Understand Hamiltonian: S^2, Sz & Constant in Time

[Cosmossos]

Hello
I saw someplace that if H is written in the total AM base as a function of S^2 and Sz then it's diagonal in that basis and the value of s^2 and Sz are constant in time.

S^2 and Sz are Its eigenvectors? no, becasue they are matrices.
why H is diagonal if it is written as function of S^2 and Sz?
And if H is diagonal why this implies that S^2 and Sz are constant in time?


Every time the H is diagonal it's eigenvalues are constant in time?

How can I tell based on the hamiltonian that values are constant in time?

thanks

 

[G01]

Hello
I saw someplace that if H is written in the total AM base as a function of S^2 and Sz then it's diagonal in that basis and the value of s^2 and Sz are constant in time.

S^2 and Sz are Its eigenvectors? no, becasue they are matrices.
why H is diagonal if it is written as function of S^2 and Sz?
And if H is diagonal why this implies that S^2 and Sz are constant in time?


Every time the H is diagonal it's eigenvalues are constant in time?

How can I tell based on the hamiltonian that values are constant in time?

thanks

When we say something like, "H is written in the angular momentum basis," what we mean is that we are using the eigenvectors of S^2 and Sz as the basis for the space, i.e. the |l,m> kets are the basis.

Once we pick these kets as our basis vectors, then we explicitly write out H. If it is diagonal then it's eigenvectors are the basis vectors of the space. So H also has the |l,m> kets as eigenvectors.


H, S^2, and Sz are all diagonal at once, therefore they must commute. Any operator that commutes with H is a constant of the motion, by the Heisenberg equation of motion. So this is why S^2 and Sz are constant if they commute with H.

 

[Cosmossos]

thanks you!