{25 marks) A lOJDg marble is gently placed on a horizontal tabletop that is LTS m wide. (a) What is the maximum uncertaintyr in the horizontal position of the marble? (b) According to Heisenberg uncertainty principle, what is the minimum uncertainty in the horizontal velocity ofthe marble ? (c) In light ofyour answer to part (b]_ what is the longest time the marble could remain on the table? Compare this time to the age of the universe= which is approximatelyl4 x 109 years . {25 marks) Consider a particle with mass m moving in a potential U = ka2 , as in a mass spring system. The total energy of the particle is E 2 p31\" in + l: xZIZ. Assume thatp and x are approximately related by the Heisenberg uncertainty principle: so pat a: h. (a) Calculate the minimum possible value of the energy E, and the value of): that gives this minimum E. This lowest possible energy, which is not zero= is called the zeropoint energy. (b) For the I calculated in part {a}, what is the ratio of the kinetic to the potential energy of the particle ? {25 marks ) A particle moving in one rimension is described by the wave function { } Ae'\". x20 1-11 x = Be\": , x