The Charged-Photon Model of the Electron, the de Broglie Wavelength, and a New Interpretation of Quantum Mechanics Richard Gauthier Santa Rosa Junior College

Abstract This article continues to develop the charged-photon model of the electron. In particular, the relativistic de Broglie wavelength of a moving electron is derived from the model. De Broglies own derivation is also summarized and compared with the present derivation. The quantum wave function of a free electron is derived from the plane wave function of a circulating charged photon. The article suggests that quantum mechanics may be reinterpreted based on considering the quantum wave functions of an atom as being descriptions of charged photons in the atom. Introduction The charged-photon model of the electron is described in Gauthier (2014). The model was introduced because a proposed circulating charged photon that models a relativistic moving electron, has several features of this electron, related to the electrons energy, momentum and spin. This is based on the relativistic energy-momentum equation E2 = p2c2 +m2c4 for an electron. The charged photon model for the electron does not include a specific model for the charged photon that models the electron. The model assumes that the charged photon obeys the relationship c = !" for its wavelength, frequency and light speed, has the usual energy and momentum relations E = h! and p = h / ! based on the photons frequency ! and wavelength ! , and has a helical trajectory and a longitudinal velocity corresponding to the velocity of the electron. The photons helical radius for a resting electron is the amplitude R = ! / 2mc from the solution to the Dirac equation for the electron. The charged photon model also includes a double-looping of the helically-moving photon around its helical axis corresponding to the zitterbewegung frequency of the electron, also found from the Dirac equation. The charged photon model predicts the de Broglie wavelength of the electron The charged photon model of the electron as first proposed in Gauthier (2014) did not contain a derivation for the de Broglie wavelength of a moving electron !deBroglie = h /" mv , because I did not then have such a derivation. Later I explored the idea that the wave number k of the circulating charged photon might be a key to such a derivation, since it is associated with the photons wave vector !k that has a component in the longitudinal direction of motion of the circulating

charged photon. This is the direction of motion of the electron that is being modeled by the circulating charged photon. The idea worked and in a few short steps I derived the de Broglie wavelength for a relativistic moving electron from the charged photon model of the electron. This derivation of the de Broglie wavelength goes as follows. Please refer to Gauthier (2014) for more information about the model that will be helpful in following this derivation. In the circulating charged photon model of the electron, the photon and the electron have energy E = ! mc2 . This equals E = h! for the charged photon. By equating these two energy terms, the frequency of the charged photon is then found to be ! = " mc2 / h . There is a charged photon wavelength ! corresponding to this charged photon frequency ! . This wavelength is ! = c /" = c / (# mc2 / h) = h /# mc . The charged photon has a wavenumber ktotal = 2! / " , which corresponds to a wave vector !ktotal pointing along the charged photons helical trajectory. For the circulating charged photon, ktotal = 2! / " = 2! / (h /# mc) = 2!# mc / h . Like the charged photons total circulating momentum !ptotal = h!ktotal , !ktotal makes an angle ! with the longitudinal direction of the circulating charged photon, where cos(! ) = v / c , and v is the velocity of the electron along the helical axis, as well as the longitudinal velocity of the circulating charged photon that is modeling the moving electron. The charged photons wave vector !ktotal has a longitudinal componentk = ktotal cos(! ) = (2"# mc / h)(v / c) = 2"# mv / h . The wavenumber component k has a longitudinal wavelength !longitudinal = 2" / k = 2" / (2"# mv / h) = h /# mv = !deBroglie . Summarizing this result, the longitudinal component of the wave vector !ktotal of the helically circulating charged photon that is a model for the moving electron, yields the relativistic de Broglie wavelength of the moving electron !deBroglie = h /" mv . The de Broglie wavelength was not explicitly designed into the charged photon model of the electron, which as previously mentioned is based on the relativistic energy-momentum formula E2 = p2c2 +m2c4 for the electron and the light-speed helical path of the charged photon described in Gauthier (2014). So it is fair to say that the electrons de Broglie wavelength is predicted by the charged photon model. Of course this was a welcome result, and increased my confidence that modeling the electron as a charged photon was on the right track. Derivation of the quantum wave function for a free electron There is another, more formal way to show how the charged photon model of the electron generates the de Broglie wavelength as well as its phase velocity vphase = c2 / v along the direction that the electron is moving, and to show why the

phase velocity of the de Broglie matter-wave is vphase = c2 / velectron . Consider an electron moving to the right along the + x -axis at velocity v . The charged photons helix winds around the electrons trajectory making an angle! with the electrons forward direction given by cos! = v / c as in the charged photon model. A photon of wavelength ! , wave number k = 2! / " , frequency ! = c / " and angular frequency ! = 2"# can be mathematically described as a plane wave ! = Aei( !k "!r#$t ) where !k is the photons wave vector pointing in the direction of the photon and ! is the photons angular frequency. The vector !r indicates the distance and direction in which the magnitude of the plane wave is to be calculated. The phase velocity of the plane wave of the photon is vphase =! / k = 2"# / (2" / $) = #$ = c . In the charged photon model, the circulating charged photon is proposed to correspond to a plane wave !(!r ) = Aei( !k "!r#$t ) which however is changing its direction because the charged photons wave vector !k is changing direction as the charged photon circulates helically around the direction that the electron is moving, in this case the + x -axis. Although the direction of the wave vector !k keeps changing with time, !k maintains its angle ! with the x -axis (this is part of the charged photon model) where cos! = v / c . So the value of!(x)along the x -axis of the circulating plane wave becomes !(x) = Aei(kxcos"#$t ) = Aei(kxv/c#$t ) since in this case, !k ! !r = kxcos" . But in the charged photon model, the value of the wave number k of the circulating photon of energy E = ! mc2 = h" and wavelength ! = c /" = c / (# mc2 / h) = h /# mc is k = 2! / " = 2! / (h /# mc) = 2!# mc / h = # mc / ! . Substituting the photons value of k into !(x) = Aei(kxv/c"#t ) gives !(x) = Aei([" mc/!]xv/c#$t ) = Aei(" mvx/!)#$t ) . This corresponds to a value kx , the x-component of the photons wave vector !k , to be kx = ! mv / ! and since kx = 2! / "x where !x is the wavelength along the x -axis, this gives !x = 2" / kx = 2" / (# mv / !) = 2"! /# mv = h /# mv , which is the relativistic de Broglie wavelength for a moving electron !deBroglie = h /" mv . The expression for !(x) becomes !(x) = Aei(kxx"#t )where kx = ! mv / ! , and the value for the corresponding phase velocity of the charged photon model of the electron becomes vphase =! / kx = (" mc2 / !) / (" mv / !) = c2 / v since !! = h" = # mc2 is the charged photons energy E . So in the charged photon model of the electron, the charged photons energy is E = ! mc2 = h" = !# and its momentum in the electrons direction is px = ! mv = h / "deBroglie = !kx we see that the charged photon has the properties of the electron that formed the basis of the quantum mechanical Schrodinger equation for an electron. The expression !(x) above corresponds to the quantum wave function !(x) = Aei(kx"#t ) for a free electron moving in the x direction whose wave number k = 2! / "deBroglie equals the longitudinal component of the charged photons wave

vector, whose! is the angular frequency of the charged photon corresponding to the photons (and electrons) total energy E = ! mc2 = !" , and whose phase velocity vphase = c2 / v where v is the velocity of the electron. Geometric relation of the de Broglie wavelength to the charged-photon model The relationship of the de Broglie wavelength of a moving electron to the wavelength of the charged photon in the charged photon model of the electron can also be shown geometrically in a simple diagram (see Figure 1 below). Consider an electron as moving horizontally to the right (along the + x -axis) with velocity v , momentum p = ! mv and total energy E = ! mc2 . The charged photon, having momentum pphoton = ! mc and energy E = ! mc2 = h" , moves in a helix around the electrons trajectory, making an angle ! (taken from the charged photon model, where cos! = v / c ) with the x -axis. (In a two-dimensional view this helix looks like two complete cycles of a sine wave that has a maximum angle ! with the x -axis.) The two cycles (in the charged photon model these two cycles are due to the zitterbewegung frequency! zitt = 2mc2 / h of the Dirac electron) of the helix corresponds to a distance along the helical path of one charged photon wavelength !photon = h /" mc (from the charged photon model.) Now unroll the helical trajectory for these two full cycles of the helix so that the two-cycle helical trajectory of length !photon = h /" mc becomes a straight-line segment of the same length !photon = h /" mc making an angle ! with the x -axis. At the top end of this line segment draw a line perpendicular to the line segment in the downward direction to the right until this line meets the x -axis. The total length of the horizontal line segment produced in this way is the de Broglie wavelength of the electron !deBroglie = h /" mv . As the speed v of the electron increases, the value of ! decreases, the wavelength !photon of the charged photon decreases, and the de Broglie wavelength decreases. The de Broglie phase wave travels to the right with velocity vphase = c2 / v as the electron travels to the right with velocity v .

Figure 1. De Broglie wavelength generated from charged photon model of electron (computer graphic thanks to Chip Akins) Comparison of the charged-photon derivation to de Broglies derivation How does the derivation of the electrons de Broglie wavelength from the charged photon model of the electron differ from de Broglies own derivation? In his derivation (de Broglie, 1924) the resting electrons energy is associated with an internal frequency given by h!o = mc2 . A larger frequency ! given by h! = " mc2 is associated with the increased energy and velocity of a moving electron. So far the two approaches are the same. But de Broglie didnt associate the larger frequency! of his moving electron with a circulating photon having a photon wavelength ! = c /" . This is the crucial difference. Rather, as described in Cropper (1970), de Broglie started by trying to reconcile the apparent contradiction between the moving electrons increasing internal frequency ! = " mc2 / h due to its increasing energy and velocity, with a relativistic time-dilation argument that predicts a decreasing clock-like internal frequency !1 = mc2 /" h of the moving electron with its increasing velocity. He assigned the increasing internal electron frequency ! of the moving electron to a traveling phase wave accompanying the electron through space and in time, such that this phase wave is always in phase with the decreased internal frequency !1 of the moving electrons internal process due to relativistic time dilation. The phase ! of the traveling phase wave is given by ! = "(t # x /u) where u is the phase velocity of the traveling phase wave. The phase ! ' of the internal time-dilated frequency !1 of the electron is given by ! ' = 2"#1t . Equating these two phases ! and ! ' gives

! = ! '2"#(t $ x /u) = 2"#1t(2" )(% mc2 / h)(t $ x /u) = 2" (mc2 /% h)t% (t $ x /u) = t /%t $ x /u = t /% 2t $ x /u = t(1$ v2 / c2 )t $ x /u = t $ tv2 / c2$x /u = $tv2 / c2u = (x / t)(c2 / v2 ) =v(c2 / v2 ) =c2 / v

where v is the constant velocity of the electron which is moving linearly through space, and can be considered to have started at x = 0 and at time t = 0 . The above particle, having internal frequency ! as it moves along with its phase velocity u , would therefore have an associated wavelength ! given by the standard wave formula: speed = wavelength ! frequency. So u = !"! = u /"= (c2 / v) / (# mc2 / h)= h /# mv = !deBroglie

The phase velocity u for a phase wave of angular frequency ! = 2"# and wave number k = 2! / "deBroglie is u =! / k = c2 / v and is always greater than c (since the electrons speed v is always less than c). But de Broglie then showed that the physically meaningful velocity for an electron, considered in analogy with a pulse of light as being composed of a group of waves with different frequencies and wave numbers, is the group velocity of these waves. The group velocity is the velocity of the local wave reinforcement regions formed due to the superposition of many waves of different frequencies and wave numbers. This group velocity w is calculated for a set of moving waves as w = d! / dk . This calculation using the electron formulas for ! and k as functions of electron velocity v gives the result

w = v , i.e. the group velocity of the electron if it is composed of a number of waves of different phase velocities is the velocity of the electron itself. De Broglie was aware that his particle-wave theory of the electron did not provide detailed answers to the question, what exactly are these particle waves?

Quoting de Broglie, The definitions of the phase wave and the periodic process were purposely left somewhat vagueso that the present theory may be considered a formal scheme whose physical content is not yet fully determined, rather than a full-fledged definite doctrine. De Broglies matter-wave hypothesis was taken up by Schrdinger (1925) partly on the recommendation of Einstein, and formed the basis of the Schrdinger equation of quantum mechanics, despite the lack of physical evidence for matter-waves of an electron at that time. The experimental evidence came a few years later with the work of Davisson and Germer (1928) and Thomson (1928) where the diffraction patterns of electrons projected onto a nickel crystal and on a thin foil closely followed the predictions of de Broglies wavelength equation. The meaning of the matter-waves that form the solutions to the Schrdinger equation, beyond their ability to correctly predict probabilities, is still controversial to this day. De Broglie never identified the electron with a circulating charged photon, nor as having any internal energy structure or wavelength associated with its proposed internal frequency that is proportional to the electrons energy. He only spoke of the electrons internal process or internal phenomenon. Now that the electron has been modeled as a circulating charged photon and the de Broglie wavelength for the electron has been derived from this circulating charged photons wave vector in a very straightforward way, it can be proposed that de Broglies internal phenomenon of the electron is a circulating charged photon, and that his matter-waves or phase waves are produced by the light waves of the helically circulating charged photon along its helical axis, which is the trajectory of the electron that is modeled by the circulating charged photon. Comparing the de Broglie hypothesis and the charged photon hypothesis for an electron Since there are now two hypotheses that predict the de Broglie wavelength for an electron, it may fairly be asked, Which hypothesis is better?, or even Is a second hypothesis for predicting the de Broglie wavelength even necessary since the de Broglie wavelength of moving particles is a well-established fact? The principle of Occams razor: If two hypotheses predict the same result, the hypothesis that makes the fewer assumptions should be accepted is relevant here, if used with care. In this case the two hypotheses make some different assumptions. De Broglies hypothesis assumes that an electron is localized in space due to the superposition of many phase waves of different speeds and wavelengths (in comparison with a pulse of light composed of light waves of different wavelengths) that together give the electron its measured velocity v as the group velocity of the localized superimposed phase waves, while nothing is said about the electrons internal state itself beyond its being a vibrational phenomenon. By assuming a harmony of phases between two measures of the phase changes of a moving

electron, taking relativistic time-dilation into account, de Broglie derived the wavelength of the moving electron known as the de Broglie wavelength. The charged photon model of the electron on the other hand is a quantum object of a specified spatial extent (based on the Dirac equation), which however decreases with the electrons speed, and is composed of a circulating charged photon. This hypothesis gives greater specificity to the charged photon model for the electron compared with the de Broglie hypothesis, by claiming that the electron is a kind of photon, despite the fact that the charged electron model lacks a specific model of the charged photon itself, beyond assuming that the wave formulas for a photon c = !" , E = h! , and p = h / ! apply to a charged photon also. Specifically, the charged photon model uses these formulas to derive a precise charged photon wavelength from the Einsteins energy content formula E = ! mc2 for a moving mass and the related Planck-Einstein formula E=h! relating energy content to frequency. Using this charged-photon wavelength and the associated charged-photon wave vector !k with a light-speed helical motion for the electrons charge derived from the Dirac equation, the relativistic de Broglie wavelength for the electron is derived from the circulating charged photon model of the electron in a few short steps. Although quantum mechanics is currently based on the de Broglie wavelength of a moving electron, de Broglies electron hypothesis is admittedly vague, a formal scheme whose physical content is not yet fully determined. It has no spatial model for the electron, no spatial model for how the de Broglie wavelength is generated, and no clear interpretation of the nature of the matter-waves or phase waves beyond showing that matter has a wave-like nature quantitatively predicted by the de Broglie equation. The equation was successfully incorporated by Schrdinger into quantum mechanics. The de Broglie equation so incorporated has been very successful in its ability to make accurate probabilistic predictions about experimental results based on treating particles as having a wave-like nature specified by the de Broglie wavelength of the particle. The charged-photon model of the electron is 1) a spatial model for the electron based on the Dirac equation that followed de Broglies work by several years, 2) predicts the de Broglie wavelength of a moving electron based on the spatial model of a helically-circulating charged photon, and 3) describes the matter-waves of the electron as charged-photon waves. The charged-photon model may also shed further light on the interpretation of the wave functions of the Schrdinger equation and on quantum mechanics in general. The charged-photon interpretation of quantum mechanics The derivation of the de Broglie wavelength !deBroglie = h /" mv from the charged-photon model of the electron has implications for the interpretation of

quantum mechanical wave functions in general, and specifically for wave functions that are solutions to the Schrdinger equation for bound or unbound electrons. The basic relationships that are built into the Schrdinger equation for non-relativistic electrons are the electrons energy E = !! and the electrons momentum p = !k = h / !deBroglie . The charged photon model for the electron contains both of these expressions, similarly defined. In the charged-photon approach to quantum mechanics, wherever the term electron occurs when using the Schrdinger equation, the term charged photon can be substituted, since the charged-photon model of the electron has all the features of the electron that are needed to solve the Schrdinger equation. For example, the Schrdinger equation for the hydrogen atom has many wave-function solutions, with each wave function corresponding to an energy level En,l ,m,s for a charged photon having quantum numbers n , l , m and s . Each of these wave functions describes a negatively-charged photon that is bound to the atom by attraction to the positively charged atomic nucleus. Each charged photon is described by its wave function, which is defined quantitatively within particular regions of the hydrogen atom. The charged-photon approach to the electron in quantum mechanics is not only for the hydrogen atom but for all atoms, molecules and chemical structures. For multiple-charged-photon atoms, the composite wave functions are more abstract and of higher dimensionality, but still are used to find the probabilities for different physical measurements of these charged photons. Each electron in any atom is a charged photon whose wave function at any time occupies a particular region of the atom or chemical structure, defined by the quantum numbers of that particular charged. The charged-photons wave functions evolve in time as described by the time-dependent Schrdinger equation, or are steady state wave functions of the charged photon in the time-independent Schrdinger equation. The charged-photon wave functions are used with the standard quantum mechanical method of calculating!"! for the appropriate wave functions and treating this as a probability density function for predicting the statistical description of experimental results. Consider how an atom emits an uncharged photon quantum mechanically. In an atom, a charged photon having a higher energy configuration with energy E1 reconfigures itself into a lower energy configuration with energy E2 , and in this process emits an uncharged photon of energy E = h! = E1 " E2 . When an atom absorbs an uncharged photon, this process happens in reverse. This process is described by quantum mechanics for the atom, but now the atom is conceptualized as containing charged photons rather than electrons. All of the photon emission and absorption interactions are between two varieties of photon -- charged and uncharged. The charged-photon quantum mechanics approach applies not only to the non-relativistic Schrdinger equation but to the relativistic Dirac equation as well.

Whenever the term electron is mentioned in association with the Dirac equation, the words charged photon may be substituted. The charged-photon model for a free electron is a spatial model of the electron based on the Dirac equations solutions for an electron, and so the model satisfies quantitatively or qualitatively many of the features of the Dirac equations electron, such as: a) unobservable speed-of-light electron velocity, b) observable sub-luminal electron velocities, c) a very small vibrational amplitude ! / 2mc at small electron velocities d) zitterbewegung (jittery motion) frequency related to the Dirac electrons mathematical rotational periodicity of 4! , e) a spin associated with the electrons vibrational amplitude, including spin up and spin down, f) a magnetic moment associated with the electrons charge, speed and vibrational amplitude, g) describing the positron as a positively-charged circulating photon The charged-photon modeling the electron does not currently provide an explanation of what is happening during the collapse of a wave function during an experiment to measure a particular observable property of an electron like its position or momentum. The charged-photon model only predicts that the same explanation would apply to the collapse of a charged photons wave function as to the collapse of an uncharged photons wave function. The proposed charged-photon interpretation of quantum mechanics is not necessarily opposed to any other interpretation of quantum mechanics, such as the Copenhagen interpretation, the pilot-wave interpretation, the many-worlds interpretation, the sum-of-histories interpretation or the decoherence interpretation. The relationship of the charged-photon interpretation to these and other interpretations needs closer examination. The emphasis of the charged-photon model, and the direction to which this approach is pointing, is that matter may be more fundamentally describable as systems of circulating photons or other light-speed particles such as gluons that circulate in such a way as to form localized energy concentrations with corresponding frequencies. These localized energy concentrations have inertial mass based on E = ! mc2 = h" , and are called matter. The fundamental quantum natures of photons and electric charge are not explained in the charged-photon model. This will require a deeper understanding of the quantum nature of the photon and electric charge than is currently available. In the meantime, in the quantum mechanics of electrons, the expression matter-waves can be reasonably replaced by charged-photon waves of various frequencies and configurations described by quantum wave functions. The charged-photon approach may lead to a somewhat more fundamental and unified description of the quantum world than is provided by the current standard model of physics and the dominant interpretation of quantum mechanics, the

Copenhagen interpretation. Not all fundamental particles lend themselves to the charged-photon approach to matter however. For example, neutrinos and the Higgs boson are fundamental particles that have mass but are neutral. But quarks might be reconceptualized as circulating charged gluons using a similar approach to the charged-photon model of the electron. And uncharged photons or other light-speed quantum particles may be able to helically circulate in ways that form neutrinos and other uncharged fundamental particles having mass. At the very least the charged-photon model of the electron indicates why it should be no surprise that, in the proper circumstances, the electron shows all the interference, diffraction, and quantum entanglement properties of the photon, such as in the well-known double-slit experiment, since in this view the electron IS a charged photon. Reconceptualizing the electron as a charged photon can also lead to new hypotheses about the charged photon, or about the uncharged photon, that can then be tested experimentally. De Broglie correctly proposed that matter has a wave-like nature. The charged-photon approach may explain why this is. Conclusion A further case is made for a helically-circulating charged photon as a model for the electron. The model predicts the electrons relativistic de Broglie wavelength in a way that was not anticipated when the model was originally formulated in the previous article (Gauthier, 2014). The simplicity of the models derivation of the de Broglie wavelength and the corresponding quantum wave function for a freely-moving electron add strong support for the model. The charged photon model suggests a new interpretation of the quantum wave functions of atomic or molecular electrons as the wave function of charged photons bound to the atom or atoms. References (1) Cropper, W. H., The Quantum Physicists and an Introduction to Their Physics,

Oxford University Press, London, 1970, p. 57-62. (2) Davisson, C. J. and Germer, L. H., Nature, 119, 558,1927. (3) de Broglie, L., Recherches sur la thorie des quanta (Researches on the

quantum theory), Thesis, Paris, 1924. (4) Gauthier, R., The electron is a charged photon, at https://www.academia.edu/9171692/The_electron_is_a_charged_photon, 2014 (5) Schrdinger, E., "An Undulatory Theory of the Mechanics of Atoms and Molecules".

Physical Review 28 (6): 10491070, 1926. (6) Thomson, G. P., Experiments on the Diffraction of Cathode Rays

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 117, No. 778. (Feb. 1, 1928), pp. 600-609.

Background information http://en.wikipedia.org/wiki/Louis_de_Broglie http://en.wikipedia.org/wiki/Matter_wave#de_Broglie_relations http://en.wikipedia.org/wiki/Waveparticle_duality http://en.wikipedia.org/wiki/Schrdinger_equation http://en.wikipedia.org/wiki/Dirac_equation Copyright 2015 by Richard Gauthier Santa Rosa, California richgauthier@gmail.com https://santarosa.academia.edu/RichardGauthier Last updated January 1, 2015