# Hamiltonian eigenvalues

Find the eigenvalues of the hamiltonian

$$H=a(S_A \cdot S_B+S_B \cdot S_C+S_C \cdot S_D+S_D \cdot S_A)$$

where S_A, S_B, S_C, S_D are spin 1/2 objects
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I rewrite it as

$$H=(1/2)*a*[(S_A+S_B+S_C+S_D)^2-(S_A+S_C)^2-(S_B+S_D)^2]$$

then i define

$$J_1=S_A+S_B+S_C+S_D$$

$$J_2=S_A+S_C$$

$$J_3=S_B+S_D$$

and uses

$$J^2_i |j_1j_2j_3;m_1m_2m_3> = (h^2) j_i(j_i+1)|j_1j_2j_3;m_1m_2m_3>$$

which gives the energies

$$E(j_1,j_2,j_3)=(h^2/2)*a*[j_1(j_1+1)-j_2(j_2+1)-j_3(j_3+1)]$$

Where j_1 is addition of four angular momentum of 1/2 which gives it values of 0 1, 2 and in the same way j_2 and j_3 have values of 0 1.

Am i doing this the right way? It doesnt feel so

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From the structure of your hamiltonian it almost looks like you could adapt transfer matrix methods, unless your spin things are vectors (I'm not clear on that). I would also say that there are restrictions on $$j_2$$ and $$j_3$$ based on $$j_1$$, but the thought process seems right.

Meir Achuz
Homework Helper
Gold Member
Your method is completely correct. Just include the a.
Did it just seem too easy?

Meir Achuz said:
Your method is completely correct. Just include the a.
Did it just seem too easy?

Thank you. Yes it seemed too easy

How about the degeneracy of the energy levels.
For example E(010)=E(001)=E(111) and then m_1 can take on 9 different values , m_2 and m_3 5 different values. So the degeneracy of this level is 3*9*5*5 ? Is it correct so far?

But then the j_i in turn are addtions of angular momentums.
Does this add even more to the degeneracy?

Meir Achuz