:53,960 --> 00:17:04,010
J So now let's do the sum over. J For example, as I do this summer over J I will get nothing here except for that particular j which is equal to Y,
137
00:17:04,010 --> 00:17:05,690
and then this will come become one.
138
00:17:06,080 --> 00:17:19,850
So this becomes the sum of a I star, a I in other words, it because the sum of I mild squared, which now that's just mathematics.
139
00:17:19,860 --> 00:17:30,770
Now we're back to physics. This is an amplitude to find this is this should be a this should be an amplitude a I a quantum amplitude.
140
00:17:31,490 --> 00:17:35,030
And we're taking a sum of the mode squares of the amplitudes.
141
00:17:35,300 --> 00:17:41,840
So this is the sum of the product, sorry of the probabilities. So that should be one because the probabilities should all add up to one.
142
00:17:44,670 --> 00:17:51,810
So my my states, I would like my states to have this normalisation condition.
143
00:17:52,880 --> 00:18:08,070
This is proper normalisation. Is that any of the state times its bra should come to one.
144
00:18:08,370 --> 00:18:11,700
Not any other complex number. That particular complex number one.
145
00:18:13,580 --> 00:18:19,700
Okay. So that's that's the basic principles of direct notation.
146
00:18:19,940 --> 00:18:21,800
Now, let's just talk about the energy.
147
00:18:21,830 --> 00:18:28,070
Let's let's have a look at this better understanding of what this physically means by having looking at energy representation.
148
00:18:30,730 --> 00:18:36,670
So supposing we in certain circumstances, for example, if you've got a particle that moves in one dimension,
149
00:18:37,540 --> 00:18:41,529
then it's then it's possible in some in some trapped in some.
150
00:18:41,530 --> 00:18:51,190
Well, then it is possible to to characterise the dynamical state of the system simply by giving the amplitude to measure.
151
00:18:52,480 --> 00:18:57,310
The possible values of the energy. So a complete set.
152
00:18:58,830 --> 00:19:07,170
So so this is this is not always the case. But for a one dimensional particle, a particle trapped.
153
00:19:09,710 --> 00:19:17,100
This is a very idealised situation, but never mind trapped. In a one dimensional potential.
154
00:19:17,100 --> 00:19:23,750
Well. We will see that.
155
00:19:23,760 --> 00:19:33,090
And I'm asserting for the moment that the a I form a complete set of amplitudes.
156
00:19:38,630 --> 00:19:48,850
Where? A mod squared is the probability of measuring the ice energy.
157
00:19:50,420 --> 00:19:58,610
The ice allowed energy, right? So the energy in this case, when we have our particle trapped inside a potential well, has a discrete spectrum.
158
00:19:58,830 --> 00:20:02,210
Remember, we introduced the idea of a spectrum. Those are the possible values of your measurement.
159
00:20:02,450 --> 00:20:06,320
You can only measure a discrete set of numbers. They're called EEI.
160
00:20:06,740 --> 00:20:09,559
There's a probability that if I would measure the energy,
161
00:20:09,560 --> 00:20:16,910
I would find the energy to be I that that's this mod square and a complete characterisation of the system.
162
00:20:16,910 --> 00:20:22,400
Complete dynamical information is provided by knowing not only these probabilities, but actually the amplitudes themselves.
163
00:20:23,630 --> 00:20:28,220
So you can think of of PSI as a vector formed by these amplitudes.
164
00:20:29,370 --> 00:20:32,729
Now, let's let's write that upside.
165
00:20:32,730 --> 00:20:40,710
The state of our system is equal. Let's let's be given some basis and let's write that it's equal to i.
166
00:20:41,730 --> 00:20:52,410
I summed over I. So out of these complex numbers, which we know and some basis, any basis we can,
167
00:20:52,710 --> 00:20:59,070
we can write a symbol like this that's just a repeat of what we've already done.
168
00:20:59,670 --> 00:21:04,229
And now let's ask ourselves, what are the meaning, what's the physical meaning of these states?
169
00:21:04,230 --> 00:21:11,670
These are this is expressing my actual state of the system as a linear combination of some states of the system that we've conjured out of nowhere.
170
00:21:13,550 --> 00:21:19,720
Right. But each one of these is, according to our formalism, corresponds to a complete set of amplitudes.
171
00:21:19,730 --> 00:21:24,770
It's it's a state of the system. Now, let's find out what these ones mean in this context.
172
00:21:25,550 --> 00:21:34,590
Suppose. We know.
173
00:21:37,200 --> 00:21:48,390
The energy is actually a three. So that implies that a three is one and a equals nought four, not equal to three.
174
00:21:48,840 --> 00:21:53,640
So supposing we happen to know that the energy is three. Then.
175
00:21:54,000 --> 00:22:01,210
Then the amplitudes must be like this. And what is that?
176
00:22:01,230 --> 00:22:08,400
What does that mean? That means Ixi. The state of our system is actually equal to three.
177
00:22:08,400 --> 00:22:11,520
Because on this. In this sum, there's only going to be.
178
00:22:12,980 --> 00:22:17,330
One non vanishing term, and that will be a three, namely one times three.
179
00:22:17,810 --> 00:22:25,520
So that tells us that this state three is actually the state of definitely being having energy three.
180
00:22:26,270 --> 00:22:32,000
And similarly for all the other ones. So a better notation or a clearer notation is.
181
00:22:37,820 --> 00:22:47,270
To write to rewrite that in a clearer notation is a cy is the sum I of a i times e i.
182
00:22:48,320 --> 00:22:56,780
This this makes it clear what we've just established that the thing is actually the quantum state of definitely being having energy.
183
00:22:57,540 --> 00:23:03,420
I. So we've discovered the physical meaning of those abstract basis vectors.
184
00:23:05,310 --> 00:23:09,270
When when these are the amplitudes to measure the different energies.
185
00:23:09,870 --> 00:23:13,620
And this is called the energy representation, right? This is the energy representation.
186
00:23:17,800 --> 00:23:23,950
This is when we express the state of our system as a linear combination of states of well-defined energy.
187
00:23:24,280 --> 00:23:30,220
This representation is and is playing an enormously important role in quantum mechanics,
188
00:23:30,790 --> 00:23:35,740
because it's how we it's by going to this representation for mathematical reasons.
189
00:23:35,950 --> 00:23:43,420
Going to this representation is how we solve the time evolution equation as we solve the quantum analogues of Newton's Laws of Motion.
190
00:23:44,260 --> 00:23:53,020
It's also as we will find a very, uh, a very abstract representation in the sense that and this may surprise you,
191
00:23:53,440 --> 00:23:59,860
no physical system ever has well-defined energy. So these quantum states are, in fact realisable in the real world.
192
00:24:00,220 --> 00:24:06,760
So this expresses a realisable state of affairs, this linear combination of states that you can never actually find anything in.
193
00:24:07,810 --> 00:24:11,920
But it's it's of enormous technical and mathematical importance.
194
00:24:15,200 --> 00:24:23,630
Let's talk now about something and we'll we'll we'll we'll we'll come back to the energy representation later on.
195
00:24:23,870 --> 00:24:28,070
But now let's move straight on to another illustration, which is back to spin a half.
196
00:24:29,870 --> 00:24:37,370
So I said that elementary particles are these tiny gyros that the the the rate at which they spin never changes,
197
00:24:37,580 --> 00:24:40,910
but the direction in which the spin is oriented does change.
198
00:24:41,420 --> 00:24:52,370
I made the point yesterday that the though you can know for certain the result of measuring the spin in one particular direction,
199
00:24:52,370 --> 00:25:03,139
for example, the component of the spin parallel to the z-axis, you cannot know the direction in which the thing is spinning,
200
00:25:03,140 --> 00:25:07,950
because even when you measure the component parallel to the z-axis with precision, you're,
201
00:25:07,990 --> 00:25:16,580
you're in deep ignorance about the about the value of the spin parallel to the x axis or the y axis.
202
00:25:16,580 --> 00:25:22,240
You only know it does have spin in those directions that you do not know the sign of this.
203
00:25:22,280 --> 00:25:32,899
You do not know how much spin is a long X or a long y, but a so so for s.
204
00:25:32,900 --> 00:25:39,200
So if we measured the spin along the Z axis and I'm going to say that this is now plus or minus a half a half.
205
00:25:39,950 --> 00:25:45,290
Now, yesterday I had an H bar here. In some sense I was using a slightly different notation, but I had an H bar there.
206
00:25:47,380 --> 00:25:52,000
I want to look at the angular momentum. H Bar has dimensions of angular momentum.
207
00:25:53,050 --> 00:25:58,720
So the angular momentum, what this means is that the if said is plus a half.
208
00:25:58,960 --> 00:26:03,490
That means the angular momentum in the Z direction is plus a half H bar,
209
00:26:04,510 --> 00:26:11,530
but it's turns out to be convenient to leave off the bar when talking about the so-called spin of said.
210
00:26:11,680 --> 00:26:16,420
Partly because you'll see that spin in quantum mechanics is.
211
00:26:17,960 --> 00:26:21,860
Really has a slightly dimensionless being.
212
00:26:21,860 --> 00:26:28,939
And partly because partly because writing we don't write any more, because we have to.
213
00:26:28,940 --> 00:26:34,910
It's just it's just economical. So that so physically there's the angular momentum is a half edge bar,
214
00:26:35,120 --> 00:26:42,110
but it's more convenient to write that as Z this abstract thing, the spin is plus a half or minus a half.
215
00:26:43,640 --> 00:26:48,350
So what do we have? We have two states. We have a we have a complete set of states.
216
00:26:53,640 --> 00:27:00,750
Followed by plus and minus. Okay, so this is the state in which I am certain.
217
00:27:00,990 --> 00:27:10,140
If I measure the spin parallel to the z-axis that I'm going to get the value a half, and this is the one where I'm certain to get minus the half.
218
00:27:11,340 --> 00:27:21,580
And the statement that's a complete set is to say that any state of my electron or whatever could be written as a plus plus.
219
00:27:22,290 --> 00:27:26,159
Actually, maybe it's better to write it this way. A minus minus plus.
220
00:27:26,160 --> 00:27:37,350
A plus plus. So since this is an easy case, there are only two components to our cat A minus and a plus.
221
00:27:37,730 --> 00:27:47,990
And just in just the same way that I might in ordinary in ordinary vectors write that oh is equal to
222
00:27:47,990 --> 00:28:03,800
is all the vector a let's say B perhaps it's better b is equal to b x e x plus b y e y plus b z e z.
223
00:28:05,660 --> 00:28:09,260
Don't need to bracket, do I know? Where?
224
00:28:09,260 --> 00:28:15,830
Here, I've got three real numbers B, B, Y and Z, which are the components of B in some particular coordinate system.
225
00:28:16,250 --> 00:28:22,700
So here I'm saying the state of our electron can be written as a linear combination of this basis vector and this basis vector.
226
00:28:22,730 --> 00:28:32,630
So these kind of map across here. But this is a simpler case insofar as it only got two components A minus and de plus rather than three components.
227
00:28:32,930 --> 00:28:37,040
So that's the analogy. Okay.
228
00:28:40,050 --> 00:28:41,910
Now we need to anticipate a formula.
229
00:28:42,630 --> 00:28:53,100
So what I what I claimed was earlier was that if you know what a minus and pluses are, what those amplitudes are, to find the spin in the Z direction,
230
00:28:53,100 --> 00:29:00,120
either up or down, then you can calculate the amplitude to find the spin in any other direction,
231
00:29:00,720 --> 00:29:04,590
either parallel to that direction or anti parallel to that direction. That's what I claimed.
232
00:29:05,010 --> 00:29:10,410
And I'm going to quote a result which which we will arrive at later.
233
00:29:10,620 --> 00:29:16,610
But we have to take it on trust for the moment. So the state if we if we have a unit vector n.
234
00:29:16,770 --> 00:29:21,040
So so let and. And it's a unit factor.
235
00:29:25,100 --> 00:29:28,280
And it's in the direction theatre and. Fine.
236
00:29:28,310 --> 00:29:35,990
Right. These are regular polar coordinates which are defining a direction by by pointing to a place on the unit sphere.
237
00:29:36,320 --> 00:29:39,080
And let n be the unit vector that points in that direction.
238
00:29:39,950 --> 00:29:51,800
Then I make the following assertion that the state of being plus along the vector n so can be.
239
00:29:52,580 --> 00:29:56,030
So this is a state of. This is a state of my electron.
240
00:29:56,270 --> 00:29:59,839
So if it's true that that's a complete set, it must be right.
241
00:29:59,840 --> 00:30:02,600
It is a linear combination of this state and this state. Right.
242
00:30:03,530 --> 00:30:09,649
And I I'm not going to say that that is sine I better just check that I'm getting this right.
243
00:30:09,650 --> 00:30:16,190
Yep. Science teacher upon to e to the i fi on to.
244
00:30:17,580 --> 00:30:26,320
Of minus. Plus costs are upon to each of the minus i.
245
00:30:27,990 --> 00:30:36,660
On to. Plus, now we will derive or at least you will drive in a problem this formula.
246
00:30:37,940 --> 00:30:41,960
We will show that it's why it's true. At the moment, we're just asserting that it is true.
247
00:30:42,650 --> 00:30:45,860
So this this is a complex number, right?
248
00:30:46,250 --> 00:30:54,890
And this is a minus. This is a complex number. And this is a plus for that particular for the for the for the quantum state, of having your spin,
249
00:30:55,700 --> 00:31:01,790
of being certain that if you measure the spin along this direction, you get the answer plus a half.
250
00:31:02,640 --> 00:31:13,280
Correspondingly, there is a minus object. Which turns out to be, cos these are over to each of the eye loops.
251
00:31:14,340 --> 00:31:26,190
Five over two minus minus sign feature over to E to the minus I fly over to plus.
252
00:31:27,580 --> 00:31:33,670
So it has it's made, of course, it's this is naturally another linear combination of this and this basis vectors.
253
00:31:34,150 --> 00:31:41,260
And now we just have different a because it's a different state, it has different a minus and different a plus.
254
00:31:43,870 --> 00:31:51,129
Now we in order to to to calculate something useful, we need to know what the bras are that belong to those.
255
00:31:51,130 --> 00:31:55,120
Right? So. So these are the kits.
256
00:31:55,120 --> 00:31:58,330
I will I will want to do something with the bras in a moment.
257
00:31:58,630 --> 00:32:03,640
So let's calculate what the bras are. So we have the bra in common.
258
00:32:03,640 --> 00:32:13,780
Plus, the rule is that we take the complex conjugate of of whatever comes in front of this, and then we change this into a bra.
259
00:32:13,870 --> 00:32:25,690
That was the rule we agreed on. So this is going to be sine theta over to e to the minus i fi over two of the bras
260
00:32:25,990 --> 00:32:36,130
minus plus cos these are over to each of the plus I high over two times the bra plus.
261
00:32:37,180 --> 00:32:40,360
So that's, that's the bra that belongs to that.
262
00:32:40,360 --> 00:32:45,880
And I want the bra that belongs to the other thing cos these are on to.
263
00:32:49,240 --> 00:32:54,310
The need to concentrate each of the minus five to.
264
00:33:06,790 --> 00:33:14,680
So there's a bit of practice in taking her mission, taking in adjoint, calculating the adjoin that belongs to a belongs to a vector, a cat.
265
00:33:16,030 --> 00:33:23,089
Now what we want to do. So let's calculate.
266
00:33:23,090 --> 00:33:33,100
Let's suppose. Let's suppose that we've just measured.
267
00:33:34,270 --> 00:33:41,860
The spin and we found the spin on the Z direction. And the result of that measurement was plus a half that in that case,
268
00:33:41,860 --> 00:33:48,820
what we will know is that the state of our electron is actually plus let's just suppose we've made the relevant measurement,
269
00:33:49,060 --> 00:34:03,700
and that's the bottom line. So what we want to find now is the amplitude that if I would measure the spin along n, I would find that it was plus on n.
270
00:34:05,740 --> 00:34:08,860
Now. I now realise that I have left out.
271
00:34:08,950 --> 00:34:12,700
Can we just cycle back to the energy representation?
272
00:34:12,700 --> 00:34:19,120
Why? I should have pointed something out. What I should have pointed out was.
273
00:34:22,280 --> 00:34:26,840
From this expression here. Well, perhaps it'd be better to be done.
274
00:34:29,080 --> 00:34:31,900
We better be done here. Let us point out at this point.
275
00:34:34,970 --> 00:34:48,080
A very simple fact that if I if I multiply this equation through by the bras e j so if I do e j times this equation,
276
00:34:48,080 --> 00:34:51,709
what that means is that I'm going to evaluate the function.
277
00:34:51,710 --> 00:34:58,640
E.J., E.J. on both sides of the equation then.
278
00:35:01,130 --> 00:35:09,800
Then what am I going to discover? I'm going to discover that, E.J. Upside is equal to AJ.
279
00:35:09,950 --> 00:35:15,980
Why is that? Because while Egypt sighs, obviously what appears on the left, what appears on the right is E.J. times all this stuff.
280
00:35:16,280 --> 00:35:20,930
But E.J. being a linear function. E.J. pops inside here and meets that.
281
00:35:21,350 --> 00:35:28,370
These are two basis factors. So they have Delta, E.J. for their E.J. on this EEI produces Delta AJ.
282
00:35:30,400 --> 00:35:37,120
So when I, when I do a so I get a Delta IJA when I do the sum of Ry, all it survives is age.
283
00:35:37,390 --> 00:35:42,280
Now, this is a fabulously I should have pointed this out. It's an obvious equation, but it's fabulously important.
284
00:35:42,280 --> 00:35:49,720
And it tells us really why we're interested in these animals here, because it means that given the state of my system,
285
00:35:49,960 --> 00:35:55,810
it enables me to recover the amplitude for measuring E.J. out of the state of the system.
286
00:35:55,810 --> 00:36:06,070
The rule is to get the amplitude for something. Take the state of your system and browse through by the bra associated with the result.
287
00:36:06,430 --> 00:36:09,850
The interesting result of your measurement. In this case.
288
00:36:09,880 --> 00:36:15,440
E J So the amplitude to find that the energies E.J. is just E.J.
289
00:36:16,510 --> 00:36:23,980
Brought into the state of our system. So when I come back to this problem here, I want to know the amplitude.
290
00:36:25,240 --> 00:36:29,110
To measure plus on end.
291
00:36:30,550 --> 00:36:36,190
So what I need to do is to calculate this by that principle.
292
00:36:42,090 --> 00:36:48,540
So what I do is I take that in plus thing this thing and I knock it into,
293
00:36:48,780 --> 00:36:56,580
I bring it into plus that will produce me a minus plus here which vanishes and a plus plus here, which is the number one.
294
00:36:56,850 --> 00:37:07,350
So I simply extract this. So this turns out to be costs feature over to each of the I find over two.
295
00:37:07,590 --> 00:37:14,069
So that's the amplitude to measure this this complex number is the amplitude to measure
296
00:37:14,070 --> 00:37:20,640
that the spin is along the vector n where C4 and phi the angles which define n,
297
00:37:21,060 --> 00:37:29,940
which means that the probability of measuring plus on n is simply cost squared phi over two.
298
00:37:30,930 --> 00:37:34,380
Does that make sense if sorry feature of two.
299
00:37:36,010 --> 00:37:47,549
Right. Because this this goes away when we take the mode square the does this make sense when theta is nought when theta is nought n coincides
300
00:37:47,550 --> 00:37:56,370
with the z axis and therefore the probability has to be one because we already know that it's certainly pointing down the z-axis.
301
00:37:57,090 --> 00:38:08,880
And guess what? It is one when theta is let's say that theatre is pi, which means that n is pointing in the direction of the minus it axis.
302
00:38:09,270 --> 00:38:15,299
We we should get the probability zero because that's the probability to find that it's pointing down the minus it axis,
303
00:38:15,300 --> 00:38:20,160
which is the same as the probability that we get minus along the plus it axis.
304
00:38:21,790 --> 00:38:28,980
And when seat is pi, lo and behold, we're looking at cost wed of cost of pi upon two squared, which is zero.
305
00:38:28,990 --> 00:38:42,130
So this does behave in a sensible way. Let's let's put three to equal to pi upon two and phi equal to nought.
306
00:38:42,160 --> 00:38:47,680
What does that imply? It implies that N is equal to x the unit vector in the x direction.
307
00:38:48,460 --> 00:38:58,120
So n becomes the x direction. What does that give me? That gives me that a that gives me that the probability for being plus on x.
308
00:38:59,780 --> 00:39:06,020
Given that I'm plus on Z is the probability the amplitude.
309
00:39:08,390 --> 00:39:16,790
Then I'm looking at I'm looking at costs of pi upon to upon to so cost pi upon four, which is one over two.
310
00:39:19,460 --> 00:39:23,420
And I have an E to the I not for each of the nothing. So that's just that, right?
311
00:39:24,700 --> 00:39:36,940
So guess what? If the spin, if we are guaranteed the answer plus a half for as I said, what's the probability of measuring plus a long X?
312
00:39:37,300 --> 00:39:41,200
The answer is a half because it's the square.
313
00:39:41,210 --> 00:39:52,360
The probability is the square of this. So p x plus is in this case equal to a half, which seems pretty reasonable because in some sense,
314
00:39:52,360 --> 00:39:58,840
knowing that the spin is along, it has a component plus a long Z doesn't really rather than minus along.
315
00:39:59,020 --> 00:40:01,389
It doesn't really help us to say anything about X.
316
00:40:01,390 --> 00:40:08,020
So we really have total uncertainty because it's the probability to be plus on x is a half the probability to minus one.
317
00:40:08,020 --> 00:40:19,450
X must also be a half. Let's let's put seats are equal to pi by two and phi equal to pi by two.
318
00:40:19,930 --> 00:40:24,850
That implies that n is equal to e y the unit vector in the y direction.
319
00:40:26,020 --> 00:40:31,120
What do we get then? Then we find that the amplitude y plus.
320
00:40:32,290 --> 00:40:38,090
Plus is. Still one upon two.
321
00:40:39,780 --> 00:40:44,520
But now we have e to the i. Pi on for.
322
00:40:52,170 --> 00:40:57,180
So the amplitude is now genuinely a complex number, whereas in the case it was a real number.
323
00:41:00,330 --> 00:41:08,309
But it means that the probability for getting plus on Y is still a half the same as it is on X,
324
00:41:08,310 --> 00:41:11,550
which again has to be the case by symmetry if you think about it.
325
00:41:19,270 --> 00:41:25,330
If we calculate the corresponding negative amplitudes, let's calculate x minus.
326
00:41:28,110 --> 00:41:32,010
The probability, let's find the amplitude that it's pointing minus on x.
327
00:41:33,420 --> 00:41:36,720
So then we have to take that n minus thing and bang it into plus.
328
00:41:37,080 --> 00:41:40,200
And what survives is the minus sign.
329
00:41:40,960 --> 00:41:46,620
Caesar over to. Well.
330
00:41:47,540 --> 00:41:54,190
Strictly speaking each of the stuff. But that's well each of the I fi over to.
331
00:41:55,610 --> 00:41:59,480
That's it. Actually, let's just make this an minus.
332
00:41:59,690 --> 00:42:04,299
All right. Then I know I can, from this formula, reduce the X and Y ones.
333
00:42:04,300 --> 00:42:08,780
That's what I want to do. I have the X minus plus.
334
00:42:10,180 --> 00:42:14,050
I have to put the feature upon two in here. That's going to be minus one over two.
335
00:42:14,620 --> 00:42:25,160
And I'm going to have that y minus. Y minus plus is going to be.
336
00:42:27,470 --> 00:42:35,330
So in that case, I'm going to be have a one of a root two. Here I have minus one over root two and then here I'll have an each of the IP upon four.