Thought experiment with electron and proton

[forcefield]

Suppose a universe with only one electron and one proton some distance apart from each others. Suppose that the motion of the particles is governed only by Coulomb's law and strong interaction. What would happen ? The electron and the proton would keep going back and forth through each others forever, wouldn't they ?

 

[Dale]

That depends on the kinetic energy of the electron relative to the proton. If the KE is high enough then the electron is not bound to the proton and will simply continue and escape off to infinity. If the KE is lower then you have an excited hydrogen atom which will release photons as the electron drops to progressively lower energy states.

 

[gordon2]

Electrons are unaffected by the strong nuclear force. They also have zero size (they're point particles). So, yes, they would keep going back and forth through each other forever, assuming they started at rest.

 

[Dale]

No, they wouldn't. They would radiate energy and go to the ground state.

 

[Nugatory]

Electrons are unaffected by the strong nuclear force. They also have zero size (they're point particles). So, yes, they would keep going back and forth through each other forever, assuming they started at rest.

This answer would be classically correct for large uncharged mases experiencing only gravitational attraction - but for an electron and a proton experiencing electromagnetic attraction, DaleSpam's answer is right.

 

[BruceW]

hmm. interesting question. It's essentially just the hydrogen atom. But once you start thinking about the quantization of the electromagnetic field, then stuff will happen like DaleSpam said, and the hydrogen atom will go to the ground state. And then a long time after that, the proton will decay too. (Since the proton is only metastable, I think I heard somewhere).

edit: ah, whoops it is the free neutron that is metastable, not the proton.

 

[dauto]

hmm. interesting question. It's essentially just the hydrogen atom. But once you start thinking about the quantization of the electromagnetic field, then stuff will happen like DaleSpam said, and the hydrogen atom will go to the ground state. And then a long time after that, the proton will decay too. (Since the proton is only metastable, I think I heard somewhere).

edit: ah, whoops it is the free neutron that is metastable, not the proton.

The chances that protons are unstable is actually very high. But its half life is known to be at least 10³³ years.

 

[BruceW]

The proton might decay into a positron and neutral muon because of Baryon number violation due to the SU(5) GUT? Seems pretty out there. But definitely a good possibility. That will be interesting if they observe it experimentally. If nothing else, it might be evidence for grand unified theories.

 

[mic*]

gordon2's response is the only one that has maintained the boundaries set by the OP. You cannot radiate (photon) energy in a universe with ONLY "one electron and one proton some distance apart from each others."

Given that this OP is suggested as a thought experiment, could the poster thereof link it to any physical events occurring, or observations that they believe such a thought experiment with the given constraints may benefit from?

 

[Dale]

You cannot radiate (photon) energy in a universe with ONLY "one electron and one proton some distance apart from each others."

Sure you can. That is the initial state. The specified universe then evolves from that initial state into a universe with a hydrogen atom in the ground state and a certain number of photons. Just because it starts in a given state doesn't mean that it remains in that state. Gordon2's response was not in agreement with the known laws of physics.

 

[mic*]

Sure you can. That is the initial state. The specified universe then evolves from that initial state into a universe with a hydrogen atom in the ground state and a certain number of photons. Just because it starts in a given state doesn't mean that it remains in that state. Gordon2's response was not in agreement with the known laws of physics.

Nice response, yet based on assumptions that are not presented in the OP. Probably it is being pedantic to say that what has been suggested is imagining a scenario where the only physical laws allowed become benign because;

- There are not multiple nucleons for the strong force to do anything.
- There has been no statement of an initial radius for Coulomb's law to be able to be applied.

Hence questioning the scientific relevance or benefit of the experiment.

I sense some merit to the conversation which may be better suited to philosophical corners, yet I remain open to being informed of scientific principles here that I do not understand.

 

[BruceW]

With the 'standard' hydrogen atom model, the hydrogen atom will either be in single energy eigenstate, or in some linear combination of energy eigenstates, and in either case, it will not 'drop down' to the ground state. But, there is also the background quantum field. This introduces fluctuations, and would allow the hydrogen atom to drop into the ground state while emitting some photons.

edit: or maybe you are saying that the OP was thinking of a proton and electron that are far enough apart that they are essentially free particles? Well this is the trivial case, where both particles just continue moving through space, unaffected by each other.

 

[forcefield]

maybe you are saying that the OP was thinking of a proton and electron that are far enough apart that they are essentially free particles?

How can that be ? Wouldn't the force between the particles always bring them together, no matter what the distance between them is ?

btw, strong force is needed to keep the quarks together.

OP

 

[Dale]

With the 'standard' hydrogen atom model, the hydrogen atom will either be in single energy eigenstate, or in some linear combination of energy eigenstates, and in either case, it will not 'drop down' to the ground state.

Why not? It happens all the time.

 

[gordon2]

Hmmm. I didn't think this would generate such a discussion. Perhaps I need to elaborate a bit:

Classically, assuming the two particles started at rest (hence zero angular momentum of the two-particle system), the two would "keep going back and forth through each other forever". The only relevant force is the coulomb interaction, so it's a trivial mathematical problem. If you were picky, you could perhaps talk about higher moments of the electric field due to the distribution of quarks in the proton (perhaps leading to non-zero angular momentum), but that's an almost negligible effect and doesn't change the essential behavior.

Quantum mechanically, there is no motion per se, only states of things like energy and momentum (and their canonical conjugates). So you can't talk about "going back and forth". You can ask about the stability of the energy quantum state, but in a two-particle system ("a universe with only one electron and one proton"), it would have to be stable since there's nothing else in the system. If you think about it, all photons are technically virtual - they have to be since a photon is by definition the mediator of the electromagnetic force. That's why the energy state would have to be stable. So given the parameters of the original question, "going back and forth forever" is about the closest analog to a stable energy and momentum state.

 

[BruceW]

Why not? It happens all the time.

yeah, but you need to introduce the idea of a second quantization (i.e. quantize the electromagnetic field).
For example, without any quantization of the electromagnetic field, if you have an electron in the n=2 state, this is an energy eigenstate of the system, and the state will simply stay the same (apart from an unimportant phase factor).
But now, if we introduce a quantized field of photons, there will be some finite probability that within a certain time, a photon will be emitted. Also, quantum field theory tells us what this probability is. This is true even if there were no photons around to begin with. So we get this spontaneous emission simply because we introduce the quantum field, not because we are saying there is a bunch of photons around (which would be stimulated emission).

p.s. I'm not certain about this stuff, but I think it's correct.

 

[BruceW]

How can that be ? Wouldn't the force between the particles always bring them together, no matter what the distance between them is ?

Yeah, that's a good point. The hydrogen atom could just be in a very high energy state, like n=1000, or n=some even larger number. So I guess in principle even if the electron and proton are very far away from each other, they still make up a hydrogen atom. I guess that when they are very far apart, you can say that they are approximately free particles. For example, for principle quantum number 'n' being very large, the difference between adjacent energy levels is very small, so you almost have a continuum of different energy levels, like you would have in 'truly free' particles.

edit: and I forgot to say, yes classically there would be a force which pulls them together. But in quantum mechanics, it's not so simple. In a hydrogen atom, there is spontaneous emission of a photon, to send the atom to a lower energy level. But I am not certain about what happens in the limit of electron and proton being very far from each other. I think that as long as we can assume that the wavelength of the emitted photon is large compared to the 'average' distance between proton and electron, then in the limit of large quantum number 'n', the probability of spontaneous emission per unit time tends to zero. (And I have been talking about dipole interaction here, I probably did not make that clear). But anyway, this is reassuring, since I would expect that when the proton and electron get very far from each other, they are almost free and are unlikely to emit any radiation due to dipole interaction, since they are hardly a hydrogen atom anymore.

edit again: The assumption that the wavelength of the emitted photon is large compared to the 'average' distance between electron and proton allows us to make a standard simplification of the equations (I think). But even if we can't make this assumption, I'm pretty sure that we still won't see much dipole radiation being emitted by the hydrogen atom when it is in a very large 'n' quantum number state.

final edit: yeah, so (I think) in quantum mechanics, we get a similar answer as we do for classical mechanics. i.e. if the electron and proton are very far away from each other, they won't get 'pulled towards each other' as much. (In a very loose sense of the word). And I would guess that in quantum mechanics, this decrease of interaction with distance is even more pronounced than it is for classical mechanics. (In classical mechanics, the force is 1/r^2). But in quantum mechanics (with field theory), I think maybe it will be even steeper than this, due to screening by virtual particles. Yes, this is a very qualitative answer, sorry for that. I suppose that just shows that I'm not 100% sure of what I'm talking about.

 

[dauto]

Hmmm. I didn't think this would generate such a discussion. Perhaps I need to elaborate a bit:

Classically, assuming the two particles started at rest (hence zero angular momentum of the two-particle system), the two would "keep going back and forth through each other forever". The only relevant force is the coulomb interaction, so it's a trivial mathematical problem. If you were picky, you could perhaps talk about higher moments of the electric field due to the distribution of quarks in the proton (perhaps leading to non-zero angular momentum), but that's an almost negligible effect and doesn't change the essential behavior.

Quantum mechanically, there is no motion per se, only states of things like energy and momentum (and their canonical conjugates). So you can't talk about "going back and forth". You can ask about the stability of the energy quantum state, but in a two-particle system ("a universe with only one electron and one proton"), it would have to be stable since there's nothing else in the system. If you think about it, all photons are technically virtual - they have to be since a photon is by definition the mediator of the electromagnetic force. That's why the energy state would have to be stable. So given the parameters of the original question, "going back and forth forever" is about the closest analog to a stable energy and momentum state.

No. Classically, accelerated charges radiate so the oscillations would damp down

 

[Dale]

yeah, but you need to introduce the idea of a second quantization (i.e. quantize the electromagnetic field).

Certainly, I was assuming a universe with identical physical laws as ours.

 

[Dale]

in a two-particle system ("a universe with only one electron and one proton"), it would have to be stable since there's nothing else in the system.

No, it wouldn't. You could even have an unstable single particle system if that single particle were unstable. The number of particles does not imply stability.

The system you mention would behave as I describe.

Even classically they would not simply oscillate. Classically an accelerating charge will radiate (the Lienard Wiechert potentials) and lose energy.

The difference between the classical and quantum description is not that one would oscillate and the other wouldn't, but rather that classically the electron will continue to lose energy until it crashes into the nucleus whereas quantum mechanically there is no available energy state below the ground state.

No matter how you look at it it is simply not correct that it winds up "going back and forth forever".

 

[forcefield]

classically the electron will continue to lose energy until it crashes into the nucleus

That is the point of my post. What is the problem with this "crashing" ?

edit: Do you mean that classically the electron would radiate ALL it's energy ?

 

[Dale]

What is the problem with this "crashing" ?

It is contrary to observation.

Do you mean that classically the electron would radiate ALL it's energy ?

All of it's kinetic and potential energy, yes.

 

[forcefield]

All of it's kinetic and potential energy, yes.

So this losing of energy would all happen before the electron is where the proton is ? I would call that soft landing instead of crash.

 

[dauto]

What's the potential energy of a system of two point particles where they are at the same point? (That's what's meant by crashing). Once you've calculated that energy you will understand what's the problem with the electron crashing into the proton.

 

[Dale]

Crash or landing, either one is fine by me. The point is that whatever you call it, it doesn't happen.

If it did happen then we wouldn't have any atoms. Electrons would continue getting closer and closer to the protons until the Coulomb forces got so large that you would get the electron and proton fusing into a neutron (inverse beta decay).

 

[BruceW]

speaking totally classically, the electron would spiral further in towards the proton, and would get arbitrarily close to the proton, releasing an arbitrarily large amount of radiation energy. clearly not something that we want in our physical theory. yay for quantum!

 

[dauto]

Crash or landing, either one is fine by me. The point is that whatever you call it, it doesn't happen.

If it did happen then we wouldn't have any atoms. Electrons would continue getting closer and closer to the protons until the Coulomb forces got so large that you would get the electron and proton fusing into a neutron (inverse beta decay).

You wouldn't form a neutron because the neutron's mass is higher than the combined electron+proton mass. What would happen is that the electron would keep falling in, emitting photons, losing energy, until the system's total energy would be negative (Including the rest masses) creating exotic matter. That bottomless pit is a catastrophic failure of classical physics.

 

[forcefield]

You wouldn't form a neutron because the neutron's mass is higher than the combined electron+proton mass. What would happen is that the electron would keep falling in, emitting photons, losing energy, until the system's total energy would be negative (Including the rest masses) creating exotic matter. That bottomless pit is a catastrophic failure of classical physics.

Do we know enough about the quarks to tell how they would classically change this analysis ? Or is that already taken into account in this analysis ?

 

[Dale]

There are no quarks classically.

 

[Dale]

You wouldn't form a neutron because the neutron's mass is higher than the combined electron+proton mass. What would happen is that the electron would keep falling in, emitting photons, losing energy, until the system's total energy would be negative (Including the rest masses) creating exotic matter. That bottomless pit is a catastrophic failure of classical physics.

OK, either way we would not have atoms. Since we observe that we do have atoms the classical analysis fails.

 

[BruceW]

yeah, it is kind of silly to talk about 'how would things work if we allowed some of quantum mechanics but not other parts'. For anything to properly make sense, either stay fully classical, or go into the quantum regime. Anything in the middle is not so useful. Well, maybe there are some exceptions, like how semi-classically, the electron in the atom feels a magnetic force from the proton, since from its viewpoint, the proton is moving around it. But really, it is better to do these things the proper (fully-quantum) way.

 

[forcefield]

No matter how you look at it it is simply not correct that it winds up "going back and forth forever".

Why not ? Are there observations that rule it out ?

 

[Dale]

Yes. Atoms at rest are observed to not emit synchotron radiation.

 

[forcefield]

Yes. Atoms at rest are observed to not emit synchotron radiation.

That seems to rule out only curved electron orbits.

 

[my2cts]

Yes. Atoms at rest are observed to not emit synchotron radiation.

You mean atoms in a stationary state.