A free particle of mass m with wave number k₁ is traveling to the right. At x = 0, the potential jumps from zero to U₀ and remains at this value for positive x.

(a) If the total energy is E = h²k²₁/2m = 2U₀, what is the wave number k₂ in the region x > 0? Express your answer in terms of k₁ and in terms of U₀.

(b) Calculate the reflection coefficient R at the potential step.

(c) What is the transmission coefficient T?

(d) If one million particles with wave number k₁ are incident upon the potential step, how many particles are expected to continue along in the positive x direction? How does this compare with the classical prediction?