Can we imagine that an electron moves on an orbit of chaos?

[thaiqi]

TL;DR

Can we tell whether hydrogen electron's motion relates to chaos?

Bohr's hydrogen atom model is outdated facing Schrödinger's wave equation. Now that wave mechanics doesn't use a concept of orbit for the electron in hydrogen atom. But can we suppose the electron is still circling around the atom core, not necessarily in circles or ellipses, but in chaos like a Lorentz attractor?

 

[Vanadium 50]

No.

 

[thaiqi]

No.

The guideline is very long.
Could you please tell me why you say no in more detail?Thanks.

 

[berkeman]

Summary:: Can we tell whether hydrogen electron's motion relates to chaos?

Bohr's hydrogen atom model is outdated facing Schrödinger's wave equation. Now that wave mechanics doesn't use a concept of orbit for the electron in hydrogen atom. But can we suppose the electron is still circling around the atom core, not necessarily in circles or ellipses, but in chaos like a Lorentz attractor?

No, of course not. What mainstream peer-reviewed journal articles have you read that would suggest that?

 

[thaiqi]

No, of course not. What mainstream peer-reviewed journal articles have you read that would suggest that?

Thanks.

 

[Vanadium 50]

Could you please tell me why you say no in more detail?Thanks.

Not until you point me to a reason you might think it's true. We don't discuss personal theories here.

 

[weirdoguy]

Now that wave mechanics doesn't use a concept of orbit for the electron in hydrogen atom.

You have to realize that it doesn't use this concept not because it could but we decided not to. It doesn't because this notion does not make sense in QM formalism. Replacing circle/ellipse with some other more complex shape does not change the formalism that forbid that kind of thinking.

 

[thaiqi]

You have to realize that it doesn't use this concept not because it could but we decided not to. It doesn't because this notion does not make sense in QM formalism. Replacing circle/ellipse with some other more complex shape does not change the formalism that forbid that kind of thinking.

Thanks.

 

[Demystifier]

But can we suppose the electron is still circling around the atom core, not necessarily in circles or ellipses, but in chaos like a Lorentz attractor?

According to the Nelson interpretation, electron has a stochastic trajectory. That's not exactly the same as chaotic trajectory, but it's pretty close.

 

[Demystifier]

You have to realize that it doesn't use this concept not because it could but we decided not to. It doesn't because this notion does not make sense in QM formalism. Replacing circle/ellipse with some other more complex shape does not change the formalism that forbid that kind of thinking.

Bohmian mechanics (which is compatible with QM formalism) allows thinking in terms of deterministic trajectories.

 

[thaiqi]

According to the Nelson interpretation, electron has a stochastic trajectory. That's not exactly the same as chaotic trajectory, but it's pretty close.

Thanks for this information.

 

[thaiqi]

Bohmian mechanics (which is compatible with QM formalism) allows thinking in terms of deterministic trajectories.

Are the deterministic trajectories circles/ellipses?

 

[Demystifier]

Are the deterministic trajectories circles/ellipses?

It depends on the quantum state (wave function). For the states with definite quantum numbers n,l,m they are circles, except for m=0 in which case the particle stays still.

 

[thaiqi]

It depends on the quantum state (wave function). For the states with definite quantum numbers n,l,m they are circles, except for m=0 in which case the particle stays still.

Thanks.

 

[akhmeteli]

Summary:: Can we tell whether hydrogen electron's motion relates to chaos?

Bohr's hydrogen atom model is outdated facing Schrödinger's wave equation. Now that wave mechanics doesn't use a concept of orbit for the electron in hydrogen atom. But can we suppose the electron is still circling around the atom core, not necessarily in circles or ellipses, but in chaos like a Lorentz attractor?

Maybe the following is not exactly what you look for, but it seems pretty close: http://www.scholarpedia.org/article/Semiclassical_theory_of_helium_atom

 

[thaiqi]

Maybe the following is not exactly what you look for, but it seems pretty close: http://www.scholarpedia.org/article/Semiclassical_theory_of_helium_atom

Thanks for the link.
I am reading it and some referred books. They are talking about what is called "quantum chaos", do you think they treat the particle as being on a deterministic/chaotic orbit?

 

[akhmeteli]

Thanks for the link.
I am reading it and some referred books. They are talking about what is called "quantum chaos", do you think they treat the particle as being on a deterministic/chaotic orbit?

My understanding (and it can be totally wrong) is that, for example, for the helium atom, most classical orbits are chaotic (and, according to the correspondence principle, quantum theory is supposed to inherit this chaotic feature), but quantum characteristics of the helium atom are defined by the few periodical classical trajectories (as stochastic trajectories do not contribute much to any specific Fourier component).

 

[thaiqi]

My understanding (and it can be totally wrong) is that, for example, for the helium atom, most classical orbits are chaotic (and, according to the correspondence principle, quantum theory is supposed to inherit this chaotic feature), but quantum characteristics of the helium atom are defined by the few periodical classical trajectories (as stochastic trajectories do not contribute much to any specific Fourier component).

I read part of it. I don't think their "quantum chaos" same as what I imagine.

 

[Sunil]

Summary:: Can we tell whether hydrogen electron's motion relates to chaos?

Bohr's hydrogen atom model is outdated facing Schrödinger's wave equation. Now that wave mechanics doesn't use a concept of orbit for the electron in hydrogen atom. But can we suppose the electron is still circling around the atom core, not necessarily in circles or ellipses, but in chaos like a Lorentz attractor?

Yes we can.

The Schrödinger equation contains a continuity equation in the configuration spacewith a velocity known as the "Bohmian velocity" because of its use in de Broglie-Bohm theory:

∂tρ(q,t)+∇(ρ(q,t)v→(q,t))=0.

In the de Broglie-Bohm interpretation, this velocity is deterministic. But there are other interpretations, like Nelsonian stochastics, which interpret it as an average velocity. So, the picture suggested by Nelsonian stochastics would be one with such a chaotic movement.

Literature:

Nelson, E. (1966). Derivation of the Schrödinger Equation from Newtonian Mechanics, Phys Rev 150(4), 1079-1085

 

[Demystifier]

Today appeared a review on chaos in Bohmian mechanics:
http://de.arxiv.org/abs/2009.05867

 

[thaiqi]

Today appeared a review on chaos in Bohmian mechanics:
http://de.arxiv.org/abs/2009.05867

I am ignorant of quantum chaos: what does the "trajectory" mean in the article? Are they trajectories of particles(e.g. electrons in an atom)?

 

[Demystifier]

I am ignorant of quantum chaos: what does the "trajectory" mean in the article? Are they trajectories of particles(e.g. electrons in an atom)?

Yes.

 

[thaiqi]

Yes.

Such that can electrons in Bohr's hydrogen atom model and others(e.g. Helium atoms) be described using trajectories?

 

[Demystifier]

Such that can electrons in Bohr's hydrogen atom model and others(e.g. Helium atoms) be described using trajectories?

Sort of, but Bohr's trajectories are not Bohm's trajectories.

 

[thaiqi]

Yes we can.

The Schrödinger equation contains a continuity equation in the configuration spacewith a velocity known as the "Bohmian velocity" because of its use in de Broglie-Bohm theory:

∂tρ(q,t)+∇(ρ(q,t)v→(q,t))=0.

In the de Broglie-Bohm interpretation, this velocity is deterministic. But there are other interpretations, like Nelsonian stochastics, which interpret it as an average velocity. So, the picture suggested by Nelsonian stochastics would be one with such a chaotic movement.

Literature:

Nelson, E. (1966). Derivation of the Schrödinger Equation from Newtonian Mechanics, Phys Rev 150(4), 1079-1085

Can anyone show more details of the Nelson's literature for me? I can't obtain it. Thanks.

 

[thaiqi]

Sort of, but Bohr's trajectories are not Bohm's trajectories.

Also Wyatt said in Quantum Dynamics with Trajectories (pp.4) :
"
However, it is important to distinguish formally exact quantum trajectory approaches from those that “simulate” quantum dynamics in terms of classical or semiclassical trajectories,
"
Why?

 

[akhmeteli]

Also Wyatt said in Quantum Dynamics with Trajectories (pp.4) :
"
However, it is important to distinguish formally exact quantum trajectory approaches from those that “simulate” quantum dynamics in terms of classical or semiclassical trajectories,
"
Why?

I gave a link to an article on semiclassics for helium in post #15 in this thread. It looks like Wyatt considers Bohm's trajectories, whereas semiclassics uses a sum over classical periodical trajectories.

 

[Demystifier]

Also Wyatt said in Quantum Dynamics with Trajectories (pp.4) :
"
However, it is important to distinguish formally exact quantum trajectory approaches from those that “simulate” quantum dynamics in terms of classical or semiclassical trajectories,
"
Why?

Are you asking why it is important to distinguish exact results from approximate ones? Because in some situations approximations give totally wrong results.