There is a very close and accurate agreement between the classical electron radius, alpha, the electron Compton wavelength and the Bohr radius. such that:
re = alpha. lambdae /2pi = alpha2 a0
Given the great accuracy ascribed to alpha in particular, this would suggest that the classical electron radius is valid.
It hasn't been ``rejected''. The classical electron radius, the Compton wavelength and the Bohr radius refer to three completely distinct physical situations. However, while it is possible to describe consistently experimental setups, thought or real, where the Compton wavelength and the Bohr radius can be defined and measured, it turns out that it isn't possible to do the same for the length scale that could be called the classical electron radius. A nice discussion about the history can be found in Feynman's lectures, http://www.feynmanlectures.caltech.edu/II_28.html, cf. also, Coleman's article,https://www.rand.org/content/dam/rand/pubs/research_memoranda/2006/RM2820.pdf
The fact that it is possible to find some expression with the dimensions of length does not, necessarily, imply that there is a physical effect associated with that length scale.
Because postmodern physics is in total confusion.
Despite the downvoting of an Anonymous researcher I confirm "postmodern physics is in total confusion".
Yes but it doesn't have to be the precise radius just the averaged or most probable radius as with the Bohr radius.
Besides if it were numerology they were worried about, why not ditch a lot of QM by the same token
On sizes of massive elementary particles there is much confusion. The classical radius of electron is about 2.82x10-15m while recently a smaller than 10-20m radius of electron has been measured.. The difference is in the ratio of a factor 105 that is extremely great comparable to the ratio between 100Km and 1m.
Equation confirming both the classical electron radius and the proton charge radius: proton to electron mass ratio Mp / Me = 4 Re / Rp α
Mp = proton rest mass
Me = electron rest mass
α = fine-structure constant
Re = classical electron radius
Rp = proton charge radius
Working from Nassim Haramein's derivation for the proton charge radius, Mark Rohrbaugh found that 4 Re / α = α^2 / π Rh, where Rh = Rydberg constant; resulting in the above alternative equation for the proton/electron mass ratio.
Sherbon, M.A. "Quintessential Nature of the Fine-Structure Constant" GJSFR 15, 4: 23-26 (2015).
Considering the preceding relation the electron radius is Re=3.4Rp. Consequently the proton radius is smaller. All this confirms we are farthest from having understood fundamental aspects of particle physics.
Despite the downvoting of an anonimous researcher I confirm "we are farthest from having understood fundamental aspects of particle physics"
From en.wikipedia.org/wiki/Classical_electron_radius
In simple terms, the classical electron radius is roughly the size the electron would need to have for its mass to be completely due to its electrostatic potential energy – not taking quantum mechanics into account. We now know that quantum mechanics, indeed quantum field theory, is needed to understand the behavior of electrons at such short distance scales, thus the classical electron radius is no longer regarded as the actual size of an electron. Still, the classical electron radius is used in modern classical-limit theories involving the electron, such as non-relativistic Thomson scattering and the relativistic Klein–Nishina formula. Also, the classical electron radius is roughly the length scale at which renormalization becomes important in quantum electrodynamics.
All this confirms we are farthest from having understood fundamental aspects of particle physics.
@Daniele
Agreed,
But a lot of this understanding can be recaptured if we re-examine two aspects of quantum physics.
1). The fundamental quantum mass should be in keeping with the fundamental action that is Planck's constant, so mq = h/c2 .It is not of concern that this mass has the dimension of [M][T] because everything in the quantum world has a frequency hence the final dimensions are [M]. So this mass now dovetails in with Planck's constant.
2). The second is velocity: a lot of the velocities involved in the wavelength of matter and spin for instance are the phase wave velocities.
For instance the deBroglie wavelength (lambda =h/p) only works if we use the phase wave velocity 137c or c/alpha
Which brings us to the original question. the radius of the free electron determines that the electron spin (h/2) means the rotational velocity must also be 137c or c/alpha.
For more information please see attached.
Article The formulation of harmonic quintessence and a fundamental e...
One of the original formulations of α, the fine-structure constant is α = v/c with v the orbital velocity of the electron. Another result of our previous equation shows:
α = (Me / Mpl) (Re / Lpl) where Mpl = Planck mass, Lpl = Planck length and again; Me = electron rest mass, Re = classical electron radius.
These relations are indeed true, and generally acknowledged to be so, as they follow directly from the definitions of the classical electron radius (e^2/(mc^2)), the Compton wave length (hbar/(mc)), the Bohr radius (hbar^2/(me^2)) and finally the fine structure constant (e^2/(hbar c)). The relations observed are quite correct, but do not necessarily point to a role of the classical electron radius in quantum mechanics. In fact, the atom does have dimensions of the order of the Bohr radius, which is 4 orders of magnitude larger than the classical electron radius.
On the other hand, all these radii are length scales associated to any particle of charge e and mass m. They correspond to various phenomena: the classical electron radius tells us at which spatial scale radiation reaction becomes important in classical electrodynamics, the Compton wavelength tells us how much we can localize the electron before the resulting momentum uncertainty leads to relativistic effects and the Bohr radius tells us how extended a wavepacket of mass m and charge e is in non-relativistic quantum mechanics.
None of these, however, are a radius of the electron, which, as far as experiments show, is a pointlike particle.
All calculations would prove a surprising thing for me: electron would be greater than proton. In other words, in a model with spherical symmetry, electron radius would be greater than proton radius. Recent experimental outcomes would prove electron radius is smaller than 10-20 m. Besides nucleus radius is smaller than 10-15 m. I think it needs to start from those data.
Likepoint particles raise many problems about mass, charge, spin and other physical properties of particles. Likepoint particles are a mathematical abstraction that sometimes myself use when it is not important to point out physical properties of single particle, like for instance in the event of electric and electronic currents. In that case it is supposed that every likepoint particle coincides with its barycentre. This mathematical model nevertheless doesn't well when we want to consider physical properties of single particles. Particles, like all massive objects, have a real space size that we don't know yet but this ignorance isn't due to a property of nature, like the Principle of Indetermination and classical Quantum Mechanics affirm, but it is due only to inadequacy of our present measuring instruments. I am sure that physical and engineering research will be able to allow a precise measurement of space sizes of electron and of other particles, in the order of experimental errors. Inter alias already now we have precise measurements of mass, charge and average lives of particles. A mass, even if littlest, cannot reside physically in a zero volume.
Questions like this one have been puzzling me from my student years. Yet another lecturer put some purely classical quantities in a combination, added Planck's constant and a mantra about "Heisenberg uncertainty" or "energy vs frequency", and, behold, we have learned something about a quantum system. In a sense, I spent all my life trying to unravel this mystery. My personal answer is, THE DIFFERENCE BETWEEN QUANTUM AND CLASSICAL MECHANICS HAS BEEN GREATLY EXAGGERATED." It does exist (cf. microscopic indistinguishability, Fermi-Dirac statistics and Bell's inequalities), but, much more often, the Hilbert-space formulation is classical mechanics in disguise.
As a mathematical fact, this is discussed in a recent paper of ours (PRA 92, 022122, 2015; available from this site). In brief, the dynamical structure of quantum and classical electrodynamics are formally identical. Since us humans only access the world through electromagnetic interactions, all quantum manifestations in real experiments may be attributed to matter. Depending on the experimental accuracy and on how elaborate the experiment is, we see more or less of the "quantumness of Nature." In the everyday life, we see none ("classical world"). With a more elaborate insight, we start distinguishing this quantumness, but it is never pure. What we see is always a mix of quantum and classical concepts.
Getting back to the initial question, the classical radius of the electron is one of the classical concepts reemerging in QED. No mystery whatsoever.
@Plimak,
Thank you for your responses.
I have come to a firm conclusion from you responses-
The classical electron radius is the is the effective (averaged) radius of the charge that a free electron generates.
It 's physical radius is a lot smaller (estimate approx 10-25m), but we are currently unable to measure its size.
When I attend to physical questions I don't raise the question if space and time are fundamental categories. From a physical viewpoint space and time are measurable physical quantities that are very useful for a satisfactory interpretation of nature in the order of physical phenomena. In the order of those often problems are born because of intrinsic difficulties due to the differences of scale. For human the natural scale of distance is the metre, for cosmological events the natural scale is the light year. for microscopic events the natural scale is the angstrom. Ratios among these scales are nevertheless alwayas the same: 105LY/1LY=105m/1m=105A/1A=105. The question is born relative to measuring thecnique and not to a mysterious conspiracy of the nature. It is manifest that human has developped before measuring techniques in concordance with the human scale, but significant progresses have been made and will be made also relative to the cosmological scale and to the nanoscopic scale. With regard to the question on the nature of physical reality I think the universe is fundamentally continuous in its fundamental structure, physical events may present sometimes quantum characteristics like in the event of orbital trajectories of electrons in atom. Those characteristics are the result of a quantization law relative to the considered physical event and aren't the result of a primary property of nature. The purpose of researchers then must be that of researching those laws of quantization.
@Worsley
Agreed. Remembering my education, the electromagnetic radius of the electron is the characteristic size of the electron electromagnetic formfactor. The latter is the structure you see when probing the electron by charged particles. This gives experimental meaning to your term "generate." If you use a neutrino instead, you'll see the weak formfactor. And so on.
Not sure what you call "physical radius." Would you please elaborate? Which interaction should be used for measuring it?
It hasn't been ``rejected''. The classical electron radius, the Compton wavelength and the Bohr radius refer to three completely distinct physical situations. However, while it is possible to describe consistently experimental setups, thought or real, where the Compton wavelength and the Bohr radius can be defined and measured, it turns out that it isn't possible to do the same for the length scale that could be called the classical electron radius. A nice discussion about the history can be found in Feynman's lectures, http://www.feynmanlectures.caltech.edu/II_28.html, cf. also, Coleman's article,https://www.rand.org/content/dam/rand/pubs/research_memoranda/2006/RM2820.pdf
The fact that it is possible to find some expression with the dimensions of length does not, necessarily, imply that there is a physical effect associated with that length scale.
This question is meaningless without precising how, in what experiment, are we supposed to measure the electron radius. The "classical radius" e^2/mc^2 does appear in formulas describing certain phenomena, in others one encounters h/mc, which one might call the "quantum" electron radius. This should not be taken too seriously. Most probably, this question does not make sense, at least at the current level of our understanding (which is quite good). Is there anything above this level, and if yes, what? - Nobody knows.
This question is meaningless in postmodern physics in which in the order of QM because of the Indeterminacy Principle positions and consequently distances have only an indeterministic meaning. Contemporary physics instead establishes electron and all particles have a size and consequently a radius into a model with spherical symmetry. Just in this question contemporary research has reached an interesting result, in concordance with Michael A. Sherbon's calculations and confirmed by my calculations, i.e. proton has a smaller radius than electron that in its turn, as per recent experimental outcomes, has a smaller radius than 10-20 m.
Electrons in particular, leptons and quarks more generally, are described as point objects in the Standard Model-and measurements at the LHC are, indeed, consistent with this statement. Protons are known not to be point objects. It is, however, not possible to calculate the proton radius from QCD directly, since the quarks are strongly coupled, in the energy range where the proton is the appropriate description-they are weakly coupled when their individual motion can be resolved. In principle it should be possible to compute the proton radius from lattice QCD, but the calculations are very challenging.
I think there is much confusion, like it often happens in postmodern physics, on the concept of point object or point particle. If point object or point particle means the object or particle has smallest sizes and volume with respect to sizes and volume of conventional objects, then it has a physical meaning. If point object or point particle means the object or particle has size and volume of a geometrical point that has zero size and volume, then it is another absurdity of postmodern physics.
Point object (or point particle, if one likes to say twice the same thing) is a term that's meaningful in classical physics, so there isn't anything ``modern'' or ``postmodern'' about it. It can be given and is given physical content by experiments that probe the structure of such objects and by theoretical descriptions that do so in a consistent way.
First of all, when saying "radius", you assume a corpuscular nature to the electron. There are no experimental results proving such a corpuscular nature. Only definite charge, spin and rest mass exist. Hence flow two intrinsic frequencies : the broglian one, mc²/h wich is spinorial and governs the interference of the electron with itself, for instance in Aharonov-Bohm experiments, and the Dirac-Schrödinger one 2 mc²/h which is electromagnetic. The later rules the Compton scattering, as it was proven by Erwin Schrödinger, and waved by P.A.M. Dirac in his Nobel lecture.
I suggest that you have a look in solid state physics, and consider the conduction electrons in a metal, say copper at room temperature. At the Fermi Energy, 7 eV, they have a group velocity of 1,570 km/s and a phase velocity of 57.2 E9 m/s, with a wave length about 4,63 Å. At 300 K, the free run is about 300 Å. So the spatial extension of a conduction electron is constantly several tens of interatomic distances. Each... And the lowest temperatures imply longest free run and spatial extension of each conduction electron, up to the entire crystal. The apexes are usually phonon-electron interactions, and the phonons cannot be small, as they are sampled on many cations.
As transactionists physicists we concluded that at the apexes, the emitting reaction and the absorbing reaction, the transversal diameter of the electron is about the fuzzy diameter of the reaction. Meanwhile, during its fly in the vacuum, the belly of the Fermat spindle remains bound by the Fermat condition : maintening a phase coherence. So we obtain electronic diffractograms by crystals, not so good than the X-Ray ones, but however useful. The electron is not a distinct "thing", it is the propagating wave or the stationary wave. There is no (magic) transmutation into some corpuscle.
Corpuscolar nature of a particle is due to its small sizes. In the event of massive particles (electron, proton, etc..) the size is caused by the mass density of particle. In the event of energy particles (photons, gamma rays, delta rays, etc..) the size is caused by the wavelength of electromagnetic nanowave that represents the quantum. It is possible to associate an equivalent wavelength to a a massive particle that can be De Broglie's wavelength or Compton's wavelength according to necessities. It is possible also to associate an equivalent mass to an energy particle. It is manifest that an electron at low speed has a De Broglie's greatest equivalent wavelength while Compton's wavelength is independent of speed. We cannot think mass of a massive particle (for instance electron) and mass of a iron ball have the same physical nature. On this account for elementary particles, whether leptonic or baryonic, in the Theory of Reference Frames I defined electrodynamic mass that is different from inertial mass and gravitational mass of ordinary bodies like an iron ball. Electrodynamic mass changes with the speed according to the relation m=mo(1-v2/2c2), it follows that electron and massive elementary particles are subject to internal transmutations whether because of interactions with other particles (for instance positron) or when they are accelerated by a force field. Electron diffraction is a physical phenomenon that is studied and explained also in the Theory of Reference Frames in which electron diffraction is due to the electrodynamic nature of particle and not only to its equivalence with an electromagnetic nanowave.
@Daniele Sasso. I do not fully undestand your answer.
How do you insert in your theory the facts of an Aharanov-Bohm experiment ?
Each electron of the electronic beam travels simultaneously above an under the micro-solenoid. There is no common phase in the beam as the electrons are fermions. So each electrons interfers only with itself.
http://deontologic.org/quantic/images/2/27/Aharanov_bohm_perspective.jpg
http://deontologic.org/quantic/images/5/58/Aharanov_bohm_coupe_transv.jpg
Daniele Sasso
Agreed
However just to qualify your statements the Compton wavelength is the wavelength of a photon at speed c with the same energy of the ( rest mass) electron.
Lamba = h/p, where p is the relativistic momentum
@Jacques Lavau,
I don't attend usually to this type of applications of electron physics. It seems to me nevertheless the Aharonov-Bohm experiment and the effect, cited by you, is based on the hypothesis that the magnetic field outside the solenoid is null, but that hypothesis isn't verified correctly in the reality because of the fact that the length of solenoid isn't infinite.
Dear Andrew, I agree fully with your clarification in regard to Compton's wavelength.
My regards.
@Daniele Sasso.
If you look at the scheme http://deontologic.org/quantic/images/2/27/Aharanov_bohm_perspective.jpg you observe that the magnetic circuit is closed by a ferromagnetic piece. The tensor field B is nearly null where the electron passes, but the vector field A remains not null, and decreases in 1/R from the solenoïd, and it is its circulation that matters.
Dear Jacques, you confirm the magnetic field B is nearly null but not null, The question is clear to me.
Pfff! indeed: what mechanism do you propose, in order to have an effect that remains non-zero even as the magnetic field tends to zero? For the magnetic field undeniably can be mad as small as desired, and the Aharonov-Bohm effect will not disappear.
And which is more : the interferences do not disappear either ! So each electron has travelled simultaneously on both sides of the beam splitter and of the micro-solenoïd, with or without the magnetizing current. It remains a variant of the Young's two-slits experiment.
The classical electron radius is definitely valid for energy calculation purposes.
But the only reason why it fits so well with the electron other parameters is that it is the lower limit of integration of an electron's energy when this energy is calculated by integrating from an upper limit located at infinity.
When calculating the complement of energy of localized particles having other levels of energy with the same method, other "classical particle radii" will of course be obtained in relation to these other energy levels.
They can always be obtained from the equation you quote when using the lambda related to any particular energy frequency.
r=alpha lambda/2pi
If interested, the following paper elaborates further on this issue:
http://www.gsjournal.net/Science-Journals/Essays/View/2257
The "problem" with quantum physics that it stems from the fact that certain things come in smallest portions, in particular that some objects do not have some properties attached at all. For example, you can say that a sample of soil came from Egypt, in fact you can buy small sacks of it as as souvenir. This soil can be discerned from sample from Scotland for example. So it does make sense to ask: "Where this piece of dirt comes from?" But, if you take one copper atom from Egypt and the other from Scotland there is no point in asking which atom came from where. Because if you give me the two atoms and I shuffle them, you will never be able to tell which one is which. You may come to the "clever" idea though to paint one of them orange and the other green, so that even after I I shuffle them you will be able to tell which is which. But - bummer - you can not do that! Because atoms do not have. or better to say, do not support color.
Similarly, an electron simply does not have a radius. It is true that in some experiments you can do one kind of "sensible" calculation and get one formula for radius (the one you quote), but in some other you may come up with another formula, and in some third with a third formula all of them being different and "valid" in specific situation, just as a photon sometimes looks like particle and sometimes like a wave. In both cases we are inquiring about properties that the object in question does not possess.
I know it may be difficult at first. Macroscopic objects may have zillions of properties each of which having infinite number of possible values: color, mass, charge, angular moment: you name it. But when it comes to elementary quantum objects, aka "elementary particles", they each have a very limited number of properties, some of which are quantized which further reduces number of possible variations. Each particle has only some properties, while all the others IT DOES NOT HAVE.
So whenever you wish to think about an electron, simply drop the idea of its radius. It does not live there.
@André Michaud.
You surreptiously use the postulate of autosimilarity of space at all scales, with unlimited extrapolations, with extrapolation of the infinitely fine topology, as postulated by the mathematicians of the 19th century.
Please exhibit an experimental validation of this postulate and of its extrapolation up to microphysics and electrons.
Thank you by advance.
As far as I know, there are some atomic limits in the nature. For instance neither atoms nor their nuclei age. In the contrary of us...
Dear Jacques
I postulate nothing of the sort. I suggest not to confuse mathematical methods used for calculation purposes with physical reality.
Integration between two limits dates back way further than the 19th century, to Newton and Leibnitz who discovered the method. Seems to me that the method of integrating energy from infinity as one limit to a convenient lower limit for purposes of calculation has been in use for quite a while.
In the case of the electron rest mass energy, it simply happens to be the so-called “classical electron radius”. Not a radius at all in fact, just a lower limit of integration, traditionally tagged “radius” because of the method, because integrating any closer to r=0 would give more energy than confirmed experimental data warrants.
Alternately, can you suggest another method?
The classical radius can also be considered as the electrical radius as it is based on its charge
Dear Andrew
Thinking about it, I recall having observed in a past analysis that this value "the classical electron radius" seemed to also exactly match what I identified as the amplitude of the transverse electromagnetic swing of the fixed kinetic energy quantum that makes up the rest mass of the electron.
So you might have a point here. Can you elaborate?
If you did already in this thread, I missed it.
I suggest you to read the "Absorber theory", Wheeler and Feynman, 1941, 1945 :
Now in line at http://authors.library.caltech.edu/11095/1/WHErmp45.pdf
Dear Jacques
Interesting paper indeed.
Note that in Gauss's time the notion of acceleration induced kinetic energy in charged elementary particles by the Coulomb force was not yet around since the electron was discovered 11 years after he described this idea of "action at a distance that would propagate with a finite velocity" (1856 by Joseph Thompson),. Even such a finite velocity of free energy was established even later by Maxwell as being c.
So Gauss did not have a full hand of cards at his disposal. He did the best with what he had. Quite remarkable in fact that he could come up with this concept at that moment.
But we know much more now about kinetic energy induction.
I note also that the paper is fields-oriented and, unless I missed it somehow, it does not directly address the energy induction process in elementary particles. As far as I can see it does not directly address calculation of energy per se either of that making up the rest mass of the electron or that which is induced in accelerating charged elementary particles.
The whole concept of radiation source versus absorber being described with its advanced and retarded fields concept due to the assumed limited velocity of the message propagation between source and absorber appears to be self-consistant. All the more so since from Wheeler/Feynman's own admission, at the limit it provides the same end results as what they name "the theory of action at a distance", but with much more complex math development to account for the advance/retarded fields that need to be involved and the associated notion of pre-acceleration.
Even in Wheeler and Feynman's time of publication (1945), it was not yet known that neutrons and protons are not elementary but composite particles made up of triads of scatterable, charged and massive elementary particles (up and down quarks). This is why he speaks of and takes into account the possible existence of "uncharged particles (page 158)". This was clarified much later in 1968 from SLAC experiments that the only massive stable elementary electromagnetic particles in existence are electron, positron, up quark and down quark.
So we know now that in physical reality, all elementary particles making up all atoms in existence are charged. No exception.
This means that in physical reality the only "accelerated" particles in existence are those that are captive of various stable states of least action electromagnetic equilibrium. All other particles are in the process of accelerating and will be until they stabilize in some electromagnetic least action states.
Regarding the so-called action at a distance that supposedly can only propagate at a finite velocity. As I mentioned, at the time of conception of the idea, the clear distinction that needs to be made between electrostatic force (Coulomb force) per se and the kinetic energy that this force induces in accelerating charged particles seemed not to have been yet completely addressed, from what I could gather.
All being considered, the Coulomb force acting between two charged particles, a force whose nature and origin are still a mystery, is known to be in permanent infinitesimally progressive decreasing or increasing action as a function of the inverse square of the distance between the particles involved, irrespective of the distance between them, which means that it is permanently present by structure and does not "travel" or "move" between the particles.
So it seems to me that the force proper does not move, but is the cause of the acceleration of charged particles towards or away from each other, or rather, it is the cause of the induction of kinetic energy in the particles, an energy that will be vectorially directed towards the other particle in case of attraction or in the opposite direction in case of repulsion, and that naturally manifests its presence by a change in velocity of particles when external electromagnetic constraints do not partially or even completely prevents such a change in velocity.
This is why the energy that progressively accumulates in an electron accelerating towards a proton, for example, can always be calculated by integrating the energy of the Coulomb equation from infinity to a0 (radius of the mean rest orbital of an electron in a hydrogen atom), where it will be captured in this well known state of electromagnetic equilibrium, a sudden stop that will force the emission of a 13.6 eV bremmsstrahlung photon of translational energy now in excess.
This is a very interesting and pertinent question Andrew. I'm currently writing up a paper that 'extends' Bohr's early quantum theory written up in 1913 in his paper "On the Constitution of Atoms and Molecules" Philosophical Magazine. Series 6 Volume 26 July 1913 p.1-25. You probably all know this was the first time a quantum method was used to solve an atomic application. He 'solved' the problem of why the electron did not lose energy and cause the annihilation of the hydrogen atom by hypothesizing that the energy related to Planck's 'theory of energy radiation' where Planck's constant supplied the energy. Second, I can't find a word against Bohr's radius even today a century after the invention of quantum theory. So it seems this radius is still accepted as gospel truth. Does anyone know if it's been measured recently?
André Michaud's comments above about the Coulomb field and the physical nature of the force between the electron and the proton are both something I'm looking at
So, quite a lot to comment on.
Let me just say for starters that I'm seeing the photon in its role as binding energy as a composite boson (in fact hydrogenic with two equal and oppositely charged internal particles) with non-negligible mass that collides with both the electron and the proton. It is by the momenta of these tiny collisions that the energy of the electron and the proton change their direction and orbit around each other in two orbits one due to their mutual electric fields the other due to their mutual magnetic fields. So each particle moves along two orthogonal directions simultaneously.
When I use Maxwell's equations for this model I get the early Bohr Theory results; I can extend his method to get both these rotational motions for each particle. This is like a deterministic quantum method
Needless to say I'm agreeing with many of the negative comments about QM and QFT.
According to quantum mechanics, the Hydrogen atom in its ground state does not emit radiation simply because there are no states with lower energy (by definition of the ground state). In classical physics the lowest energy states corresponds to the electron falling on the proton. One might say that it is the uncertainty principle that forbids the electron to fall on the proton.
Hi Christian
What is amazing then is how did I get the solutions via electrodynamics (I won't say classical electrodynamics) using Maxwell's equations
That’s a fair request and comment. There’s a book published by Pan Stanford in 2012 called Self-Field Theory: A New Mathematical Description of Physics. I’m currently writing up a 2nd edition of it while writing up the extension of Bohr’s early quantum theory as a chapter in the book that will also be sent to be published in a suitable journal. If I were you I’d wait until the new edition was published in about 18 months.
In the SFT formulation the main equations boil down to two equations that are identical to the inequality statement of Heisenberg’s Uncertainty Principle except they are equality statements.
Having obtained the early Bohr results for the radius and frequency via the Maxwell-Lorentz Eqns I have turned to Bohr’s early work and am trying to understand how he got these correct results using the Coulomb equation in what is ostensibly an electromagnetic application!!??
And I think I’ve worked it out; it’s because all fields including electric and magnetic fields act via these small collisions. So rather than the fields being just an ‘action at a distance’ the force is NOT an action at a distance (although the amplitudes of the fields do depend on distance and time). Rather the tiny momenta between electron and photon, and proton and photon are a 'near-field' effect regardless of how far away they originated.
Michael,
This is an interesting way of putting it, but I believe it is wrong. What I think is that electron cannot fall on proton because there is no such system. And if system does not exist it cannot become. If you think of it you cannot just drop electron on the surface of proton. Even though electron and proton attract each other, as electron closes in it will start to radiate energy and finally end up in the ground state just if there is an infinitely stiff impenetrable bubble around the proton. You cannot even close electron onto a proton by force. If you accelerate it, the two will simply bounce of each other or generate a soup of particles that will spread in all directions..You cannot arrive to a still system called "electron happily sits on the surface of proton attracted by electrostatic force". I bet there is no solution in QM that would allow it, probably because it is not symmetric and maybe simply because it is not on God's list of allowed systems.
Christian, you are right of course. But I was referring to the classical picture in which, in the ground state electron is orbiting around the proton at an incredibly large distance comparing to the classical radius of electron which brings in the suspicion that electron can come any closer to the proton. However ground state IS the closest distance. For all it counts classically, it is zero.
@Ed
Correct, but that then begs the question what is the relativistic radius of the free electron
Hello Andrew and all: Thank you. You ask, "what is the relativistic radius of the free electron"? The answer depends on coherence, on collective effects. It can be 1 Km long. It is certainly not a point particle, it is a quantum wave. At some high energy, it can behave like a point particle, it is an approximation.
Classical electron radius $r_{e}$ can also be defined by magnetostatic criteria which reveal that $r_{e}$ might have a remarkable relationship with the magnetic flux quantum $\Phi_{0}$ (fluxon):
\begin{eqnarray*}
r_{e}= \frac{\mu_{0}\hspace{2}\mu_{B}}{\Phi_{0}} =\mu_{0}\hspace{2}\mu_{B} K_{J} = \frac{e^{2}}{4 \hspace{2}\pi \hspace{2}\epsilon_{0}\hspace{2} m_{e}\hspace{2} c^{2}}
\end{eqnarray*}
$\mu_{0} = (\epsilon_{0} \hspace{2}c^{2})^{-1}$= magnetic vacuum field constant, $\mu_{B} = e \hspace{2}\hbar/ 2 m_{e}$ = Bohr's magneton, $\Phi_{0} = h / 2\hspace{2} e$ = magnetic flux quantum (fluxon) $K_{J}$ = Josephson's constant.\\
Above relationship is based on a single-fluxon dipole-field with a point-like dipole $\mu_{B}$. In the equatorial plane of such dipole, a circle of radius $r_{e}$ precisely marks off one fluxon outside of that circle.
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