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In elementary QM, an electron is typically viewed as a cloud around a proton. The idea is that it's position can only be determined once a measurement is made. The probability that the electron will be found in a certain region is determined by it's wave function.

For background see the question Is it that electron of an atom can be found anywhere in the space?

Assume a hydrogen atom in the ground state.

Now, there is chance, however small, that the electron will be found very far away from the proton. So far away, that it cannot reasonably considered under the proton's influence.

In other quantum situations, a particle can tunnel though a potential barrier if the barrier is finite. This allows a particle to escape as in beta decay. This is not exactly that situation, but the further away electron is found, the less influence the proton field will have on it. Is there a point where the electron becomes unbound? And is it the measurement process that causes this to happen? Or if we observe enough non-interacting, isolated hydrogen atoms, will we observe that some of the protons no longer have a bound electron if we wait long enough?

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  • $\begingroup$ As Julian's answer suggests, the electron might go 'way out in the probability tail and then get captured by some other nucleus. Sans competing energy wells, it'll always be bound by the original proton's field. $\endgroup$ – Carl Witthoft Jul 17 '14 at 12:03
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The problem with your example is that in the tunnel effect, energy is in the end conserved (the particle tunneled to a barrier but very quick, within the time allowed by the uncertainty principle, but emerges at the other end with the law of conservation of energy satisfied. In your example, the electron would end up in a state of more energy (if you do not find clear why let me know) and this difference in energy, will happen beyond what is allowed by the uncertainty principle, because it will become permanent.

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  • $\begingroup$ So the case where the wave function is an eigenstate of energy, it will remain. Higher eigenstates (further on average from the proton) would have different energies. This is what you mean by energy not conserved. You need something to from one energy state to another such as a photon. However, an unbound state could have the any energy. For the case of a tunneling, the barrier can be of any width or height (not infinte) and there would still be some amplitude on both sides of the barrier. Additionally, if you measure the position, you loose all information about energy of the electron $\endgroup$ – yalis Jul 17 '14 at 4:04
  • $\begingroup$ yes exactly!... $\endgroup$ – user16007 Jul 17 '14 at 4:07
  • $\begingroup$ In a sense, the probability that the ground state bound electrons can be found very far away allows for the interactions between atoms to form molecules, crystals, and lattices in general. Matter as we found it. $\endgroup$ – anna v Jul 17 '14 at 4:18
  • $\begingroup$ @annav But it is not due to the tunnel effect, just to the tail of the probability distribution $\endgroup$ – user16007 Jul 17 '14 at 4:20

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