Electron far away from the nucleus of ground state H atom

[Dinosky]

The wave function of an electron in ground state hydrogen atom is non-zero at points far away from the nucleus. Interpreted as the probability of finding the electron, this is telling us that there is a finite probability of finding an electron far away from the nucleus even if it is at ground state.

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)?

 

[dauto]

Yes, the electron goes into regions that would be forbidden in classical mechanics due to negative kinetic energy. That's how Tunnel effect works.

 

[Dinosky]

What would be observed for a particle with negative kinetic energy? Or put it the other way, how can one conclude from an observation of the particle that it possesses negative K.E.?

 

[Nugatory]

What would be observed for a particle with negative kinetic energy? Or put it the other way, how can one conclude from an observation of the particle that it possesses negative K.E.?

Easy - you observed it in a position where the potential was such that the only way of making E_tot=E_kin+E_pot balance is to assign E_kin a negative value.

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.

 

[Dinosky]

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?

 

[USeptim]

Hi Dinosky,

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.


Sergio

 

[eudo]

Just for comparison, this is similar to the measurement of angular momentum. If you have an electron that is spin up with respect to the z axis, then measure the angular momentum along the X axis, you put it into a new state. If you then measure its angular momentum along the z axis again, you have a 50% chance of now getting spin down. But doesn't this violate conservation of angular momentum? No, because the measurement must have transferred angular momentum to the electron.

 

[USeptim]

In fact, as long as the wave function is no collapsed but some interaction with an external source, the electron will stay "everywhere" when there is wave function with a "electron density" of $|ψ|^{2}$, this density applies both to the electron mass and charge.

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)$|ψ|^{2}$ + $|∇ψ|^{2}$ / 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.

 

[Dinosky]

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)$|ψ|^{2}$ + $|∇ψ|^{2}$ / 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)$|ψ|^{2}$ has the value of EPE of electron at Bohr radius and the value of the second term $|∇ψ|^{2}$ / m consistent with the KE of a "classical" electron moving in that orbit.

The interpretation of $|ψ|^{2}$ 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...)