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Prior to observation, the electron can be found anywhere (from inside the nucleus to the ends of the universe), but once its position is determined the answer is precise (albeit its momentum is not due to the uncertainty principle).

I have several questions related to this idea

First, how do you actually determine the position of the electron without "kicking" it out of the atom?

Second, if you were able to determine its position very precisely, wouldn't its momentum be so high that it would exceed the speed of light? (or does it just become more massive? Either way, it doesn't seem like it could remain bound to the nucleus.

Third, if you were able to determine its position, how does your knowledge of its position degrade with time? It would appear that to get back to its original probability distribution (over all space) it would need a great deal of time, again so as not to violate the speed of light (unless it can pop in and out of existence far, far away).

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  • $\begingroup$ Second, if you were able to determine its position very precisely, wouldn't its momentum be so high that it would exceed the speed of light? The relativistic equation relating velocity to momentum doesn't give $v>0$ for any finite $p$. $\endgroup$ – Ben Crowell Nov 11 '14 at 3:13
  • $\begingroup$ Hi @JohnG, you really should restrict yourself to one question/concept at a time. It helps you get a precise answer and it also helps avoid people closing your question because it's too broad. $\endgroup$ – Brandon Enright Nov 11 '14 at 3:19
  • $\begingroup$ thank you for the advice- I will attempt to parse out my one million questions in quantum packets $\endgroup$ – JohnG Nov 12 '14 at 15:57
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First, how do you actually determine the position of the electron without "kicking" it out of the atom?

When talking of quantum mechanical entities, as the atom and the electron are, one has to keep clearly in mind that our well validated models that allow us to probe their behavior are probabilistic, the probability given by the square of the wave function.

The wavefunction is a function of (x,y,z,t) . The way it has been validated is by making probability distributions and checking them against the data. The only way of measuring positions for an electron in the atom is by the electron interacting. This might be by its being kicked off and measured, giving one point eventually in the probability distribution under measrurement, or in fitting weak scattering data, for example, like light through a crystal, or x-rays , the interferences of light giving information of position . Again a statistical distribution. In this case the end result is a probing of the atom's position as a whole, as the electron orbitals define the size of the atoms.

Second, if you were able to determine its position very precisely, wouldn't its momentum be so high that it would exceed the speed of light? (or does it just become more massive? Either way, it doesn't seem like it could remain bound to the nucleus.

If you only determine the position, it could be as precise as your measurement capabilities. The Heisenberg uncertainty constrains one only if both momentum and position are required together.

Third, if you were able to determine its position, how does your knowledge of its position degrade with time? It would appear that to get back to its original probability distribution (over all space) it would need a great deal of time, again so as not to violate the speed of light (unless it can pop in and out of existence far, far away).

Again, please note that experiments are one off for individual interactions. One photon goes through the crystal and interacts with the field of the electron and is registered as one point in a probability distribution. Or one electron is kicked off and its track is measured and projected back to its position , as in this recent expreriment, giving the distribution of the electron in the hydrogen orbitals.

hydrogen orbitals

hydrogen orbitals

In the case of photons probing non destructively the atom there is no way that one can know what an individual electron is doing in its orbital after that slight interaction. So there is no degradation detectable with time as the electron is still in its orbital.

In the case of scattering electrons off the hydrogen atom, the process is completely destructive of the atom, the electron flies off and is detected in an appropriate detector system, and the hydrogen becomes an ion, a proton seeking for an electron from the environment to return to neutrality.

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  • $\begingroup$ Thank you for the link to this recent experiment - very interesting and helpful. $\endgroup$ – JohnG Nov 12 '14 at 15:45
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First, how do you actually determine the position of the electron without "kicking" it out of the atom?

This is in the context of non-relativistic quantum mechanics (QM).

In QM, an ideal measurement of position requires that, immediately after the measurement, the electron have a definite position. However, a state of definite position is necessarily a superposition of all energy states including bound and unbound states.

Thus, after an ideal position measurement, the electron is neither bound or unbound (it is in a superposition of bound and unbound states) so we can't say that it is definitely 'kicked out' of the atom.

Second, if you were able to determine its position very precisely, wouldn't its momentum be so high that it would exceed the speed of light?

Again, immediately after an ideal position measurement, the electron does not have a definite momentum but is, instead, in a superposition of momentum eigenstates. If an ideal measurement of momentum is made immediately after the position measurement, the outcome of the measurement is completely uncertain. One might measure a very high momentum or a very low momentum.

Third, if you were able to determine its position, how does your knowledge of its position degrade with time?

The state of definite position is, as stated before, a superposition of all energy states and, thus, evolves in time. It's not that our knowledge degrades with time, it's that the localization 'degrades' with time.

Immediately after the position measurement, the probability density is maximally localized. As the state evolves, the probability density becomes less localized.

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  • $\begingroup$ Thank you for your answer - I was not thinking of the electron and proton together as a quantum "object". This idea leads to more questions, which I will have to think through carefully before formulating. $\endgroup$ – JohnG Nov 12 '14 at 15:46
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Very high momentum doesn't mean speeds in excess of lightspeed - electrons approach infinite momentum as they approach lightspeed.

The uncertainty principle means that very shortly after a precise measurement of position the huge uncertainty in momentum means you no longer know where it is, only where it was. You also don't know where it came from before that.

A similar principle applies with photons. A small antenna can detect where a photon arrives, fairly accurately, but has a wide beam so can't tell which direction the photon came from, so has a large uncertainty in momentum (which is a vector). A large aperture antenna has a narrow beam and can tell which direction a photon came from but can't localise it except that it was somewhere in its aperture.

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