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2. *Problem 6.9 Consider a quantum system with just three linearly independent states. The Hamiltonian, in matrix form, is 0. H = Vo 1 where Vo is a constant and e is some small number (e « 1). (a) Write down the eigenvectors and eigenvalues of the unperturbed Hamiltonian (e = 0). %3D (b) Solve for the exact eigenvalues of H. Expand each of them as a power series in E, up to second order. (c) Use first- and second-order nondegenerate perturbation theory to find the ap- proximate eigenvalue for the state that grows out of the nondegenerate eigen- vector of H. Compare the exact result from (b). (d) Use degenerate perturbation theory to find the first-order correction to the two initially degenerate eigenvalues. Compare the exact results.
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Transcribed Image Text:2. *Problem 6.9 Consider a quantum system with just three linearly independent states. The Hamiltonian, in matrix form, is 0. H = Vo 1 where Vo is a constant and e is some small number (e « 1). (a) Write down the eigenvectors and eigenvalues of the unperturbed Hamiltonian (e = 0). %3D (b) Solve for the exact eigenvalues of H. Expand each of them as a power series in E, up to second order. (c) Use first- and second-order nondegenerate perturbation theory to find the ap- proximate eigenvalue for the state that grows out of the nondegenerate eigen- vector of H. Compare the exact result from (b). (d) Use degenerate perturbation theory to find the first-order correction to the two initially degenerate eigenvalues. Compare the exact results.
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