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  2mU0a22 >> 1 the transmission coefficient can be written as  T ≈ (16E/U0)exp[-2(2mU02)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∫R1R2 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) = mω2x2 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.