Question

A system with an unperturbed Hamiltonian H_0 is subject to a perturbation H' with H_0 = E_0 egin{pmatrix} 15 & 0 & 0 & 0 \ 0 & 3 & 0 & 0 \ 0 & 0 & 3 & 0 \ 0 & 0 & 0 & 3 end{pmatrix}, and H' = alpha E_0 egin{pmatrix} 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 0 end{pmatrix}. Where alpha is the strength of perturbation. 1. Find out the energy eigenvalues and eigenstates of the unperturbed Hamiltonian. 2. Write down the total Hamiltonian of the system in matrix form using H = H_0 + H'. 3. Find out the energy eigenvalues of the total Hamiltonian (H). 4. In calculating part (1) you will observe that H_0 has one nondegenerate eigenvalue and a threefold degenerate eigenvalues. Keeping this point in mind and just focusing the nondegenerate value of H_0 find the first-order correction in this specific energy. (Hint: E_n^1 = langle Psi_n^0 | H' | Psi_n^0 angle)

          A system with an unperturbed Hamiltonian H_0 is subject to a perturbation H' with
H_0 = E_0 egin{pmatrix} 15 & 0 & 0 & 0 \ 0 & 3 & 0 & 0 \ 0 & 0 & 3 & 0 \ 0 & 0 & 0 & 3 end{pmatrix}, and H' = alpha E_0 egin{pmatrix} 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 0 end{pmatrix}.
Where alpha is the strength of perturbation.
1. Find out the energy eigenvalues and eigenstates of the unperturbed Hamiltonian.
2. Write down the total Hamiltonian of the system in matrix form using H = H_0 + H'.
3. Find out the energy eigenvalues of the total Hamiltonian (H).
4. In calculating part (1) you will observe that H_0 has one nondegenerate eigenvalue and a threefold degenerate eigenvalues. Keeping this point in mind and just focusing the nondegenerate value of H_0 find the first-order correction in this specific energy. (Hint: E_n^1 = langle Psi_n^0 | H' | Psi_n^0 
angle)

A system with an unperturbed Hamiltonian H0 is subject to a perturbation H' with
H0 = E0 eginpmatrix 15 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3 endpmatrix, and H' = alpha E0 eginpmatrix 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 endpmatrix.
Where alpha is the strength of perturbation.
1. Find out the energy eigenvalues and eigenstates of the unperturbed Hamiltonian.
2. Write down the total Hamiltonian of the system in matrix form using H = H0 + H'.
3. Find out the energy eigenvalues of the total Hamiltonian (H).
4. In calculating part (1) you will observe that H0 has one nondegenerate eigenvalue and a threefold degenerate eigenvalues. Keeping this point in mind and just focusing the nondegenerate value of H0 find the first-order correction in this specific energy. (Hint: En^1 = langle Psin^0 | H' | Psin^0
angle)

Added by James B.

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