1. Calculate the expectation value for the ‘F’ operator given the
(25 pts) Using the Bohr model, calculate the radius for the electron in the n=1 energy level
for a hydrogen atom (1 proton and 1 electron) and a helium cation (2 protons, 2 neutrons
and 1 electron). Explain why the radii are either the same or different.(20 pts) Explain the
difference between an eigenvalue and an expectation value (<3-5 sentences).(25 pts) What
is the kinetic energy for a proton with the following wave function? (Hint: the ‘m-1’ in the
wave function is the unit of ‘meter’ to the power of -1)(20 pts) Calculate the expectation
value for the ‘F’ operator given the following wave function and relationships.(10 pts) The
particle in a 1D box has an infinite potential, V = ∞, at x < 0 and x > L. A modified particle in
a 1D box additionally has an infinite potential at exactly x = L / 2. Usingequations, show how
this modifies the eigenfunctions and energy levels.|Ψ(x)⟩ = 13 cos(3m −1 x)|ψ⟩ = c|φ1⟩ +
2c|φ2 ⟩̂F|φ1⟩ = 1 ⋅ |φ2⟩̂F|φ2⟩ = 0c* = c⟨φi | φj ⟩ = 0 when i ≠ j⟨φi | φj ⟩ = 1 when i = j