1. Representations of the Nuclear Potential

2. The nucleus can be thought of as a spherical finite potential well that is created
and filled by the nucleons themselves. Even though closely packed, the nucleons
are quasi-free and they will occupy the energy levels within this well from the lowest
energies upward, constrained by the Pauli Exclusion principle. Like electrons
filling orbital states in atoms, there are quantum numbers nlm associated with these levels, which nucleons can fill partially or completely. When a given “shell” is filled
(see below) the corresponding nucleus will be much more stable than its “nearby” isotopes. This is the basis of the “shell model” of nuclei.

5. Alpha decay is a quantum tunneling process
The illustration represents an attempt to model the alpha decay characteristics of polonium-212, which emits an 8.78 MeV alpha particle with a half-life of 0.3 microseconds. The Coulomb barrier faced by an alpha particle with this energy is about 26 MeV, so by classical physics it cannot escape at all. Quantum mechanical tunneling gives a small probability that the alpha can penetrate the barrier. To evaluate this probability, the alpha particle inside the nucleus is represented by a free-particle wavefunction subject to the nuclear potential. Inside the barrier, the solution to the Schrodinger equation becomes a decaying exponential. Calculating the ratio of the wavefunction outside the barrier and inside and squaring that ratio gives the probability of alpha emission.
The half-lives of heavy elements which emit alpha particles varies over 20 orders of magnitude, from about a tenth of a microsecond to 10 billion years. This half-life range depends strongly on the observed alpha kinetic energy which varies only about a factor of two; from about 4 to 9 MeV. This extraordinary dependence upon kinetic energy suggests an exponential process, and is modeled by quantum mechanical tunneling through the Coulomb barrier.