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Unformatted text preview: Physics 828: Problem Set 5 Due Friday, February 15, 2008 at beginning of class 1. Consider the onedimensional harmonic oscillator with a quartic per turbation. The Hamiltonian is then given by H = p 2 2 m + 1 2 kx 2 + x 4 . (1) where k is the spring constant and is a parameter governing the size of the perturbing potential. (a) Calculate the perturbed energy levels to first order in . Hint: use the raising and lowering operators for the harmonic oscillator. (b). For a given , estimate the quantum number n where the first order perturbation becomes inaccurate. (c). Write down formally the second order term to the energy correc tion. Explain which matrix elements in the sum will be nonzero, and why. (d). What is the sign of the secondorder correction for the ground state (n = 0)? 2. Consider a spin1/2 particle in a timeindependent external magnetic field B x x + B z z . Assume that the Hamiltonian is H = S B , where > 0 and B x > 0....
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 Winter '08
 STROUD
 Physics, mechanics

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