Ground State Energy: What Regulates & Why is it Constant?

[snapback]

an interesting approach to the problem of hydrogen ground state can be found here:

http://arxiv.org/abs/quant-ph/0307154"

cheers


oops, sorry, have not seen that nuby already mentioned ZPF.. but that paper might be related to his post 18

 

[JeffKoch]

"Holds" the binding force (potential?) at around -27.2 eV , and electron kinetic energy at +13.6 eV.

If you're referring to the particular numbers, they're in arbitrary units but 13.6 comes from the rest mass of the electron in eV and the fine structure constant. So the question really boils down to why the dimensionless fine-structure constant is what it is, and I don't think anyone has an answer to that question.

 

[Modey3]

nuby,

1.) Why does the electron energy remain constant in ground state hydrogen, as well as the average size of the atom?

You don't need to go into particle physics to answer this question! You have a coulombic potential between the electron and proton. This is a quantum mechanical system! If you solve the Hamiltonian for this system you get a set of eigenvalues which correspond to stationary energies. Stationary energies are energies which correspond to a minimum in the 3-dimensional energy surface. For the hydrogen atom there are 3 quantum numbers: n, l, ml which describe the wave function for the system. I am disregarding the spin quantum number since the electron spin isn't effect by the Hamiltonian. The restriction is that these quantum numbers must be integers. These integers correspond to stationary solutions. If by some perturbation that n went from n = 2 to n = 2.1. The electron would spontaneously go back to n = 2 because that is a stationary state - it's at the bottom of the hill.

modey3

 

[nuby]

How does the zero-point field come into the equation?

 

[snapback]

Random oriented zero-point field overpowers the directed (along the line between electron and proton) attraction between proton and electron at short distances and makes, sloppy speaking, the electron "lose its way on its fall towards proton"

To my knowledge ZPF is not included explicitly in the Hamiltonian approach where you rely (by assumption) on the fact that a hermitian operator has a lower bound in its spectrum. By "explicitly" I mean: ... well ... you can always discuss what the physics behind the existence of a lower bound of a hermitian operator is.

What about positronium, why does that thing annihilate ? Here the interaction between the components is also coulombic, at least at large distances.

 

[Redbelly98]

How does the zero-point field come into the equation?

That's a rather complicated calculation. There are even people with Ph.D.'s in physics who do not know how to do it.

The effect is quite small, though. For the n=2 level of hydrogen it's a few micro eV.

 

[snapback]

That's a rather complicated calculation. There are even people with Ph.D.'s in physics who do not know how to do it.

The effect is quite small, though. For the n=2 level of hydrogen it's a few micro eV.

I bet you are referring to Lamb-shift (fine structure of the n=2 level) ?
Please note, Lamb-shift affects the n=2 level, the discussion here is about the ground state (n=1).

What regulates the ground state energy of a hydrogen atom? Why is it constant (more or less)?

 

[Redbelly98]

I bet you are referring to Lamb-shift (fine structure of the n=2 level) ?
Please note, Lamb-shift affects the n=2 level, the discussion here is about the ground state (n=1).

Please note, I did explicitly say n=2 in my post. Also, discussions often shift or expand in scope by the time you get to the 30th post in a thread.

I wanted to make the point that the effect of vacuum fluctuations:

• Is not a simple matter to calculate.
• Is small compared to the overall hydrogen energy. I don't know the value for the ground state, so I quoted the amount of the Lamb shift for n=2*** to give a sense of it's small contribution to the overall energy

If you happen to know how much the ground state is affected by vacuum fluctuations, or a simple way to estimate it, I would be interested in knowing that.

Regards,

Mark

*** EDIT: 1057 MHz or 4.371 x 10^-6 eV

 

[snapback]

Please note, I did explicitly say n=2 in my post. Also, discussions often shift or expand in scope by the time you get to the 30th post in a thread.

I simply wanted to stay within the ground-state topic (well, it might be a little stubborn after 30th post ;-))

Sadly, I cannot help you with any estimation how much the ground state is affected by vacuum fluctuations. I'm aware of a http://arxiv.org/abs/quant-ph/0307154v1" , where it has been attempted to track the "behavior of a classical charged point particle under the influence of only a Coulombic binding potential and classical electromagnetic zero-point radiation" numerically. This calculation was done within the framework of classical stochastic electrodynamics (SED). No energy values are visible in this paper. And ... "55 days of CPU time for all runs" does not sound like a handy estimation ;-)

Nevertheless, don't you think it would be somehow amazing if vacuum fluctuations would be responsible for both effects: the energetically tiny Lamb shift at n=2 and the prevention of catastrophic collapse at n=1 ?

Regards

snapback

 

[Redbelly98]

Nevertheless, don't you think it would be somehow amazing if vacuum fluctuations would be responsible for both effects: the energetically tiny Lamb shift at n=2 and the prevention of catastrophic collapse at n=1 ?

Regards

snapback

Absolutely. I never meant to imply otherwise. My main point, that it is a small effect, addresses this earlier post (not the same one I quoted in Post #36):

I guess I'm wondering if the ground state electrostatic potential (or electron) interacts with the zero-point-field, and if the ZPF dictates the ground state energy?

A more direct answer would be: No. ZPF, also known as vacuum fluctuations, cause a small perturbation to the energy of the hydrogen atom.

 

[snapback]

If ZPF is not resonsible for the stability of the ground state, then we would need another force than electromagnetic force to oppose the steady attraction between electron and proton. As I see it: with electromagnetics gone, the number of suitable forces for prevention of the Hydrogen collapse is dramatically reduced (at least in the framework of standard model).

 

[Redbelly98]

If ZPF is not resonsible for the stability of the ground state, then we would need another force than electromagnetic force to oppose the steady attraction between electron and proton. As I see it: with electromagnetics gone, the number of suitable forces for prevention of the Hydrogen collapse is dramatically reduced (at least in the framework of standard model).

But, who says another force is required to prevent collapse? It's basic quantum mechanics that only discrete states are possible. electrostatic + kinetic energy are enough to explain what happens. Everything else--ZPF, gravity, and even magnetic interactions--are minor perturbations.

 

[snapback]

basic QM allows the calculation of measurement results (=discrete states) out of some postulates but gives no physical explanation why discrete states or the ground state exist. QM is a (very useful) calculational tool but it gives us no hints why Hydrogen is stable. But of course, if one supports the Copenhagen interpretation, then the results of QM calculations cannot be further scrutinized.

 

[Marco_84]

Having hamiltonian operators bounded from below
(i.e. we can have stables ground states) it isn't a quantum mechanical necessity.
It is an assumption already made during the XIX century.
It just states that the world we observe isn't collapsing on his-self.

I hope i was clear.

Marco

 

[snapback]

Having hamiltonian operators bounded from below
(i.e. we can have stables ground states) it isn't a quantum mechanical necessity.
It is an assumption already made during the XIX century.
It just states that the world we observe isn't collapsing on his-self.

I hope i was clear.

Marco

I regret, that your point is not clear to me. As I see it: there should be a physical mechanism behind any assumption.

 

[Marco_84]

I regret, that your point is not clear to me. As I see it: there should be a physical mechanism behind any assumption.

In fact, i wrote:

It is an assumption already made during the XIX century.
It just states that the world we observe isn't collapsing on his-self.

"This is the physical mechanism behind".

marco

 

[snapback]

I understand that there might be as many meanings of what a "physical mechanism"is, as there are physicists walking on the earth, so maybe I'm still missing your point.

In fact, i wrote:

It is an assumption already made during the XIX century.
It just states that the world we observe isn't collapsing on his-self.

"This is the physical mechanism behind".

marco

But what was in the 18th century, before that "assumption" was made ? Was there a "different" physical mechanism that prevented collapse ?

To me, the stability of Hydrogen is not merely an "assumption" but rather an empirical fact.

 

[Marco_84]

basic QM allows the calculation of measurement results (=discrete states) out of some postulates but gives no physical explanation why discrete states or the ground state exist. QM is a (very useful) calculational tool but it gives us no hints why Hydrogen is stable. But of course, if one supports the Copenhagen interpretation, then the results of QM calculations cannot be further scrutinized.

I'm sorry but my english is not so good :D.

What I'm trying to say is more general than just talking about H atom.

I said 19 century because during that period Hamilton developed his mathematical tools such Hamilton equations and so on... Obviously not only him.

Systems that we observe are usually stable, so the "assumption" comes from the observation, it is an empirical fact! And we don't need QM to assert this.

If you think a bit i was the "theoretical" instability of H atom that made Bohr and Sommerfield to introduce the quantized orbit.

QM is built on Observation, i.e. a System is described by its Observables!

Marco

 

[snapback]

Hi marco,

no problem about the English.. I think each of us two is missing the other's point, but to circumvent this we discuss ;-) ...

You wrote:

If you think a bit i(t) was the "theoretical" instability of H atom that made Bohr and Sommerfield to introduce the quantized orbit.

I want so see an "explanation" of the so called "quantized orbit" in physical terms: charges, forces, moving things, rotating things, whatsoever.

I stated this question already before: why does the electrostatic attraction of opposite charges lead to an annihilation in case of electrons and positrons but not in case of electrons and proton. At long distances the force between the charges is the same in both cases, isn't it ? What makes the difference at short distances ?

Please do not use words like "antiparticle" or "positron is different from proton" to discribe the situation at short distances ;-). Please try to argument with notions like "field, force, charge..." ;-)

Cheers & good weekend

 

[ZapperZ]

I stated this question already before: why does the electrostatic attraction of opposite charges lead to an annihilation in case of electrons and positrons but not in case of electrons and hydrogen. At long distances the force between the charges is the same in both cases, isn't it ? What makes the difference at short distances ?

Please do not use words like "antiparticle" or "positron is different from proton" to discribe the situation at short distances ;-). Please try to argument with notions like "field, force, charge..." ;-)

Cheers & good weekend

Electron and hydrogen do not have any electrostatic attraction between them.

Your question is more appropriate to be posted in the High Energy/Particle Physics forum, not here.

Zz.

 

[Marco_84]

Hi marco,

no problem about the English.. I think each of us two is missing the other's point, but to circumvent this we discuss ;-) ...

You wrote:


I want so see an "explanation" of the so called "quantized orbit" in physical terms: charges, forces, moving things, rotating things, whatsoever.

I stated this question already before: why does the electrostatic attraction of opposite charges lead to an annihilation in case of electrons and positrons but not in case of electrons and hydrogen. At long distances the force between the charges is the same in both cases, isn't it ? What makes the difference at short distances ?

Please do not use words like "antiparticle" or "positron is different from proton" to discribe the situation at short distances ;-). Please try to argument with notions like "field, force, charge..." ;-)

Cheers & good weekend

I bet you wanted to say proton not H ;)

Well now the question changed a little bit, in any case follow what zapper z is suggesting to you, search there (high energy physics)..

To answer you question you nedd more than QM, actually the most recent (and confirmed theory) is QFT more properly the SM.

best regards
marco

 

[snapback]

yes marco thanks for the hint: I meant proton but my typing fingers were too fast ;-)

Anyway, ZapperZ is suggesting the shift the discussion to another forum ... well ... why not, my question is obviously enormously different from what we had before in post #14

these might make more sense.

1.) Why does the electron energy remain constant in ground state hydrogen, as well as the average size of the atom?

2.) Why don't protons and anti-protons interact like protons and electrons?

Thanks in advance

Oh well ... I asked about electrons and positrons...

 

[snapback]

I do no think that a physical answer to stability of the groundstate of Hydrogen (or positronium) will be found somewhere inside of those above mentioned QM or QFT books (at least not in the conventional textbooks). Most of QM or QFT books deal with tools and recipes for calculation.

It seems to me that J. S. Bell's view about Copenhagen QM is perfectly adequate to summarize present status of discussion:
"... We emphasize not only that our view is that of a minority, but also that current interest in such questions is small. Thy typical physicist fells that they have long been answered, and that he will fully understand just how if ever he can spare twenty minutes to think about it" , Bell, J. S. "Speakable and Unspeakable in Quantum Mechanics", Cambridge University Press, Cambridge, 1993

Good luck

 

[yuanyuan5220]

The ground state energy is a relative quantity. It is often defined as zero. what concerns us is the symmetry and gap.