Problem 6: ore n and at are the annihilation and creation operators of a simple tiarn he mmber is the total angular niomentum quantum nmumbor: Prove where d an oscillator satistying the usual simp...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian of a particle of a simple Harmonic oscillator potential in one dimension can be expressed in term of the creation and annihilation operators àt and à, respectively, as: or with In >, n = 0,1,..) are the nth eigenstates of the above Hamiltonian. Part A A.1. Show that the energy levels of a simple harmonic oscillator are E,' Aw (nti), n=0, 12, A.2. Calculate...
6. The energy levels of a harmonic oscillator with angular frequency w are given by 2 (a) Suppose that a system of N almost independent oscillators has total energy E^Nhw 2 Mhw. Show that the number of states with exactly this energy equals the number of ways of distributing M identical objects among N compartments and that this number 1S MI(N 1) Hint: Consider the number of distinct arrangements of a set of M objects and N -1 partitions (b)...