(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Two spin-half particles with spins S1 and S2 interact with a spin-dependent Hamiltonian H=λS1*S2 (the multiplication is a dot product and is a positive constant). Find the eigenstates and eigenvalues of H in terms of |m1,m2>, where (hbar)m1 and (hbar)m2 are the z-components of the two spins.

2. Relevant equations

Sx |m>=1/2(Sp-Ss) |m>

Sy |m>=1/2i(Sp+Ss) |m>

Sz |m>=(hbar)m |m>

Sp=(hbar)√[s(s+1)-m(m+1)]

Ss=(hbar)√[s(s+1)-m(m-1)]

3. The attempt at a solution

S1*S2=S1xS2x+S1yS2y+S1zS2z

S1x=S2x=S1y=S2y=0. I said this because the problem only mentioned z-component and most problems only talk about Sz.

H|m1,m2>=λSz1Sz1|m1,m2>

H |1/2,1/2> = λ*(hbar)^2 (1/2)(1/2) |1/2,1/2> = λ*(hbar)^2/4 |1/2,1/2>

H |-1/2,1/2> = λ*(hbar)^2 (-1/2)(1/2) |1/2,1/2> = -λ*(hbar)^2/4 |-1/2,1/2>

H |1/2,-1/2> = λ*(hbar)^2 (1/2)(-1/2) |1/2,1/2> = -λ*(hbar)^2/4 |1/2,-1/2>

H |-1/2,-1/2> = λ*(hbar)^2 (-1/2)(-1/2) |-1/2,-1/2> = λ*(hbar)^2/4 |-1/2,-1/2>

Is this my final answer? Am I close? Or was I completely off.

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# Spin-dependent Hamiltonian of two particles

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