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Science · Advanced Physics
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Please write clearly the formulas and concepts behind it. Thanks a lot.

4. A four-level system has the Hamiltonian: λ, ε 0 ε, ελ, 00 where ε and ε, are non-zero small (real) perturbation of similar magnitudes. problem we use the notations for unperturbed levels: In this 0 0 4 0 (a) Assuming λ¡ (J = 1, 2, 3, 4) are all distinct, find the first order correction to eigen-energies for all the four levels due to ε and ε, perturbation. (b) Continue (a), find the second order correction of the energies for 1) and 2). (7 marks) (c) From (b) indicate a condition that perturbation results are accurate. (d) If λ!=Α, and λ. λ4, use degenerate perturbation theory to obtain the first order mar (10 marks) correction of the energies for the states |1) and 2) mar (e) Based on (d), if λι-λ, and the system is initially in the state |1), find the expectation value of the operator | 1〉 〈2 as a function of time approximately. mar

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