Exploring Spin in 3-Particle Systems

In summary: So we're supposed to use the Clebsch-Gordon coefficients? How do they work?The Clebsch-Gordon coefficients work by combining the states of two particles to get a third state. So if you have a system of two spin-1/2-particles, you can combine them to get a third state that has the total spin of both particles combined.
  • #1
broegger
257
0
Hi,

I have to find out the possible total spins for a three-particle system composed of spin-1/2-particles. My guess is that there are two possible spins; 1/2 (one up, the others down or vice versa) and 3/2 (all up or all down), but I'm not sure.

In my book they show how to find the total spin of a system composed of two spin-1/2-particles, but I don't understand the derivation. He talks about triplets and singlets (what is that!?) and apparently the state,

[tex]\tfrac1{\sqrt{2}}(|\uparrow\downarrow\rangle + |\downarrow\uparrow\rangle[/tex],​

represents a system of total spin 1. How come? I don't get it.

Also, another question: Is the total spin of a spin-1/2 particle s = 1/2 or is it slightly bigger (like for orbital angular momentum, where the total is always bigger than the z-component). I would think that it is, since if it is 1/2 you would know the direction of the spin vector completely (Sx = 0, Sy = 0, Sz = +/-1/2), which would violate the uncertainty principle.
 
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  • #2
What did u make of Clebsch-Gordan theorem and the C-G coefficients...?

There's one way to do it.Use the 2 1/2 spins case & compose it with a spin 1/2 case.Instead of 4,u'll have 8 states...

Daniel.
 
  • #3
Huh? We aren't suppose to use the Clebsch-Gordon coefficients (we skipped that part).
 
  • #4
Composing spins (and angular momenta in general) is done starting with the theorem of Clebsch-Gordan...Read it and compute

[tex] \mathcal{E}_{\frac{1}{2}}\otimes \mathcal{E}_{\frac{1}{2}}\otimes \mathcal{E}_{\frac{1}{2}} [/tex]



Daniel.
 
  • #5
Can anyone give a more intuitive explanation? Am I right in my initial guess?

And what about my last question?

dextercioby said:
Composing spins (and angular momenta in general) is done starting with the theorem of Clebsch-Gordan...Read it and compute

[tex] \mathcal{E}_{\frac{1}{2}}\otimes \mathcal{E}_{\frac{1}{2}}\otimes \mathcal{E}_{\frac{1}{2}} [/tex]

I'm not familiar with that notation or the Clebsch-Gordan theorem. We're not supposed to use that (trust me).
 
  • #6
There are 3 irreducible representations (3 irreducible spaces) spanned by the vectors given by the C-G theorem...

Daniel.
 

Related to Exploring Spin in 3-Particle Systems

1. What is the significance of studying spin in 3-particle systems?

Studying spin in 3-particle systems allows us to gain a deeper understanding of the fundamental properties of matter and the interactions between particles. It also has practical applications in fields such as quantum computing and materials science.

2. What is spin and how is it measured?

Spin is an intrinsic property of particles that describes their angular momentum. It is measured using a device called a Stern-Gerlach apparatus, which can detect the direction of a particle's spin.

3. How does spin affect particle behavior in 3-particle systems?

Spin plays a crucial role in determining the overall properties and behavior of a 3-particle system. It affects the energy levels, interactions, and stability of the system, and can also determine the likelihood of certain outcomes in particle collisions.

4. What is the difference between spin-1/2 and spin-1 particles?

Spin-1/2 particles, such as electrons, have two possible spin states (up or down) while spin-1 particles, like protons, have three possible spin states (up, down, or neutral). This difference affects the behavior and interactions of these particles in 3-particle systems.

5. How do scientists use mathematical models to explore spin in 3-particle systems?

Scientists use mathematical models, such as the Pauli spin matrices and the Clebsch-Gordan coefficients, to describe and predict the spin states and behaviors of particles in 3-particle systems. These models are based on quantum mechanics and have been experimentally validated to accurately represent spin phenomena.

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