# Spin Matrices for Multiple Particles

• tomdodd4598
In summary, the three spin matrices for three particles with the same spin are:σ ⊗ I ⊗ I,I ⊗ σ ⊗ I andI ⊗ I ⊗ σ.
tomdodd4598
I have two questions, but the second is only worth asking if the answer to the first is yes:
Are the spin matrices for three particles, with the same spin,
σ ⊗ II,
I ⊗ σ ⊗ I
and
II ⊗ σ
for particles 1, 2 and 3 respectively, where σ is the spin matrix for a single one of the particles?

I know that the spin matrices for two particles are
σ ⊗ I and
I ⊗ σ
for particles 1 and 2 respectively, so I am guessing that the same is the case for when there are more than two particles.

Yes, it is.

Nice - thanks for the quick answer ;)

What I was wanting to ask was whether the following is a way to generalise the spin matrices for each of N particles:

Where w is the direction, a is the matrix/particle number, N is the number of particles and dim(V) = (2s + 1)^N, where s is the spin of each particle (1/2 for spin 1/2, 1 for spin 1 etc.).

You have to check again what the notations used in that expression mean. For instance, does the first ##I_{dim(V)}## means that it is actually a series of (Kronecker) product of ##a-1## individual identity matrices? If yes, then ##dim(V) = 2s+1##, that is, the dimension of the spin space corresponding to a single particle.
I think that is indeed the case there since the dimensions of the first and second ##I## are denoted identically.

Ah yes, I made a mistake there - yeh, the subscript of the identity matrices should be just 2s+1, and the a-1 and N-a under the brackets are the number of Kronecker/tensor products to have - for example, for 8 spin 1 particles, the 3rd matrix would be:

...which is a rather large matrix...

There is a problem though:

If I add together the three spin matrices (z-direction) for 3 spin 1/2 particles, I get:

But surely it should be:

It doesn't really matter as long as you remember, which eigenvector (when you have to calculate it) belongs to which eigenvalue. For example the eigenvector ##(0,0,0,1,0,0,0,0)^T## belongs to eigenvalue ##-\hbar## in the upper matrix, but belongs to ##\hbar## in the lower one.

## 1. What are spin matrices for multiple particles?

Spin matrices for multiple particles are mathematical representations used in quantum mechanics to describe the spin properties of particles. They are a set of matrices that correspond to different spin states of a particle and are used to calculate the probability of observing a particular spin state during an experiment.

## 2. How do spin matrices for multiple particles differ from single particle spin matrices?

Single particle spin matrices only describe the spin properties of a single particle, whereas spin matrices for multiple particles take into account the spin interactions between multiple particles. This allows for a more accurate description of the spin behavior of a system with multiple particles.

## 3. What is the significance of spin matrices for multiple particles in quantum mechanics?

Spin matrices for multiple particles play a crucial role in quantum mechanics as they are used to describe the spin properties of particles, which are important for understanding the behavior of subatomic particles and their interactions.

## 4. How are spin matrices for multiple particles calculated?

Spin matrices for multiple particles are calculated using a combination of the Pauli matrices and the tensor product. The Pauli matrices represent the spin states of individual particles, and the tensor product combines them to describe the spin states of multiple particles.

## 5. Can spin matrices for multiple particles be used to predict the spin behavior of a system?

Yes, spin matrices for multiple particles can be used to predict the spin behavior of a system. By using these matrices, scientists can calculate the probability of observing a particular spin state, which can be compared to experimental data to validate the predictions.

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