# Exploring Spin in 3-Particle Systems

• broegger
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.
broegger
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,

$$\tfrac1{\sqrt{2}}(|\uparrow\downarrow\rangle + |\downarrow\uparrow\rangle$$,​

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.

Last edited:
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.

Huh? We aren't suppose to use the Clebsch-Gordon coefficients (we skipped that part).

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

$$\mathcal{E}_{\frac{1}{2}}\otimes \mathcal{E}_{\frac{1}{2}}\otimes \mathcal{E}_{\frac{1}{2}}$$

Daniel.

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

$$\mathcal{E}_{\frac{1}{2}}\otimes \mathcal{E}_{\frac{1}{2}}\otimes \mathcal{E}_{\frac{1}{2}}$$

I'm not familiar with that notation or the Clebsch-Gordan theorem. We're not supposed to use that (trust me).

There are 3 irreducible representations (3 irreducible spaces) spanned by the vectors given by the C-G theorem...

Daniel.

## 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.

• Quantum Physics
Replies
7
Views
925
• Quantum Physics
Replies
61
Views
2K
• Quantum Physics
Replies
1
Views
668
• Quantum Physics
Replies
8
Views
782
• Quantum Physics
Replies
2
Views
1K
• Quantum Physics
Replies
2
Views
257
• Quantum Physics
Replies
12
Views
1K
• Quantum Physics
Replies
12
Views
2K
• Quantum Physics
Replies
1
Views
1K
• Quantum Physics
Replies
1
Views
866