€ 2/ here 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). (b) Solve for the exact eigenvalues of H. Expand each of them as a power series ir 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 H0, Compare the exact result from (b).
€ 2/ here 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). (b) Solve for the exact eigenvalues of H. Expand each of them as a power series ir 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 H0, Compare the exact result from (b).
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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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