Consider the one-dimensional system of a particle of mass m in a uniform gravitational field above an impenetrable plane. Take the potential energy to be infinite at the plane and locate the plane at z=0.
(a) Plot the potential energy of the particle. What is the Hamiltonian? Sketch roughly the ground-state wave function.
(b) Use an appropriate trial wave function to estimate the ground-state energy.
(c) What is the average position ⟨z⟩ of the particle above the plane?