Quantum chem 7

(See attached file for full problem description)

Wavefunction

1.) A particle of mass m is confined to a one-dimensional potential well with infinite potential walls. The well extends from 0≤ x ≤ a. At time t = 0, the normalized wavefunction is Ψ(x, t=0) = (8/5a)1/2 [ 1 + cos (πx/a)] sin (πx/a) What is the wavefunction at a later time t = t0? (Hint: Any wavefunction Ψ(x,t) can be expanded in terms of the basis wavefunctions Ψ(x,t) = Σn An(t) Ψn(x,0) where An(t) = A0 (0) exp(-iEnt2π/h). See attached file for full problem description.

Energy Eigenfunctions of a particle

2.) Find for the first three energy eigenfunctions of a particle in an infinite dimensional well. (Hint: You might want to use the energy eigenvalues to avoid a lot of detailed calculations.)

Tunneling in Chemistry

4.) Find two examples of tunneling in chemistry from the literature. Explain where quantum mechanical tunneling is involved and reference the literature source.

Commutator of Wave Function

5.) Find the commutator of [x , d/dx ] for the wave function Ψ(x). See attached file for full problem description.