The Dynamics of Wave-Particle Duality

. Both classical and wave-mechanical treatments of monochromatic wave-like features may be faced in terms of exact trajectories, mutually coupled by a “Wave Potential” function ( encoded in the structure itself of all Helmholtz-like equations ) which is the cause of any diffraction and/or interference process. In the case of Wave Mechanics, an energy-dependent Wave Potential establishes a bridge between de Broglie's matter waves and the relevant particles by piloting them along their path by means of an energy-preserving "gentle drive”, allowing an insight into the dynamical mechanism of wave-particle duality. Thanks to a numerical treatment which is not substantially less manageable than its classical counterpart each particle is seen to "dance", under the action of the Wave Potential, "a wave-mechanical dance around the classical motion".


-Introduction
As is well expressed in Ref. [1], "the knowledge of several routes and their connections is always helpful when traveling through the quantum territory". The authors of Ref. [1] had in mind, however, a "territory" crossed by routes declaredly leading to Bohm's Mechanics [2][3][4][5][6][7], i.e. to probability-based guidance-laws and to their hydrodynamic flow-lines. In order to describe, on the contrary, a (non-Bohmian) dynamical set of exact point-particle trajectories running as close as possible to the classical ones, without resorting to wave-packet particle representations, we follow here a route beginning from the very foundations of Wave Mechanics [8,9], i.e.
1. de Broglie's seminal conjecture p = k , to which we owe the concept of matter waves and wave-particle duality [10,11]; 2. Schrödinger's time-independent equation [12][13][14][15], bypassing (with its eigen-fuctions and eigen-values) the heuristic prescriptions of the "old" Quantum Mechanics: an equation whose energy-dependence suggests a crucial analogy with Classical Mechanics, to be exploited in the present paper; 3. Schrödinger's time-dependent equation, whose energy-independence suggests a probabilistic distribution over the entire eigen-energy spectrum; and finally 4. Davisson-Germer's experimental test [16] of the real existence of de Broglie's matter waves in physical space.
We demonstrate, to begin with, that any Helmholtz-like equation is associated with a characteristic Hamiltonian set of "rays" in physical space, allowing an exact, trajectory-based approach to any kind of classical wave-like process, beyond any geometrical optics approximation. Passing then to the wave-mechanical case, de Broglie's relation p = k clearly tells us that the "Helmholtz rays" (with wave vector k ) of de Broglie's matter waves coincide with the dynamical trajectories of the associated point-particles with momentum p , thus laying the foundations of Wave Mechanics. Each particle is guided along its path by the energypreserving "gentle drive" of a matter wave having a well-established [16] physical existence. Our approach reaches the double goal of a clear insight into the mechanism of wave-particle duality and of a reasonably simple computability: a necessary condition for any theory of practical interest. An experimentally tested Hamiltonian description of wave-like features holding beyond the eikonal approximation [17] was first contrived in Refs. [18][19][20] for the propagation of electromagnetic waves in thermonuclear plasmas. This method was applied until recently [21] to the microwave diagnostics installed on the TORE SUPRA fusion facility, and extended in Ref. [22] to the treatment of the Schrödinger equation. The general demonstration of the equivalence between classical and wave-mechanical Helmholtz-like equations and exact, trajectory-based Hamiltonian systems was given in Refs. [23][24][25][26][27], and led to extensive application both in relativistic electrodynamics and in the analysis of experimental arrangements and devices employed for light transmission and guiding [28][29][30]. Let us finally remind that the equivalence between the time-dependent Schrödinger equation and Hamiltonian mechanics was demonstrated in Ref. [31] in terms of symplectic geometry, without giving rise, so far, to numerical applications. We derive here the exact Hamiltonian systems concerning, respectively, classical electromagnetic waves (Sect.2), non-relativistic matter waves (Sect.3) and relativistic matter waves (Sect.4). Examples of numerical computation are given in Sect.5, and a comparison is made in Sects.6 and 7 with the hydrodynamic approach of the Bohmian theory.

-The case of classical waves
We shall refer in the present Section, in order to fix ideas, to a stationary, isotropic and (generally) inhomogeneous dielectric medium, sustaining a classical monochromatic electromagnetic wave whose electric and/or magnetic field components are represented in the form where () u  r, is assumed to satisfy the Helmholtz equation and with a (time-independent) refractive index  n n(r, ) . The time-independence of eq. (2) does NOT mean, of course, that no physical change is expected. Just as for the usual laws of Dynamics, once suitable boundary and initial conditions are assigned the ensuing motion, occurring in stationary external conditions, is exactly described. If we perform in fact, into eq.(2), the quite general replacement and a "ray" velocity ray 0 c k / k v= is implicitly defined. It is easily seen that, as long as ) (which we call "Wave Potential"), represents a newly discovered intrinsic property encoded in any Helmholtz-like equation. It is seen to couple together the whole set of ray-trajectories, causing (thanks to its frequency dependence) any diffraction and/or interference process. , i.e. by the "ray-trajectories" themselves) one obtains, then, the amplitude  R(r, ) over the next wave-front. The iteration of this procedure allows to build up, step by step, both the "Helmholtz trajectories" along which the "rays" are channeled and the time-table of the "ray motion" along them. We shall see at the end of Sect.4 what the "rays" are conveying. When, in particular, the space variation length L of the amplitude  , R(r ) satisfies the condition 0 k L >> 1, the Wave Potential term ( , ) Wr may be dropped, thus removing any raycoupling. The "rays" will therefore propagate independently of each other, without any diffraction and/or interference, according to the eikonal equation

-The case of non relativistic de Broglie's matter waves
Let us pass now to the case of non-interacting particles of mass m and total energy E, launched with an initial momentum into an external force field deriving from a time-independent potential energy V(r) . The classical dynamical behavior of each particle is described, as is well known [17], by the time-independent Hamilton-Jacobi (HJ) equation where the basic property of the HJ function ( , ) S r E is that the particle momentum is given by The classical HJ surfaces S(r,E)= const , perpendicular to the momentum of the moving particles, pilot them along stationary trajectories, according to the laws of Classical Mechanics. The origin of Wave Mechanics was due to considerations based on the analogy between the Maupertuis and Fermat variational principles in physical space [17], inducing Louis de Broglie [10,11] to associate (in a single physical entity) each moving particle with a "matter wave" of the form, suggested by eqs.
under Planck's condition according to the basic conjecture laying the very foundations of Wave Mechanics thanks to its merging of Maupertuis's and Fermat's principles. We have therefore, thanks to eqs. (6) and (11)- (14), the relations showing that the HJ surfaces = const

S(r,E)
represent the phase-fronts of matter waves, while maintaining the piloting role played in the classical eq.(11). Eq. (14) provides both the structure (15) and the "guidance equation" of de Broglie's matter waves, addressing the particles with momentum p along the wave-vector k . Point-particles move, in other words, along stationary trajectories orthogonal to the phase-fronts of matter waves with successive step was performed by Schrödinger [12][13][14][15]), with the suggestion of viewing the laws of Classical Mechanics -represented here by eq.(10) -as the eikonal approximation of a suitable Helmholtz-like equation. Reminding that eq.(9) provides the eikonal limit of the Helmholtz equation (2), this suggestion simply amounts to performing, into eq.(2), the replacement, making use of eqs. (9)-(14), leading to the Helmholtz-like equation which is the well-known time-independent (and energy-dependent) Schrödinger equation, holding for stationary matter waves. It's an eigen-value equation admitting in general both continuous and discrete energy and eigen-function spectra, which replace the heuristic prescriptions of the "old" quantum theory [8].
The real existence of de Broglie's matter waves in physical space was established in 1927 by the Davisson-Germer experiments [16] on electron diffraction by a nickel crystal. Having in mind the dynamics of particles with an assigned total energy, just as it usually occurs in classical physics, let us now apply to the Helmholtz-like energy-dependent equation (17) the same procedure leading from the Helmholtz eq.(2) to the Hamiltonian ray-tracing system (5), by simply replacing eq.(15) into eq.(17) and separating real and imaginary parts. After having defined the energy-dependent "Wave Potential function" and the energy function we get now the dynamical Hamiltonian system (20) providing, in strict analogy with the Helmholtz ray-tracing system (5), a stationary set of pointparticle trajectories, together with the time-table of the particle motion along them. These trajectories are mutually coupled, once more, by the "Wave Potential" function W(r,E) of eq. (18), acting orthogonally to the particle motion and exerting an energy-conserving "gentle drive" which is the cause, thanks to its "monochromaticity", of any diffraction and/or interference process. Notice that this energy-dependence makes W(r,E) basically different (in spite of its formal analogy) from Bohm's energy-independent "Quantum Potential" ) (with its derivatives) over the next wave-front, and so on. Notice that the Wave Potential (18) depends on the profile of the transverse particle distribution ) , but not on its intensity, which may be quite arbitrarily chosen.

4.-The case of relativistic de Broglie's matter waves
Let us finally pass (maintaining the same notations of the previous Sections) to the relativistic dynamics of particles with rest mass 0 m and assigned energy E, launched, as in the previous case, into a force field deriving from a stationary potential energy V(r) , and moving according to the (relativistic) time-independent Hamilton-Jacobi equation [8,9] After having repeated de Broglie's logical steps (11) (n k ) (22) thus obtaining the time-independent Klein-Gordon equation [8,9] holding for de Broglie's relativistic matter waves associated with particles of total energy E. By replacing eqs. (15) into eq.(23) and separating once more real and imaginary parts, eq.(23) leads [26] to the dynamical Hamiltonian system and a Wave Potential ( , ) couples and guides, once more, without any wave-particle energy exchange, the particle trajectories, acting orthogonally to the particle motion. It is interesting to observe that the first of eqs.(24) turns out to coïncide with the "guidance formula" presented by de Broglie in his relativistic "double solution theory" [32],  (23), thanks to eq.(13), reduces to the form structurally analogous to eq.(2), which may be viewed therefore as a suitable Klein-Gordon equation holding for massless point-particles. Einstein's first historical duality (conceived for the electromagnetic radiation field [33]) is therefore included in de Broglie's wave-particle mechanism.

-Numerical examples
We compute in the present Section, for a number of different experimental set-ups, the stationary sets of particle trajectories provided by the integration of the Hamiltonian system (20), which is a direct consequence, as we know, of Schrödinger's energy-dependent eq.(17). The very plausibility of the numerical results plays in favor of the underlying philosophy. No kind of measurement perturbation is taken into account, and the geometry is assumed, for simplicity sake, to allow a computation limited to the (x,z)-plane. Because of the orthogonality between the wave front and the particle momentum we shall make use, over the (x, z) plane, of the identities We show in Fig.1 the diffraction of a Gaussian beam of the initial form  . We plotted on the right-hand side the initial and final transverse intensity profiles of the beams, and, on the left-hand side, the relevant trajectory pattern.  , by the force field, while continuing its diffractive widening due to the Wave Potential. We shall consider now 1) the case of a potential barrier of the form (where the parameters B z and d determine the position of the peak and of the distance between the flexes, respectively), and 2) the case of a "step-like" potential of the form where α and S z determine, respectively, the slope and the flex position of the continuous line connecting the two asymptotic levels where ()  , on the other hand, the beam overcomes the top of the barrier and undergoes a strong acceleration beyond it. We omit in both cases, for brevity sake, the relevant plots, limiting ourselves to the remarkable case 0 E / V  1 (Fig.5).  6. Step-like potential: case As a further applicative example, let us come to a collimated matter wave beam launched into a potential field ( , ) V x z representing a lens-like focalizing structure, of which we omit here, for simplicity sake, the analytical expression (see, for instance, Refs. [18] and [22]). Fig.7 and Fig.8 present, in this case, the results obtained by neglecting, respectively, and by taking into account, in eqs. (25), the Wave Potential term W(r,E) . Its diffractive effect is seen to replace the point-like focus of geometrical optics by a finite focal waist, strictly reminding the case of the Luneburg lenses considered in Refs. [29,30]. Potential. Fig.9 shows, in its turn, the progressive sharpening of the beam intensity while reaching its finite waist. A final application of eqs. (20), clearly showing the diffractive role of the Wave Potential, is obtained by applying to a collimated Gaussian beam, launched along the z-direction, a potential of the form which we assume to be shared by the whole beam. We show in Figs.10 and 11, respectively, the launch of the beam from z = 0 and its successive oscillations around such a position. The beam is seen to oscillate between the positions while progressively widening under the diffractive action of the Wave Potential.

-Born's Wave Function and Bohm's approach
Even though, as we have seen, the wave-mechanical dynamics in stationary external fields is already adequately described by the energy-dependent Schrödinger equation (17), one more interesting equation may be obtained from eq.(17) itself, making use of (12) and (13) turns out to be, as is well known [8,9], a (deterministically evolving) solution of eq.(36). Such a function is a weighted average over the whole set of eigen-functions ( , ) n rt  , where the coefficients cn (in duly normalized form) may represent either a set of experimental results (in view of a statistical treatment) or an ad hoc mathematical assembling, in view of the construction of a particular "packet" of wave-trains. Each eigen-function is acted on by its own, energy preserving, Wave Potential n W(r,E ) , and has therefore its own trajectories, provided by eqs. (20). In Born's words [35], indeed, "it's very attractive to interpret a particle of matter as a wave-packet due to the superposition of a number of wave trains. But this tentative interpretation comes up against insurmountable difficulties, since a wave packet of this kind is in general very soon dissipated". Born himself proposed however, for the function (38), an interpretation [36] going much beyond a simple superposition, assuming it to represent the most complete possible description of the physical state of a particle whose energy is not determined, in the form of a simultaneous permanence (before observation) in its full set of eigenstates, according to the (duly normalized) probabilities 2 n c . Since eq.(36) is not, in itself, an ordinary-looking wave equation, its solution (38) is not, in itself, a wave. Such a mathematical object was called "Born Wave-Function", and became the pivot of Standard Quantum Mechanics (SQM), an intrinsically probabilistic theory which became the pillar of any further development of microphysics.
We find it useful to remind here a well known comment written by Jaynes [37] in 1989: "Our present quantum mechanical formalism is an omelette that nobody has seen how to unscramble, and whose unscrambling is a prerequisite for any future advance in basic physical theory". The pre-eminent role and the probabilistic nature assigned to the energy-independent Schrödinger equation (36) -scrambling all eigen-energies together -suggested, indeed, to extend the same nature to the energy-dependent eq.(17), thus compromising the specific guiding role of the energy-dependent matter waves (14,15) conceived by de Broglie and experimentally identified by Davisson and Germer.
An attempt to rescue this guiding role, while maintaining the properties of Born's Wave-Function, was performed in Bohm's approach [2,3], where ψ(r,t) was written in the form interpretation, was assumed to represent, in Bohm's words [2], "the probability density for particles belonging to a statistical ensemble". We made use here of the term G(r,t), instead of Bohm's original term S(r,t), in order to avoid its instinctive a priori identification with the Hamilton-Jacobi function of eqs. (14) and (15). Bohm's expression (39) was "shaped", in fact, on eq.(15) (i.e on de Broglie's definite-energy matter waves, whose physical reality was established by the Davisson-Germer experiments) with the aim of depicting Born's "Wave-Function" as a single pilot wave, hopefully sharing and generalizing the nature and experimental evidence of de Broglie's guiding waves.

=-2m
formally similar to (but utterly different from) our "Wave Potential" function W(r,E), whose energy-dependence makes it apt to describe diffraction and/or interference processes. The exploitation of Bohm's Quantum Potential was faced, for instance, by means of iterative solutions leading to complex quantum trajectories [39], but didn't lead in general to easily practicable numerical procedures. The equation set of TAB.I, bypassing the use of Q( ) r,t , represents therefore a more tractable approach.