Notation for two-particle spin states in addition of angular momenta

[Coffee_]

1. In Griffiths, in the section where he discusses the 'total' spin of two spin 1/2 particles (''Addition of angular momenta'') he starts of using a notation new to me. Namely $\chi=\chi_1\chi_2$ where I know what $\chi_1$ and $\chi_2$ represent independently, which has been discussed in the previous chapter. They were a representation of the spin relative to a basis before, and hence vectors. How to read this notation, is it a product? Is it some union-type of thing?

 

[DEvens]

It is kind of a sloppy notation.

Think what chi(x) means for a single particle. It means when you take chi*(x) times chi(x) you get the probability density function, that is, the probability that the particle will be found in the range dx of x.

What will you write down for two particles? Think about a combined wave function for particle 1 and particle 2. So chi1(x1) gives, after the same process, the probability density function for particle 1 to be in dx1 of x1, and chi2(x2) gives the same for particle 2 in dx2 of x2.

So what function would you need so that when you took chi* times chi you got the probability that particle 1 was in dx1 and particle 2 was in dx2?