(8) Quantum tunneling can be treated as a particle with energy E interacting with a "step" potential energy function where U = 0 for x < 0 and x > L, and U = U₁ for 0 ≤x≤L, where the energy E
(8) Quantum tunneling can be treated as a particle with energy E interacting with a "step" potential energy function where U = 0 for x < 0 and x > L, and U = U₁ for 0 ≤x≤L, where the energy E
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Transcribed Image Text:(8) Quantum tunneling can be treated as a particle with energy E interacting with a "step"
potential energy function where U = 0 for x < 0 and x > L, and U = U₁ for 0 ≤x≤L, where
the energy E<Uo. This situation is drawn in part (a) of the next problem. By applying
boundary conditions at x = 0 and x = L this problem can be solved. From that full solution,
one finds that for the case where E is well below the barrier (E <Uo) the transmission
coefficient T can be approximated by
E
T≈ 16- -(1
Uo
E
-)e
-2KL
2m(Uo - E)
where k =
Uo
In this problem we use this formula to treat radioactive decay, which we model as tunneling
through a rectangular barrier. We will return to this topic in Chapter 12. Note the units here
are fm and MeV instead of nm and eV.
(a) For an alpha particle (a helium nucleus) with E = 10 MeV, calculate the transmission
coefficient T through a 13 fm wide (nuclear Coulomb) barrier with a height Up = 15
MeV.
(b) Find T if the barrier height is increased to U₁ = 20 MeV
(c) Find T if instead the barrier width is doubled (and the height is back to 15 MeV). Note
the difference between the answers to (b) and (c)
(d) For the case in (a), calculate the number of alpha particles you would observe per second,
if the frequency with which the alpha particles were hitting the barrier was 1.1 x 1021 s-1
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