p828aps5 - Physics 828 Problem Set 5 Due Friday at...

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Physics 828: Problem Set 5 Due Friday, February 15, 2008 at beginning of class 1. Consider the one-dimensional harmonic oscillator with a quartic per- turbation. The Hamiltonian is then given by H = p 2 2 m + 1 2 kx 2 + λx 4 . (1) where k is the spring constant and λ is a parameter governing the size of the perturbing potential. (a) Calculate the perturbed energy levels to first order in λ . Hint: use the raising and lowering operators for the harmonic oscillator. (b). For a given λ , 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 non-zero, and why. (d). What is the sign of the second-order correction for the ground state (n = 0)? 2. Consider a spin-1/2 particle in a time-independent external magnetic field B x ˆ x + B z ˆ z . Assume that the Hamiltonian is H = γ S · B , where γ > 0 and B x > 0.
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