(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. 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 ≈ 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.
(b) E << U0 and 2mU0a2/ħ2
sinh(x) = ½(ex - e-x) ≈ ½ex, if x >> 1.
T ≈ 4E(U0 - E)/[¼U02exp[2(2m(U0 - E)/ħ2)1/2a] + 4E(U0 - E)]
≈ 4EU0/[¼U02exp[2(2mU0/ħ2)1/2a] + 4EU0)]
(c) Assume that
T ≈ exp[-2∫R1R2 dr (2m(U(r) - E)/ħ2)1/2].
R1 = R0, R2 = U0R0/E,
R2 >> R1.
∫R1R2 dr (U(r) - E)1/2 = ∫R1R2 dr (U0R0/r - U0R0/R2)1/2
= (U0R0)1/2∫R1R2 dr (1/r - 1/R2)1/2.
∫R1R2 dr (1/r - 1/R2)1/2 = ∫R1R2 dr [(R2 - r)/(rR2)]1/2
= √R2[(R1/R2 - (R1/R2)2)1/2 + π/2 - tan-1(R1/(R2 - R1))1/2
--> √R2 π/2 if R2 >> R1.
T ≈ exp[-√(2m/E) πU0R0/ħ) = e-λ', λ' = √(2m/E) πU0R0/ħ
e-λ' is the transmission coefficient.
To calculate the escape probability per second, we have to multiply T by the rate of the alpha particle hitting the barrier, which is approximately v/R0.
escape probability ≈ √(2m/E)/R0 * e-λ' = λ = 1/τ.
This is a reasonable order of magnitude approximation as long as the escape probability is very small.
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.