A simple model of the helium atom consists of two electrons attached by harmonic springs to a nucleus of zero charge at the origin. If we label the two electrons as 1 and 2, and assign them coordinates r1 and r2, momenta p1 and p2, then the classical energy for this "helium" atom is E=2mep12+2mep22+21k(r12+r22)+40r1r2e2 where k is the spring constant, p1=p1,p2=p2,r1=r1,r2=r2, and 0 is the permittivity of free space. e. Suppose now that one of the electrons is a spin-up electron and the other is a spin-down electron. Based on your proposed eigenfunction for part b, and considering your answer for part c, what conditions must be imposed on F(R) and G(r) in order to satisfy the Pauli exclusion principle? Write down the full eigenfunction (x1,x2) in terms of F(R),G(r) and the appropriate spin wave functions. N.B. There are two possible answers to this question! Give both of them. f. Show that the time-independent Schrdinger equation for G(r) is separable in spherical coordinates (r,,). If G(r)=R(r)f()g(), derive the radial equation satisfied by R(r)