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  3. consider the hamiltonian h, namely, h=a(jz,1 +jz,

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

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Physics

Consider the Hamiltonian H, namely,

H=A(Jz,1 +Jz,2)+BJ1 ·J2

for a system of two particles in which the first particle hasa 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 yourmatrix diagonal?

b) Determine the eigenvalues of the matrix you found ina).

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? Whatenergy eigenvalues do you find? Do your energies agree with thosefound in the other basis? Should they?

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