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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 $E$.

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

  • 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)

  • 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)

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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.

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  • $\begingroup$ when the measurement of the position is made, the particles energy is not well-defined but can we say that it is a lest bigger than the potential of the potential barrier ? (I think it is the case since the particle juste after the measurement acts like a classical particle). If the energy of the electron can theoretically (without actually measuring it) be less than the potential, wouldn't that be absurd ? $\endgroup$ – Jbar May 8 '14 at 8:03
  • $\begingroup$ I disagree with your notion that, after the measurement, the particle "acts like classical particle". A classical particle would have well defined energy as well as position. I suppose you are thinking that the more localized the wavefunction, the more classical the particle. This is not true. You might say that after the measurement the position of the particle becomes a more classical property, but at the same time, energy has become a less classical property of the particle in that it has become less well defined. $\endgroup$ – jensa May 8 '14 at 8:45
  • $\begingroup$ To answer your question - No, we can not say that the particle has an energy larger than the potential of the barrier. The particle will be a linear combination of energy eigenstates with some energies below the potential barrier. This must be the case since there must be an overlap with the original energy eigenstate (the wavefunction before the measurement). $\endgroup$ – jensa May 8 '14 at 8:48
  • $\begingroup$ You may look at it like this - First you knew the energy and QM tells you there is a chance the particle is in the classically forbidden region, after the measurement you know the particle is inside the barrier and QM tells you there is a chance it has an energy below the classically allowed limit. It's equally "absurd". $\endgroup$ – jensa May 8 '14 at 8:58
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One possible view on this is that while the average energy is given by $\int \psi^*\hat{H}\psi dV$, the actual energy value fluctuates in time around this value; the electron receives energy and gives it back again to fluctuating electromagnetic fields (background radiation), which are always present (in this view). This is motivated by stochastic electrodynamics, where background electromagnetic radiation has been used with some success to explain several microscopic phenomena (Casimir forces, thermal radiation, stability of the atom) as alternative to quantum theory.

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  • $\begingroup$ (I'm not familiar with stochastic electrodynamics so I can't say that I have understand well what you are said). But I don't see why the energy of the particle fluctuates ... the particle has a well defined energy in my example (the wave function is a eigenfunction of the Hamiltonian). So the electron take some energy when it wants to enter the classical forbidden region and then gives it back ? So the energy of the electron is not conserved but the energy of the system {electron + background} (without the measurement apparatus) is conserved. Doesn't that contradicts the answer above ? $\endgroup$ – Jbar May 8 '14 at 8:17
  • $\begingroup$ You are assuming the common view of quantum measurement theory, that energy of the particle has numerical value only when the $\psi$ function we use to describe it equals Hamiltonian eigenfunction, otherwise there is no definite value. Under this assumption, energy is constant or does not have meaning unless you measure it at some time and my answer is indeed incomprehensible. This view is the scheme of von Neumann postulates and is sometimes useful, mainly for spins. $\endgroup$ – Ján Lalinský May 8 '14 at 9:44
  • $\begingroup$ But quantum measurement theory is by no means the only one way to describe and explain experiments or to make sense of Schroedinger's equation for positions. Try to think of experiment where energy (value of classical Hamiltonian) of microscopic particle system such as electron or atom was reliably measured and eigenvalue of the Hamiltonian was found - I do not know any. $\endgroup$ – Ján Lalinský May 8 '14 at 9:45
  • $\begingroup$ On the other hand, even without hypothetical zero-point field of stochastic electrodynamics, there is thermal EM radiation everywhere due to charged particles forming neutral matter. This EM radiation acts on charged particles and makes them move randomly. If you assume physical quantities always have value (in contradiction to the quantum measurement scheme) it is most natural to expect every real system's energy fluctuates. $\endgroup$ – Ján Lalinský May 8 '14 at 9:47

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