By measuring the energies of the emitted alpha particles, how can we say that they have energies less than the height of the nuclear potential well when they are inside the well? How can we be sure that alpha particles are not momentarily acquiring energies greater than the height of the nuclear potential well and coming out without requiring the mechanism of quantum tunneling?
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The idea that a nucleus is a bunch of nucleons sharing kinetic energy with a would be alpha particle is entirely classical. In the limit the half life goes to infinity, the nucleus becomes a stationary state: it never changes. That doesn't mean there isn't probability current circulating to make orbital angular momentum, nor that it is static. They whole thing is one multi particle entangle state of position, spin, and isospin wave functions.
If you look at the picture from alpha tunneling:
You can imagine there is some operator that connects the nuclear wave function with the free particle alpha wave function outside the nucleus, after that, all that matter is the the density of final states is non-zero (e.g.: the free alpha has positive energy, but it need not be high enough to so-called go over the barrier), and then there is a finite transition probability per unit time.
JEB
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