2nd order perturbation calculation for a system involving spins

[Thunder_Jet]

Hello!

I am answering a problem which involves spins in the hamiltonian. The hamiltonian is given by H = B(a1Sz^(1) + a2Sz^(2)) + λS^(1)dotS^(2). The Sz^(1) and Sz^(2) refers to the Sz of the 1st and 2nd spins respectively. B is the magnetic field and the others are just constants. The question is to use the 2nd order perturbation theory to approximate the energy eigenvalues, given that the second term is the perturbing potential. Also it says that the problem is solvable exactly, so that the answer can be verified. I haven't been exposed to spins and its hamiltonian or eigenvalues. Please suggest a detailed way of attacking this proble. Thanks a lot!

 

[azntoon]

In order to solve this problem using second order perturbation theory, you will need to first calculate the zeroth order energy eigenvalues of the Hamiltonian. This can be done by setting λ=0 and then calculating the energy eigenvalues for each of the basis states (e.g., Sz^(1)=+1/2, Sz^(2)=+1/2). This will give you the unperturbed energy eigenvalues. Once you have the zeroth order energy eigenvalues, you can then calculate the first order corrections to the energy eigenvalues using the following formula: E_n = E_n^(0) + λΣ_{m≠n}(|<n|H_1|m>|^2/(E_n^(0)-E_m^(0))),where H_1 is the perturbing potential of the Hamiltonian, E_n^(0) is the zeroth order energy eigenvalue of the nth state, and the summation is over all states m except for the nth state. In this case, H_1=S^(1)dotS^(2). Finally, you can calculate the second order corrections to the energy eigenvalues using the following formula: E_n = E_n^(0) + λΣ_{m≠n}(|<n|H_1|m>|^2/(E_n^(0)-E_m^(0))) + λ^2Σ_{m,l≠n}(|<n|H_1|m><m|H_1|l>|/(E_n^(0)-E_m^(0))(E_n^(0)-E_l^(0))),where the second summation is over all states m and l except for the nth state. Once you have calculated the energy eigenvalues using second order perturbation theory, you can then compare them to the exact energy eigenvalues to check your work.