Explain the difference between an eigenvalue and an expectation value

Physical chemistry foe biochemistry II

(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. Using equations, show how this modifies the eigen functions and energy levels.

|Ψ(x)⟩ = 1

3 cos(3m −1 x)

|ψ⟩ = c|φ1⟩ + 2c|φ2 ⟩

̂

F|φ1⟩ = 1 ⋅ |φ2⟩

̂

F|φ2⟩ = 0

c* = c

⟨φi | φj ⟩ = 0 when i ≠ j

⟨φi | φj ⟩ = 1 when i = j