# Four spin 1/2 particles at the Vertices of tetrahedron

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• Diracobama2181
In summary, for a tetrahedron with four spin (1/2) particles, there are three separate energy levels at l=2, l=1, and l=0. To find the degeneracy of each level, you can use the representation theory of SU(2) and take the tensor product of four spin 1/2 states. This will result in a decomposition into different irreducible representations, with a total of 15 states (including a singlet state) for l=2, 9 states for l=1, and 2 states for l=0. It is important to note that different copies of irreps may have different energies and that the tetrahedral symmetry is just a subgroup of
Diracobama2181
For a tetrahedron with four spin (1/2) particles, I know there are three separate energy levels at $$l=2,l=1,and l=0$$. My question is how I would go about finding the degeneracy of each level. I know that the number of states must be $$2^4$$. Any clues on where to start would be appreciated. Thank you.

Are you familiar with the representation theory of SU(2)?

Unfortunately not. It still be interested in your explanation though.

Are you familiar with how two spin-1/2 couple to form a triplet (spin 1) and a singlet (spin 0) state?

Yes I am. I know you can essentially form separate pairs, and hence two groups of spin 1 and spin 0 states. Would I just take a tensor product of these two states?

Your full space is the tensor product of four spin 1/2 states. You can split them into different irreps product by product. So first do two 1/2, then do another 1/2 with the results from the first two, then do another 1/2 with the results from the first three. This will give you the decomposition into different irreps.

Diracobama2181
So $$(1/2)\bigotimes(1/2)\bigotimes(1/2)\bigotimes(1/2)$$? When I work it out, I get $$2\bigoplus1\bigoplus1\bigoplus1\bigoplus0$$ which would give me 5 states for l=2, 9 for l=1, and 2 for l=0. Thanks!

Almost. You are missing a singlet state (5+9+1=15, which is not 16). Edit: I see now that you said two l=0, so I assume this was just an error in the writing of the direct sum.

Note that, in general, different copies of irreps may have different energies even if they correspond to the same irrep. For example, the singlet states could a priori have different energies without violating the tetrahedral symmetry.

It is also the case that the tetrahedral symmetry is just a subgroup of rotations. In general you should check that this restricted symmetry does not split the irreps further.

Also note that the standard notation is to use the dimensionality of the representation, not the spin, ie, ##2\otimes 2\otimes2\otimes 2 = 5\oplus 3\oplus 3 \oplus 3 \oplus 1 \oplus 1##.

Diracobama2181

## 1. What is a tetrahedron and how does it relate to spin particles?

A tetrahedron is a three-dimensional shape with four triangular faces. It is commonly used in geometry and can represent the arrangement of four spin 1/2 particles at its vertices. Each vertex of the tetrahedron can be associated with a spin state of the particle, making it a useful model for studying spin interactions.

## 2. What are spin 1/2 particles and how do they behave?

Spin 1/2 particles are quantum particles that have a spin quantum number of 1/2. This means they have two possible spin states: spin up and spin down. These particles also exhibit properties of both particles and waves, and can be described by wave functions.

## 3. How do the spin particles interact with each other in a tetrahedron configuration?

In a tetrahedron configuration, the spin particles can interact with each other through a process called entanglement. This means that the state of one particle is dependent on the state of the other particles, even if they are separated by a large distance. The specific interactions between the particles will depend on their individual spin states and the overall spin state of the system.

## 4. Can the spin states of the particles in a tetrahedron be changed?

Yes, the spin states of the particles in a tetrahedron can be changed through various methods such as applying external magnetic fields or using quantum operations. These changes can affect the entanglement between the particles and alter the overall spin state of the system.

## 5. What are the potential applications of studying four spin 1/2 particles at the vertices of a tetrahedron?

Studying this system can provide insights into the behavior of quantum particles and the phenomenon of entanglement. It can also have potential applications in quantum computing and communication, as well as in understanding the properties of complex materials and systems.

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