1. (a) Consider a Hamiltonian H = Ho+AH' where (A << 1) is a small parameter. The unperturbed Hamiltonian Ho satisfies f

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1 A Consider A Hamiltonian H Ho Ah Where A 1 Is A Small Parameter The Unperturbed Hamiltonian Ho Satisfies F 1 (126.33 KiB) Viewed 74 times

1. (a) Consider a Hamiltonian H = Ho+AH' where (A << 1) is a small parameter. The unperturbed Hamiltonian Ho satisfies following eigenvalue equation: Hol >= E>, where all the eigenvalues are non degenerate. i. Calculate the first order corrections to the n-th energy eigenvalues and n-th eigenfunctions. ii. Calculate the second order corrections to the n-th energy eigenvalues. = (b) Suppose ) and ) are two degenerate orthogonal eigenvectors of the Hamiltonian Hº with same eigenvalue Eº. Let the system is perturbed by a Hamiltonian H' which does not com- mute with HD. So the system is now defined by the full Hamiltonian H HO+AH', where A << 1. Using degenerate perturbation theory calculate the eigenvalues of H. How does these eigenvalues vary with A. For simplicity you may assume that the (H'|V?) = 0 for i = a, b. (c) Consider a charged particle in the one-dimensional harmonic oscillator potential. Suppose we turn on a weak clectric field (E), so that the potential energy is shifted by an amount V'-qEx. Calculate the first and second order change in the energy for the n-th eigenstate. (5+4)+5+6=20