Problem 1:

(a)  Calculate the transmission coefficient for a particle with mass m and kinetic energy E < U₀ passing through the rectangular potential barrier
U(x) = 0 for x < 0, U(x) = U₀ > 0 for 0 < x < a, and U(x) = 0 for x > a.

(b) Show that for E << U₀ and  2mU₀a²/ħ² >> 1 the transmission coefficient can be written as  T ≈ (16E/U₀)exp[-2(2mU₀/ħ²)^1/2a].

(c) Many heavy nuclei decay by emitting an alpha particle.  In a simple one-dimensional model, the potential barrier the alpha particles have to penetrate can be approximated by
U(r) = 0 for r < R₀,  U(r) = U₀R₀/r for r > R₀,
where R₀ is the radius of the nucleus and U₀ is the barrier height for r₀ = R.  The energy E of the alpha particle can be assumed to be much smaller than U₀.  For a non constant potential barrier the expression for the transmission coefficient found in part (b) can be used as a guide.  Assume that for E << U₀, we have T ≈ T ≈ exp[-2∫_R1^R2 dr (2m(U(r) - E)/ħ²)^1/2].
The integration limits R₁ and R₂ are determined as solutions to the equation U(r) = E. 
Calculate the alpha transmission coefficient and the decay constant λ, i.e. the decay probability per second.

Problem 2:

Let U(x) = ∞ for x < 0, U(x) = ½mω²x² for x > 0.  Use the WKB approximation to find the energy levels of a particle of mass m in this potential.  Compare the WKB energies with the exact energies for this potential.