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Good morning, welcome to Lecture 14. Today we'll consider composite systems and if you

recall the first axiom of quantum mechanics that I presented in the first lecture, the

first axiom stated that for each quantum system there is an underlying Hilbert space.

And we already saw the L2 of R3 for one particle in three dimensions. We saw C to the 2j plus

one for a spin-j system and so on. And today we ask the question, well, assume you have

two quantum systems, say one particle and another particle, then you understand each

of these quantum systems individually. Each of them will have an underlying Hilbert space.

The question is, what is the underlying Hilbert space of the quantum system if you look at

them both together as one system? So the two-particle system, if you understand the one-particle

system, what is the two-particle quantum mechanical system? So question. So if we have a quantum

system, one, number one, that has an underlying Hilbert space H1 with an inner product that

belongs to that H1, and you have a second quantum system, number two, which might or

might not have the same Hilbert space with the same inner product, but let's say it's

an entirely different quantum system. So it may have a different Hilbert space with a

different inner product. The question is, what is the Hilbert space

of the composite system? Well, in classical mechanics you know. So depending on the

formulation, you would say the state space of classical mechanics, well, the Hilbert

space isn't quite the state space, but we'll come to that again. But in classical mechanics,

the phase space of a composite system is just the product of the manifolds, the Cartesian

product, if you want, of the individual systems. Or if you want to formulate this with the

idea that the phase space is a vector space, which it isn't, you could say it's the direct

sum of the spaces. And in fact, if you, in classical mechanics, have two particles, you

know the state of one particle, and separately you know the state of the other particle,

and then you take this information together, then you have all possible states of the

combined system, because you know one and you know the other. Now in quantum mechanics

this is very different. In quantum mechanics, in general, you do not understand all the

states of a composite system by understanding the states of one part of the system and the

states of the other part. It's just the case. So it may be natural, may seem natural, you

see the answer will be, the stable will be this is not true, it may seem natural to model

the Hilbert space of the composite system as the, now I use a word which I'm going to

explain in a second and which will play a role later on in another context, so we have

it here, as the direct sum. And what is the direct sum? So assume, so the direct sum is

the new Hilbert space that has this name. But now what is this? Well the Hilbert space

certainly is at least a set, and this would be just the set of all the pairs, phi and

psi, psi and phi, where psi is from the first Hilbert space and phi is from the second.

And one could also induce an inner product on this and blah blah blah. Now the thing

is, this would be exactly what I said before, what is not the case. This would be if you

know the element in the Hilbert space H1, which corresponds to a pure state, at least

the pure states can be built from that, and an element in the second Hilbert space, and

you look at all the pairs, so you have the information about the one state and the

information about the other state. I lose state here loosely, I'll comment on that again.

Then you would understand all, if this was really the Hilbert space of the composite

system, you would understand the composite system by knowing the individual state of

the parts of the system. This is not the case. This however is not the case.

What is the case could be taken as an addendum to the first axiom. What is the case may count

as a refinement of the statement of the first axiom. Because in the first axiom we said

to each quantum system there corresponds a Hilbert space, but I didn't tell you in any

way how you arrive at that Hilbert space. Now the addendum to the first axiom, one is

that if a quantum system is composed of two, and I should add and hence by induction by