How can get the eigenvalues of the two spin entangle

[nicxm]

there is a equation for two spins in entangled
Hamiltonian: H=J($$\vec{\sigma}^1\cdot\vec{\sigma}^2+\vec{\sigma}^2\cdot\vec{\sigma}^1$$)+B($$\vec{\sigma}^1_z+\vec{\sigma}^2_z$$)

where $$\vec{\sigma}^i=(\sigma^i_x,\sigma^i_y,\sigma^i_z)$$ are the pauli matrics for the ith (i=1,2) spin. J is the exchange constant ,

My difficulties: how can we solve the Hamiltonian to get the four eigenvalues?

thank you for helping me, and i want a detail reprentation.

 

[malawi_glenn]

Simply let H act on your four different spin states.

The triplet S_tot = 1, M=+1,0,-1

and the singlet S_tot = 0, M =0

Good luck