Solving Spin-Spin Interaction for Total Spin Commutation

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In summary: Therefore, each component of the total spin commutes with the hamiltonian and we have shown that the total spin operator $\sum_i S_i$ is a conserved quantity. In summary, by expanding the commutator and using the properties of the spin operators, we can show that each component of the total spin operator commutes with the hamiltonian, proving that the total spin is a conserved quantity.
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Diracobama2181
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How do I go about showing that the total Spin component commutes with the hamiltonian
Let there be four spin (1/2) particles at the corners of a tetrahedron, coupled such that the total hamiltonian is given by $$H=\sum_{i\neq j} S_{i} \cdot S_{J}$$.
How would I go about showing that each component of the total spin $$\sum_{i} S_{i}$$ commutes with the hamiltonian.

Work so far:
I know
$$\sum_{i} S_{i}^{x}=S_{1}^{x}+S_{2}^{x}+S_{3}^{x}+S_{4}^{x}$$
so $$[\sum_{i} S_{i}^{x},H]=[S_{1}^{x},H]+[S_{2}^{x},H]+[S_{3}^{x},H]+[S_{4}^{x},H]$$
From there, I can use $$H=S_{1}S_{2}+S_{1}S_{3}+S_{1}S_{4}+S_{2}S_{3}+S_{2}S_{4}+S_{3}S_{4}$$
I can also use $$S_{i}\cdot S_{j}=S_{i}^{x}S_{j}^{x}+S_{i}^{y}S_{j}^{y}+S_{i}^{z}S_{j}^{z}$$

As you can see, it gets rather messy. I just want to know if my reasoning is correct thus far and or if I am over complicating my solution.
Thank you.
 
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A:Yes, you are on the right track. What you need to do is expand out the commutator:$$[\sum_i S_i^x,H]=\sum_{i \neq j}[S_i^x,S_i\cdot S_j] + [S_j^x,S_i\cdot S_j].$$Using the fact that $[S_i^x,S_i\cdot S_j] = 0$ (this follows from the definition of the dot product and the fact that $S^x$ is linear in $S$), we see that the above equals$$[\sum_i S_i^x,H] = \sum_{i\neq j}[S_j^x,S_i\cdot S_j].$$Now using the fact that for any two $2\times 2$ matrices $A,B$ we have $[A,B]= AB - BA$, we can expand out the commutator on the right hand side to get$$[\sum_i S_i^x,H] = \sum_{i\neq j}(S_i\cdot S_j S_j^x - S_j^x S_i\cdot S_j).$$Finally, using the fact that $S^x$ is hermitian, we see that the above equals$$[\sum_i S_i^x,H] = \sum_{i\neq j}(S_i\cdot S_j S_j^x - S_i\cdot S_j S_j^x) = 0.$$The same argument can be applied to show that $[\sum_i S_i^y,H] = 0$ and $[\sum_i S_i^z,H] = 0$.
 

Related to Solving Spin-Spin Interaction for Total Spin Commutation

What is spin-spin interaction?

Spin-spin interaction is a phenomenon in quantum mechanics where the spins of two or more particles interact with each other, causing changes in their respective energies and states.

Why is it important to solve spin-spin interaction for total spin commutation?

Solving spin-spin interaction for total spin commutation is important because it allows us to accurately predict and understand the behavior of particles in quantum systems. It also has practical applications in fields such as quantum computing and nuclear magnetic resonance.

What methods are used to solve spin-spin interaction for total spin commutation?

There are several methods used to solve spin-spin interaction for total spin commutation, including perturbation theory, mean field theory, and variational methods. Each method has its own advantages and limitations, and the choice of method depends on the specific system being studied.

What is total spin commutation?

Total spin commutation is a mathematical concept in quantum mechanics that describes the behavior of a system when the total spin of its particles is conserved. This means that the total spin of the system remains constant even as individual spins may interact and change.

What are some real-world examples of spin-spin interaction?

Spin-spin interaction can be observed in many physical systems, such as in atoms, molecules, and nuclei. For example, it is responsible for the splitting of spectral lines in atomic and molecular spectra, and it is also used in nuclear magnetic resonance imaging to study the structure of molecules in the human body.

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