Hydrogen atom - Superposition of States

exercise 3_4936

Let φ nℓm ( r ) denote the properly-normalized energy eigenfunctions of the Hydrogen atom (H-atom) with principle quantum number n and angular momentum quantum numbers and m . Consider an electron in the following state which is a superposition of H-atom eigenstates:

ψ ( r )= C [ φ 100 ( r )+4 210 ( r ) − 2 Ö 2 φ 21−1 ( r )]

(a) Find the normalization constant C . Can C be complex, real or imaginary?

(b) What is the expectation value of the Hamiltonian for the H-atom in this superposition of states? Calculate your answer in units of eV.

In parts (c) and (d), find the following expectation values:

(c) Find á L 2 ñ .

(d) Find á L z ñ .

(e) What is the probability of finding the electron in the φ 210 ( r ) state?

(f) If the electron is in the φ 100 ( r ) state, find the radius at which the radial probability density is a maximum. How does this compare with the Bohr radius? What is the significance?

(g) Find the wavefunction ψ ( r , t ) at some later time t .