In a region of space, a particle has a wave function given by ψ(x) = Ae^(-x^2/2L^2) and energy h^2/2mL^2, where L is some length.

a. Find the potential energy as a function of x, and sketch V vs. x. What classical system has this potential energy function?

b. Find the kinetic energy of the particle as a function of x.

c. Show that x = L is the classical turning point.

d. Since the angular frequency ω of a classical harmonic oscillator is related to the spring constant k and mass m via ω = √(k/m), the potential energy of the oscillator can be written as V(x) = 1/2 mω^2x^2. Compare this with your answer to part (a), and show that the total energy for this wave function can be written as E = 1/2 hω.