Can Hydrogen Atoms on Earth and the Moon Share Electrons?

[tasiz]

Is it possible to find a hydrogen atom which atomic nucleus is on Earth and electron is on moon?

 

[sysprog]

Is it possible to find a hydrogen atom which atomic nucleus is on Earth and electron is on moon?

In principle I see no reason why a single hydrogen atom couldn't be ionized to such a distance in empty space; however, a free proton on Earth and a free electron on the moon aren't going to remain part of the same hydrogen atom for long.

 

[PeroK]

Is it possible to find a hydrogen atom which atomic nucleus is on Earth and electron is on moon?

No. A hydrogen atom is, by definition, an isolated system. There is no way to associate a "free" electron on the moon with a "free" proton on Earth. Elementary particles are not stamped with serial numbers!

 

[Dr_Nate]

Is it possible to find a hydrogen atom which atomic nucleus is on Earth and electron is on moon?

In quantum mechanics we view the electron as a wave function, which means that it is distributed throughout space. The wave function quickly decays with distance from the nucleus. In principle, the atomic wave function doesn't go to zero, but practically it gets close enough.

That being said, I do tell my students all the time that a tiny tiny portion of the electrons in your body are on the moon.

 

[PeroK]

That being said, I do tell my students all the time that a tiny tiny portion of the electrons in your body are on the moon.

Which is, of course, utter nonsense!

Not least because the probability that even a single electron is more than $1m$ from its host atom is so small that you wouldn't expect even a single electron to be found that far if you observed all the atoms in your body for the lifetime of the universe.

 

[Cthugha]

That is a "standard" problem in Rydberg physics, which deals with highly excited states of atoms. Hydrogen has plenty of excited states described by the principal quantum number n. The radius of the hydrogen atom (for high-n states the atom becomes almost classical - strictly speaking it is the distance between proton and electron that has the highest probability) scales as $r=n^2 a_b$, where $a_b\approx 53\,pm$ is the Bohr radius of hydrogen in its ground state. The distance between Earth and moon is about $380*10^6\,m$ and a short calculation shows that you only need a principal quantum number $n=2.67*10^9$ to get a Rydberg hydrogen atom of the correct radius.

However, also the binding energy of the Rydberg state shows a scaling with n and goes as $E_b=\frac{13.6 eV}{n^2}$, which gives a binding energy of about $1.9*10^{-18}\,eV$. So, if the thermal energy $E_t=k_B T$ is below this, the atom would be stable. Keeping in mind that the Boltzmann constant $k_B$ is approximately $86\,\frac{\mu eV}{K}$, the temperature where the atom becomes unstable will be about 22 Femtokelvin. That does not sound too realistic to me.

 

[Dr_Nate]

Which is, of course, utter nonsense!

Not least because the probability that even a single electron is more than $1m$ from its host atom is so small that you wouldn't expect even a single electron to be found that far if you observed all the atoms in your body for the lifetime of the universe.

I think maybe you misinterpreted me. I'm not saying a small number of electrons are on the moon. I'm saying that the wave of each electron reaches the moon.

 

[PeroK]

I think maybe you misinterpreted me. I'm not saying a small number of electrons are on the moon. I'm saying that the wave of each electron reaches the moon.

If you have a truly isolated electron, say, then it makes sense to talk about its wave function extending indefinitely. Similarly, if you have a larger isolated object. The probability of finding one particle a long way away is small, but finite.

If, however, there are other objects (everything on Earth and the Moon) in your system, then (to me anyway), it makes no sense to talk about "your" particles being on the Moon. In QM terms, you have to consider the larger system of all particles. If you find an apparently free electron on the Moon, then how do you associate it with an ion in your body on Earth?

You might find an ion in your body, but why would you associate the Moon electron with that ion, rather than with any other ion in the system?

 

[PeroK]

That is a "standard" problem in Rydberg physics, which deals with highly excited states of atoms. Hydrogen has plenty of excited states described by the principal quantum number n. The radius of the hydrogen atom (for high-n states the atom becomes almost classical - strictly speaking it is the distance between proton and electron that has the highest probability) scales as r=n2abr=n2ab, where ab≈53pmab≈53pm is the Bohr radius of hydrogen in its ground state. The distance between Earth and moon is about 380∗106m380∗106m and a short calculation shows that you only need a principal quantum number n=2.67∗109n=2.67∗109 to get a Rydberg hydrogen atom of the correct radius.

However, also the binding energy of the Rydberg state shows a scaling with n and goes as Eb=13.6eVn2Eb=13.6eVn2, which gives a binding energy of about 1.9∗10−18eV1.9∗10−18eV. So, if the thermal energy Et=kBTEt=kBT is below this, the atom would be stable. Keeping in mind that the Boltzmann constant kBkB is approximately 86μeVK86μeVK, the temperature where the atom becomes unstable will be about 22 Femtokelvin. That does not sound too realistic to me.

One could argue that for an isolated hydrogen atom, the electron can be a long way away.

If, however, the atom is not isolated, then these calculations are meaningless, as the electron has trillions of other atoms, molecules and ions to interact with.

 

[Dr_Nate]

If you have a truly isolated electron, say, then it makes sense to talk about its wave function extending indefinitely. Similarly, if you have a larger isolated object. The probability of finding one particle a long way away is small, but finite.

If, however, there are other objects (everything on Earth and the Moon) in your system, then (to me anyway), it makes no sense to talk about "your" particles being on the Moon. In QM terms, you have to consider the larger system of all particles. If you find an apparently free electron on the Moon, then how do you associate it with an ion in your body on Earth?

You might find an ion in your body, but why would you associate the Moon electron with that ion, rather than with any other ion in the system?

I should clarify that I'm not answering OP's exact question. I agree with what you said about the free electron and serial numbers. I'm giving an answer to a question that I think OP might have intended.

And, you are right that things get messy when we talk about any large system. And from any practical standpoint the vast vast majority of the wave function is near the nucleus. However, I'm unaware of any wave function that decays to exactly zero without an infinite potential, so even with the mess of a large system I think it's still reasonable to say that the wave functions are distributed through space all the way to the moon...and beyond!

 

[Cthugha]

One could argue that for an isolated hydrogen atom, the electron can be a long way away.

If, however, the atom is not isolated, then these calculations are meaningless, as the electron has trillions of other atoms, molecules and ions to interact with.

That is a standard comment and seems straightforward, but it is simply wrong and has been studied in the literature hundreds of times. As long as the space in between is filled with particles is effectively charge-neutral, highly excited states are pretty stable. You can even fit a whole BEC inside a single atom and they survive just well.

Published version:
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.120.193401

Popular version:
https://physicstoday.scitation.org/do/10.1063/PT.6.1.20180604a/full/

Of course in practice the background temperature and the presence of few ions would kill the atom, but the notion that you can treat these states on the level of interactions of the constituent particles only leads to completely wrong conclusions.

 

[PeroK]

That is a standard comment and seems straightforward, but it is simply wrong and has been studied in the literature hundreds of times. As long as the space in between is filled with particles is effectively charge-neutral, highly excited states are pretty stable. You can even fit a whole BEC inside a single atom and they survive just well.

Published version:
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.120.193401

Popular version:
https://physicstoday.scitation.org/do/10.1063/PT.6.1.20180604a/full/

Of course in practice the background temperature and the presence of few ions would kill the atom, but the notion that you can treat these states on the level of interactions of the constituent particles only leads to completely wrong conclusions.

The Earth-Moon system is not a BEC.

 

[Cthugha]

The Earth-Moon system is not a BEC.

Your claim was that these calculations are meaningless if there are huge numbers of other atoms, molecules and ions in between the particles that make up the bound state. I fully agree for ions, but the scientific consensus is: neutral matter (not plasmas of course) inside does not destroy the bound state. This is not an opinion, it is the standard position. Feel free to write rebuttals to the literature on Rydberg atoms if you disagree.

 

[PeroK]

Your claim was that these calculations are meaningless if there are huge numbers of other atoms, molecules and ions in between the particles that make up the bound state. I fully agree for ions, but the scientific consensus is: neutral matter (not plasmas of course) inside does not destroy the bound state. This is not an opinion, it is the standard position. Feel free to write rebuttals to the literature on Rydberg atoms if you disagree.

Okay, but we started talking about the Earth-Moon case and atoms in a human body. That was the context of the debate.

My point is simply that the physics you refer to is not relevant to the case under discussion.

 

[Cthugha]

My point is simply that the physics you refer to is not relevant to the case under discussion.

Well, the point was that even if you keep interaction with matter aside, the binding energy of the resulting state is so incredibly low that even in vacuum in outer space thermal ionization will break it, so that such huge distances are simply unrealistic. This was in response to the OP, who was not talking about bodies or matter at all, but just about distances comparable to the earth-moon distance. If that is not relevant to the OP, this is fine, but I cannot judge that.

 

[Dr_Nate]

Well, the point was that even if you keep interaction with matter aside, the binding energy of the resulting state is so incredibly low that even in vacuum in outer space thermal ionization will break it, so that such huge distances are simply unrealistic. This was in response to the OP, who was not talking about bodies or matter at all, but just about distances comparable to the earth-moon distance. If that is not relevant to the OP, this is fine, but I cannot judge that.

Can you explain how you are picturing the electron to be on the moon while in a highly excited state? Are you perhaps picturing the highest probability density or an expectation value for the location?

I ask because I don't see such a big fundamental difference between the ground state and some other excited state of an atom.

 

[Cthugha]

Just solve the radial part of the standard hydrogen problem in quantum mechanics for different principal quantum numbers. For l=0, you will get a wavefunction consisting of two factors. One scales as $e^{-\frac{r}{n}}$ and the other is a sum of terms similar to: $a_0+a_1 r^1 +a_2 r^2 + ... + a_{n-1}r^{n-1}$ terminating at n-1, which is determined by Laguerre polynomials. For the ground state the latter term is just a constant, while for higher proncipal quantum numbers, it is a polynomial in r, which shifts the maximum of the occupation probability of the electron further away from 0 with n. A typical wavefunction for the 35S state is shown e.g. on this wiki page:
https://en.wikiversity.org/wiki/Rydberg_Atoms/Quantum_Defect_Theory

Just to emphasize it: The typical distances you routinely achieve in experiments on atoms or excitons in semiconductors are several micrometers, which is still pretty remarkable compared to the tiny Bohr radius.

Feel free to pm me in case of further questions. I do not intend to hijack this thread.

 

[Dr_Nate]

I understand what you mean. You are referring to the maximum in the probability density.

 

[PeroK]

Not least because the probability that even a single electron is more than $1m$ from its host atom is so small that you wouldn't expect even a single electron to be found that far if you observed all the atoms in your body for the lifetime of the universe.

I must confess that I made this statement without checking the numbers! So, I thought I ought to back it up with some calculations.

Take the ground state of hydrogen, as an example. The probability that the electron is found further than $1m$ is approximately of the order of $e^{-3 \times 10^{10}} < 10^{-10^{10}}$.

If we take $10^{28}$ atoms in the human body, an experiment every Planck time for, say, $10^{15}$ years, then we hardly make a dent in that number in terms of finding a single case of the electron more than $1m$ away.

In conclusion, I think it's safe to say:

The probability of an electron from any atom on Earth being found on the Moon, by purely QM probabilistic calculations, at any time during the lifetime of the Solar system is almost indistinguishable from zero.

 

[A. Neumaier]

Is it possible to find a hydrogen atom which atomic nucleus is on Earth and electron is on moon?

That's called a $H^+$ ion. You can match it with an arbitrary electron on the moon and get an instance of a hydrogen atom. But such a nonlocal atom will be difficult to ''find'', because one usually searches only locally.

because the probability that even a single electron is more than 1m from its host atom is so small

? How small it is depends on the state of the hydrogen atom. It is easy to find states where this probability is one!

Take the ground state of hydrogen, as an example.

But the question was not about the ground state, but about the existence of a state with the required property!

A hydrogen atom is, by definition, an isolated system.

Such hydrogen atoms do not exist at all!

 

[PeroK]

How small it is depends on the state of the hydrogen atom. It is easy to find states where this probability is one!

But the question was not about the ground state, but about the existence of a state with the required property!

The question then becomes how likely one is to find an atom in this state.

In my opinion, this whole thread is all a mis-statement about how QM applies to ordinary macroscopic bodies. A human body is exchanging matter with the environment all the time. Not through incredibly unlikely QM "fluctuations" but because of macroscopic processes like breathing, skin and hair loss etc.

Trying to say something like "part of you might be on the moon" just cuts across so many issues, theoretical and experimental, as to be meaningless.

There is no way to identify an electron found outside your body with an atom inside your body. Something larger you could do with DNA analysis. But, each electron doesn't have your DNA inside it.

 

[EPR]

In my opinion, this whole thread is all a mis-statement about how QM applies to ordinary macroscopic bodies.

Does it? There was a thread about the name of the 'cut' between the quantum domain and macro objects which IFRC was started by a mentor(Bhobba?).
This is a great thread btw.

 

[PeterDonis]

There was a thread about the name of the 'cut' between the quantum domain and macro objects which IFRC was started by a mentor(Bhobba?).

Please note that thread is in the Quantum Foundations and Interpretations forum. Discussions of the "cut" you mention belong there.

Please keep this thread focused on the predictions and models of basic QM, as discussed in the guidelines:

https://www.physicsforums.com/threads/guidelines-for-quantum-physics-forum.978328/

 

[A. Neumaier]

The question then becomes how likely one is to find an atom in this state.

I gave a prescription how to get lots of them: pair any $H^+$ ion on Earth with any lone electron on the moon, and you have one!

Not that this would something sensible to do in usual applications. But it is permitted by the quantum formalism. It is not very different from having an entangled 2-photon system with the photons detected kilometers apart.

 

[DaTario]

Considering the atom in its non ionized form and the references to the Earth and moon in the OP as just a way to set an objective distance, the isolated atom (located very far from everything), with its electron in its fundamental state will be described by $\Psi_{1,0,0}(r,\vartheta, \varphi, t )=\Psi_{1,0,0}(r,\vartheta, \varphi ) e^{-iE_1 t/\hbar}$, where:

and

(taken from wikipedia, and I suspect there is a cube missing in the term $(n+l)!$)

So, evaluating this expression and taking its modulus square, we see that the probability of detecting its electron $d_{e-m} = 384.400 km$ away seems to be incredibly small (see #19), allowing one to think that FAPP it may considered correct to say that this situation cannot be observed. In a somewhat similar sense we could say that a broken cup of tea being observed to suffer spontaneous reconstruction cannot be observed.

 

[tasiz]

I gave a prescription how to get lots of them: pair any $H^+$ ion on Earth with any lone electron on the moon, and you have one!

Not that this would something sensible to do in usual applications. But it is permitted by the quantum formalism. It is not very different from having an entangled 2-photon system with the photons detected kilometers apart.

you got to the point I wanted to discuss, so yes is the answer to my question.

 

[tasiz]

Thank you to everyone who participated in this discussion. You all helped me understand more about the topic.

 

[vanhees71]

No. A hydrogen atom is, by definition, an isolated system. There is no way to associate a "free" electron on the moon with a "free" proton on Earth. Elementary particles are not stamped with serial numbers!

To the contrary they are indistinguishable, and that doesn't mean as in our macroscopic world that they are very similar and thus not easily to distinguish, but they are in principle and strictly indistinguishable. You can only say "there's one electron with a large probability to be found at the moon and one electron with a large probability to be found on Earth" but you cannot tell them apart after some time when they come together and are likely to be found at the same place (they can never be strictly at the same place, because they are fermions though).