Ground state of hydrogen: We claimed that the ground-state wave function of the hydrogen atom is ψ 100(r) = Ce^-r/a0 where a0 is the Bohr radius and C is a normalizing constant.

(a) A small element of space with spherical polar coordinates (r, θ,Φ ) spans the interval (dr, dθ, dΦ) in the three coordinates. What is its volume?

(b) In terms of a0 and C, what is the quantum mechanical probability of the electron in a hydrogen atom being in this small volume, assuming that the nucleus is at the origin and the atom is in its ground state?

(c) Integrate over all r, θ ,Φ, and make use of the normalization condition on the wave function to calculate C.

(d) Verify that the wave function is indeed a solution of the Schrodinger equation with energy -13.6 eV.