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Okay. I guess we should. We should get going.
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So we have on we have today to discuss some unfortunately rather formal stuff.
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And tomorrow we will do something that's physically more interesting,
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the Einstein Podolsky Rosen experiment, but which will draw heavily on what we're going to do today.
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So what we want to say is this face up to the fact that many of the systems that we want to apply quantum mechanics to come in parts.
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So for example, the hydrogen atom consists of an electron and a proton.
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And to know the state as a hydrogen atom, you want to know the state of the electron and you want to know the state of the proton.
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Both. A diamond consists of on the order of so condensed matter.
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Physics is about things like diamonds, which a diamond would contain ten to the 23 or whatever carbon atoms.
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And to know the state of the diamond officially you would need to know the state of the ten to the 23 carbon atoms.
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So it's going to be important to move forward towards applying quantum mechanics to any non-trivial and really interesting system.
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We're going to have to learn how to describe systems that come in parts.
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And this turns out to be quantum mechanics is its own way of doing this,
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which is actually very elegant and powerful, but at least some surprising results.
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The hydrogen in a hydrogen atom, the electron, of course, is strongly interacting with the proton,
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its electrostatic, the attracted towards the proton and in a carbon atom,
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sorry, in a diamond, the carbon atoms are obviously very tightly coupled to each other by covalent bonds or whatever.
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So there's a there were springs as it were. There are there are things connecting the different paths.
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But it turns out that the quantum mechanics of a system made up of two objects is non-trivial to it.
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It is non-trivial trivially different from the quantum mechanics of the two isolated things, even if you just logically consider them to be the same.
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So when we do angular momentum, we will. In the coming weeks we will find that very strange and interesting results arise just
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because we put two gyroscopes in a box with no physical connection between the two of them,
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and start asking questions about what's the angle dimension of the box, as opposed to what is the case of the individual gyros.
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So knowing the state of the well defined states of the box turns out to be very different from knowing well-defined states of the individual gyros.
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So the there, there's a, there's a what we're talking about today is putting things logically together to make compound
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systems and that may or may not be springs connecting physically connecting these things.
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All right. So the central problem of see, the central thing we have to address is if we have a system A and a system B,
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so we have we have two distinct systems and this one, let's say, has states I All right.
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So this indicates which system we're talking about.
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Then there's supposed to be a semicolon here and there's an index here which tells us which of this system states we are addressing.
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And we have a system B and it will have states something like this.
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And what we want to know is so how are we supposed to write the states of the compound system,
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the system that you get by considering A and B together.
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So this might be the electron. This might be the proton. What's the state of the compound system, which we call a hydrogen atom, for example?
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All right. Because if we know how to add, if we know how to compound a system with system B, to make a combined system, we can compound another one.
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We can we can use the same rule to add another element of a bigger system, ABC and so on and so forth.
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And we can do eventually we can build up a a diamond of ten to the 23 carbon atoms, the central once you know how to add two systems by power,
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by repeating this process, adding more and more systems, you can put any number of systems together.
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So this is the central problem that we have to address.
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So the states, what we first of all, we just write some formal stuff, the states of the compound system,
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maybe this is when you logically think of the system in A with B as one system or of well,
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some of the states of the system may be written like this a B, semicolon, i j and we write it symbolically as a semicolon.
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Oops. I b semicolon.
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J Sorry, the semi carried on the j looked too similar to what is this?
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Let's just ask ourselves, what does this mean?
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This means the state of the compound system where the state where where the subsystem is and its ice state and the subsystem B's and it's J State.
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Right. So this is we know we have to know what this means.
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And I think we do know what this means. I've just I've just given words that give meaning to that.
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And on the right hand side, we have a symbolic multiplication, and we don't need to worry too much.
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You'll see as we go on that we don't need to worry too much about what exactly we mean by this multiplication.
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But this is this is just a symbolic product of efficiency.
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It's a tensor product, but we don't want to frighten everybody.
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This is a this is a symbolic multiplication of a cat on a cat.
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Right. We'll find out how to interpret that as we go along.
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Then we will obviously, we can have the bras that must be associated bras, since this is a state of a system,
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it has a bra which will be i j oops is equal to of course the logical product of the bras.
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And we, we, we give meaning to this thing by explaining what happens when this goes onto this.
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So when this goes on to this, we should get a complex number. So to give to give meaning to all this, I need to explain what what this is.
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I primed j primed on a b a b semicolon i j is equal to right.
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So we need to give this should be a complex number to give meaning to all this hocus pocus.
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I need to explain which complex number. The complex number it is is this complex number a a primed.
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A I times.
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PJ primed PJ.
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So what I've written on the right makes perfectly is completely well-defined because this is a complex number and this is a
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complex number and we can multiply complex numbers and we get a complex number which which gives meaning to this on the left.
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And that's really, really, really the we that's the essence of giving meaning to this thing here.
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Because, remember, we only want these cats.
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What we want with cats is in order to calculate amplitudes, which squares are going to be the probabilities for give us our predictions.
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So as long as you know how to get an amplitude out of a cat, you know enough about the cat to get on with it, right?
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So we've given meaning to the process by which we extract amplitudes out of cats, which is this borrowing through business,
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because that leads to the experimental predictions, which are the whole point of the theory.
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Okay. Why is this? Why does this make sense? Why is this a sensible definition?
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Right. This is the definition of what we mean by these animals. Why is it a sensible definition?
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Well, it says that the probability of getting.
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Of measuring the results, ie primed and primed, given that we're in this state, is equal to obviously the mod square of this horrible thing.
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A b i primed j primed a b i j mod square.
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Right. That's how we would interpret this complex number.
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And according to this formula, this is equal to the product of the probability associate.
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This is the probability with for the system may be the probability system way of getting the result.
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I primed times the probability of getting the result j Primed right.
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Because if I take the mod square of both sides, the mod square of this product is the product of the mod squares.
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The mod squares on this side, by definition, all these probabilities that.
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So the probability this says the probability that if I take measurements of my combined system,
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I find that A is in the I prime state and J is in the B is in the J prime state is simply the product of the is the
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product of the probabilities that the A system is in the prime state and the B system is in the J prime state.
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If you make individual measurements so that that makes perfect sense and it's motivating.
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This is why we write the case of the compound system like this.
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This multiplication rule, this symbolic multiplication is inherited from this law for multiplying probabilities and probability theory.
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K. Now that having said that, and everything's nice and simple,
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we have to make the point that I now want to show that not all states this is the thing that's surprising of a be of the form a.
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That's what I want to now establish. That's that this is that it's not true that all states of the system or of this form.
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Okay. So let's let's so-and-so, for example, consider it, consider two, two state systems.
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So I'm going to do a concrete example to illustrate this general and very fundamental principle.
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We're going to have to two systems. We're going to have a who's going to have states plus and minus these a complete set.
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So we're considering the simplest, non-trivial example and B is going to have the states up and down.
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Right. This is just a notation that enables us to by using a plus sign and a minus sign for a and an up an arrow in a down arrow for B,
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I avoid the necessity of writing down these pesky a b labels.
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Right? Let's now consider let the state of AA be a plus plus.
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A minus minus. So this is a general state.
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Okay. By taking a linear combination of the two basis factors of my two basis states, the my two state system.
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I write down a general state by choosing these amplitudes to be whatever you like.
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You can make any state of whatsoever and let the state of be.
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Similarly b b up. Up plus.
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B down, up, down. Then what's the state?
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Now let's have a look at the state. AB the state of AB that we get.
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Well it's going to be this thing bracketed into this thing.
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A plus plus plus minus minus B plus plus plus B.
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Oops. Sorry, sorry. This has this has the up and the down states B subscript down, down.
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And when you multiply this out, you get a disgusting mess, right?
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Because you get a plus plus.
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Oh, sorry, sorry.
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Hey, plus B down B up of plus up plus A plus, B down of plus and down plus a minus B up of minus an up plus a minus B down of minus and down.
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So my. So this state is now along.
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It's it's now linear combination of four states.
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And it is strongly suggesting that these four states are basis states for the compound system.
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And indeed, we will show that they are times amplitudes, which are these products of those individual amplitudes?
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And these amplitudes have well-defined meanings, right?
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So for example, A minus B plus is the amplitude that A will be found minus, and B, what do they say?
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Oh. So take the mode square of this.
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You get the probability of that. The experiment to measure A's property and B's property would be these particular values.
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But but what I'm trying to show is that this state is not the most general state.
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Okay. And the way I'm going to do that is I'm going to calculate the probability that B may do this the same way I've got it here that B is up,
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given that A's in the plus state. All right.
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So this is the kind of. So if this is a reasonable question, we've measured and found, today is up.
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And I now want to know. Okay, so suppose I measure B's property, will I?
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Sorry. I found the days. Plus, will I find that B's property is up or down?
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This is going to be the probability that that given that I am where I am, b will be found to be up.
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Okay. Well, this is equal to the probability simply that we have up and the probability for being up and plus over the probability that A is plus.
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Now, why is that? If I would move this here, then this would say that the probability of being up.
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And plus is the probability of being plus times the probability that we get up.
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Given that we have. Plus this is this is a very important result from statistics.
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This is classical probability theory.
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This is known as Bayes Theorem, but it's really a trivial rearrangement of the rule for multiplying probabilities.
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The probability to be OP and plus is the probability for being plus times the probability if you are plus that that you are up.
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So this is not doing quantum mechanics.
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This is just a rule of probability theory, which now plays a very important role in statistical inference in all in,
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in, in all the sciences, physical and social.
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All right. So what is what is that?
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That's the probability that we are up and plus over the problem.
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This is the having plus on a comes in we can have plus in a in two ways with a composite system,
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we can have it either with B down or B up and they are mutually exclusive events.
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So I can have that probabilities. So this probability on the bottom is.
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P up plus plus.
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P down plus. So what is this?
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This is equal to one over dividing through one plus p down plus of a p up plus.
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What about this? Let's go back to that expression up there.
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What's this probability? What is this probability in terms of those amplitudes?
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P down plus. P down.
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Woops, down and plus is equal to is equal to A-plus p down and p up plus going up there p up plus is a plus B up.
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So these a plus is cancel. Oh we need to take the mod square of this whole thing of course.
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Right. But these, the crucial thing is those things cancel.
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So this is in fact equal to be down.
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Plus this is equal to be this ratio.
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So what's what's the point? The point is that this probability is actually we've just shown it's independent of a plus and a minus.
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So this probability does not depend on the state of a.
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What does that mean physically? Firstly, it means that the systems are not correlated.
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I've just calculated one specific conditional probability,
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but you can calculate any other conditional probability and you'd find the same thing that the probability
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of any state of be is independent of what you assume about what the result of measuring a and so on.
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These are uncorrelated systems.
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So what we conclude from this is that when the state of a B is a product of a state of a times A, state of B, the systems are uncorrelated.
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That's an important physical assumption.
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Now, for example, if you have a hydrogen atom, is the location of the proton correlated with the location of the electron?
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Well, of course it is, because if the hydrogen atom is here,
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you can be pretty damn certain the electron lies within a few nanometres, or if you will, within a matter a metre of the proton.
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If the protons over here, you can be pretty sure that the electron is within the nano metre of the proton and it's over here.
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The electron and the proton are very strongly correlated because they're, you know,
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there's physics, there's, there's a piece of Hamiltonian which is, which is correlating them.
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So, so we don't.
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Yeah. So we do expect systems to be correlated and that means we do not expect systems in
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general to have way functions that look like to have states that look like that. So let me see.
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The point is that the but I'm not going to go through the demonstration.
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I think that I said so let's go back up some way.
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Let's go back to let's go back to here.
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So if these objects form a complete set of states of A and these objects form a complete set of states for B,
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then it's not hard to persuade yourself that it's right that these objects form a complete set for AB.
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All right. So this is a complete set if these complete for their respective subsystems.
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I want us this telling us this is telling us that any state of the system, including correlated states which as I've tried to argue in natural states,
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states in which the two subsystems are correlated, they must be writable as linear combinations of these objects.
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So the conclusion here is. But this.
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Put that back and start over here. So any state of Abby can be written as Abby equals the some c a j someday by some division of states a.
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I b j.
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These states describe uncorrelated states in which the two subsystems are uncorrelated, but this may be correlated, probably is correlated.
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So the way quantum mechanics introduces correlations between subsystems is by taking linear combinations of uncorrelated states.
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We just had such a linear combination of uncorrelated states here.
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Right. And it turned out that in this case, that was still an uncorrelated state,
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because this was simply an expansion in terms of some basic states of a state which is which was already a product of just two states.
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So the point is that the general state cannot be written.
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This thing in general cannot be written like that. Even though when you see a long list of basis states, it may, you know,
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with certain complex numbers in front, it may be that that that the state can be written thus.
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So whether this thing can be written as a as a product of two separate states depends on.
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On these numbers. Now, we haven't got time to go into what property it is of these numbers,
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which which ensures that you can do a decomposition like this into one correlated states, which makes this state uncorrelated.
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And when these are correlated, but you can find a complete account of it in the book there.
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I think some and there are there are problems investigating this.
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But the point is that if you in this concrete example here, right, this is one of the CS, this is another of the CS, another of the seas.
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Another of the seas. And these CS are not general.
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They they have the property. You could arrange those in a two by two array of, of objects.
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And if you uh, this, this matrix of this two by two matrix is sort of a degenerate matrix.
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It's a special matrix is not the general one that you get by making choosing these numbers independently.
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So correlations go in like that.
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And in quantum mechanics, when you say that two states, a two systems are correlated, you actually usually use the word entangled.
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Entangled is just the same things as quantum mechanical jargon for correlated.
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And what it means is if a compound system of two subsystems are entangled, it means the state of the compound system cannot be written in that form.
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It has to be written in this form. The and these numbers and these these numbers do not have the property that requires them.
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They have to have to enable them to be to be expressed as products of of individual amplitudes of the individual systems.
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So that's doing a bit of quick counting. Suppose there are and basis states.
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Of A and and a B.
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All right. So there and there are M values that I can take and there are any values that J can take.
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So then there'll be m times and amplitudes.
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C i j So to specify a general state of the system, you need to specify and numbers CIJ To specify a state, but to specify a you need just m numbers.
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A I and to specify B you need an amplitudes b j.
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So to specify a general state of the form a b you need to m plus n amplitudes.
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So M plus N is generally much less than men.
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If you got it with two, two, and this little example M was two and was two.
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So this number was four and this number was four. But supposing.
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So they're the same. But supposing that this number was eight and this number was eight, then this would be 16 and that would be 64.
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So. So usually most systems are not two state systems usually.
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So is what this is telling us is that in a general state of the system, there's very much more information than than there is in here.
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And why is that? Because it's specify a general state of the system. You have to specify all the correlations between the subsystems.
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And there are a lot of possible correlations. This is not a problem only for quantum mechanics.
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This would be a problem if we're were doing statistical physics. Classical statistical physics correlations have nothing to do.
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I mean, not directly to do with quantum mechanics. There are a logical problem that arises in all physical inference also in the classical world,
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and correlations are very hard to handle in the classical.
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In classical probability theory. They're actually easier in this apparatus here because quantum mechanics pulls this amazing
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trick correlated states of the system or obtained are understood as quantum interference.
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Try to sum like this is a quantum interference between uncorrelated states of the system.
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When you're doing classical probability theory, you aren't able to pull that trick, and it's much harder to specify correlations.
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So correlations are important in both the classical world and the quantum world,
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but they're actually easier to handle in the quantum world than the classical world
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because of the strange way in which quantum mechanics compounds these amplitudes.
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Does this quantum interference.
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The quantum interference is how quantum mechanics handles correlations, because each has its own completely unique way of handling correlations.
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Oh, the the results can be surprising, right?
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But they can be ones that that raise eyebrows.
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And the Einstein Podolsky Rosen experiment is an example.
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Let's try and pin these ideas bit by by looking at a concrete example of the atom.
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So in the position representation. What do we want to know?
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A complete set of amplitudes are going to be things like X.
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So this is. So let's let's make this the electron wave function.
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And we're going to have we're going to have also.
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So we'll call this XY, therefore, and we will have XP times, a big U.
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This will be a proton wave function, right?
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Which gives you the amplitude to find the proton at the point XP.
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This gives you the amplitude to find the electron at the point x e and we and supposing these things have
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subscripts on them ui and you j so this might be the amplitude to find the electron at the point xy given.
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So this might be an UI given that the energy of the,
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the energy of the electron is E-I and this might be the amplitude to find the proton somewhere given that the protons energy is e.g. say right,
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then what is the state a state of the atom would be.
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Sorry. XP.
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XP. So what is this?
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This is. This is a state of the hydrogen atom in which the proton has this energy.
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The electron has this energy. And that gives me a state of the logically coupled pair of proton and electron.
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And this, as I say, is not going to be a very realistic state of the of the hydrogen atom,
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because it's going to give us this is going to give this this says that the electron and the proton are uncorrelated.
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And I've just tried to persuade you that the electron and the proton are very strongly correlated.
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Consequently, their way functions can't. This isn't going to be a realistic, useful way function for hydrogen atoms as found in lab.
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So what do we have to do? A more realistic state?
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Might be XY, XP, shall we say, Kai, for a new label, which would be some some CIJ of xy ui xp big u j.
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But what are these? This is a boring function of X with a label.
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I lose a set of functions of x e which have labels are in return complex values.
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And then this complex number is multiplied on this complex number, which is a function of XP.
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A member of a family of of of functions with labels.
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J Here is an amplitude, another complex number, at least complex number together.
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And you get this complex number and this.
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So any state of a hydrogen atom must be rewritable like this.
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But realistic states are not reachable like that because. Because of this correlation of the proton and the electron.
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Okay. Now we need to revisit the collapse.
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Oops. Of wave function. Function.
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So what happens when we make measurements on compound systems?
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We know that when we make measurements, what happens when we make measurements on a single system and we have to extend these ideas?
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So suppose let's go back to our state of our systems.
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So we go back to the two state system to two state system A and B and consider consider this particular state upside,
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which is equal to a times plus up plus minus brackets, B up, a plus, C down.
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Supposing this is what we have, this is pretty much written down at random.
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It is a well defined state of the system because it's the sum of three of the four basic states that we were discussing.
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Right. It's the sum of of plus up, minus up and minus down.
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This is the amplitude that if you would measure A and you would measure B, you'd find that A was was plus and B was up.
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This is the amplitude for finding that A is minus and B is up, etc., etc.,
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etc. But I've written this, but this one down, this state is, as it turns out, entangled.
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That is to say,
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you won't be able to write this as a product of a state A and a state B so this is more realistic than the states that I was discussing before.
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Okay. Okay.
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Now, suppose. Suppose we measure. So so let's measure.
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Measure state SES subsystem a. If we get.
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Plus then after measurement the theory says right.
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The dogma is I'm not going to justify this.
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I'm stating this as a as a conjecture that the state of the system as it is now goes to a PSI primed, which is equal to plus up.
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So how does the system let's just remind ourselves what collapse the wave function was all about in the one state system and the one single system.
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Sorry. If we got a single system we wrote up, psi was equal to the sum.
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And let us say n for example.
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And we measured e and got the answer e m Then Abassi went to the state.
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M Right after the measurement it was in this state.
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So I'm making I'm stating that in this more complicated scenario where we have a two,
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we have a composite system, we measure only one of the subsystems, we get a certain answer.
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It goes to that state, which is consistent what we had over there, because we, we,
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we found the answer plus so we throw away everything times minus but the whereas over there is simply m
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the coefficient up there of plus was not just a complex number a which was giving me the probability.
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It was also times the state of B and the state of P just gets copied down.
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So what does this say? This. So this is what the theory claims is that that goes to that.
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It doesn't explain how this happens. This is the problem of measurement.
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But the there's a physical implication of this, which is that you're now a measurement of B is guaranteed to produce or to find up.
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Right. Because this thing is something times up.
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There is now zero amplitude to find down. You're certain not to find down.
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You are certain to find up even. Right. If on the other hand we get minus four k then.
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The new state is equal to minus.
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Sorry. Sorry. The new state is equal to yes. Minus brackets.
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B up plus C down, properly normalised.
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So over the square root of of B squared plus C squared.
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So this is what the theory claims, that if if you get the minus thing,
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then your new state is essentially the coefficient of of minus and minus itself all properly normalised.
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And now so if we get minus, there is now uncertainty as to what the result of a measurement on B will be.
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So it's so now measurement. A B yields, for example, up with probability be squared over the square root of B squared plus C squared.
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So we now apply the same old rules about the probability of measuring.
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But the interpretation of the amplitudes. Right. Because we are certain to get minus.
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If we measure with, we measure a again.
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But if we measure B, we can get two outcomes either up or down. And the probabilities are like that.
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So that's a that's a that's a conjecture. That's a statement, a theoretical statement about how the interpretation of the theory works.
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And we just have to accept it and see whether it leads to proper experimental predictions.
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So in the last minutes, we have unfortunately, a big topic to discuss, which is operators for composite systems.
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So we've talked exclusively so far about the cats.
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But we know that operators play a very important role with every measurable quantity.
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There's going to be an operator and we need to know how this behaves. So we found that the cats of the subsystems were multiplied.
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This rule was inherited from the multiplication of probabilities of successive events.
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The, the operators add.
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So for example, if we have two free particles they and B are both free particles,
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then AJ is equal to p a squared the momentum of a squared over twice the mass of A and HB the Hamiltonian operator is equal to B squared over to be.
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So what's the Hamiltonian of the combined system?
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H a b is equal to AJ plus HB.
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In other words, it's squared over 2ma plus P squared over two MP.
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And that's sort of saying the energies of the combined system is the sum of the energies of the individual bits.
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How does the operator pee?
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We now need to explain how an operator P.A. operates on one of these states here.
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Okay. So when P.A. hits a i b j what we have so so this is a states of the combined system
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and this is an operator which has to operate on the state of the combined system.
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And what does it do? It produces P.A. operating on a high, which is a well-defined state of A symbolically times b, j if p b works on this thing.
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If PB ignores this, it passes through this as if PBE was just an ordinary, complex number and homes in all this its target.
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So this is this is simply a I times p b b j this is a well defined state of B gets to be symbolically multiplied by this well defined state of A.
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And there you are. So, for example, what would the expectation value?
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A. B i. J of h ab.
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In this case here, let's just make sure that we get some sense out of this.
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Sorry. AB a.j. So what does that mean?
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That means i b i.
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Sorry. J j brackets h a plus h b close brackets i.
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B j. So this operator ignores that because it's a b operator and homes in on that.
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This operator operates on this. And then we have the other things come in on the other side.
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And this this gives me a i.
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P sorry. Hey, i b j bga plus.
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So that comes from this. This this because because that passes through this operator as if it were just this was just a number bangs into that.
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Plus, correspondingly, we can have a i i b j hb b.j.
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This, of course, is going to be the number one.
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This is going to be a the expectation value of the energy of a this is the number one and this is the expectation value of a fee.
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So we find that the expectation value of the energy of the combined system is low and behold the sum of the energies of individual bits.
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I think that makes physical sense.
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If it makes that makes physical sense when the Hamiltonian takes that simple form, if it's just the sum of the individual bits.
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But for, for example, for hydrogen.
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The Hamiltonian h is equal to p electron squared over two massive electron plus p proton squared over to the mass of
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a proton minus the charge on the electron squared over four pi epsilon nought x electron minus x proton in modulus.
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All right. Because the energy of the hydrogen atom is the sum of the kinetic energy of the
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electron and the kinetic energy of the proton and an interaction energy of the two.
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Right. Because they electrostatic they attract each other.
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So so this is equal to h electron plus h proton, these being the hamiltonians of the free electron in the free proton plus an interaction Hamiltonian.
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And the thing about this interaction Hamiltonian is that it depends on operators belonging both to the first subsystem and the second subsystem.
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And the consequence of that is that h e comma h interaction commentator is not equal to nought because the because the p
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the electron momentum operator sitting inside here has a bone to pick with the electron position sitting inside here.
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And similarly, of course, HP comma h interaction is not equal to zero.
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So without that interaction we would have that the.
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So what's the important point about this is that the Hamiltonian of the hydrogen
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atom does not compute with the hamiltonians of the electron and the proton.
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You cannot know the energy. So generically you do not expect to be able to know the energy of the hydrogen atom if you know the
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energy of electron because they don't compute and it's the interaction that stops them computing.
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Well, we're going to have to stop, unfortunately, that at that point, but we're pretty nearly done.
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I'll just write down one final statement, which is that the operators of different subsystems always compute.
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Right. So for example, p proton comma x electron is precisely nothing,
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etc. We do not have to worry about non vanishing comet cases of operations that belong to different subsystems.
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Okay.