[Intro QM] A bit confused on spin of system notation

In summary, Griffiths discusses the notation ##\chi=\chi_1\chi_2## in the section on the 'total' spin of two spin 1/2 particles. This notation represents a combined wave function for particle 1 and particle 2, where ##\chi_1## and ##\chi_2## represent the spin relative to a basis. When multiplied, this notation gives the probability density function for the particles to be found in a specific range of x. It is a sloppy notation but is used to represent the combined probability of both particles being in specific ranges of x.
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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?
 
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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?
 

1. What is spin notation in quantum mechanics?

Spin notation in quantum mechanics is a mathematical representation of the intrinsic angular momentum of a quantum system. It is denoted by the symbol S and is measured in units of ħ/2, where ħ is the reduced Planck constant.

2. How is spin notation represented in quantum mechanics?

Spin notation is typically represented by an arrow pointing in the direction of the spin, with its length proportional to the magnitude of the spin. The direction of the arrow represents the spin's orientation in space.

3. What is the significance of spin notation in quantum mechanics?

Spin notation is significant in quantum mechanics because it is a fundamental property of particles and is used to describe their behavior in certain physical systems. It also plays a crucial role in determining the energy levels and interactions of particles.

4. Are there different types of spin notation in quantum mechanics?

Yes, there are different types of spin notation in quantum mechanics, such as spin up and spin down notation, which indicate the direction of the spin in relation to a chosen axis. There is also spinor notation, which is used to describe the spin of particles with half-integer spin values.

5. How is spin notation related to the Pauli exclusion principle?

The Pauli exclusion principle states that no two particles can have the same set of quantum numbers. Spin notation is one of these quantum numbers, and it plays a crucial role in the principle by limiting the number of particles that can occupy a particular energy state, due to the restrictions on the possible spin values.

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