QUESTION 1: Consider the Hydrogen atom eigenstate = Nre-r/(2a) sin e-io. (a) By considering the integral o foo√20 ||²² sin(0)d0dodr, find the normalisation constant N of, assuming that it is a positive and real number. (b) Show that is an eigenfunction of Iz =-ih. Hence, find (L2) for the state (this does not require any integrations). (c) Derive the expectation values (r) and (2) and the uncertainty Ar for the state . Hints: you may use without proof the formula for e-r/c = n!cn+1, and for the angular integra- tion it is useful to use the identity (sin 0)3 = sin (1 - (cos 0)²).

QUESTION 1: Consider the Hydrogen atom eigenstate = Nre-r/(2a) sin e-io. (a) By considering the integral o foo√20 ||²² sin(0)d0dodr, find the normalisation constant N of, assuming that it is a positive and real number. (b) Show that is an eigenfunction of Iz =-ih. Hence, find (L2) for the state (this does not require any integrations). (c) Derive the expectation values (r) and (2) and the uncertainty Ar for the state . Hints: you may use without proof the formula for e-r/c = n!cn+1, and for the angular integra- tion it is useful to use the identity (sin 0)3 = sin (1 - (cos 0)²).