3. The Hamiltonian for a particle of mass M constrained to move on a ring is H =- 21 dp 0 and the Schrdinger equation is HO(@)=E() where I = Ma is called the moment of inertia, Misthe mass of the particle, and a is the radius of the ring. is the angular coordinate that specifies the location of the particle on the ring and thus lies on the interval OS 521. (a) By substituting in to the Schrdinger equation show that the solutions to this equation are ( )= A come 21E # where m= (b) The boundary conditions in this case are that we wave function be continuous, i.e., that (0)=0(2x). This condition says that since o=0 and = 2x are the same point on the ring, the probability distribution at that point can have only one value. Use this boundary condition to show that m=0, #1, #2,.. # 21 so that the allowed energies are E.. = m? with m=0,+1,2,... 2 Md (c) Show that the normalization constant is A= 727, and does not depend on m. (d) Show that the resulting eigenfunctions 0.6) m= 0, 1,2,. are orthogonal: d,)=0 20 min