Answer Happy

(Q1) (20 pts) Partition Function for 3-Dimensional Har- monic Oscillator Consider a system consisting of N non-interacting and indistinguishable particles with mass m. The ith particle is under an effect of a confining potential V(x, y, z): V(x, y, z) = mw²(x² + y² +2²) Hence, the Hamiltonian for the ith particle becomes: H = +Vi(2,34,2) P² 2m (a) (2 pts) Write down the Hamiltonian H for the system of N particles and provide the partition function ZN of N particles expressed in the configuration (or phase) space. (Leave it in an integral form.) (b) (3 pts) Reexpress Zy as a function of single particle partition functions Z₁ and clearly show that ZN = ZN/N!. (c) (6 pts) Find Z₁ by calculating the integrals. Find Zy. (d) (6 pts) Calculate the expected internal energy E of the N particle system using ZN. (e) (3 pts) Calculate the expected internal energy E of the N particle system using equipar- tition theorem and compare it with the result in part(d).