Consider a hydrogen atom. the energy eigenstates of the atomic electron are usually described by wave functions v nem(r). relate each of n, l, and m to the eigenvalue of a specific operator by giving the eigenvalue equation for this operator acting on nem(r). [6] state which values each of n, l, and m can take. (ii) the ground state of the hydrogen atom is described by a wave function that is proportional to e-r/a, where ao is real and positive and has the dimensions of a length. state what n, l, and m are for the ground state, and determine the correctly normalized wave function vnem(r) for this state. (iii) for an electron in the ground state, determine the probability of finding the electron in an infinitesimal shell between the radii r and r + dr, and then the radius of the shell where it is most likely to be found. (iv) determine the expectation value of the distance of the atomic electron from the nucleus for an electron in the ground state. (v) a helium atom can approximately be considered as two electrons interacting with the nucleus as if they each were in a hydrogen-like atom of their own. write down the approximate hamiltonian for this system. for the ground states of ortho- and parahelium, give expressions for the spatial parts of the wave functions in terms of hydrogenic wave functions vnlm for a single electron. [4] state which of these is the true ground state of the helium atom. (vi) give an expression for the additional term in the hamiltonian that must be considered for describing the helium atom fully. state how it can be considered in a calculation of the exact ground-state energy, and very briefly explain the effect it would have on the ground-state energy [you are not expected to carry out the calculation.