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