Transition of free to bound electrons

[xortdsc]

Hi,

I wonder if there is an elegant way of how to picture (or describe at all) the transition of free electrons (non-quantized, point-like charges) into let's say the bound ground state of the hydrogen atom (in which it becomes quantized and cloud-like).
How does the transition occur ?
When does the transition occur (at a specific distance ?) ?

Can somebody give some insights to this ?

Cheers.

 

[mfb]

I moved your thread to quantum physics, there is no high-energy physics in the question.

"Electron" implies you use them as quantized objects - you can give the number of electrons (here: 1).

Free electrons show the same "cloud-like" behavior as bound electrons, I don't see the transition you mean.

 

[xortdsc]

sure the charge is quantized, but I'm worried about momentum. isn't it that a free electron can have an arbitrary momentum where a bound electron quantizes its momentum (leading to the fixed orbitals, disallowing arbitrary distances between nucleus and electron) ? or do i misinterpret something here ?

 

[mfb]

The energy is quantized for bound states, the momentum is not (otherwise the electron would be in a momentum eigenstate - and fly away).
Those orbitals are all spread out in terms of their radial distance. There is no fixed distance.

Free electrons can have a well-defined energy, too. If you look at a transition between free and bound states, and measure the released energy, this is always the case.

 

[xortdsc]

right. talking about "distances" in a bound state was non-sense.
so when you say energy is quantized i'd assume for let's say for 1s orbital the energy is a fixed constant value (being lower than when the electron is free). and this constant energy is "invested" in momentum or couloumb potential energy (and constantly converted between the two). Is that conceptually correct ?

 

[xortdsc]

Free electrons show the same "cloud-like" behavior as bound electrons, I don't see the transition you mean.

But a free electron is a point-like charge. How is this cloud-like as in the orbital ?

 

[mfb]

But a free electron is a point-like charge. How is this cloud-like as in the orbital ?

It is as point-like or not-point-like as it is in an orbital in an atom. If you look close enough (with x-rays, for example), it will always look like a point-charge. If you look with low-energetic interactions, it will look more like something spread out.

 

[xortdsc]

It is as point-like or not-point-like as it is in an orbital in an atom. If you look close enough (with x-rays, for example), it will always look like a point-charge. If you look with low-energetic interactions, it will look more like something spread out.

ah okay, so even in the orbital it can appear to be point-like ? that was new to me. thanks for the info.

 

[mfb]

Well, after that measurement it is not in that orbital any more - the interaction with a high-energetic photon will kick it out.

 

[Jano L.]

But a free electron is a point-like charge. How is this cloud-like as in the orbital ?

In common theories, electron is always point-like irrespective of whether in an atom or far from it. What is cloud-like is some marginal probability density for its position. This probability density (or wave function) is not the electron, only some auxiliary mathematical device for describing it.

If the electron had some some size, it would have parts and state of these parts would enter into the description via different kind of Hamiltonian or in some other way, changing the equation of motion for the electron. So far, everything seems to point to point electron:-)

 

[mfb]

This probability density (or wave function) is not the electron, only some auxiliary mathematical device for describing it.

Be careful, this depends on the interpretation of quantum mechanics. There is no "right" and "wrong" answer to the question whether the wavefunction is real or not.

The electron has no size in the way a proton has a size, sure, and even in interpretations where the wavefunction is a physical object it is said that the electron is point-like, but that does not mean you could describe the electron with "there it is".

 

[Jano L.]

Be careful, this depends on the interpretation of quantum mechanics. There is no "right" and "wrong" answer to the question whether the wavefunction is real or not.

In a sense, yes. I did not intend to go into "right answer" or "real" with its notoriously confusing semantics. But I think we can agree that the purpose of the physical theory is to describe atomic systems, hydrogen atom, water molecules etc., not the associated wave functions (that is perhaps the purpose of interpretations). From the point of view of physicist, wave functions just serve to understand atoms and molecules, which we know exist and had meaning independenty of any wave functions (from kinetic theory, chemical laws, ...)

he electron has no size in the way a proton has a size, sure, and even in interpretations where the wavefunction is a physical object it is said that the electron is point-like, but that does not mean you could describe the electron with "there it is".

I am not sure about that - can you explain why? de Broglie - Bohm theory seems to do just that.

 

[xortdsc]

I am not sure about that - can you explain why? de Broglie - Bohm theory seems to do just that.

Yep, I'd be interested in that, too. I always thought it was common understanding that the electron is not point-like, but in a distributed state when in an orbital. Though I could take it as an argument that after measuring it, it appears point-like as it is leaving the orbital state due to the energy injected for probing it.
But would that mean there is no way to check the real state (point vs cloud) of an electron in an orbital and it's shape is just a theory ?

 

[mfb]

In a sense, yes. I did not intend to go into "right answer" or "real" with its notoriously confusing semantics. But I think we can agree that the purpose of the physical theory is to describe atomic systems, hydrogen atom, water molecules etc., not the associated wave functions (that is perhaps the purpose of interpretations). From the point of view of physicist, wave functions just serve to understand atoms and molecules, which we know exist and had meaning independenty of any wave functions (from kinetic theory, chemical laws, ...)

The purpose of physical theories is to explain observations. Observations of particle properties are not better/more real/whatever than observations of wave properties. We know interference exists and has a meaning independent of any particles.

he electron has no size in the way a proton has a size, sure, and even in interpretations where the wavefunction is a physical object it is said that the electron is point-like, but that does not mean you could describe the electron with "there it is".

I am not sure about that - can you explain why? de Broglie - Bohm theory seems to do just that.

dBB does not work without the pilot wave, which is similar to the wave-function of other interpretations.

But would that mean there is no way to check the real state (point vs cloud) of an electron in an orbital and it's shape is just a theory ?

It is a quantum-mechanical wavefunction (at least in some interpretations).

 

[Jano L.]

...Observations of particle properties are not better/more real/whatever than observations of wave properties.

I agree, but I think we miss each other's point. "Observation of particle property" of electron happened when Thomson did his experiments with cathode rays in magnetic field and Millikan measured electric charge of the electron, later in observing tracks in vapor and bubble chamber, always adding further confidence to the idea that single electron can be thought of as a particle. "Observation of wave property" of electron or heavier particle occurred when people looked at the interference pattern composed of impacts of many such particles. There is no interference pattern for one electron, there is just single spot. There is no "observation of wave property" for single electron. The "wave property" of ensemble does not disprove that electron can be modeled as a point.

We know interference exists and has a meaning independent of any particles.

How would you observe interference pattern of electrons without registering positions of the arrival electrons?

dBB does not work without the pilot wave, which is similar to the wave-function of other interpretations.

Yes, but this is irrelevant for the point I made - the particles are at definite positions in dBB, pilot wave does not change that. So why do you think we cannot "describe the electron with "there it is"" ?