solution3 - 8. A system with Hamiltonian H (0) is sented by...

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8. A system with Hamiltonian H (0) is subject to a perturbation H (1) , which in a certain ONB can be repre- sented by the following matrices h H (0) i = ε 0 00 0 0 0 ε 0 00 0 ε 0 0002 ε 0 0 000 02 ε 0 h H (1) i = 0 0 0 0 0 000 0 i 000 i 0 (a) Find the new energies of this system, correct to f rst order in the perturbation. From the diagonal elements of H (0) we see that the unperturbed system has two energies, ε 0 which is triply degenerate and 2 ε 0 which is doubly degenerate. These degeneracies are not removed by the diagonal elements of H (1) . Thus we cannot apply the formula’s of nondegenerate perturbation theory. We must follow the basic rule of degenerate perturbation theory: diagonalize the submatrix representing the perturbation in each degenerate subspace of H (0) . In the 3-dimensional subspace of H (0) with energy ε 0 we diagonalize the matrix h H (1) i 1 = 0 0 0 0 0 det ³h H (1) i 1 λ ´ = ¯ ¯ ¯ ¯ ¯ ¯ λ 0 λ 0 λ ¯ ¯ ¯ ¯ ¯ ¯ = λ 3 +2 2 λ which has three distinct roots: λ =0 , ± 2 , giving new energies ε 1 = ε 0 2 ε 2 = ε 0 ε 3 = ε 0 + 2 In the 2 dimensional subspace of energy 2 ε 0 we diagonalize the matrix h H (1) i 2 = μ 0 i i 0 det ³h H (1) i 2 λ ´ = ¯ ¯ ¯ ¯ λ i i λ ¯ ¯ ¯ ¯ = λ 2
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solution3 - 8. A system with Hamiltonian H (0) is sented by...

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