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Know. So the big idea introduced yesterday was that of a quantum amplitude,

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a complex number whose mode square gives you the probability for the outcome of some experiment, some measurement.

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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,

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then you could calculate the amplitudes for any experiment that you might you might conceive any measurement that you might make.

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And I made the point that quantum mechanics is all about going from the amplitudes in some

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complete set to calculating these other amplitudes for the outcomes of other experiments.

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So this is there is a there is a very powerful analogy here between so a knowledge of the state,

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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.

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And there's a very good analogy here between the way that we identify points in space and the coordinates of vectors.

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So we can use many different coordinate systems, many numbers to identify one in the same point in space.

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So the points in space are primitive notion in the sense the three numbers we use to identify them depend on well preference.

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And you might use their medical order systems.

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We might use many different Cartesian quarters systems, we might use public borders we have and the corners we use to identify a given

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point depend on the problem we're trying to solve may be most efficient use.

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Furthermore, this may be most efficient to use a particular Cartesian coordinate system, whatever.

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So we want and we find it very useful to have the concept of a position vector.

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Oh what we understand to be taken away from ANZ you said the three numbers.

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But it's more than a set of three numbers. It's really an equivalence class itself.

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It's freedom. It's because every different court and system, we have a different set of three numbers.

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On the same point we introduced the concept of a can of CI.

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So this symbol effectively characterises the symbol stands for the state of our system, the dynamic,

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the state of our system and you can think of it in symbolically stands for a one another.

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We have a new one, so it stands for a 1.334.1.1.

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Right. So we don't know how many quantum amplitudes we need in order to characterise our system.

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So it just goes on to. But the power of the notation is, is the power that we get from position vectors.

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Instead of writing all this, if we write all this stuff down, then we are committing ourselves to a particular coordinates,

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to a particular coordinate system, if you like, to a complete particular set of complete amplitudes.

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Whereas what we really want to do is focus on the dynamical state of our system.

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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.

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We might find it convenient to use instead the amplitudes with the different,

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different possible measurements of the momentum or the position or whatever.

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We leave that flexible by using by using, excuse me, that we have by using this symbol, said Kett.

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And of course, that is sorry. That is the back end of brackets.

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We will have browse in a moment. Okay.

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Now we know what it is. We can if we have got two cats.

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Supposing this stands for. These are not another dynamical state of the system.

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And let it be defined. Let it be in some particular system.

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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.

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When something can happen by two different routes, it makes sense to define the object.

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We know what this object is. It is A1 plus B1 comma, a two plus B2, two comma and so on.

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So if you add two cats that that says the dynamical state of the system, which is described by the amplitude,

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the first amplitude being the sum of the amplitudes from the individual bits,

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the second amplitude being the sum of the amplitudes, the second amplitudes for the individual bits and so on.

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Right. So just as you add two vectors, if you add two vectors,

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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.

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That's what we do with Cat. So we know to add cats now and we also know what it is to multiply cats.

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We can define a new cat by primed being, which we write like this Alpha PSI, which is just some complex number.

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We define this to be the cat alpha, a one comma, alpha a two comma, and so on.

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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,

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which has amplitudes, alpha times, the original alpha amplitude in every slot so we know how to add.

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These things are not to multiply these things by complex numbers.

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It follows that cats form a vector space.

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So you I guess you've been you've encountered this idea with in Professor Ashley's lectures, right?

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That the elements of a vector space for a mathematician,

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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.

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So that can form part of a vector space. We'll call this vector space Big V.

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You from those lectures, I hope know that what you get.

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No, let's let's let's let's get those lectures.

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I hope you've met the idea of a basis. A set of basis.

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Cats. What is a set of basis? Cats. It's set of objects.

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I well, you like this, which is such that any cat can be written as a linear combination.

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Whatever you need. It's a set of cat such that any cat does.

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For example, the dynamical state of our system can be written as a linear combination of these cats.

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Right. Then we have the idea of an adjoined space.

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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.

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Valued. Complex valued.

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Functions on cats. Mathema a mathematician would say.

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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.

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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

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numbers are going to be the all important amplitudes for something to happen,

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for something to be measured. Right.

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And that's you know, we completely focus the whole all this mathematical power is only there to help us to calculate these amplitudes,

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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.

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A probabilistic prediction for what some experiment is going to is going to yield.

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Okay. So so we're interested in these complex valued functions.

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I'm just I'm just saying that they're going to turn out to be the amplitudes.

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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.

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But these mean the same thing, a bracket opening, sort of angular bracket opening this way F of CI.

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This thing here means the function F evaluated on its side means that it is a complex number.

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It is going to be interpreted as an amplitude for something to happen.

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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.

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So we've got cats which define dynamical states of our system and we've got bras which are

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functions on the dynamical states of the system which extract the all important amplitudes,

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the cats form of vector space because it's a vector space, it must have bases like that up there.

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And the bras also form a vector space, as I hope you've discovered in in Professor Isserlis lectures.

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So the brass. Form the Adjoint space.

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Often called V primed. Why do they form a vector space?

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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.

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Let's call it H for originality. Right.

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What do you want? In order to. In order to give meaning to this, I need to know what H does.

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What H does to any state website.

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I want to know. Function is defined by the value it takes on any on any possible argument.

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So I need to know what age of CI is, what number that is, and I define it to be efficaci plus skip CI,

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which of course is a perfectly well defined expression because this is a complex number.

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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.

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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.

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So I define the g primed meaning alpha g.

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By the rule. G primed of upside is alpha g of upside case of gain.

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This is perfectly well-defined because. That's just a complex number.

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And so this multiplication is well defined. So now I know what g primed what value it takes in every CI.

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So this is so this is the point that this is, this is the basic principle that establishes that the functions,

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the linear, complex valued functions on a vector space form, a vector space, the adjoined space.

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And we're going to be working extensively with both the cats and the pros.

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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.

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And so if we. And how do we how do we define this?

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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.

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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,

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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.

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So, so this do this this equation defines.

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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.

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This is a function and it's defined such that two on two is one and two on anything else equals nought.

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So that is a perfectly good rule which defines the values that the function j takes in every element of the basis.

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And again from Professor S this lectures.

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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.

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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.

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We want to say, supposing Abassi is equal to the sum I.

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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.

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This is the funny part, right? So so far I hope I think everything's been I hope everything's been fairly straightforward.

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But now I'm saying associated with the state of our system.

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I want to find a function on states and the function in question is defined by this rule that it's a complex conjugate.

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Times I. The bra i. So given that my state of my system is a certain linear combination of the basis states,

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I'm saying that the function associated with that state of the system is a certain linear combination of the functions,

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these functions which are associated with the basis states. Why do we do that?

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One reason we do that is in order that we can evaluate this important number of science.

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I. So let's have a look at that number.

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That is the sum. I write this out as a sum, a I star I sum of I.

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And then I have to write the this one out as a sum, a j of j.

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So I'm summing over J. These are just dummy labels, right? So I'm entitled to call one J and one I.

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So it's a sum over J is one to how many we need and i's one to have a many we need.

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This is a this is a linear function, right?

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We're evaluating this linear function on this dirty great sum.

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But because it's a linear function, the dirty great sum can be taken out side.

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So I can write this is the sum of I and now J being one two, whatever it is of a I star,

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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?

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J So it is nothing and less I equals.

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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,

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and then this will come become one.

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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.

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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.

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And we're taking a sum of the mode squares of the amplitudes.

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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.

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So my my states, I would like my states to have this normalisation condition.

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This is proper normalisation. Is that any of the state times its bra should come to one.

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Not any other complex number. That particular complex number one.

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Okay. So that's that's the basic principles of direct notation.

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Now, let's just talk about the energy.

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Let's let's have a look at this better understanding of what this physically means by having looking at energy representation.

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So supposing we in certain circumstances, for example, if you've got a particle that moves in one dimension,

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then it's then it's possible in some in some trapped in some.

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Well, then it is possible to to characterise the dynamical state of the system simply by giving the amplitude to measure.

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The possible values of the energy. So a complete set.

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So so this is this is not always the case. But for a one dimensional particle, a particle trapped.

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This is a very idealised situation, but never mind trapped. In a one dimensional potential.

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Well. We will see that.

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And I'm asserting for the moment that the a I form a complete set of amplitudes.

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Where? A mod squared is the probability of measuring the ice energy.

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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.

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Remember, we introduced the idea of a spectrum. Those are the possible values of your measurement.

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You can only measure a discrete set of numbers. They're called EEI.

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There's a probability that if I would measure the energy,

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I would find the energy to be I that that's this mod square and a complete characterisation of the system.

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Complete dynamical information is provided by knowing not only these probabilities, but actually the amplitudes themselves.

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So you can think of of PSI as a vector formed by these amplitudes.

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Now, let's let's write that upside.

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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.

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I summed over I. So out of these complex numbers, which we know and some basis, any basis we can,

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we can write a symbol like this that's just a repeat of what we've already done.

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And now let's ask ourselves, what are the meaning, what's the physical meaning of these states?

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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.

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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.

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It's it's a state of the system. Now, let's find out what these ones mean in this context.

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Suppose. We know.

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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
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Then the amplitudes must be like this. And what is that?

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What does that mean? That means Ixi. The state of our system is actually equal to three.

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Because on this. In this sum, there's only going to be.

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One non vanishing term, and that will be a three, namely one times three.

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So that tells us that this state three is actually the state of definitely being having energy three.

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And similarly for all the other ones. So a better notation or a clearer notation is.

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To write to rewrite that in a clearer notation is a cy is the sum I of a i times e i.

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This this makes it clear what we've just established that the thing is actually the quantum state of definitely being having energy.

183
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I. So we've discovered the physical meaning of those abstract basis vectors.

184
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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
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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
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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
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But it's it's of enormous technical and mathematical importance.

194
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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.

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But now let's move straight on to another illustration, which is back to spin a half.

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So I said that elementary particles are these tiny gyros that the the the rate at which they spin never changes,

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but the direction in which the spin is oriented does change.

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I made the point yesterday that the though you can know for certain the result of measuring the spin in one particular direction,

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for example, the component of the spin parallel to the z-axis, you cannot know the direction in which the thing is spinning,

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because even when you measure the component parallel to the z-axis with precision, you're,

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you're in deep ignorance about the about the value of the spin parallel to the x axis or the y axis.

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You only know it does have spin in those directions that you do not know the sign of this.

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You do not know how much spin is a long X or a long y, but a so so for s.

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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.

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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.

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I want to look at the angular momentum. H Bar has dimensions of angular momentum.

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So the angular momentum, what this means is that the if said is plus a half.

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That means the angular momentum in the Z direction is plus a half H bar,

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but it's turns out to be convenient to leave off the bar when talking about the so-called spin of said.

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Partly because you'll see that spin in quantum mechanics is.

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Really has a slightly dimensionless being.

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And partly because partly because writing we don't write any more, because we have to.

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It's just it's just economical. So that so physically there's the angular momentum is a half edge bar,

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but it's more convenient to write that as Z this abstract thing, the spin is plus a half or minus a half.

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So what do we have? We have two states. We have a we have a complete set of states.

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Followed by plus and minus. Okay, so this is the state in which I am certain.

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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.

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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.

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Actually, maybe it's better to write it this way. A minus minus plus.

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A plus plus. So since this is an easy case, there are only two components to our cat A minus and a plus.

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And just in just the same way that I might in ordinary in ordinary vectors write that oh is equal to

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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.

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Don't need to bracket, do I know? Where?

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Here, I've got three real numbers B, B, Y and Z, which are the components of B in some particular coordinate system.

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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.

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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.

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So that's the analogy. Okay.

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Now we need to anticipate a formula.

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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,

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either up or down, then you can calculate the amplitude to find the spin in any other direction,

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either parallel to that direction or anti parallel to that direction. That's what I claimed.

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And I'm going to quote a result which which we will arrive at later.

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But we have to take it on trust for the moment. So the state if we if we have a unit vector n.

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So so let and. And it's a unit factor.

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And it's in the direction theatre and. Fine.

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Right. These are regular polar coordinates which are defining a direction by by pointing to a place on the unit sphere.

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And let n be the unit vector that points in that direction.

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Then I make the following assertion that the state of being plus along the vector n so can be.

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So this is a state of. This is a state of my electron.

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So if it's true that that's a complete set, it must be right.

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It is a linear combination of this state and this state. Right.

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And I I'm not going to say that that is sine I better just check that I'm getting this right.

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Yep. Science teacher upon to e to the i fi on to.

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Of minus. Plus costs are upon to each of the minus i.

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On to. Plus, now we will derive or at least you will drive in a problem this formula.

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We will show that it's why it's true. At the moment, we're just asserting that it is true.

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So this this is a complex number, right?

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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,

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of being certain that if you measure the spin along this direction, you get the answer plus a half.

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Correspondingly, there is a minus object. Which turns out to be, cos these are over to each of the eye loops.

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Five over two minus minus sign feature over to E to the minus I fly over to plus.

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So it has it's made, of course, it's this is naturally another linear combination of this and this basis vectors.

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And now we just have different a because it's a different state, it has different a minus and different a plus.

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Now we in order to to to calculate something useful, we need to know what the bras are that belong to those.

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Right? So. So these are the kits.

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I will I will want to do something with the bras in a moment.

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So let's calculate what the bras are. So we have the bra in common.

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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.

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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

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minus plus cos these are over to each of the plus I high over two times the bra plus.

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So that's, that's the bra that belongs to that.

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And I want the bra that belongs to the other thing cos these are on to.

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The need to concentrate each of the minus five to.

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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.

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Now what we want to do. So let's calculate.

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Let's suppose. Let's suppose that we've just measured.

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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,

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what we will know is that the state of our electron is actually plus let's just suppose we've made the relevant measurement,

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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.

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Now. I now realise that I have left out.

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Can we just cycle back to the energy representation?

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Why? I should have pointed something out. What I should have pointed out was.

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From this expression here. Well, perhaps it'd be better to be done.

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We better be done here. Let us point out at this point.

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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,

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what that means is that I'm going to evaluate the function.

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E.J., E.J. on both sides of the equation then.

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Then what am I going to discover? I'm going to discover that, E.J. Upside is equal to AJ.

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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.

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But E.J. being a linear function. E.J. pops inside here and meets that.

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These are two basis factors. So they have Delta, E.J. for their E.J. on this EEI produces Delta AJ.

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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.

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Now, this is a fabulously I should have pointed this out. It's an obvious equation, but it's fabulously important.

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And it tells us really why we're interested in these animals here, because it means that given the state of my system,

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it enables me to recover the amplitude for measuring E.J. out of the state of the system.

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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.

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The interesting result of your measurement. In this case.

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E J So the amplitude to find that the energies E.J. is just E.J.

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Brought into the state of our system. So when I come back to this problem here, I want to know the amplitude.

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To measure plus on end.

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So what I need to do is to calculate this by that principle.

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So what I do is I take that in plus thing this thing and I knock it into,

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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.

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So I simply extract this. So this turns out to be costs feature over to each of the I find over two.

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So that's the amplitude to measure this this complex number is the amplitude to measure

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that the spin is along the vector n where C4 and phi the angles which define n,

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which means that the probability of measuring plus on n is simply cost squared phi over two.

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Does that make sense if sorry feature of two.

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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

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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.

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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.

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We we should get the probability zero because that's the probability to find that it's pointing down the minus it axis,

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which is the same as the probability that we get minus along the plus it axis.

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And when seat is pi, lo and behold, we're looking at cost wed of cost of pi upon two squared, which is zero.

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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.

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What does that imply? It implies that N is equal to x the unit vector in the x direction.

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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.

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Given that I'm plus on Z is the probability the amplitude.

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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.

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And I have an E to the I not for each of the nothing. So that's just that, right?

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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?

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The answer is a half because it's the square.

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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,

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knowing that the spin is along, it has a component plus a long Z doesn't really rather than minus along.

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It doesn't really help us to say anything about X.

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So we really have total uncertainty because it's the probability to be plus on x is a half the probability to minus one.

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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.

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That implies that n is equal to e y the unit vector in the y direction.

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What do we get then? Then we find that the amplitude y plus.

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Plus is. Still one upon two.

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But now we have e to the i. Pi on for.

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So the amplitude is now genuinely a complex number, whereas in the case it was a real number.

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But it means that the probability for getting plus on Y is still a half the same as it is on X,

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which again has to be the case by symmetry if you think about it.

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If we calculate the corresponding negative amplitudes, let's calculate x minus.

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The probability, let's find the amplitude that it's pointing minus on x.

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So then we have to take that n minus thing and bang it into plus.

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And what survives is the minus sign.

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Caesar over to. Well.

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Strictly speaking each of the stuff. But that's well each of the I fi over to.

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That's it. Actually, let's just make this an minus.

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All right. Then I know I can, from this formula, reduce the X and Y ones.

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That's what I want to do. I have the X minus plus.

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I have to put the feature upon two in here. That's going to be minus one over two.

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And I'm going to have that y minus. Y minus plus is going to be.

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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.