Let us consider two boxes of different energies that are separated by a barrier and box 1 is filled with many electrons. The barrier allows the electrons to tunnel from box 1 to the other side with a 1% probability. In the event that an electron in a given time frame does tunnel through the barrier, does this affect the probability that a different electron teleports through the barrier during the same time frame, or is this probability statistically independent? Therefore in the extreme case, do we know if the probability of ALL electrons tunneling is equal to P(1 electron tunneling)^(Number of electrons). Let us assume that the change in energy due to electrons filling/leaving certain energy levels is negligible.
To calculate tunneling, you use continuity conditions of the wave function in the Classically prohibited region.
This wave function allows caclulation of probability density.
Anything about tunneling is then probabilistic.
If you conduct experiments under identical circumstances of many different systems, you would definitely see dispersion of results.
The subject of tunneling time is rather controversial yet. Better understood is a transmitance T from one side of a barrier to another (T+R=1) where T=t*t , t calculated quantum mechanically.
There are cases I believe of stimmulated tunneling, where if one charge tunnels, a second is more likely to do so, but many body problems are hard to deal with.
In superconductivity you would always find the Cooper pair to tunnel (Josephson junction)
The simplest picture is independent tunneling of separate charges, for a normal Fermi electron gas.
There are essentially two ways of looking at this issue. One is Juan Weisz's way - totally correctly making use of wave functions and probability densities.
Another way is to look at the physical mechanism whereby quantum tunneling happens. In a nutshell, to tunnel through an energy barrier, a particle must borrow energy from a nearby virtual particle, and relinquish back that energy to the virtual particle within the alloted permissible time frame (as per Heisenberg's relationship linking virtual energy to permissible duration of virtual existence.)
To tunnel, the particle needs the presence of a suitable virtual particle nearby: by suitable is meant capable of fleetingly lending out the exact right amout of energy needed for the tunneling event to occur. (It is, by the way, the exact selfsame mechanism that allows any individual atom of a radioactive element to decay: there is an energy barrier that must be hurdled before decay can occur, and borrowing virtual energy from a suitable nearby virtual particle is what enables overcoming that barrier.)
The amount of energy that must be borrowed depends on the particle or entity (atom, etc.: anything can tunnel) that must tunnel. If the requisite energy is high, then the occurence of matching virtual particles is statistically rarer (which is why some half-lives are long, and some are short.)
If the energy to be borrowed is low, then separate occurences of tunneling (or equivalently radioactive decay) remain statistically independent of anything that happens around - as is for instance the case with carbon-8, which needs so very little energy to decay: there are so many low-energy virtual particles popping up all the time within the quantum foam that no interference with the environment occurs.) When however the requisite tunneling or decay energy is high, then suitable matching virtual particles are much rarer (think osmium-186). In that case: an atom of, say, Osmium-186 could conceivably snap up the rare suitable such virtual particle and thus interfere with and prevent another nearby atom from having its chance at decay - until the next nearby suitable virtual particle pops up, and as such statistical independence would not hold.
For any particular particle or atom, the best is to do the math. You can extrapolate the statistics of the appearance of suitable-energy virtual particles from the measured values of half times, see e.g. the table at https://en.wikipedia.org/wiki/List_of_radioactive_nuclides_by_half-life
The probability is changed since the set is depleted by one when a second electron moves
The Tunnel Effect Is Independent From Which Side On The Potential Well Particle Is Incident.
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