Problem 1 [6] Compute the eigenvalues of Hˆ = pˆ 2 2m + 1 2 mΩ2xˆ 2 + λxˆ using two different methods: 1. Complete the square in1 2mΩ 2x 2+λx (that is, write the term as 1 2mΩ 2 (x− x0) 2+C withsuitable constants x0 and C) and use the exact eigenvalues En = (n+1 2 )¯hω of a harmonic oscillator with potential V (x) = 1 2mω2x 2. 2. Apply second-order perturbation theory in λ
Problem 2 [2] Compute the eigenvalues of the matrix Hˆ = 2 λ λ 3− 2λ ! and Taylor expand them to second order in the real numberλ
