1. and Quantum Mechanical Tunneling
Barrier Penetration
and Quantum Mechanical Tunneling

2. The potential "barrier" defined
…but what if we “turn it upside down”?
This is a finite potential barrier.
We’ve learned about this situation: the finite potential well… I II
III U E
-L/2
L/2
When we solved this problem, our solutions looked like this…
What would you expect based on your knowledge of the finite box?

5. wave packet approaching a barrier

6. Pedagogical exercise: consider time independent case
(pure momentum states)
to the left of the barrier
to the right of the barrier
Instructive to consider the probability of transmission and reflection…
R+T=1 of course…

8. Wavefunction for Square Barrier

9. Transmission Coefficient:

10. Transmission Coefficient for Arbitrary non-Square Barriers:

11. Application: alpha decay

12. Why does the half life vary more than the kinetic energy?

13. alpha decay
U(r) r E R
kinetic energy of escaping alpha particle
Separation of centers of alpha and nucleus at edge of barrier
9.1 fm
Height of barrier
26.4 MeV
Radius at which barrier drops to alpha energy
26.9 fm
Width of barrier seen by alpha
17.9 fm
Alpha's frequency of hitting the barrier
1.1 x 10^21/s

14. Decaying Black Holes

16. Hawking Radiation
Mass of Sun: 10^{66} years. Mass of billion tons, size of proton, only 10 billion years.