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According to quantum mechanics it should be possible. But can it happens when it has so small probability to occur? also if it can happens that means that energy must be provided in order to the electron escape the attractive force of the nucleus (ionization energy)?

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closed as unclear what you're asking by Aaron Stevens, ZeroTheHero, Cosmas Zachos, Kyle Kanos, Jon Custer Feb 4 at 13:55

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  • $\begingroup$ Google "Rydberg atoms". $\endgroup$ – Lewis Miller Feb 3 at 2:47
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The wavefunction for every atomic orbital extends to infinity. But it decreases exponentially so a few dozen atomic radii away it is negligible. (By “atomic radius” I mean the expectation value $\langle r \rangle$. For Rydberg atoms, this can be many Bohr radii.)

“Can happen” is different from “happens with a experimentally reasonable probability”. If the probability of finding the electron past a certain radius is, say, $10^{-1000}$, you’re not going to find it there.

If you could easily measure where an electron is, and you found an electron some distance away from the nucleus, at a low-probability radius, it would not mean that the electron needed more energy to get there. For example, the ground state of hydrogen is -13.6 eV. This means that it would take 13.6 eV of extra energy to ionize the atom so that the electron escapes to infinity. But without any additional energy, you might find the electron at, say, 3.2 Bohr radii rather than at 0.9 Bohr radii. (These are random positions, to emphasize that an orbital does not constrain the electron to a particular radius.)

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It's a good question. If the electron suddenly appeared far from the hydrogen atom then wouldn't that violate conservation of energy? Yet the wavefunction appears to suggest this is possible.

The confusion arises because of a misunderstanding about what it means for the electron to be found far from the atom. When we talk about, for example, the $1s$ orbital of a hydrogen atom this does not mean the electron is a point particle and the wavefunction gives the probability of finding the point particle. In a hydrogen atom the electron is not a point particle. It is delocalised over the region around the atom and it does not have a position - it has an average position, but that's all. What the wavefuntion gives us is the probability that the interaction with the electron will happen at a position.

To clarify this let's consider a concrete example of an experiment. Suppose we fire another electron at high speed towards our hydrogen atom in the hope of colliding with the electron in the atom.

Hydrogen atom

In this experiment we fire in an electron (the red line) and it collides with and ejects the electron from the atom (the green line). The collision happens at some distance $r$ from the proton.

The wavefunction tells us how likely the collision is. If we fire our electron close to the nucleus where the electron density is high (small $r$) it will be more likely to collide and ionise the atom than if we fire it so it passes far from the nucleus (large $r$). However in principle the collision can take place for arbitrarily large values of $r$, although in practice the probability of collision quickly falls below detectable levels.

But it is very important to understand that when we talk about a collision at some distance $r$ we do not mean the electron in the hydrogen atom was present at that point at the moment of the collision. The collision is an interaction between two wavefunctions not two point particles. When we say the collision happened at the distance $r$ we mean the interaction between the two wavefunctions happened at that distance.

If we measure the total energies before and after the collision we find that the total energy after is 13.6eV less than the total energy before, and this is of course the energy required to ionise the hydrogen atom. This energy change is independent of where the collision happened - it always has the same value of 13.6eV.

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  • $\begingroup$ What bothers me is how we know if an electron "belongs" to an atom if it can be found in the space of another atom ? also definition of ionization energy states that is "the minimum amount of energy required to remove the most loosely bound electron, the valence electron". But even at infnity probability cannot be zero. Does this mean that the atom is not ionized ? I know i am missing something...Thanks in advandance. $\endgroup$ – ado sar Feb 4 at 22:37

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