# Energy levels in atoms & speed of interaction

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DrDu
Maybe because we use different units: I think in cgs units. So there is no permittivity or permeability constant. In SI units, I would change both electron charge and epsilon so as to keep the instantaneous Coulomb force constant.

mfb
Mentor
Maybe because we use different units: I think in cgs units. So there is no permittivity or permeability constant.
You are hiding the constants in other unit definitions then, and there is still ##\epsilon_0##, although it is replaced by ##\frac{1}{c^2}## everywhere because ##\mu_0## gets fixed to 1 (electromagnetic CGS, otherwise swap the two constants and mess around with 4pi). The conclusion does not change.

Also, the very fact that you need to specify a unit system shows that there is no change of physics behind it. Physics does not depend on the unit system used to describe it.

Jano L.
Gold Member
Most calculations of hydrogen like atoms only consider the instantaneous Coulomb potential. The effect of the finite speed of the EM field is called retardation. It is fully taken into account when solving the Dirac equation for the hydrogen atom. Retardation effects can also be described using the Dirac-Breit hamiltonian. Have a look e.g. at Landau Lifshitz, vol. 3.
Retardation is not taken fully into account by the Dirac equation for the hydrogen atom, because the hamiltonian description uses only one time variable and there is no way to make it work with retarded times of electron and proton. At most magnetic force is accounted for by the additional momentum and ##c## -dependent terms, but this still does not even approximately describe the retardation - the interaction described by Hamiltonian is instantaneous.

DrDu
Retardation is not taken fully into account by the Dirac equation for the hydrogen atom, because the hamiltonian description uses only one time variable and there is no way to make it work with retarded times of electron and proton. At most magnetic force is accounted for by the additional momentum and ##c## -dependent terms, but this still does not even approximately describe the retardation - the interaction described by Hamiltonian is instantaneous.
You are right, but the point is that there is little to retard with the field of a static nucleus. So the Dirac equation is not a good example to demonstrate the effect of finite propagation of force.
More relevant is probably the inclusion of the Dirac Breit interaction into the Hamiltonian of a multi electron atom, but I doubt it has much effect.

Jano L.
Gold Member
You are right, but the point is that there is little to retard with the field of a static nucleus. So the Dirac equation is not a good example to demonstrate the effect of finite propagation of force.
More relevant is probably the inclusion of the Dirac Breit interaction into the Hamiltonian of a multi electron atom, but I doubt it has much effect.
The nucleus in the hydrogen atom is not static; it is heavier so it moves less than the electron, but it does move in the non-relativistic theory (its momentum can be described in a probabilistic way the same way electron's momentum can if the ##\psi## function is that of two-body system). If the EM field is to obey Maxwell's equations, the field of the nucleus cannot be that of a static charge distribution.

The Breit interaction term is a first order correction to the Hamiltonian of two charged particles interacting via Coulomb forces that gets it closer to the fully relativistic description between the charged particles. It does describe magnetic interaction of the particles, so in this respect it is more accurate, but it does so only through first term of an infinite series in position, velocity and acceleration of the particles. Retardation is not taken into account by the Breit terms. The resulting Hamiltonian describes theory that (although containing speed of light), has instantaneous interaction and is not Lorentz invariant.

You still don't get it.

The "second" is a UNIT OF TIME measurement. I can express that unit of time as "unit of oscillation of pendulum" if I so wish without using the "second". Another alien civilization will probably define a unit of time as the rate of change of the zoodoniun compound under the exertion of Ethon. It STILL will not change the period of time! ...
Ok, but how you compare time intervals/periods in different universes? (You seem to forget that my aim is to find how energy levels in an alternative universe - with Fc instead of c for the speed of light / force carrier particles - would be, compared with the same atom energy levels in our universe). If we use the second, it is based on Cs atom, on its energy levels, so if there is a change in energy levels in the alternative universe, the second will be also changed, with the same rate. If we don't use the second, how we compare time intervals in 2 universes?

That's why I proposed a computer model/simulation where the second would be the same for both atoms, the computer second.

If we want to understand how force carriers speed really affects energy levels in atom, we should use something else, something that deals with force carriers traveling between electrons and protons/nucleus ... maybe a computer model where the speed of force carriers is considered as distances on the display (pixels) per computer second, and not as is considered in real life, where the time is given by the atom(s) in the same model we investigate ... In this manner we can see on the same display 2 atoms with different (computer related/defined) speeds for force carriers.

What would happen with the speed of electrons around the nucleus if the speed of the photons "become" (is, in the alternative universe) 1m/s (in our units)? I don't think that they will have the same speeds as in this universe, because they cannot travel faster than light ... So, if the electrons would "slow down", what would happen with energy levels and the second in that alternative universe?

The nucleus in the hydrogen atom is not static ...
Thank you Jano L. for your input. Many thanks also for DrDu and mfb.

ZapperZ
Staff Emeritus
Ok, but how you compare time intervals/periods in different universes? (You seem to forget that my aim is to find how energy levels in an alternative universe - with Fc instead of c for the speed of light / force carrier particles - would be, compared with the same atom energy levels in our universe). If we use the second, it is based on Cs atom, on its energy levels, so if there is a change in energy levels in the alternative universe, the second will be also changed, with the same rate. If we don't use the second, how we compare time intervals in 2 universes?

That's why I proposed a computer model/simulation where the second would be the same for both atoms, the computer second.
I propose using the speed of light!

You are again forgetting that we use Cs atom because it is highly reliable and accurate! It has nothing to do with time being defined by atoms! "One second" is arbitrary. Another alien race will define time differently. You compare by looking at how they differ over the same time event, such as the half-life of something.

We already have different "time measuring" systems. Chinese and Muslim calendars are different than the Gregorian calendar. Has this been a major problem?

You also ignored the last part of my response. I guess that in itself answers it.

Zz.

I propose using the speed of light!
Which one, the one in our universe or the one in the alternative universe? And how?

You are again forgetting that we use Cs atom because it is highly reliable and accurate! It has nothing to do with time being defined by atoms! "One second" is arbitrary. Another alien race will define time differently. You compare by looking at how they differ over the same time event, such as the half-life of something.
How you compare the "time event, such as the half-life of something" in the alternative universe (with another speed for force carriers), with the same event in our universe? You assume that the "event" is not altered by the change of force carriers speed? Why?

ZapperZ
Staff Emeritus
Which one, the one in our universe or the one in the alternative universe? And how?

How you compare the "time event, such as the half-life of something" in the alternative universe (with another speed for force carriers), with the same event in our universe? You assume that the "event" is not altered by the change of force carriers speed? Why?
Which "force carrier"? You are only proposing changing the EM interaction, aren't you? So why would the weak force be affected? I'm simply using your faulty logic in which you think by changing one fundamental constant, you keep everything else the same (which, if you are able to follow what has transpired in this thread, is a fallacy!).

This thread has gone WAY beyond speculation.

Zz.

Which "force carrier"? You are only proposing changing the EM interaction, aren't you?
The speed of all force carriers and the speed of light "become" (is, in the other universe) Fc, instead of c.

I'm simply using your faulty logic in which you think by changing one fundamental constant, you keep everything else the same (which, if you are able to follow what has transpired in this thread, is a fallacy!).
I was able to follow that, in order to find the answer to my question, we cannot just use the equations proposed, with Fc instead of c.
So another approach is needed, something that deals with force carriers moving between electrons and protons. Do you know such an approach?

This thread has gone WAY beyond speculation.
What speculation? I just want to see if/how the atoms are influenced by the speed of force carriers.

mfb
Mentor
The speed of all force carriers and the speed of light "become" (is, in the other universe) Fc, instead of c.
As discussed, this alone does not make sense.

You need measurements you can do within the universes, with answers that are dimensionless. Those are the only measurements you can reasonably compare between different physics. Usually those measurements are ratios. The fine-structure constant is something like a ratio, the ratio between different transitions in atoms is such a ratio (and it depends mainly on the fine-structure constant).

As discussed, this alone does not make sense....
Well, I'm not convinced that it is impossible to see if/how the atoms are influenced by the speed of force carriers. I agree that to change from c to Fc in QM equations doesn't make sense, but it must be a way to see how the atoms are influenced by the speed of force carriers in one (our) universe, as I said, probably by making a computer model where force carriers travelling between electrons and protons are considered. Maybe it would be easier to consider in such a way (by using force carriers) something else, like a particle decay mediated by the weak force. What would happen with the half-life of a particle if weak force carriers "slow down"?

mfb
Mentor
That speed on its own has no physical relevance, unless you have something to compare it with.
Give photons a mass, and you do change the energy levels.
What would happen with the half-life of a particle if weak force carriers "slow down"?
More massive W and Z? Most lifetimes should increase (for particles where weak decays are relevant).

ZapperZ
Staff Emeritus
Well, I'm not convinced that it is impossible to see if/how the atoms are influenced by the speed of force carriers. I agree that to change from c to Fc in QM equations doesn't make sense, but it must be a way to see how the atoms are influenced by the speed of force carriers in one (our) universe, as I said, probably by making a computer model where force carriers travelling between electrons and protons are considered. Maybe it would be easier to consider in such a way (by using force carriers) something else, like a particle decay mediated by the weak force.
When you have made such a model and published it in PRL, do let me know.

Zz.

More massive W and Z? Most lifetimes should increase (for particles where weak decays are relevant).
How an increase in mass for W and Z bosons would affect their speed? You can calculate, for instance, what mass they should have in order to decrease their speed to 1/3 of their usual/normal speed? And then, you can calculate how their new mass would affect the lifetime of a particle where weak decays are relevant?

By the way, I made a mistake saying that "the speed of all force carriers and the speed of light "become" (is, in the other universe) Fc, instead of c". I ignored that W and Z bosons have speeds lower than c, because they have mass. Sorry for that.

Give photons a mass, and you do change the energy levels.
You can calculate how a massive force carrier photon, traveling with 1/3 c, would affect an atom (energy levels, electron speeds, atomic radius)?

mfb
Mentor
How an increase in mass for W and Z bosons would affect their speed? You can calculate, for instance, what mass they should have in order to decrease their speed to 1/3 of their usual/normal speed?
They do not have a "usual/normal" speed, but if you keep the energy constant, a larger mass means particles are slower. In the rare cases where you have real W and Z, otherwise speed is not a meaningful concept.
And then, you can calculate how their new mass would affect the lifetime of a particle where weak decays are relevant?
I would have to look up the formulas, but it is possible to calculate that.
You can calculate how a massive force carrier photon, traveling with 1/3 c
You cannot fix the speed to some number, the speed of massive objects depends on their energy, and virtual particles (which would be relevant for the atom energy levels) do not even have a well-defined speed.
A non-zero photon mass would change the field of the nucleus to a Yukawa potential, those have different energy eigenstates than 1/r potentials.

You cannot fix the speed to some number, the speed of massive objects depends on their energy, and virtual particles (which would be relevant for the atom energy levels) do not even have a well-defined speed.
I thought that force carrier photons are virtual particles and do have a well-defined speed, c.
Anyway, my intention was/is to see how a change in force carriers speed would affect the atom, so, if we cannot "fix" the speed by changing the mass, maybe this mass increase/addition is not the solution to my problem.

They do not have a "usual/normal" speed, but if you keep the energy constant, a larger mass means particles are slower. In the rare cases where you have real W and Z, otherwise speed is not a meaningful concept.
I would have to look up the formulas, but it is possible to calculate that.
Although you/we cannot tell exactly how much week force carriers speed would decrease if their mass increases, it is interesting to see/calculate how the lifetime of a particle (where weak decays are relevant) would be affected.

mfb
Mentor
I thought that force carrier photons are virtual particles and do have a well-defined speed, c.
Virtual particles do not have a position or speed. And you can argue that they do not exist at all.

The muon lifetime for example can be expressed as
$$\tau = \frac 1 \Gamma$$
with the decay width
$$\Gamma=\frac{G_F^2 m_\mu^5}{192\pi^3}I\left(\frac{m_e^2}{m_\mu^2}\right)$$
which uses the Fermi coupling constant
$$G_F=\frac{\sqrt{2}}{8}\frac{g^2(\hbar c)^3}{m_W^2}$$
where mW is the W boson mass.
If you increase the W mass, you decrease the coupling constant, which decreases the decay width, which increases the lifetime. The lifetime is proportional to the fourth power of the W mass if we keep all other constants the same.

...
If you increase the W mass, you decrease the coupling constant, which decreases the decay width, which increases the lifetime. The lifetime is proportional to the fourth power of the W mass if we keep all other constants the same.
Thank you.

Virtual particles do not have a position or speed. And you can argue that they do not exist at all.
Yes, I just read 2 articles about that (1 and 2), but still, is the EM force "carried" faster or slower than c?

mfb
Mentor
Changes in the field do not propagate faster than c.

Changes in the field do not propagate faster than c.
Of course not faster then c, but how fast?
And how? If virtual (force carrier) photons do not exist at all, how is EM force transmitted?

mfb
Mentor
Of course not faster then c, but how fast?
With c. This does not change if you give photons a mass, but changes that spread out that fast will be negligible then.
And how? If virtual (force carrier) photons do not exist at all, how is EM force transmitted?
Via changes in the field. The field is the fundamental concept, particles (both real and virtual) are just things we make up to make calculations easier.

With c. ...
Via changes in the field. The field is the fundamental concept, particles (both real and virtual) are just things we make up to make calculations easier.