I How to study a particle with several features?

[Tio Barnabe]

Usually textbooks on QM deals with systems with a single feature. For example, we could analyse electron spin. In such case the state vector is a (function?) only of the corresponding "spin variable" for spin, etc...
But suppose I'm interested in say, study about electron spin and also its position. In such a case, the state vector would need to contain information of both spin and position, correct?
How can we proceed in this case? Should we consider spin part as being in a space while position part in another? And make the (appropriate, according to the rules) product between these spaces? If the answer is, yes, I have the following question: What would be the dimension of the resulting product space?

For instance, the electron spin space would be 2 dimensional, while the position space would be 3 dimensional. Would the dimension of the resulting product space be 5?

Also, based on my questions above, if someone could indicate me some good lectures on web about the related math of the Hilbert space, such that I could finely understand how to mathematically describe what I asked, I will appreciate.

I have found excellent books about Hilbert Space in the uni library, but unfortunately none of them deals with the situation above, i.e. the rules, algebra, etc, when we have to consider product spaces.

 

[zonde]

In QM electron can be in superposition of positions and this superposition is not some other new position but rather part of probability amplitude here and part of probability amplitude there. So position space is infinite dimensional i.e. for each position it's own dimension. For spin it's only three dimensions as superposition of spin states is new spin state in certain direction.

 

[Tio Barnabe]

Oh, yes. I realized that! The components of a state vector can even be written as the probability amplitude for the state to be in each of the positions, if I remember well... As there are infinite possible positions, the space is infinite dimensional.

I'd like to receive a answer for my other questions, though.

 

[DrClaude]

Yes, you can build up the full Hilbert space by considering a direct sum the tensor product of a Hilbert space for position/momentum, a Hilbert space for spin, etc. You do the same when considering more than one particle.

I don't know of any online resource, but if your library has a copy of Greiner's Quantum Mechanics - An Introduction, it has a good chapter on the subject.

 

[Tio Barnabe]

Thanks dr Claude

Greiner's Quantum Mechanics - An Introduction, it has a good chapter on the subject.

do you know what chapter is that?

 

[A. Neumaier]

Yes, you can build up the full Hilbert space by considering a direct sum of a Hilbert space for position/momentum, a Hilbert space for spin, etc.

No, not a direct sum!

One needs a tensor product of the position Hilbert space and the spin Hilbert space. This means that wave functions are functions of the position coordinate and the spin index.

 

[DrClaude]

No, not a direct sum!

You are right, of course.

 

[DrClaude]

do you know what chapter is that?

Chapter 16

 

[Talisman]

Have you seen examples that consider the joint state of, say, particle 1's spin and particle 2's spin? Their joint state comes from the tensor products of the individual state spaces in exactly the same way as the first particle's spin and any other commuting observable.