Spin of system of n particles

In summary: However, the determination of all possible spin values for a given system of particles using these diagrams is still an open problem.
  • #1
Xian
25
0
When there are two particles with spin s1 and s2 the composite system has possible spins s1+s2 all the way down to |s2-s1| (using integer steps). The azimuthal quantum number goes from s1+s2 to -s1+s2 (using integer steps).

Now my question is if we had three particles (or more) how would we calculate their possible composite spin?
 
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  • #2
Hi Xian,

You add the spins one at a time. For example, [tex] 1/2 \otimes 1/2 = 0 \oplus 1 [/tex] and [tex] 1/2 \otimes 1/2 \otimes 1/2 = (0 \oplus 1 )\otimes 1/2 = 1/2 \oplus 1/2 \oplus 3/2 [/tex]. Basically you're doing the analog of distributing multiplication over addition. There are several different ways to group the spins in the 3 spin case, and the different choices are related by 6j symbols. Hopefully you can now figure out the generalization to N spins.

Hope this helps.
 
  • #3
Xian said:
When there are two particles with spin s1 and s2 the composite system has possible spins s1+s2 all the way down to |s2-s1| (using integer steps). The azimuthal quantum number goes from s1+s2 to -s1+s2 (using integer steps).

Now my question is if we had three particles (or more) how would we calculate their possible composite spin?

The maximum value of the spins is the same (i.e. the sum of all of the spins), and assuming , all values represented by integer steps (or half integer, if you are mixing fermions and bosons), are possible down to the minimum value. However, the minimum value (which obviously cannot be less than zero) is more complicated to determine in general. Some simple cases are easier to determine, such as the case of N particles each with the same spin s, in which case the minimum total spin will be zero for even N, and s for odd N. Where things really get complicated is with the degeneracies of the different total spin values. I know that analytical expressions exist for up to 4 spins (9-J symbols), but beyond that I am not sure.
 
  • #4
There exists a relatively easy way to obtain all the irreducible representations for the spin of n particles using so called "Weyl diagrams", at least for identical particles which is possibly the most important case. It makes use of the fact that the irreducible representations of the group SU(2) can be labeled by Young type diagrams pertaining to the Young type diagrams of the permutation group S_n of the n particles.
 

What is spin in quantum mechanics?

Spin is an intrinsic property of particles in quantum mechanics that describes their angular momentum and orientation in space. It is a fundamental property, similar to mass and charge, and cannot be changed or controlled.

How is the spin of a system of particles determined?

The total spin of a system of particles is determined by adding together the individual spins of each particle. This can be done using mathematical operators and equations in quantum mechanics.

Can the spin of a system of particles change?

No, the spin of a system of particles is conserved, meaning it cannot change over time. However, individual particles within the system can change their spin, which affects the overall spin of the system.

What is the significance of spin in particle physics?

Spin is an important property in particle physics as it helps distinguish between different types of particles and how they interact with each other. It also plays a crucial role in determining the stability and behavior of atoms and molecules.

What is the relationship between spin and angular momentum?

Spin is often referred to as the "intrinsic" or "internal" angular momentum of a particle, as it is not due to its physical rotation. However, spin and angular momentum are related, and both play a role in describing the motion and behavior of particles.

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