all questions pls

4. Quantum mechanics Answer ALL parts. Three solutions to the Schrdinger equation for the hydrogen atom with common orbital angular momentum quantum number 1 are given below Ua re-1/200 sin Ae-id VE' Cre-1/200 uc cos e re-r/200 sin Detio Vz = where an is the Bohr radius and C is a constant. (a) Show that all three wavefunctions are eigenfunctions of the n operator, where , -ihlaq. Hence, give the values of the magnetic quantum number m for each wavefunction. [4] (b) Show that Yr is an eigenfunction of the 12 operator, where 1 a a 1 a 2 = -? sin e + sino a ae sin e 262 and hence determine the value of l. [4] (c) Show that the total probability density for the sum |4al? + [46]? + |4c|2 is spherically symmetric (i.e. has no angular dependence). [3] (d) The radial distribution function of a wavefunction is defined as 2 P(r) = So So Vlr sin babad. =0 =0 For the wavefunction Ub, find the most probable distance of the electron from the nucleus. [4]