# Simple harmonic oscillators-Quantum mechanics

 P: 59 1. The problem statement, all variables and given/known data An ion in a harmonic ion trap sees a potential which is effectively that of a simple harmonic oscillator. It has a natural oscillation frequency given by v = 1 MHz. Ignoring any internal excitations, it is known to be in a superposition of the n = 0, 1 and 2 SHO energy states. A measurement is then made and it is found to be in the n = 2 level. a)What is the energy of the ion after the measurement has been made? 3. The attempt at a solution Why is the answer E_n = (2n+1)/2 $$\hbar\omega$$ I do not understand the (2n+1) / 2 Thanks!
 P: 599 Hi, The average energy in the nth state (or in the phonon picture: number of phonons in a mode associated with frequency $$\omega$$) for a single harmonic oscillator is given by: $$E_n=\frac{2n+1}{2}\hbar\omega=(n+\frac{1}{2})\hbar\omega.$$ where $$h\nu=\hbar\omega.$$