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collapse and revivalenter image description here

There are a few things that I quite didn't understand about the tunneling of alpha particles.

  1. Where does the kinetic energy of the alpha particle comes from? Is it because of the electromagnetic repulsion between protons?

  2. There's a small probability of finding the particle inside the Coulomb barrier. What would that mean? Would the particle return to the nucleus or repelled away by the electromagnetic force?

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  1. Where does the kinetic energy of the alpha particle comes from? Is it because of the electromagnetic repulsion between protons?

Kinetic and potential energies do not commute - hence, according to the uncertainty principle, the particle always have some kinetic energy. E.g., a particle bound in an infinite square well has only kinetic energy - different at every energy level.

However, even classically, being in a bound state does not mean zero kinetic energy - e.g., a planet rotating around the Sun has kinetic energy, yet not sufficiently high to escape. Also, as per classical physics, kinetic energy increases as the particle moves to the regions with lower potential energy.

  1. There's a small probability of finding the particle inside the coulomb barrier. What would that mean? Would the particle return to the nucleus or repelled away by the electromagnetic force?

It means that, when measuring the position of the particle, one could also get values where it is inside the barrier. We consider a particle as a wave packet - i.e., a superposition of the Hamiltonian eigenstates - which is initially localized within the nucleus. As the system evolves, at times $t>0$ the superposition is no more localized and we have finite probability of finding particle elsewhere, including inside the barrier and outside the nucleus. In principle, since every eigenstate evolves periodically as $e^{-iEt/\hbar}$, one could expect that after some time all the states return to the situation that we had at $t=0$ - this is known as collapse and revival of the wave function. However, incommensurability of the periods makes the revival time very long... and when we take continuum limit for energy states, it becomes infinite - so particle never really returns to the initial state (although we always have non-zero probability of finding it in the nucleus).

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