Energy levels in atoms & speed of interaction

[DanMP]

Hello,

I wonder how the speed of interaction affects the energy levels in atoms. We know that electrons in atom are attracted to the protons in the nucleus through the electromagnetic force. Photon, the force carrier for the electromagnetic force, moves in empty space with the speed c (the speed of light).

My question is: if the speed of interaction would have been different than c, let's say Fc (F>0), how this would affect the atoms and energy levels in atoms? Can someone re-calculate one energy level using Fc instead of c, and then compare the result with the normal energy level (the one with c as the speed of interaction)?

 

[mfb]

There is no meaningful way to change the speed of light. How would such a change look like? We define the length of a meter as the distance light travels in 1/299... second. This definition can always be applied. You can ask "what would happen if everything is suddenly half its size" - which somehow would look like a doubled speed of light. The result would be a massive explosion of everything solid and liquid, to relieve the pressure coming from the compression.

The speed of light is more like a conversion factor between units. It is not a fundamental parameter you could change in a meaningful way.

 

[DrDu]

The speed of light has but little importance on the electronic structure of atoms. In fact, most calculations assume that $c=\infty$, as the speed of the electrons is usually much smaller than the speed of light. There are some notable effects of the finiteness of the speed of light for very heavy elements like gold, mercury or uranium. E.g. the color of gold is due to relativistic effects. If not, gold would appear white like silver. Confer
http://www.insp.upmc.fr/webornano/ressources/2009/pdf_dijon/02_Pyykko.pdf

 

[DanMP]

There is no meaningful way to change the speed of light. How would such a change look like? ...

Don't see it like a change. Consider an alternative universe where the speed of force carriers is different than in our universe. Let's say that F is close to zero and that we can see a gold atom in this alternative universe. How this atom would behave compared with a gold atom in our universe? Can you calculate the differences in size, energy levels, speed of the electrons, etc.?

The speed of light has but little importance on the electronic structure of atoms. In fact, most calculations assume that $c=\infty$, as the speed of the electrons is usually much smaller than the speed of light. ...

This is strange. The speed of light (and force carriers speed) is not infinite.

There is no c in energy levels calculation?

 

[ZapperZ]

.There is no c in energy levels calculation?

Try solving the hydrogen atom yourself. Do you get any c in there?

What you are being told is that since the electrons in these light atoms move so slow when compared to c, the interactions can accurately be described accurately as instantaneous. The finite speed of light has no effect.

Zz.

 

[mfb]

Let's say that F is close to zero

That does not make sense. If you have two universes, you cannot even calculate that F because it has no physical meaning. It would be a simple conversion factor between units, like the factor between miles and kilometers. Does the universe change if you increase the length of a mile relative to a kilometer? Can you measure the conversion factor between miles and kilometers without using references on Earth? No.

 

[DanMP]

Try solving the hydrogen atom yourself. Do you get any c in there?

c is present there (https://en.wikipedia.org/wiki/Hydrogen_atom#Energy_levels) but not only in one place (there are constants depending on c), so it's impossible for me to find exactly how c influences the energy levels. I thought that in this forum there are physicists able to do that.

What you are being told is that since the electrons in these light atoms move so slow when compared to c, the interactions can accurately be described accurately as instantaneous. The finite speed of light has no effect.

Yes, in this universe, but in an alternative universe, where the speed of force carriers would be close to zero, things could be different ...

 

[ZapperZ]

c is present there (https://en.wikipedia.org/wiki/Hydrogen_atom#Energy_levels) but not only in one place (there are constants depending on c), so it's impossible for me to find exactly how c influences the energy levels. I thought that in this forum there are physicists able to do that.

I mean as in the SPEED of light, not c as a constant, conversion factor (the "c" in E=mc² does not imply something is moving with speed c, for example). And yes, physicists on this forum are able to do this because this is a standard example and exercise in into QM classes.

Yes, in this universe, but in an alternative universe, where the speed of force carriers would be close to zero, things could be different ...

But here's the thing. If the peed of light holds the same significance in that other universe, then you can't tell too much differences within context with this, if we want to play with such speculation. This is because c, whatever its value, will still be the fundamental constant in which a lot of other constants depend on.

Maybe you should try to learn how it works in this universe first before attempting to make up other universes?

Zz.

 

[DanMP]

That does not make sense. If you have two universes, you cannot even calculate that F because it has no physical meaning. ...

We don't calculate F. F is given.

Anyway, you can see it like this: we have, side by side, 2 gold atoms, one normal and the other with a different speed for force carriers (it is a thought experiment). If one atom emits a photon (electronic transition), would the other absorb it? Are the energy levels the same?

OR, we make a computer simulation for one atom, including the finite speed of force carriers, c. Then, on the same screen, we project another simulation, for the same atom but with Fc instead of c. For F=1, the atoms should look/behave identical. What would we see if we change F?

 

[ZapperZ]

We don't calculate F. F is given.

Anyway, you can see it like this: we have, side by side, 2 gold atoms, one normal and the other with a different speed for force carriers (it is a thought experiment). If one atom emits a photon (electronic transition), would the other absorb it? Are the energy levels the same?

OR, we make a computer simulation for one atom, including the finite speed of force carriers, c. Then, on the same screen, we project another simulation, for the same atom but with Fc instead of c. For F=1, the atoms should look/behave identical. What would we see if we change F?

This is a fallacy that most people have, that in physics, you can change one thing without affecting another. It is why one cannot learn physics in bits and pieces.

It makes scenarios such as this meaningless, because the rules aren't clear and some can change on a random whim.

Zz.

 

[DanMP]

This is a fallacy that most people have, that in physics, you can change one thing without affecting another. It is why one cannot learn physics in bits and pieces.

We actually do learn (and use) physics in bits and pieces. Sometimes we consider light as a wave (refraction), sometimes as particles (Compton effect). And we don't use relativity and QM at the same time ...

It makes scenarios such as this meaningless, because the rules aren't clear and some can change on a random whim.

What is not clear?

 

[mfb]

We don't calculate F. F is given.

But F has no meaning. It's like proposing a parallel universe where a mile is exactly two kilometers long. Who cares?

Anyway, you can see it like this: we have, side by side, 2 gold atoms, one normal and the other with a different speed for force carriers (it is a thought experiment). If one atom emits a photon (electronic transition), would the other absorb it? Are the energy levels the same?

"What would the laws of physics predict if we violate the laws of physics?" does not have a useful answer.

There is one thing you can change - and DrDu was probably thinking about that: the fine-structure constant. It is a dimensionless constant, so it is independent of your unit system. In our universe it is about 1/137. If you change this, you change the energy levels - the binding energies are (to a good approximation) proportional to this constant.
Note how many different ways there are to express it in terms of some physical constants that have units. The speed of light for example appears in the numerator in some ways and in the denominator in others. You cannot just say "increase the speed of light" without context, it doesn't mean anything.

 

[DanMP]

But F has no meaning. ...You cannot just say "increase the speed of light" without context, it doesn't mean anything.

So, it's impossible/meaningless (or very hard) to calculate how the speed of force carriers affects the atom. This may be caused by the fact that "speed" is something defined and measured using atoms?

 

[mfb]

So, it's impossible/meaningless (or very hard) to calculate how the speed of force carriers affects the atom. This may be caused by the fact that "speed" is something defined and measured using atoms?

It is meaningless, if the force carrier is massless (and therefore the speed of light in relativity and the speed of electromagnetic interaction are the same). You just do not have an indepedent reference to compare with.
A photon with mass would change energy levels.

 

[DrDu]

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.

 

[DanMP]

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.

As far as I understand, those equations take into consideration the relativistic effects in order to obtain more accurate results in QM calculations for the atom. It is interesting but not very helpful. Those equations use/contain time (t), but time (the second) is defined based on energy levels:

since 1967 the second has been defined as the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom

(https://en.wikipedia.org/wiki/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.

 

[ZapperZ]

As far as I understand, those equations take into consideration the relativistic effects in order to obtain more accurate results in QM calculations for the atom. It is interesting but not very helpful. Those equations use/contain time (t), but time (the second) is defined based on energy levels:
(https://en.wikipedia.org/wiki/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.

Sorry, but you got this all wrong.

We define "1 second" as the frequency of transition in Cs atom, but TIME isn't defined that way! "Time" is a dimension. "1 second" is a UNIT of measurement of that time. There's nothing that says that another civilization can't define another unit of measurement for time.

So the fact that the unit "second" is defined using atomic transition has nothing to do with us using time-dependent dynamics.

Zz.

 

[DanMP]

Sorry, but you got this all wrong...

Sorry for mixing notions. What I meant was that when we calculate/express the energy levels we use the second (to express time intervals or speeds), and the second depends on energy levels in Cs atom.

 

[ZapperZ]

Sorry for mixing notions. What I meant was that when we calculate/express the energy levels we use the second (to express time intervals or speeds), and the second depends on energy levels in Cs atom.

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!

You really should not be suggesting how physics should be done without actually learning physics itself first. Do you really think what you are doing here is rational?

Zz.

 

[DrDu]

So here is your original question:

My question is: if the speed of interaction would have been different than c, let's say Fc (F>0), how this would affect the atoms and energy levels in atoms? Can someone re-calculate one energy level using Fc instead of c, and then compare the result with the normal energy level (the one with c as the speed of interaction)?

In a relativistic quantum chemistry package you can do exactly this. You could change c to another value and see how this influences the energy levels in atoms and molecules. c is nothing else but the propagation speed of the electromagnetic field (the carrier of force).

 

[mfb]

In a relativistic quantum chemistry package you can do exactly this. You could change c to another value and see how this influences the energy levels in atoms and molecules. c is nothing else but the propagation speed of the electromagnetic field (the carrier of force).

You cannot change c without changing anything else, as c is related to various other physical constants. So what else do you change? To change relative energy differences, you have to change the fine-structure constant, the only thing that actually changes the physics (apart from electron to nucleus mass ratios and negligible corrections from other things).

 

[DrDu]

Hm yes, if I change x, it will also change the value of $x^2$.

 

[mfb]

I am not talking about powers of c.
$$c=\frac{1}{\sqrt{\epsilon_0 \mu_0}}$$
Change c, and you have to change one constant on the right side as well. Which one do you change? Or both - in which way?
$$\alpha = \frac{e^2}{4 \pi \epsilon_0 \hbar c}$$
Same question. And you have to change at least one more constant, because it is impossible to get away with changing just c and one other variable.
$$E=mc^2 \text{(at rest)}$$
What do you change? The rest energy of the electron or its mass?

The Rydberg energy can be expressed as $R_\infty=\frac{\alpha^2 m c^2}{2} = \frac{\alpha^2 E_0}{2}$. One expression has the speed of light in it, one has not. So does R change if you change c?

If you decide to keep $\alpha$ constant, all you do is an effective re-definition of the units. You do not change the physics.

 

[DrDu]

I thought it was clear from the OP's question, that we didn't want to change the properties of the electron or the nuclei, including their charges, and masses. I think this pretty much fixes the problem.

 

[mfb]

I thought it was clear from the OP's question, that we didn't want to change the properties of the electron or the nuclei, including their charges, and masses. I think this pretty much fixes the problem.

But you cannot! That is the whole point. You cannot change the numerical value of the speed of light without changing anything else. The equations I posted prove that. You have to choose other physical parameters as well. And unless you change dimensionless constant, all you do is re-define measurement units. "A mile is now two kilometers long" is not new physics, just a different way to describe the same physics.

 

[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]

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.]

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.]

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.