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19matthew89

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## Homework Statement

Hi,

I must find eigenvalues and eigenvector of this Hamiltonian, which describes a system of two 1/2-spin particles.

H = A([itex]S_{1z}[/itex] - [itex]S_{2z}[/itex]) + B([itex]S_{1}[/itex] · [itex]S_{2}[/itex])

where [itex]S_{1}[/itex] and [itex]S_{2}[/itex] are the two spins, [itex]S_{1z}[/itex] and [itex]S_{2z}[/itex] are their z-components, and A and B are constants.

## Homework Equations

Since it's easy finding eigenvalues and eigevectors of ([itex]S_{1}[/itex]· [itex]S_{2}[/itex]) in total spin base (I can write it as (0.5)([itex]S^{2}[/itex]-[itex]{S_{1}}^{2}[/itex]-[itex]{S_{2}}^{2}[/itex]) I've thought I should use this base, but [itex]S_{1z}[/itex] and [itex]S_{2z}[/itex] do not commute.

## The Attempt at a Solution

I've tried to solve this problem in this way. I know you can write this Hamiltonian using the basis {|++>; |+->; |-+>; |-->} and, if I know how H acts on this basis I can write a matrix which represents the action of H. So I computed how H acts using the basis {|++>; |+->; |-+>; |-->} to estimate the action of A([itex]S_{1z}[/itex] - [itex]S_{2z}[/itex]) and the basis {|11>; |10>; |1-1>; |00>} to estimate the action of B([itex]S_{1}[/itex] · [itex]S_{2}[/itex]) and the I've correlated the two bases using Clebcsh-Gordan coefficients. Is this way of proceeding correct?

At the end I've found a matrix in {|++>; |+->; |-+>; |-->} which, obviously, is not diagonal. How can now find eigenvalues and eigenvector? Diagonalizing? Is there a faster way to get a diagonal matrix?

Thanks

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