**Problem 1:**

(a) Calculate the transmission coefficient for a particle with
mass m and kinetic energy E < U_{0} passing through the rectangular potential
barrier

U(x) = 0 for x < 0, U(x) = U_{0 }> 0 for 0 < x < a, and
U(x) = 0 for x > a.

(b) Show that for E << U_{0} and 2mU_{0}a^{2}/ħ^{2}
>> 1 the transmission coefficient can be written as
T ≈ (16E/U_{0})exp[-2(2mU_{0}/ħ^{2})^{1/2}a].

(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 < R_{0}, U(r)
= U_{0}R_{0}/r for r > R_{0},

where R_{0} is the radius of the nucleus and
U_{0} is the
barrier height for r_{0} = R. The energy E of the alpha particle can
be assumed to be much smaller than U_{0}. 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 << U_{0}, we have T ≈
T ≈ exp[-2∫_{R1}^{R2 }dr (2m(U(r) - E)/ħ^{2})^{1/2}].

The integration limits R_{1}
and R_{2} 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) = ½mω^{2}x^{2} 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.