- #1
JohanL
- 158
- 0
Find the eigenvalues of the hamiltonian
[tex]
H=a(S_A \cdot S_B+S_B \cdot S_C+S_C \cdot S_D+S_D \cdot S_A)
[/tex]
where S_A, S_B, S_C, S_D are spin 1/2 objects
_________________________
I rewrite it as
[tex]
H=(1/2)*a*[(S_A+S_B+S_C+S_D)^2-(S_A+S_C)^2-(S_B+S_D)^2]
[/tex]
then i define
[tex]
J_1=S_A+S_B+S_C+S_D
[/tex]
[tex]
J_2=S_A+S_C
[/tex]
[tex]
J_3=S_B+S_D
[/tex]
and uses
[tex]
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>
[/tex]
which gives the energies
[tex]
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)]
[/tex]
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 doesn't feel so
[tex]
H=a(S_A \cdot S_B+S_B \cdot S_C+S_C \cdot S_D+S_D \cdot S_A)
[/tex]
where S_A, S_B, S_C, S_D are spin 1/2 objects
_________________________
I rewrite it as
[tex]
H=(1/2)*a*[(S_A+S_B+S_C+S_D)^2-(S_A+S_C)^2-(S_B+S_D)^2]
[/tex]
then i define
[tex]
J_1=S_A+S_B+S_C+S_D
[/tex]
[tex]
J_2=S_A+S_C
[/tex]
[tex]
J_3=S_B+S_D
[/tex]
and uses
[tex]
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>
[/tex]
which gives the energies
[tex]
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)]
[/tex]
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 doesn't feel so
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