I understand that the question is long so even partial answer is appreciated

2m = 9 410 3D central potential problem allows. In particular, the eigenstates of the hydrogen atom can be written as Ynm(r), where for a given n the allowed values of I are 0,1,2,..., 1 1, while for a given l the allowed values of m are -1,-1+1,..., 1 1,1. In a typical central potential problem the degeneracy in the energy spectrum is (21 + 1). However, for the hydrogen atom the degeneracy is n?, as the energy eigenvalues only depend on n. In this problem we will find out that this large degeneracy in the hydrogen energy spectrum is due to a hidden SO (4) symmetry in the Hamiltonian. In contrast, for a typical 3D central problem the symmetry group is SO(3). To show this, we introduce the Laplace-Runge-Lenz (LRL) vector, which is given by 1 k M -(P x L - Lxp) --r, e? where k and L is the usual angular momentum operator. Also note that the Hamiltonian for the hydrogen atom is p2 k Hr) = 2m where r = V x2 + y2 +22. (a) Show that the newly defined LRL vector M satisfies the following relations, [H, Lj] = 0, [H,M;]=0, [L, M;] = iijk M In addition, we also have [M;,M;] =- iHeijxL-H(r). 26 You are not required to prove this relation in order to get the full points for this problem, as the derivation is a bit lengthy. However, if you are able to include such a derivation, you will earn an additional 5 points as bonus. (b) Now consider the bound states of hydrogen atoms, where we have E