How do point charges in a conductor move and stop?

[feynman1]

The probability distribution has to go to zero at some stage, past a notional boundary. Any model must include this, surely?

If electrons must stay on the boundary under electrostatic equilibrium, shouldn't the prob function collapse to 1 on the boundary?

 

[sophiecentaur]

My point was that beyond what might be called the boundary, the probability has to approach zero. It can’t be less than zero.

 

[Lord Jestocost]

Thanks for the suggestion. I did, but don't get what it implies. Could you give any qualitative result?

Friedel oscillations

Have a look at Figure 2a in http://venables.asu.edu/qmms/PROJ/metal1a.html.
The figure depicts the electron density at a metal surface in the Jellium model.

 

[nasu]

Take 1D. An electron is put in a 1D conductor with a potential V=0 -1<x<1 and very high elsewhere. Schrödinger's solution suggests the prob distribution on the well boundaries -1 and 1 is the least (and 0 for an infinite well). Why does this result contradict charges staying on the boundary of a conductor?

A single electron does not model the situation in a metal which is not neutral. You need to take into account the interaction between electrons. This is what makes the extra charge to go on the surface. Maybe if you put two electrons in a potential well you get some of the features you are looking at.

 

[Herman Trivilino]

Let's speak in the classical context (non quantum). We assume that point charges move in a conductor following Newtonian mechanics. How do point charges move along the boundary of the conductor and how do they stop (equilibrium) in the end?

I think you will get the answer you're seeking if, instead of looking at single electrons, you look at collections of electrons. Imagine a differential element of charge $dq$. It is a collection of electrons that is macroscopically small enough to be treated as a differential element, but it is microscopically large enough to contain a large number of electrons.

Loosely speaking, the differential elements $dq$ repel each other until they get as far apart from each other as they possibly can. Then the net force on each element is zero.

 

[feynman1]

I think you will get the answer you're seeking if, instead of looking at single electrons, you look at collections of electrons. Imagine a differential element of charge $dq$. It is a collection of electrons that is macroscopically small enough to be treated as a differential element, but it is microscopically large enough to contain a large number of electrons.

Loosely speaking, the differential elements $dq$ repel each other until they get as far apart from each other as they possibly can. Then the net force on each element is zero.

Thanks but can the differential element treatment show if electrons will stay on or bounce off the boundary before equilibrium?

 

[feynman1]

A single electron does not model the situation in a metal which is not neutral. You need to take into account the interaction between electrons. This is what makes the extra charge to go on the surface. Maybe if you put two electrons in a potential well you get some of the features you are looking at.

Suppose there are infinitely many electrons in the 1D conductor so that the conductor is already equipotential V=0. Then put another electron in subject to a potential V=0 -1<=x<=1 and very high elsewhere. Would the result differ? The prob function on the boundary would still be small.

 

[nasu]

The potential in a metal is periodic, and is due to the interaction of electrons with the ion cores. The wave function of the electrons are combinations of Bloch functions. But these are obtained by using periodic boundary conditions. The surface effects are not teated in "simple" models that describe the bulk properties of the solid. I don't think you can get the macroscopic properties of conductors with the simple models that you propose. Maybe someone with experience in surface physics could help.

 

[Herman Trivilino]

Thanks but can the differential element treatment show if electrons will stay on or bounce off the boundary before equilibrium?

Yes.

 

[feynman1]

Yes.

how?

 

[Herman Trivilino]

how?

Have you tried to understand this by reading a book? I don't understand what you mean by "bounce off". Do you mean they move from the surface towards the interior? Any movement is due simply to the repulsion by other elements $dq$.

 

[feynman1]

Have you tried to understand this by reading a book? I don't understand what you mean by "bounce off". Do you mean they move from the surface towards the interior? Any movement is due simply to the repulsion by other elements $dq$.

You seemed to be saying that charges are repelled and get away from each other. But that doesn't show how they move when approaching a conductor wall, whether they stay on the wall or bounce back from the wall...

 

[sophiecentaur]

You seemed to be saying that charges are repelled and get away from each other. But that doesn't show how they move when approaching a conductor wall, whether they stay on the wall or bounce back from the wall...

I’m not sure why you seem determined on an inadequate model in order to increase your feeling an understanding of this topic. Why not look for a more up to date approach which can handle more situations reliably?
That would, of course, involve a lot of hard graft.

 

[Herman Trivilino]

But that doesn't show how they move when approaching a conductor wall, whether they stay on the wall or bounce back from the wall...

What is a "bounce"? Isn't it just a repulsion?

 

[feynman1]

What is a "bounce"? Isn't it just a repulsion?

I don't know. Just guessing. Repelled by what?

 

[Delta2]

In quantum physics there is no such thing as a solid boundary or a solid wall.

 

[sophiecentaur]

I don't know. Just guessing. Repelled by what?

Your guess should include the knowledge that the actual position of an electron when it's in the bound state (i.e. in a metal) is not defined. There is no 'mechanical' language that can describe what you want. That's why we use QM.
This thread is not going anywhere because there is no satisfactory answer in your terms. Learn a bit of QM and the problem will become clear.

 

[Herman Trivilino]

I don't know. Just guessing. Repelled by what?

My goodness! Each differential element of charge $dq$ is repelled by every other element of charge.

Instead of guessing, crack a book.

Or at least examine your own thinking processes. What led you to "guess" bouncing off the surface? Unless you are willing to examine why you made that guess you might as well be asking why they don't bounce off the moon.

 

[Vanadium 50]

Have you tried to understand this by reading a book?

He is a faculty member looking for PhD students. So he read the books.

That of course leads to the next question - why he wants to consider a model that is neither classical (continuous charge) nor quantum mechanical, and why he thinks this model will produce anything but nonsense.

How many times have we said "Drude model"? Water...duck...back...

 

[feynman1]

He is a faculty member looking for PhD students. So he read the books.

That of course leads to the next question - why he wants to consider a model that is neither classical (continuous charge) nor quantum mechanical, and why he thinks this model will produce anything but nonsense.

How many times have we said "Drude model"? Water...duck...back...

Can we consider electron-ion interaction in Drude model? What potential is appropriate?

 

[nasu]

Can we consider electron-ion interaction in Drude model? What potential is appropriate?

Have you ever studied solid state physics? Or at least browsed through an introductory textbook like Kittel? It's really hard to guess your background from your questions.

 

[anuttarasammyak]

I have read a related article last month.

VirginiaTech210915
https://vtx.vt.edu/articles/2021/09/science-electrons_interactions_heremans_lab.html
When electrons flow through a conductor — such as the copper wires in our phone chargers or the silicon chips in the circuit boards of our laptops — they collide with material impurities and with each other in a tiny atomic frenzy. Their interaction with impurities is well known.
Yet, while understanding how electrons interact with each other is fundamental to understanding the physics, measuring the strength of these interactions has proven to be a tricky challenge for physicists.
A team led by Virginia Tech researchers has discovered that by creating a specific set of conditions, they could quantify electron-electron interactions more precisely than ever. Their findings expand upon existing physics theories and can be applied to improving electronic devices and quantum computers. They recently published their findings in the journal Nature Communications.
https://www.nature.com/articles/s41467-021-25327-7

 

[anuttarasammyak]

Let's speak in the classical context (non quantum). We assume that point charges move in a conductor following Newtonian mechanics. How do point charges move along the boundary of the conductor and how do they stop (equilibrium) in the end?

Another approach from the other end is superconducting current in a ring. Non stop. Genuine QM effect.