Reconciliation of the Wave-Particle Duality and Formulation of the Photoelectric Equation from the Laws of Capacitance Lane M Davis ©2014 [email protected] Abstract When the Planck constant was applied to the photoelectric equation in 1905, it introduced the greatest paradox of modern physics: the wave- particle duality of light. If the energy of a photon can be explained to arise from its wave nature, then the particle regime becomes obsolete and irrelevant. By modeling the photon’s wave as a capacitor, the wave nature of light becomes sufficient to explain and calculate the energy of the photon, as well as all other aspects of behavior, thereby reconciling the wave-particle duality. Math is the language of the universe. All governing laws are related in ways that should make logical and mathematical sense. This author proposes that if a constant has to be injected into an equation as a fudge- factor, it is not necessarily an unsolvable mystery, but that the reason the constant is in the equation is just not yet understood. Planck’s constant is empirically measured to produce incredibly accurate results in equations where it is injected, but there has never been postulated a coherent logical reason why the constant is in the equations – until now. Keywords: Wave-Particle Duality, Photoelectric Equation, Quantum Op- tics, Planck’s Constant, Fine Structure Constant, Quantum Mechanics, Pho- ton, Capacitor, Energy, QED, Quantum Transition Ad-hoc classical wave. In a classical wave, the energy associated with the wave corre- Injections lates to its amplitude[17]. With a light wave, it seemed like the amplitude was irrelevant, as it Prior to 1905, the photoelectric effect did not matter how much light was was a baffling mystery[3] – deeply tied shined onto metal – larger wave- to what was dubbed the ultraviolet lengths would not eject any electrons catastrophe[10]. It was widely believed at all[4]. Curiously, at shorter wave- that light was a wave after the observa- lengths, even with only small ampli- tion of diffraction experiments[16][11], tudes of light being shined, the pho- but this wave did not behave like any tons imparted enough energy to eject 1 Lane M Davis ©2014 all rights reserved electrons[4]. In 1900, by observing in the first place[5]. Planck vehemently a pattern in blackbody radiation and disagreed with Einstein’s conclusion making the then-outlandish assump- that energy was inherently quantized tion that energy came in discrete pack- or particle in nature, and saw Ein- ets, Max Planck was able to give stein’s formulation as “nothing more a formula which described the ob- than a mathematical trick”[3]. served distribution of energy among This author contends that Planck the emitted wavelengths[3], which led was right all along – that the particle him to formulate the Planck Radiation nature of the wave is just a mathemat- formula[19], which introduced a new ical artifact that was hitherto misun- constant, h. Planck did not realize the derstood, and that there is a logical applications for his constant and did reason why Planck’s constant is a con- not truly believe in the discrete nature version factor between the frequency of energy; he believed it to be an arti- of a photon and its energy, and that fact of a flawed mathematical model[3]. the photoelectric equation can be for- Everything changed in 1905. With mulated logically by the application of his revolutionary idea of a fixed ve- commonly known laws. locity for light, Einstein was able to give every wavelength a corresponding frequency[23]. He then took Planck’s newly discovered constant, and in- Capacitance of jected it ad-hoc[4] into a very simple equation: E = hf. the Photon Einstein did not understand how, or why[1], but when he multiplied Fig. (1) Planck’s constant by the frequency of light, it produced the experimen- tally proven energy levels for all known wavelengths. He could not explain why Planck’s constant is the conver- sion factor between the frequency of light and its energy[1], but it undoubt- edly worked. Quantum optics was born. The idea that the energy of a photon came from its frequency in- troduced the wave-particle duality[25]. The energy in this regime came from The first step to achieving this is to the wave’s frequency, instead of the model the photon as a parallel plate ca- wave’s amplitude, which gave the pho- pacitor. Undoubtedly, the photon has ton a particle nature even though no net charge, but carries a positive it behaved like a wave in diffraction and negative charge at different times experiments[25]. This ad-hoc solution as it propagates through space[13][28]. was heralded as one of the biggest The photon also displays another char- breakthroughs of the century, and Ein- acteristic inherent to capacitors – it is stein eventually won his only Nobel essentially connected to positive and prize for this simple equation, despite negative terminals; the atom emitting the fact that it could not be ratio- energy (positive lead) and the atom nalized or explained by anyone why absorbing energy (negative lead). De- Planck’s constant was in the equation spite these two similarities, it may 2 Lane M Davis ©2014 all rights reserved seem illogical that this fundamental as- traveling in a line. Fig. (2) Is a horri- pect of nature could or should be com- ble representation of reality, although pared to a capacitor. At face value, a it does display some of the physical photon does not seem to exhibit any attributes and is an easy model for of the characteristics of a set of plates. the Luddite stuck in the Newtonian This misunderstanding is because the regime to digest. The uncertainty prin- photon is usually visualized from the ciple tells us this model is not accu- frame of reference of an outside ob- rate; that a particle has neither a per- server, where it propagates through fectly defined history nor a perfectly both time and space. It is necessary to defined location[24]. Single slit Fraun- visualize what the photon experiences, hofer diffraction shows that a photon not what the outside observer experi- does indeed have a width, as when the ences from the photon. slit is closed tighter than one wave- From the frame of reference of the length, the wave interferes with itself, photon, many things change, and the and produces strange but mathemati- system can then be examined electro- cally logical results[16]. Therefore, the statically. For instance, the photon is photon not only has a wavelength, but traveling at the speed of light, there- it also has a width. It also has a dis- fore because of the implications of spe- tance between the positive and nega- cial relativity and the Lorentz trans- tive components (Fig. 2). The dis- formation, the photon does not expe- tance between the positive and nega- rience time[22]. Time does not exist tive charge peaks of the sine wave is for the photon. Because there is no half a wavelength. time, this means the photon occupies When the photon is viewed as both the positive and negative states frozen in time, it does display the char- simultaneously. In reality it is alter- acteristics of a plate capacitor; two nating between the positive and nega- oppositely charged surfaces both with tive charge sinusoidally through space- a length, a width, and separated by time, as an outside observer would see a distance. It must be noted that (if one could hypothetically observe a the photon is positive for half of its photon without interfering with it in length, and negative for half of its some way), but from the frame of ref- length (Fig. 2); so the length of each erence of the photon there is no time, plate is half a wavelength. It must only space, so it is indeed both pos- also be noted that the photon does itively and negatively charged at the not become even semi-localized until same time. This fits the first aspect it interacts with a particle[26]. From of a plate capacitor; having a charge emission, it travels through spacetime differential. as nothing more than a superposition of possibilities, propagating according Fig. (2) to Schrödinger’s wave equation[21] un- til the wavefunction collapses upon absorption, where these three dimen- sional properties take shape. Know- ing the three properties of length, width, separating distance, and that the dielectric in this case is space- time itself[29], with a vacuum permit- tivity of ε0, we can calculate for the Another misconception is that a capacitance of the photon using the photon is a one dimensional particle standard equation for an ideal paral- 3 Lane M Davis ©2014 all rights reserved lel plate capacitor with the vacuum as tion, and plugging in e for the charge the dielectric[18]. Q, produces: ε A e2 C = 0 (1) Eα = (4) D 2ε0λ 1 It would suffice to just solve for Plugging in 2 λ for length, λ for 1 E to calculate the correct energy lev- width, and 2 λ for distance, and then canceling, we produce: els for the entire electromagnetic spec- trum, but something very interesting happens when we put this equation in C = ε λ (2) 0 terms of frequency, which is possible c because f = λ [23]. Solving for E, and converting wave- Energy of the length to frequency by substituting c λ = f , it produces: Photon e2 E = f (5) The standard equation for calculating 2ε0cα the potential energy stored in an ideal The terms in the brackets equals capacitor in terms of capacitance is Planck’s constant, h.