Extra energy in quantum tunneling

In quantum tunneling, the probability of finding an electron inside the potential barrier is non zero . So we can actually find an electron which had an energy $E$ in a place where classically it should have an energy bigger than $E$.

So if we find an electron in this potential barrier what will its energy be ?

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  The energy of the electron is conserved so its $E$ and the fact that its $KE \lt 0$ is just a weird fact of quantum mechanics. (I don't like this answer)

  
  


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  Since we know the electron is there we can treat it as a classical particle and because of that it's energy has rose to a value bigger than the potential of the potential barrier. The energy of the electron is not conserved unless the electron manages to pick up energy from nowhere.

  
  

These are the answers I have in mind, I'm practically sure none of them is correct ...

So how can we interpret this fact from an energy point of view ? The question may be silly but I'm just beginning quantum mechanics and I'm having a difficult time trying to understand it.

I hope someone can help me.

(Sorry for my bad english)

Answer

Short answer: Position and energy are not compatible observables, meaning you can not determine them both at the same time, much like position and momentum are non-compatible observables.

Long answer: If you know the energy of your particle, that means it's wavefunction is an eigenfunction of the Hamiltonian (a solution to the time-independent Schrödinger equation). This wavefunction will be spread out over the system with non-zero components in the classically forbidden region, i.e. there is a finite probability to find the particle in this region.

To actually find it there, you must perform a measurement. The measurement will collapse the wavefunction to one which is localized around the point where you happen to find it (let's assume we do find it in the classically forbidden region). The new wavefunction now is no longer an eigenfunction of the Hamiltonian, and the particle therefore does not have a well-defined energy. To determine the energy of the particle you would have to perform an energy measurement. This measurement would collapse the wavefunction into an eigenstate of the Hamiltonian, which would again be spread out over the system, i.e. the particles position would now be undetermined. Furthermore, the energy you would measure would likely be different from the original energy of the particle (before position and energy measurements).

As for energy conservation: When you introduce a measurement apparatus the system is no longer closed, and energy conservation does not apply unless you consider the total system, including the measurement apparatus.