- #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 doesnt 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 doesnt feel so

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