Using -e^2/(Re-Rp) potential in Shrodinger equation for hydrogen atom means that proton and electron are point particles. Why do we think, that on atomic size they are point-like? Just guess, that give exact spectrum?
We know that it doesn't work good for heavy nuclei with charge Z*e, where Z~137. But we have no idea how to fix this, not going to secondary quantization.
A mistake:
it is not THE electron or THE proton as such but the potential energy between the two entities as it shows up in the S-equation. The solution is what counts and that is NOT one with point-particles
Peter, I'm asking if there is more accurate potential for Schrodinger equation, giving Coulomb in small Z approximation.
Dear Eugene,
In one atom of hydrogen or even atoms with higher atomic number, it is easy to assume a Coulomb interaction or potential between the charges. What is more surprising is that this interaction works also in the case of having a solid with extended Bloch (Wannier) bands of energy. The reason is that the Thomas-Fermi screening length is of the order of the lattice constant of the lattice which allow to have free electrons without interaction among them.
I hope this helps.
Dear Daniel,
You are going to secondary quantisation :) Check my question. Do we have self consistent principle about charges interaction in quantum mechanics?
Dear Eugene,
No. I'm thinking in Schrodinger's equation of couse. No second order quantization operators, i.e. I was thinking in a Hilbert space and no in Fock functional space. But that is not fundamental question. What is important is why the Coulombic potential is enough for calculating most of the electronic interactions. Notice that there are many case where this is not possible as in the case of the strongly correlated electrons and in such a case the concept of band is difficult to use. Even models as Hubbard, Mott, Anderson,...are not simple to apply.
Another point of view is that Fermi gas is possible to use with great success in most of the physical materials. Why?
By the way, it is well known of how you cannot have elements Z higher or equal to 137. This is due to the relativistic limit of the velocity of light and the structure constant.
Dear Daniel,
Crystal spectrum with bands is not closely connected with Coulomb. Any interaction which gives lattice structure will give you almost free electrons.
Daniel, you definitely can have Z>137 :) Just use correct Hilbert space. On some Zs you will see spectrum reordering. In fact it's better to go to Dirac equation and 137 first limit will change to something about 120, then will be other critical numbers.
Eugene, no. That is not possible and it is a simple calculation of the spin-orbit coupling. Dirac spinors are a good representation but this doesn't solve this problem. The periodic table is finite! It is one of proofs that I do every year to my students in Electronic Structure subject.
Very interesting. I'll find references for you about calculations with any Z.
Check "Principles of advanced mathematical physics" Richtmyer, vol. 1. Ch. 10.16, 10.17
Dear Eugene,
Let me to recommd you the books:
1. Patrik Fazekas. Lecture Notes on Electron Correlation and Magnetism. World Scientific (1999).
2. Anisimov, Vladimir; Yuri Izyumov. Electronic Structure of Strongly Correlated Materials. Springer.(2010)
Where you can find the limits of the Coulomb law in a lattice.
I hope this helps.
Thank you for references.
As for periodic table - this is another story: big Z gives you nonzero size of proton and Dirac equation gives you solution for any Z.
We have some corrections to atomic spectra due to finite size of nuclei.
Electron in QFT is considered as point-like.
Actually point-like electron is a mystery from the point of view of general relativity: electron has huge spin angular momentum, it parameters correspond to exact solution of Maxwell-Einstein equations with loop singularity with diameter about Compton length. See http://www.mathnet.ru/php/presentation.phtml?option_lang=rus&presentid=14910
Sorry, what is loop singularity?
Why "Electron in QFT is considered as point-like"?
We can consider one electron in a cube. It will be smudged to all space, no point-like.
I don't agree with "We have some corrections to atomic spectra due to finite size of nuclei", because for big Z we will have spectrum where it isn't exist for pure Coulomb.
Dear Eugene,
I would like to ask you what is the model of the electron and the proton in your opinion?
The second point is that the potential energy of the electron in the field of the proton which you brought in your question is valid in the space outside the volumes occupied by the particles irrespective of their radius if they are assumed to be spheres. So, the solution will be valid also in space where this equation is valid.
Best wishes
Dear Eugene,
I would like to ask you what is the model of the electron and the proton in your opinion?
The second point is that the potential energy of the electron in the field of the proton which you brought in your question is valid in the space outside the volumes occupied by the particles irrespective of their radius if they are assumed to be spheres. So, the solution will be valid also in space where this equation is valid.
Best wishes
Dear Eugene,
You write " Why do we think, that on atomic size they are point-like? "
That's not really what "we" generally think. What we really think is that they "behave" mathematically point-like during scattering encounters, just like the Earth and the Moon can be mathematically considered to behave point-like when calculating their trajectories even in classical mechanics.
Dear Andre,
Do you mean, that Coulomb field is just zero approximation?
Dear Eugene,
I mean that in physical reality, the Coulomb interaction is not an approximation, but that it provides the exact value of energy induced in charged particles as a function of the distance separating them.
This is put in perspective in this recently published paper, as well as in all older style undergrad reference books that explain the true foundations of classical physics and electromagnetism:
http://file.scirp.org/pdf/JMP_2018042716061246.pdf
See also references 20 and 36 in the reference section.
In fact, energy levels can be calculated as precisely between charged particles as a function of the axial distances separating them as they can be with classical mechanics between large massive bodies.
Best Regards
André
First of all, yes, it is possible (in the spirit of A.A. Vlasov's ideas) to describe the electron via self-interacting cloud with the total potential being the sum of the contribution of the non-local kernel K(|r-r'|) and the local Coulomb potential of the nucleus, like this:
U(r,t) = -Ze2/r + \int K(|r-r'|) W(r',p',t) dr' dp'
where W(r,p,t) is the electron's Wigner function. However, if you wanted to include the nucleus in the same non-local picture, you would have to go over to the 5-dimensional Kaluza formalism, because non-identical non-local particles cannot be described by a single distribution function in a 4-dimensional theory. But in Kaluza's theory the non-local cloud supported in the neighbourhoods of two different values of 5-velocity would correspond to two different "particles", i.e. with different charge to mass e/m ratio.
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