Please help with the QM-related question below:

Consider the spherically-symmetric infinite potential well, with potential V ( 7 ) = 1 o oErza r > a . (5) (a) Write the wave function as un,e,m(r, 0, 4) = Rn,e(r) Ye,m(0, $), with the Ye,m the spherical harmonics. Show that the radial wave function satisfies the differential equation d' Rn,e 2 dRin,l e( e + 1 ) \\ Rin,l = 0 , dr2 r dr + kne - 1 2 (6) where kn,e is a function of the energy En,e. Also derive the relation between kn,l, En,e, and the mass of the particle m. (b) Manipulate the differential equation above to show that the solutions may be given in terms of the solutions to the Spherical Bessel Function equation: dx2 + 2x dy + [ x2 - e(6 + 1)] y= 0. (7 ) There are two linearly independent solutions to this equation, which are denoted by je(x) and ye(x). Through Rayleigh's formula, they may be written as je ( 20 ) = ( -2) e ( 10) sin x x (8) ye ( 20 ) = - (-2)e (Id cos x Before imposing boundary conditions and the normalization condition, write the general solution for Rn,e in terms of the je and the ye. (c) Compute and plot the je and ye for ( = 0, 1. Argue that the ye-type solutions are not in the Hilbert space because they are not normalizable. (d) Denote the nth zero of je(x) by Zn,e (that is, je(zn,() = 0 for n = 1, 2, 3, ...). Impose the boundary condition at r = a to solve for kn,e in terms of the zn,e. Then, write down the equation for the energy levels En,e in terms of the zn,l. (e) You may find a table of Bessel function zeros here: https://mathworld. wolfram. com/ BesselFunctionZeros . html. Use this table to compute the numerical values of the energy levels for the ground state and the first three excited states, assuming a = 1 fm and m = 1 GeV/c2, with c the speed of light. Note that this is a reasonable approximation for the energy levels of nucleons in a typical atomic nucleus. Give the degeneracies of these states