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Question 01: Consider a 1-dimensional quantum system of one particle in which the particle is under a potential V(x) = {mwºr?, with m being the mass of the particle and w being a parameter (you may take it as angular fre- quency) with inverse dimension of time. The particle may be found in the region <<<0. What is the averaged position at which the particle may be found while it is in the ground state? What is the uncertainty in the position in the ground state? What is the uncertainty in the momentum in the ground state? See if Heisenberg uncertainty principle is varified at the ground state? • Calculate the probability that the particle make a transition to the first excited state under the operator 0 =qf. Varify that the lowest two states of the system are mutually orthonormal. Question 02: A certain physical system has energies En = n with n= 1, 2, 3, 4, 5, ... to be occupied by its atoms. Calculate the partition function for the system. On average what state is expected, i.e. in the form of n, at T = 1k? Calculate average energy for the system using the Boltzmann distribution. You may approximate the system by continuous n for your calculation. n.