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By this question, a particle can tunnel to inside a potential barrier. One aspect of this process that was not addressed in the answers was the conservation of energy. Motivated by this question, I am under the impression that having a particle inside a potential barrier would make for a higher energy state and, therefore, increase the energy of the system. Where does this energy come from?

There is already a similar question in here but it has a roundabout answer. (Saying that the energy is not well defined because the position is defined does not answer the question. It may be not well defined but it sure is higher than before.)

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The simplest physics description of potential barrier tunneling can be found here:

tunneling

According to classical physics, a particle of energy E less than the height U0 of a barrier could not penetrate - the region inside the barrier is classically forbidden. But the wavefunction associated with a free particle must be continuous at the barrier and will show an exponential decay inside the barrier. The wavefunction must also be continuous on the far side of the barrier, so there is a finite probability that the particle will tunnel through the barrier.

Note it is the same energy level occupied by the particle inside or outside the barrier, it is the wavefunction $Ψ$ that is affected, i.e.the probability $Ψ^*Ψ$ of finding the particle at a particular x, in this one dimensional representation.

It is all quantum mechanics after all, a matter of probabilities, there is no change in energy so there is no problem.

Edit after comment:

In the image above, the energy is negative counted from the top of the well and continues negative counted from the top of the well,or positive, if the zero is defined at the bottom of the well.

, I am under the impression that having a particle inside a potential barrier would make for a higher energy state

A change of energy states can happen only with the exchange of energy, so it will no longer be tunneling, but another quantum mechanical proccess.

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    $\begingroup$ I don't think this really addresses the confusion of the OP, which seems to be "If you measure the particle, and you find it in the negative region, then where did the energy come from to get it to that location?" $\endgroup$ – Acccumulation Aug 9 at 21:50
  • $\begingroup$ @Acccumulation my answers says it is in the same energy state it was before tunneling, In the image above, it is negatve counted from the top of the well and continues negative counted from the top of the well. $\endgroup$ – anna v Aug 10 at 3:38
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The issue is that position eigenstate and energy eigenstates are incompatible. That is, if position is measured, the state then becomes a superposition of energy eigenstates. And vice versa.

So it doesn’t make sense to ask what the energy is in a localised region of space. The system doesn’t have those properties simultaneously.

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