(a) Calculate the transmission coefficient for a particle with
mass m and kinetic energy E < U0 passing through the rectangular potential
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.
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.
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.