Question

Imagine a harmonic oscillator with Hamiltonian H=p^2/2m+½x^2 For simplicity, we will assume that m=ℏ=1. First, we set up our system in the first excited state of this Hamiltonian. Second, we turn on an extra potential, V_ex(x)=x^4. Third, before the added potential has a chance to change the system state, we measure the energy.By expressing H and V_ex in terms of the raising and lowering operators, evaluate the average energy we would see if we repeated this whole process many times.

Answer 1