# For given Hamiltonian, is spin conserved?

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In summary: In (b), you should find the eigenvalues of H and the corresponding eigenstates. The eigenstates will be expressed in terms of the eigenstates of the total spin.
Leicester Fantasy
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A system consisting of two spins is described by the Hamiltonian (b>0)
H = aσ1 ⋅ σ2 + b1z - σ2z)
where a and b are constants.
(a) Is the total spin S = ½ (σ1 + σ2) conserved? Which components of S, if any, are conserved?
(b) Find the eigenvalues of H and the corresponding eigenstates in terms of the eigenstates of the total spin S.

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Thank you for your attention sir. I'm first time PH, so I made a mistake sorry.
I can't understand how do I know the spins of the system if there is given a Hamiltoninan.
In (a), I think that there's no term for time, so the Hamiltoninan does not change along the time. But I don't know the how to solve this problem.
In (b), should I use some spinors? How do I express the eigenstates? matrix, vector, or ket notation? would you give me a example, please?

For (a), you are actually asked to calculate ##[H,S_x]##, ##[H,S_y]##, and ##[H,S_z]##. Then find which of them vanish, which ones do not. Here ##S_i## for ##i=x,y,z## are the component of the total spin.

## 1. What is spin conservation?

Spin conservation refers to the principle that the total spin of a system remains constant, even as the individual particles within the system may change their spin states.

## 2. How does spin conservation relate to the Hamiltonian?

The Hamiltonian is the operator that describes the total energy of a system. For a given Hamiltonian, the spin of the system is conserved if the operator commutes with the total spin operator.

## 3. Why is spin conservation important in physics?

Spin conservation is important because it is a fundamental property of particles and plays a crucial role in many physical phenomena, such as magnetic properties and quantum states.

## 4. Can spin conservation be violated?

No, spin conservation is a fundamental law of nature and cannot be violated.

## 5. How is spin conservation measured?

Spin conservation can be measured by observing the spin states of particles before and after interactions, and comparing the total spin of the system. Conservation of spin implies that the total spin will remain constant throughout the interaction.

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