Bohr radius of the electron. It is combination of fundamental contants, while its analogues mentionrd abive can include non-fundamental values, depending on particular system.

Literally, the Bohr radius is 0.53 Ångström. However, the experimentally determined radius of the hydrogen atom is 1.06 Ångström.

This discrepancy seems to have been ignored by the quantum mechanistics proponents and the CODATA experts.

Thank you for noticing it.

The Bohr radius of the hydrogen atom is the expectation distance of the electron in the central field of the nucleus and like the semi-major axis of classical mechanics, it defines the binding energy of the electron. If we consider the motion to have two degrees of freedom, as the Lagrangian formalism requires, then when we multiply the Bohr radius by 4π we get the de Broglie wavelength for the two components (the period is two cycles, not one, as we have to have a wave crest and a trough)

I note one other answer has said the size of a hydrogen atom is 106 pm, 53pm, which is the Bohr radius. Actually this distance is hard to determine but collisional experiments suggest just under 170 pm. However, the Bohr radius is the ex0pectation distance inn the potential field, and is an average of the potential interactions. The electron has to go more than twice the average to compensate for the inverse distance of the potential.

Ian Miller. At the Bohr radius, neither the energy balance nor the law of conservation of energy are observed.

And the mathematical calculation is carried out with the violation of the principles of mathematics.

How do you explain the violation of the energy imbalance in the Bohr atomic model?

How do you explain why the cause-and-effect relationship is violated in the Bohr atomic model?

Bakhodir Khonnazarovich Tursunbaev

The Bohr radius is simply a distance, and because Bohr assumed r = h/2πp, and because in the de Broglie wave theory λ = h/2πp you should see that there is a relationship between the Bohr radius and the de Broglie wave length, and the de Broglie wave length is the same as the wave length of Schrödinger's ψ.

I am not endorsing the Bohr model - it is wrong, not the least because it predicts the wrong orbital angular momentum. However, I am unaware of any violation of mathematical principles that Bohr used. He made a wrong assumption, but after that the mathematics were clear and valid, from the point of mathematics, if not physics. There is no energy imbalance in that model, and I am unaware of any faulty cause and effect relationship, other than the reluctance of the electron to spiral into the nucleus as per Maxwell's equations, but the same problem actually occurs with Schrödinger's ψ.

Ian Miller

Hello.

Sorry, the Bohr radius is not just a distance, but a very significant physical quantity that determines the size of an atom of a chemical element. Each atom of an element has its own specific radius.

The existing relationship between the Bohr radius and the de Broglie wavelength is not proof of the correctness of the Bohr model of the atom.

The mathematically beautifully written theories of de Broglie waves and Schrödinger ψ-waves are devoid of physical meaning. Quantum mechanics, instead of correcting Bohr's errors, continued to make new errors, introducing new imaginary concepts into the theory. As a result, the theory of quantum mechanics turned into an absurdity based on probability theory.

Physics before quantum mechanics was an exact science of nature, and now it has become a competitor to astrology.

For discussion, I suggest you read the book: Atom: Fundamental Errors and Methods for Correcting Them

Bakhodir Khonnazarovich Tursunbaev

Hello,

The Bohr radius is quoted (https://en.wikipedia.org/wiki/Bohr_radius) as 52.9 pm. That is a distance because its only dimension is length. Of course if the value is inserted into other equations it gives more information and of course it varies from atom to atom. It is like the semimajor axis of orbital mechanics - it defines the energy of the system, but its value is different for each orbiting body.

I have never suggested the Bohr model was coirrect; my point was that by being equal to the de Broglie wave length meant that it gave the correct energy for the hydrogen atom (leaving aside things like the Lamb shift.)

The de Broglie waves and Schrödinger ψ-waves are not devoid of physical meaning, although I accept that linking them to probability is actually wrong. See the open access paper "Miller, I. J. (2024). The Wave Function Must Represent a Physical Wave. J Math Techniques Comput Math, 3(7), 01-02."

Meanwhile, I claim all computations in chemistry other than for H2 are wrong because they use the wrong wave functions. Evidence: "I. J. Miller 1987. The quantization of the screening constant. Aust. J. Phys. 40 : 329 -346." and a broader physical explanation and examples of its use in my ebook, "The Covalent Bond From Guidance Waves" (https://www.amazon.com/dp/B07GCDYDRR).

So up to a point I agree with you that physics has gone off track, at l;east here.

Ian Miller

Hello.

Here is the problem. Despite theoretical justification and experimental evidence, it is claimed that the radius of the hydrogen atom is 1.06 angstroms, although Wikipedia, CODATA and quantum mechanics claim that it is 0.53 angstroms. Modern scientists promote this point of view despite experimental data, which raises doubts about the reliability of the information. Chemists already 100 years ago experimentally determined that the ionization of the hydrogen atom is 13.6 eV, and the Bohr potential well is 27.2 eV. When a proton captures an electron, it emits a photon with an energy of 13.6 eV. However, in the Bohr model, the question arises as to where half of the energy of the Bohr potential well went. It is also worth paying attention to the use of an incorrect reference frame in the Bohr model, which can be verified mathematically and physically.

Bakhodir Khonnazarovich Tursunbaev

Hello

It depends on what you mean by the radius. The Bohr radius is the position in the potential field that gives the energy of the level. However, the size of the atom is the space it occupies more or less exclusively, and that would normally be the van der Waals radius, which can be obtained by colliding two hydrogen atoms together and analysing the outcome. Unfortunately, this sort of experiment has its difficulties so you could also collide helium atoms with hydrogen atoms. Whichever you do, the problem is the collisions have to be soft or the electron distributions penetrate. In my opinion the best approach is from Mantina, M., Chamberlin, A. C., Valero, R., Cramer, C. J., Truhlar, D. G. 2009. Consistent van der Waals Radii for the Whole Main Group. J. Phys. Chem. A 113: 5806 – 5812. This gives a hydrogen radius in an extrapolated collision of near zero velocity of 168 pm.

If we think of the Bohr radius as the "average" position, the 106 pm is just lazy doubling, but the electrons go further, mainly because the force, being inverse square with distance, attenuates. Regarding the Bohr potential of 27.2 ev, twice the ionization energy, that is from the virial theorem (easily derived from the Lagrangian see Landau and Lifshitz "Mechanics"). The virial theorem has for U (total energy), V (potential energy) and T (kinetic energy) given by T = -U = -V/2. Basically the missing "potential energy" is the offset by the kinetic energy.

Ian Miller

Hello. Dear Ian Miller.

In order for our discussion to be truly fruitful, it is necessary to follow the academic rules.

In 1913, when Niels Bohr published his famous article, which later formed the basis of quantum mechanics, it did not consider either the average radius of the hydrogen atom or the concept of probability. Therefore, we will use Bohr's article as the basis for our discussion, since quantum mechanics built on these errors can eventually crumble like a house of cards.

Bohr's article discusses methods for determining the radius of the hydrogen atom, the main quantum numbers, the postulates of the theory, as well as methods for calculating the radii of stationary orbits and the energy levels of these orbits.

Isn't that right?

We argue that Bohr made the following fundamental errors:

1. An error in the method for calculating the radius of the electron orbit (a mathematical error);

2. An error in the method for calculating the radius of stationary orbits of the electron;

3. Violation of the energy balance of a closed system (the law of conservation of energy);

4. Incorrect use of the reference frame;

5. Mathematically incorrect calculation of the potential energy of an electron in orbit and incorrect interpretation of the physical meaning of these calculations and other aspects.

These errors require detailed analysis and discussion to understand how they affect the development of quantum mechanics and its modern interpretations.

I cannot understand how the Bohr model of the atom with such errors was accepted as the basis of quantum mechanics?

Maybe some self-respecting scientist will explain and answer this question?

Bakhodir Khonnazarovich Tursunbaev

Hello,

The reason why the Bohr model was accepted was that it correctly predicted the energy levels of the hydrogen atom ground and excited states, and further, Sommerfeld extended this to include minor influences such as relativity, he correctly assessed that it was the action generated per period that was quantized, and he correctly obtained the rule for transitions, i.e. which were forbidden.

The big flaw was that the theory predicted magnetic effects that were not observed. At the time, nobody knew the van der Waals radius of the hydrogen atom, so it is wrong to criticise them for not getting that right, and anyway they did not predict a solid radius; they said that was where the electron would be, but applying basic electric field theory to the atom meant atoms could never get that close together. The theory is criticized for not predicting the chemical bond but in my view current theory doesn't do much better. They could not explain why the electron did not spiral into the nucleus although they might have had they thought more deeply about Lagrangian mechanics.

Yes, it was wrong, it had flaws that Bohr recognized, but it was an advance.

Ian Miller

Since you are confident that the Bohr model correctly predicted the energy levels of the ground and excited states of the hydrogen atom, could you, for the sake of clarity, mathematically verify the potential, kinetic, and total energies of an electron in an orbit of radius 0.53 angstroms (relative to the chosen frame of reference)?

For the sake of fairness, we could compare your results with those of the Bohr model.

Additionally. In 2013, an experiment was conducted to visualize the structure of the hydrogen atom (DOI: https://doi.org/10.1103/PhysRevLett.110.213001). In the experiment, the hydrogen atom was excited by a laser with an energy of E = 5.413 * 10 (– 19) J (wavelength 365-367 nm, 3.383 eV). And note the result of the experiment.

Now tell me if Bohr's predictions about the energy levels correspond to the results of the experimenter.

Good luck with your calculations and analysis!

Bakhodir Khonnazarovich Tursunbaev

See https://en.wikipedia.org/wiki/Bohr–Sommerfeld_model for the calculation Arnold Siommerfeld made that calculate relativistic energies corresponding to those obtained by Dirac, and which are experimentally verified, except for the Lamb shift, which requires quantum electrodynamics. The Bohr model did give the correct energies within the limits of what was in it. However, overall it was wrong. There is nothing difficult about the calculations once you see how it is done, but they are too turgid to reproduce here, especially when they are there for all to see elsewhere.

I have never stated that the Bohr model is correct - merely that it did give the correct energies because the Bohr radius, when multiplied by 2π gave the correct de Broglie wavelength for the momentum of one electron at that distance from the nucleus. Bohr and Sommerfeld made the mistake of thinking the electron followed a reproducible trajectory, which it does not.

Having said that, the standard thinking of all other atoms, in my opinion, is wrong. See I. J. Miller 1987. The quantization of the screening constant. Aust. J. Phys. 40 : 329 -346. Then, if you are up to it, see the consequences in my ebook "The Covalent Bond From Guidance Waves". (https://www.amazon.com/dp/B07GCDYDRR) My main claim here is we can get fairly accurate chemical bond energies from analytical functions, thus for the molecules P2, As2, Sb2 we get energies within about 2 kJ/mol and for P2 a very accurate bond distance (the others are unmeasured). Standard computational methods won't get anywhere near this without a battery of assignable constants, which must get the right answer otherwise the constants are adjusted. And NONE of these constants ever applies more than for the special case of them, which hardly makes them constant.

Ian Miller

Bakhodir Khonnazarovich Tursunbaev

Thanks for the scientific discussion and comments. Certainly it is useful and of great scientific importance. But my question is which is more important and more accurate: the Bohr radius of the electron, the gap, or the exciton?

Nidhal Moosa Abdul-Ameer

In one sense, I would say that at east in chemistry the Bohr radius, but not because of the Bohr theory, but rather because it defines a useful length unit. That means if you have the solution of the Schrödinger equation for hydrogen, and provided you know the characteristics of a different element's orbitals, from the Principle of Mechanical Similarity you can directly get the energies for other elements and you don't have to solve increasingly hideous equations. The advantages should be obvious as shown in my ebook "The Covalent Bond from Guidance Waves". However, anyone not interested in chemical bonds will get no value from the Bohr radius so it becomes a matter of opinion.

Nidhal Moosa Abdul-Ameer

Hi, sorry for the delay in replying.

Errors in the Bohr atomic model (and in the Bohr radius) led to such imaginary concepts as the gap or the exciton, among others. This is due to problems of quantum mechanics with theoretical mechanics and electrodynamics (Newtonian, Maxwellian, etc.).

Quantum mechanics is a unique theory - the only "science" without paradigms for which the paradigms are Bohr's postulates and calculations, despite the fact that Bohr carried out calculations according to the canons of Newtonian mechanics.

You can verify this by delving into Bohr's method used to mathematically calculate the parameters of the hydrogen atom and energy levels, and interpreting the physical meaning of these calculations.

If you are interested in the real physical value of the distance between the proton (nucleus) and the orbit of the electron around the proton, then for scientific purposes its value (without probability) is 1.06 angstroms. This is called the radius of the electron orbit.

Reference: Nigmatov H. et al. A NEW INTERPRETATION OF THE RESULTS OF THE EXPERIMENT ON VISUALIZING THE STRUCTURE OF THE HYDROGEN ATOM (64-68) // Eurasian Union of Scientists - publication of scientific articles in the monthly scientific journal. Physics and Mathematics. 20.08.2020; 76(2):64-68. 10.31618/ESU.2413-9335.2020.2.76.904