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But if an electron is really found at a point where it electrical potential energy is greater than the ground state energy, would it be a violation of energy conservation (unless it then possesses negative kinetic energy)?

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- Thread starter Dinosky
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- #1

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But if an electron is really found at a point where it electrical potential energy is greater than the ground state energy, would it be a violation of energy conservation (unless it then possesses negative kinetic energy)?

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Nugatory

Mentor

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Easy - you observed it in a position where the potential was such that the only way of making E

OK, that's a totally unhelpful answer because it's just restating your original question....

Here's what's going on: the total energy of the system (for this problem, that's the electron and the measuring apparatus) is conserved. If we find the electron at classically forbidden location, there has to have been be a net transfer of energy from the measuring apparatus to the electron, and the negative KE of the electron is a convenient book-keeping device for accounting for this transfer. However, the negative KE isn't "real' in the sense that the electron was floating around in the classically forbidden region with some definite KE that happens to be negative; the electron didn't have a definite position and KE until we measured it.

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That means if we are going to really detect such negative KE electron, it would inevitably absorb enough energy from the measuring device to turn it into a state with positive KE to be observable... could this be "proved" using quantum theory? (e.g. showing that using photons of insufficient energy to pay off the negative KE, it is impossible to interact with the electron in that region, or something like that...)

Or are we just satisfied to accept it, as this is the only "reasonable" way things could have happened?

Or are we just satisfied to accept it, as this is the only "reasonable" way things could have happened?

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This is very interesting question. I cannot answer it but I would like to comment that in QM you can break the conservation of energy for an interval of order ΔE ≈h / T.

Anyway as I have read from QED, there must be always a way to compensate this difference of energy, I don't know how can it be done in this case.

Best regards,

Sergio

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As a test, I once took the wave function of the first S level of the hydrogen atom and integrated through the space:

e * V(x)[itex]|ψ|^{2}[/itex] + [itex]|∇ψ|^{2}[/itex] / m

Where V(x) is the potential at any point and ∇ψ would be de density of momentum. The result of the integration gave me -13.6 eV, consistent with the fact that the electron is everywhere.

So, probably you will not be able to find the electron very far from the nucleus until you can give it the necessary energy.

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As a test, I once took the wave function of the first S level of the hydrogen atom and integrated through the space:

e * V(x)[itex]|ψ|^{2}[/itex] + [itex]|∇ψ|^{2}[/itex] / m

Where V(x) is the potential at any point and ∇ψ would be de density of momentum. The result of the integration gave me -13.6 eV, consistent with the fact that the electron is everywhere.

It would be interesting to know that the first term e * V(x)[itex]|ψ|^{2}[/itex] has the value of EPE of electron at Bohr radius and the value of the second term [itex]|∇ψ|^{2}[/itex] / m consistent with the KE of a "classical" electron moving in that orbit.

The interpretation of [itex]|ψ|^{2}[/itex] as probability density of "finding" and electron at a point is weird as the act of "finding" also depends on the other party interacting with it at the spot which, as the discussion implies, must have sufficient energy for the interaction to occur. So the electron is not simply "there" with a certain probability....

It is also strange to think of the possibility of successfully interacting with the electron at far-away points given that there is sufficient energy to pay off the negative KE, although such probability is so small (well, if there is a large no. of H atoms and these probabilities all add up...)

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