Consider particles of mass m and energy E moving along the x-axis under the influence of a potential V(x). The particles have energy E > 0 and the potential had the from
V(x) = 0 for x < 0,
V(x) = V0 for x ≥ 0,
Where V0 is a constant such that V0 > E.
(i) For x < 0 show that there are solutions of the Schrödinger equation of the form Ψ(x) = exp(ikx). Determine the two possible values of k and use the appropriate operator to determine the corresponding values of the momentum,
(ii) A beam of particles travels in the positive x-direction from negative x. Describe qualitatively what happens to the particles in the beam according to classical mechanics and according to quantum mechanics.
(iii) For positive x consider electrons with E = 1eV and the potential with V0 = 2eV. If Ψ(x) is the wave function at the point x, where x is measured in units of 10-10m, determine |Ψ(1)/Ψ(0)|2. What does this quantity represent?