Viewing quantum mechanics through the prism of electromagnetism
VViewing quantum mechanics through the prism ofelectromagnetism
Ankit Pandey, Bill Poirier, Luis Grave-de-PeraltaFebruary 10, 2021
In this paper, we demonstrate novel relationships between quantum mechanics and theelectromagnetic wave equation. In our approach, an invariant interference-dependent electro-magnetic quantity, which we call “quantum rest mass”, replaces the conventional role of theinertial rest mass. In the ensuing results, photons, during interference, move slower than thespeed of light in vacuum, and possess de Broglie wavelength. Further, we use our electromag-netic approach to examine double-slit photon trajectories, and to arrive at the Schrodingerequation’s results for a particle in an inﬁnite square well potential.
We begin this discussion with the following question. What does a standing wave look like whenit is boosted in a given direction? One could expect the intuitive answer that we should simplysee a standing wave moving in the direction of the boost. This, indeed is the correct answer if weare talking about a Galilean boost of a non relativistic standing wave. But what if the questionwas about an electromagnetic wave that was Lorentz-boosted? The answer, in this case, is shownin Fig 1. V C V V Printed by Wolfram Mathematica Student Edition
Figure 1: The ﬁgure at the top represents an electromagnetic standing wave. In the bottom ﬁgure,we see the result of a Lorentz-boost acting upon it. Apart from the intuitive forward motioncorresponding to the velocity of the boost, the wave also gets modulated by a superluminal wave,shown in red.Interestingly, the standing wave gets modulated by a long superluminal (faster than light) wavewhen seen from the boosted frame of reference. This superluminal wave is more widely known inthe context of phase waves obtained from the Klein-Gordon equation . If we assume that thestanding wave was created by a single photon of frequency ( ω ) oscillating inside a cavity, then thesuperluminal wavelength is found to obey: λ sup = hm (cid:48) v (1)1 a r X i v : . [ qu a n t - ph ] F e b here m (cid:48) = Ec = ¯ hωc (2)Now this formula looks like the de Broglie wavelength equation, but the quantity m (cid:48) here isnot related to the inertial rest mass; the latter is always zero for light. However, rest mass can bedeﬁned in more ways than one. We are going to start out by deﬁning it in a way which yields thevalue shown in Eq. 2 for standing waves of light. In general, rest mass can be derived from the momentum four vector as follows : m = P µ P µ c (3)We note that if standing waves of light are assumed to correspond to a particle being at rest, thenthe particle mass indeed corresponds to Eq. 2: P µ = (cid:18) ¯ hωc , , , (cid:19) ⇒ m = ¯ h ω c (4) ⇒ m = ¯ hωc (5)Standing waves of light are the superposition of a forward-moving and a backward-moving si-nusoidal wave. The average momentum of these two sinusoidal waves is zero. By deﬁning standingwave photons to be at rest, we are associating photon trajectories with this average momentum.To distinguish m from inertial rest mass, we shall call it the “quantum rest mass”.More generally, we can consider two waves of equal amplitude but diﬀerent frequencies ω + and ω − , moving in opposite directions. This superposition is what we call a “bidirectional wave”.It should be understandable that there will exist a boosted frame of reference f (cid:48) in which thesefrequencies are equal. To calculate the quantum rest mass of photons in this case, one can move tothe f (cid:48) frame of reference and use Eq. 4. Alternatively, it can be shown that one can also calculatethe quantum rest mass of a bidirectional wave from an arbitrary frame of reference by simply usingEq. 6, where the wave is assumed to live in the x-axis. P µ = ¯ h c ( ω + + ω − , ω + − ω − , , (6)It should be noted that the quantum rest mass of photons depends on interference. Wherethere is no interference, the quantum rest mass of photons is zero, just like the inertial rest mass.To drive home this point, we explain the double slit pattern in the 2-dimensional space with ourapproach. In the double slit experiment, two wavefronts intersect each other. The angle of intersection variesat each point ( x, y ) of space. Let us say that their wave vectors intersect at the angle θ ( x, y ) . Thefollowing equations describe the quantum rest mass of the photon, and its corresponding velocity,“locally” at ( x, y ) : m ( θ ) = ¯ hωc sin (cid:18) θ (cid:19) (7) v ( θ ) = c cos (cid:18) θ (cid:19) (8)2he local velocity points in the direction mid-way between the individual wave vectors. Note thattheta can only vary between 0 and π . The “local wavelength” corresponding to the above quantities,given by λ sub ( θ ) = hm ( θ ) v ( θ ) , (9)lies in the direction perpendicular to the local velocity.The amplitude contributed by a given slit decays by r as a function of distance r from a givenslit. Due to this, it is understandable that in regions very close to either of the two slits (theregions represented by red and blue colors in Fig 2), the contribution of the other slit is com-paratively negligible. The trajectories here move radially outwards from the two slits, and havezero quantum rest mass. On the other hand, in the green region, the amplitudes contributed byboth the slits are rather similar. Here, Eq. 9 holds true, and thus, the trajectories point radi-ally outward from the midpoint between the slits, as can be derived from the comment below Eq. 8.Figure 2: Double slit interference can be studied using quantum rest mass of light, as explained inthe text.Please keep in mind that θ is not the angle extended at the origin of Fig 2, but the anglebetween the wave vectors intersecting at a given point. It remains relatively constant for smallmovements perpendicular to the velocity vector, for example, along the arc shown in the ﬁgure.Let us use the quantum rest mass to ﬁnd the spacing between two bright fringes on a screen faraway from the slits. For small vertical displacements along this arc from its center in the ﬁgure,we can approximate: sin θ d D (10)where D is the distance of the arc from the origin, and d is the distance between the slits. We canalso approximate cos θ = 1 . The distance between two consecutive bright fringes along the verticalaxis is half the local wavelength, which is calculated from Eq. 9. It is found to be equal to λ sub ( θ (0 , D ))2 = Dλd (11)Where λ = πcω . This result is the same as that obtained by the traditional interference method. The above method can be used further to numerically draw massive photon trajectories or toplot the photon quantum rest mass as a function of space. It is easily noted that the trajectoriesare the most massive (in terms of quantum rest mass) around the region where the red, blue andgreen regions in Fig 2 are closest to each other. 3 Particle in a box solution via bidirectional waves
In this section, we derive a Schrodinger-like  “particle in an inﬁnite square well” solution in 1dimension using bidirectional waves. Physically, we aim to model a massless cavity, which in turnis placed inside a much larger inﬁnite square well potential. The cavity, representing a particle, isassumed to move with the velocity v in either direction. To model this situation with electromag-netic waves, we consider the superposition of two bidirectional waves moving with the velocity vopposite to each other inside an inﬁnite square well. Let us call this superposition F(x,t). Whenthis function is plotted and animated with time, two distinct frequencies can be visually observed:A higher frequency causing standing-wave-like oscillations, and a much lower frequency. The latterfrequency causes slow broad oscillations of the waveform between being sine-like and cosine-like,as shown in Fig 3Figure 3: The “lower frequency” mentioned in the text corresponds to slow oscillations of thewaveform between a broad sine function and a broad cosine function. The frequency of theseoscillations is mentioned in the ﬁgure.It should be kept in mind that the “particle” is not represented by the waveform itself, but by thecavity. Although we have drawn the waveform of light all along the inﬁnite square well, we needto remember that the photon is constrained within the cavity. Let us assume that the cavity is oflength L. If the cavity’s instantaneous location is assumed to be at an arbitrary location x, thenthe waveform only exists within the boundary of the cavity, i.e, between x − L/ and x + L/ .Nevertheless, the waveform inside this region will be the same as F(x,t) in this region. Hence,F(x,t) should be thought of as the “internal structure of the particle", as opposed to being ananalogy of its Schrodinger-like wavefunction.To derive a Schrodinger like wavefunction from F(x,t), we can proceed as follows. Consider thetwo functions between which the system oscillates in Fig 3. We shall hereby call these functionsas its two “internal states”, namely the Sine state and the Cosine state. For simplicity, we alsoassume that the cavity is much smaller than the well. In Fig 4, we plot the amplitude of thesestates as a function of the cavity’s position as it moves with its constant velocity v. It is foundin this ﬁgure that the wave has the correct de Broglie wavelength that is expected from thequantum mass m . Note that the sine and cosine states here bear resemblance to the real andimaginary components of the Schrodinger’s wavefunction of a particle. Thus, it can be easilyobserved that by superimposing Fig 4 with the corresponding function for the cavity moving in theopposite direction, and requiring that the superimposition vanishes at the boundaries, we recoverthe Schrodinger equation’s solutions for particle in an inﬁnite square-well. Printed by Wolfram Mathematica Student Edition
Figure 4: The horizontal axis represents the displacement (x) of the cavity. The two axes per-pendicular to the x-axis represent the amplitudes of the Sine and Cosine states as a function ofx.
The quantum rest mass of light, as deﬁned in this paper, provides alternative insights into variouselectromagnetic phenomena involving the interference of light. Beyond electrodynamics itself, this4oncept also helps provide unique approaches towards understanding and appreciating quantummechanics from a relativistic semi-classical perspective.
B. Poirier acknowledges a grant from the Robert A. Welch Foundation (D-1523). We also thankIgor Volobouev and Maik Reddiger for their valuable opinions and discussions on these topics.