2. Consider a particle of mass m is subjected to a one-dimensional potential of the following form: if -a < x < 3a; otherwise. 0, (a) Find out the energy of the particle in the first excited state. (b) Sketch the probability density for the first excited state in the entire range of the potential. (c) Suppose you add another (non-interacting) particle in the potential, how much change you will get in the energies of the ground state and first excited states? 3. Consider a single particle quantum system whose wave function is given by Useful constants and formulae: (x) = expikx (ik ( 2. 4d where k is the wave number of a plane wave and d is some constant. (a) Is the wave function normalized? If not, then do it. (b) How much uncertainty do you expect if you measure position of the particle? + [+ d, e-6 = 8 [1] [2] [2] [2] [3]