Consider the Hamiltonian H, namely, H=A(Jz,1 +Jz,2)+BJ1 ·J2 for a system of two particles in which...

Consider the Hamiltonian H, namely,

H=A(Jz,1 +Jz,2)+BJ1 ·J2

for a system of two particles in which the first particle has a spin of 3/2 (j1), and the other has a spin of 1/2 (j2).

a) Compute the Hamiltonian matrix in the |j1j2;m1m2⟩ basis. (Hint: Be sure to rewrite ˆˆˆˆ

the J1 · J2 operator in terms of J±,i and Jz,i.) Is your matrix diagonal?

b) Determine the eigenvalues of the matrix you found in a).

c) Now compute the Hamiltonian matrix in the |j1j2;JM⟩ basis, where J and M are the ˆ2 ˆ ˆˆˆ

eigenvalues associated with the J and Jz operators, respectively. (Recall J ≡ J1 +J2.) Is your matrix diagonal? What energy eigenvalues do you find? Do your energies agree with those found in the other basis? Should they?

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genius_generous answered 3 months ago

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