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Both classical and wave-mechanical monochromatic waves may be treated in terms of exact ray-trajectories ( encoded in the structure itself of Helmholtz-like equations ) whose mutual coupling is the one and only cause of any diffraction and interference process. In the case of Wave Mechanics, de Broglie’s merging of Maupertuis’s and Fermat’s principles (see Sect ion 3) provides, without resorting to the probability-based guidance-laws and flow-lines of the Bohmian theory, the simple law addressing particles along the Helmholtz rays of the relevant matter waves. The purpose of the present research was to derive the exact Hamiltonian ray-trajectory systems concerning, respectively, classical electromagnetic waves, non-relativistic matter waves and relativistic matter waves. We faced then, as a typical example, the numerical solution of non-relativistic wave-mechanical equation systems in a number of numerical applications, showing that each particle turns out to “dances a wave-mechanical dance” around its classical trajectory, to which it reduces when the ray-coupling is neglected. Our approach reaches the double goal of a clear insight into the mechanism of wave-particle duality and of a reasonably simple computability. We finally compared our exact dynamical approach, running as close as possible to Classical Mechanics, with the hydrodynamic Bohmian theory, based on fluid-like “guidance laws”.

As is well expressed in Ref. [

Having in mind, on our part, a set of exact particle trajectories running as close as possible to the ones of Classical Dynamics, the route adopted in the present paper starts from the very foundations of Wave Mechanics [

1) de Broglie’s founding principle p = ℏ k , to which we owe the very concept of “wave-particle duality”, associating for the first time a “matter wave” (of wave-vector k ) with a particle of momentum p , being ℏ = h / 2π and h the Planck constant [

2) Schrödinger’s time-independent equation [

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 diffraction test 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, bypassing any geometrical optics approximation.

Passing then to the wave-mechanical case, de Broglie’s relation p = ℏ 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 particles with momentum p , thus laying the foundations of Wave Mechanics. Each particle is guided along its path by the energy-preserving “gentle drive” of a matter wave having a well-established physical existence [

An experimentally tested Hamiltonian description of wave-like features holding beyond the eikonal approximation [

We derive here the exact Hamiltonian systems concerning, respectively, classical electromagnetic waves (Section 2), non-relativistic matter waves (Section 3) and relativistic matter waves (Section 4). Examples of numerical computation are given in Section 5, and a comparison is made in Sections 6 and 7 with the hydrodynamic approach of the Bohmian theory.

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

ψ ( r , ω , t ) = u ( r , ω ) e − i ω t , (1)

where r ≡ ( x , y , z ) and u ( r , ω ) is assumed to satisfy the Helmholtz equation

∇ 2 u + ( n k 0 ) 2 u = 0 , (2)

with

∇ _ ≡ ∂ / ∂ r ≡ ( ∂ / ∂ x , ∂ / ∂ y , ∂ / ∂ z ) ; ∇ 2 = ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ∂ z 2 ; k 0 ≡ 2π λ 0 = ω c (3)

and with a (time-independent) refractive index n ≡ n ( r , ω ) . The time-independence of Equation (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 Equation (2), the quite general replacement

u ( r , ω ) = R ( r , ω ) e i φ ( r , ω ) , (4)

with real amplitude R ( r , ω ) and real phase φ ( r , ω ) , and separate real from imaginary parts, Equation (2) is seen to provide [

{ d r d t = ∂ D ∂ k = c k k 0 d k d t = − ∂ D ∂ r → = ∇ [ c k 0 2 n 2 ( r , ω ) − W ( r , ω ) ] ∇ ⋅ ( R 2 k ) = 0 k ( t = 0 ) ≡ k 0 = ω / c (5)

where

k = ∇ φ , (6)

D ( r , k , ω ) ≡ c 2 k 0 [ k 2 − ( n k 0 ) 2 − ∇ 2 R R ] , (7)

W ( r , ω ) ≡ − c 2 k 0 ∇ 2 R ( r , ω ) R ( r , ω ) , (8)

and a “ray” velocity v r a y = c k / k 0 is implicitly defined. It is easily seen that, as long as k ≡ | k | = k 0 , we’ll have v r a y ≡ | v r a y | = c . The function W ( r , ω ) (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. Its gradient ∇ W turns out to be orthogonal to k ≡ ∇ φ , thus modifying the direction, but not the amplitude, of the wave vector k itself. The time-integration of the Hamiltonian system (5) requires the full knowledge of the amplitude R ( r , ω ) on an assigned starting wave-front, together with the ray positions r ( t = 0 ) and of the corresponding wave vectors k ( t = 0 ) , orthogonal to the wave-front. Thanks to the third of Equations (5) (expressing the constancy of the flux of R 2 ∇ φ along any tube formed by the field lines of the wave vector field k = ∇ φ , i.e. by the “ray-trajectories” themselves) one obtains, then, the amplitude R ( r , ω ) 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 Section 4 what the “rays” are conveying.

When, in particular, the space variation length L of the amplitude R ( r , ω ) satisfies the condition k 0 L ≫ 1 , the Wave Potential term W ( r , ω ) may be dropped, thus removing any ray-coupling. The “rays” will therefore propagate independently of each other, without any diffraction and/or interference, according to the eikonal equation [

k 2 ≅ ( n k 0 ) 2 . (9)

Let us pass now to the case of non-interacting particles of mass m and total energy E, launched with an initial momentum p 0 (with p 0 = 2 m E ) 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 [

( ∇ S ) 2 = 2 m [ E − V ( r ) ] , (10)

where the basic property of the H-J function S ( r , E ) is that the particle momentum is given by

p = ∇ S ( r , E ) . (11)

The classical H-J surfaces S ( r , E ) = c o n s t , perpendicular to the momentum of the moving particles, pilot them along stationary trajectories, according to the laws of Classical Mechanics.

Let us remind now [

ψ = u ( r , ω ) e − i ω t ≡ R ( r , ω ) e i [ φ ( r , ω ) − ω t ] (12)

under Planck’s condition

ω = E / ℏ , (13)

according to the basic conjecture

p = ℏ k (14)

laying the very foundations of Wave Mechanics [

u ( r , E ) ≡ R ( r , E ) e i ℏ S ( r , E ) , (15)

showing that the H-J surfaces S ( r , E ) = c o n s t represent the phase-fronts of matter waves, while maintaining the piloting role played in the classical Equation (11). Equation (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 are driven, in other words, along stationary trajectories orthogonal to the phase-fronts of matter waves with

λ ≡ 2π k = 2 π ℏ / p . The successive step was performed by Schrödinger ( [

[

( n k 0 ) 2 → k 2 ≡ p 2 ℏ 2 = 2 m ℏ 2 ( E − V ) (16)

leading to the Helmholtz-like equation

∇ 2 u ( r , E ) + 2 m ℏ 2 [ E − V ( r ) ] u ( r , E ) = 0 . (17)

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 [

The real existence of de Broglie’s matter waves in physical space was established in 1927 by the Davisson-Germer experiments [

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 Equation (2) to the Hamiltonian ray-tracing system (5), by simply replacing Equation (15) into Equation (17) and separating real and imaginary parts. After having defined the energy-dependent “Wave Potential function”

W ( r , E ) = − ℏ 2 2 m ∇ 2 R ( r , E ) R ( r , E ) . (18)

and the energy function

H ( r , p , E ) ≡ p 2 2 m + W ( r , E ) + V ( r ) (19)

we get now the dynamical Hamiltonian system

{ d r d t = ∂ H ∂ p ≡ p m d p d t = − ∂ H ∂ r ≡ − ∇ [ V ( r ) + W ( r , E ) ] ∇ ⋅ ( R 2 p ) = 0 p ( t = 0 ) ≡ p 0 = 2 m E (20)

providing, in strict analogy with the Helmholtz ray-tracing system (5), a stationary set of particle 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 Equation (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” Q ( r , t ) = − ℏ 2 2 m ∇ 2 ψ ( r , t ) ψ ( r , t ) [

(20) expresses the constancy of the flux of R 2 ∇ S along any tube formed by the field lines of p = ∇ S ( r , E ) , i.e. by the trajectories themselves. In complete analogy with Section 2 the launching values r ( t = 0 ) and p ( t = 0 ) of each particle must be supplemented, in the time-integration of the system (20), with the assignment of the amplitude R ( r , E ) (and its derivatives) over a starting wave-front, orthogonal to p ( t = 0 ) at each point r ( t = 0 ) . The third of equations (20) provides then, step by step, the function R ( r , E ) (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 R ( r , E ) , but not on its intensity, which may be quite arbitrarily chosen.

Let us finally pass (maintaining the same notations of the previous Sections) to the relativistic dynamics of particles with rest mass m 0 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 [

[ ∇ S ( r , E ) ] 2 = [ E − V ( r ) c ] 2 − ( m 0 c ) 2 . (21)

After having repeated de Broglie’s logical steps (11)-(14), we shall assume, with Schrödinger, that the relevant matter waves satisfy a Helmholtz-like equation of the form (2), reducing to standard Mechanics―represented now by Equation (21)―in its eikonal approximation, and perform therefore, into Equation (2), the replacement

( n k 0 ) 2 → k 2 ≡ p 2 ℏ 2 = [ E − V ( r ) ℏ c ] 2 − ( m 0 c ℏ ) 2 (22)

thus obtaining the time-independent Klein-Gordon equation [

∇ 2 u + [ ( E − V ( r ) ℏ c ) 2 − ( m 0 c ℏ ) 2 ] u = 0 , (23)

holding for de Broglie’s relativistic matter waves associated with particles of total energy E.

By replacing Equations (15) into Equation (23) and separating once more real and imaginary parts, Equation (23) leads [

{ d r d t = ∂ H ∂ p ≡ c 2 p E − V ( r ) d p d t = − ∂ H ∂ r ≡ − ∇ V ( r ) − E E − V ( r ) ∇ W ( r , E ) ∇ ⋅ ( R 2 p ) = 0 p ( t = 0 ) ≡ p 0 = ( E / c ) 2 − ( m 0 c ) 2 (24)

where

H ( r , p , E ) ≡ V ( r ) + ( p c ) 2 + ( m 0 c 2 ) 2 − ℏ 2 c 2 ∇ 2 R ( r , E ) R ( r , E ) (25)

and a Wave Potential

W ( r , E ) = − ℏ 2 c 2 2 E ∇ 2 R ( r , E ) R ( r , E ) (26)

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 Equations (24) turns out to coïncide with the “guidance formula” presented by de Broglie in his relativistic “double solution theory” [

∇ 2 u + [ ℏ ω − V ( r ) ℏ c ] 2 = 0 , (27)

structurally analogous to Equation (2), which may be viewed therefore as a suitable Klein-Gordon equation holding for massless particles. Einstein’s first historical duality (conceived for the electromagnetic radiation field [

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 Equation (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

( p x ∂ ∂ x + p z ∂ ∂ z ) R = 0 (28)

∇ 2 R = ( p / p z ) 2 ∂ 2 R / ∂ x 2 . (29)

We show in

The two heavy lines in

x w 0 = ± 1 + ( λ 0 z π w 0 2 ) 2 . (30)

We shall consider now

1) the case of a potential barrier of the form

V ( z ) = V 0 exp [ − 2 ( z − z B ) 2 / d 2 ] (31)

(where the parameters z B 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

V ( z ) = V 0 { 1 + exp [ − α z − z S w 0 ] } − 1 (32)

where α and z S determine, respectively, the slope and the flex position of the continuous line connecting the two asymptotic levels where V ( z → − ∞ ) = 0 and V ( z → ∞ ) = V 0 .

We plot in

In the case of the potential barrier (31) the beam gradually widens under the diffractive action of the Wave Potential, and is stopped and thrown back when E = V ( z ) < V 0 , with a behavior quite similar to the one of

E / V 0 ≫ 1 , 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 E / V 0 ≅ 1 (

Here, in a narrow region around the peak position z = z B ≡ 10 4 w 0 , both the external force F z ( z ) and p z are very close to zero. The beam motion basically occurs, under the action of its Wave Potential, along the x-axis, and is evanescent in the z-direction. After this brief interlude, the beam is strongly accelerated along z for z > z B .

In the case of the step-like potential (32), the beam gradually widens under the diffractive action of the Wave Potential, and is stopped and thrown back, once

more, for E = V ( z ) < V 0 , with a behavior similar to the one of

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. [

in Equations (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. [

A final application of Equations (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

V ( z ) = m ω 2 z 2 / 2 , (33)

suggested by the classical case of particles harmonically oscillating around z = 0 , with a period T = 2 π / ω , under an elastic force − m ω 2 z . The particles moving along the beam axis are standard linear oscillators [

E n = ( n + 1 / 2 ) ℏ ω , (34)

which we assume to be shared by the whole beam. We show in

oscillations around such a position. The beam is seen to oscillate between the positions

z ± ≅ ± 2 E n / m ω 2 , (35)

while progressively widening under the diffractive action of the Wave Potential.

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 Equation (17) itself, making use of (12) and (13), in the form

i ℏ ∂ ψ ∂ t = − ℏ 2 2 m ∇ 2 ψ + V ( r ) ψ (36)

which is the standard energy-independent Schrödinger equation. Referring now, in order to fix ideas, to a discrete energy spectrum of Equation (17), and defining both the eigen-frequencies ω n ≡ E n / ℏ and the eigen-functions

ψ n ( r , t ) = u n ( r ) e − i ω n t ≡ u n ( r ) e − i E n t / ℏ , (37)

any linear superposition (with arbitrary constant coefficients c n ) of the form

ψ ( r , t ) = ∑ n c n ψ n ( r , t ) , (38)

turns out to be, as is well known [_{n} (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 W ( r , E n ) , and has therefore its own trajectories, provided by Equations (20). In Born’s words [

Born himself proposed however, for the function (38), an interpretation [

We find it useful to remind here a well known comment written by Jaynes [

An attempt to rescue this guiding role, while maintaining the properties of Born’s Wave-Function, was performed in Bohm’s approach [

ψ ( r , t ) ≡ ∑ n c n ψ n ( r , t ) = R ( r , t ) e i G ( r , t ) / ℏ (39)

with real R ( r , t ) and G ( r , t ) . The function R 2 ≡ | ψ | 2 ≡ ψ ψ * , in agreement with the SQM interpretation, was assumed to represent, in Bohm’s words [

Since however, as we said before, the function ψ ( r , t ) is not, in itself, the solution of a usual wave equation, let us look for its effective nature and relevance. By applying simple analysis to Equation (39) we get

∇ G ( r , t ) / m ≡ ℏ m i Im ( ∇ ψ ψ ) ≡ ℏ 2 m i ψ * ∇ ψ − ψ ∇ ψ * ψ ψ * . (40)

Reminding therefore that the quantity J ≡ ℏ 2 m i ( ψ * ∇ ψ − ψ ∇ ψ * ) represents, in SQM, a probability current density [

∇ G ( r , t ) / m ≡ J / R 2 (41)

is seen to coincide with the velocity v p r o b ( r , t ) at which “the probability density is transported” [^{2} and with the fluid-like equations (which we omit here for simplicity) obtained by Bohm [

The basic equations of the hydrodynamic Bohmian theory, together with the ones of our dynamical system (20), may be summarized, respectively, in the following

The first equation of

d r d t = ℏ 2 m i ψ ∗ ∇ ψ − ψ ∇ ψ ∗ | ψ | 2 |
---|

i ℏ ∂ ψ ∂ t = − ℏ 2 2 m ∇ 2 ψ + V ( r ) ψ |

d r d t = p m |
---|

d p d t = − ∇ [ V ( r ) + W ( r , E ) ] |

∇ ⋅ ( R 2 p ) = 0 |

The equations of

Q ( r , t ) = − ℏ 2 2 m ∇ 2 ψ ( r , t ) ψ ( r , t ) , formally similar to (but utterly different from) our

“Wave Potential” function W ( r , E ) , whose (monochromatic) 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 [

The Hamiltonian system of

In conclusion, the approach of the present paper is seen to provide a consistent wave-mechanical extension of Classical Dynamics. Starting from de Broglie’s and Schrödinger’s foundations of Wave Mechanics, we avoid (in agreement with Born’s original caution alert [

In the light of the emergence of “weak” experimental measurements, made possible by an increasingly powerful technology [

The authors declare no conflicts of interest regarding the publication of this paper.

Orefice, A., Giovanelli, R. and Ditto, D. (2018) The Dynamics of Wave-Particle Duality. Journal of Applied Mathematics and Physics, 6, 1840-1859. https://doi.org/10.4236/jamp.2018.69157