How do we derive the orbital angular momentum observable?

How do we check if a given vector observable is in fact an angular momentum?

See if it commutes with <math>\hat L^2</math>

What is the definition and physical interpretation of the center of mass coordinate system?

Center of mass coordinates assume that the collective center of mass is at rest.

 The total momentum <math>\sum_i P_i = 0</math>

How do we show that the MC motion is decoupled from the relative motion?

What is the justification for focusing on the relative motion Hamiltonian?

The center of mass (CM) is decoupled from relative motion, so the Hamiltonian can be written in terms of CM coordinates. This cuts the number of dimensions of the problem in half, simplifying the mathematics.

What are the mathematical consequences and physical interpretation of the central potential V(r)?

The central potential is that which describes the potential in the Hydrogen atom: V(r) = -e^2/r. The potential depends only on r, so the Hamiltonian commutes with L^2 (the sqaured angular momentum operator). This fact allows the separation of variables for solving the SE and lets us know that the Hamiltonian and L^2 share a common set of eigenfunctions (namely, the spherical harmonics).

What are the energy eigenfunctions and eigenvalues for the hydrogen atom?

What are the relations between the different quantum numbers?

See this Wikipedia page.