Answer Happy

Consider the potential for a two-dimensional isotropic harmonic oscillator of frequency w and recall that the steady states Unm are given by the product of the steady states Un and Um of two one-dimensional oscillators with the same frequency, and that the energy associated with Unm is En = (1+n+2)ħw, with ñ=n+m, where ground states are counted from zero. a) How many states of a particle share the energy En? Remember that this is the degeneracy di associated with the energy En. b) Suppose you now place two non-interacting particles in this potential and write down all distribution sets of this system with total energy 4hw. Remember that a distribution set is described by listing its occurrence numbers, which in this case is the number of particles Nñ with energy Eń. c) Using a direct count, determine the number of ways in which each of the distribution sets in part b) can be realized for the cases in which the particles are i) distinguishable, ii) identical bosons, iii) identical fermions. In no case consider the spin. d) For each case of c), calculate the total number of states of two particles that have total energy 4ħw and use this number, together with the results of the previous parts, to calculate the probability that when measuring the energy of one of these two random particles, we obtain E = {hw.