PreprintPDF Available
Preprints and early-stage research may not have been peer reviewed yet.

Abstract and Figures

De Broglie's problem: can a particle have a permanent localization in space? is solved in terms of the general properties of Helmholtz-like equations, within the standard frame of de Broglie's and Schrodinger's Wave Mechanics, avoiding any further hypothesis.
Content may be subject to copyright.
Journal of Applied Mathematics and Physics, 2018, 6, 2621-2634
http://www.scirp.org/journal/jamp
ISSN Online: 2327-4379
ISSN Print: 2327-4352
DOI:
10.4236/jamp.2018.612218 Dec. 27, 2018 2621 Journal of Applied Mathematics and Physics
Primary Assumptions and Guidance Laws in
Wave Mechanics
Adriano Orefice*, Raffaele Giovanelli, Domenico Ditto
Department of Agricultural and Environmental Sciences (DiSAA), University of Milan, Milan, Italy
Abstract
In an article written by Louis de Broglie in 1959 (30
years after the Nobel
prize rewarding his foundation of Wave Mechanics), the most challenging
problem raised by the Bohr, Heisenberg and Born Standard Quantum Me-
chanics (SQM) was pointed out in the renunciation to describe
a permanent
localization in space
,
and therefore a well-defined trajectory
for any moving
particle. This challenge is taken up in the present paper, showing that de
Broglie’s
Primary Assumption
=pk
, predicting the wave-
particle duality,
does also allow to obtain from the
energy-dependent
form of the Schrödinger
and/or Klein-Gordon equations the
Guidance Laws piloting particles along
well-defined trajectories. The
energy-independent
equations, on the other
hand, may only give riseboth in SQM and in the Bohmian approach—t
o
probabilistic descriptions,
overshadowing the role of de Broglie
s
matter waves
in physical space
.
Keywords
Helmholtz Equation, Wave Potential, Hamilton-Jacobi Equations,
Wave Mechanics, de Broglie’s Duality, Matter Waves, Guidance Laws,
Schrödinger Equations, Klein-Gordon Equations
1. Introduction
We translate here the beginning of a little known de Broglie
s paper
:
L
interprétation de la Mécanique Ondulatoire
[1],
marking his abandonment
of the acceptance
,
lasted
30
years
,
of the interpretation of Born
,
Bohr and Hei-
senberg of Quantum Mechanics and the return to his own original interpreta-
tion. The french text is reported in Appendix I
.
In my first works on Wave Mechanics, dating back to 1923, I had clearly
perceived that it was necessary, in a general way, to associate with the movement
How to cite this paper:
Orefice, A., Gi-
ovanelli
, R. and Ditto, D. (2018)
Primary
Assumptions and Guidance Laws in Wave
Mechanics
.
Journal of Applied Mathema
t-
ics and Physics
,
6
, 2621-2634.
https://doi.org/10.4236/jamp.2018.612218
Received:
November 20, 2018
Accepted:
December 24, 2018
Published:
December 27, 2018
Copyright © 201
8 by authors and
Scientific
Research Publishing Inc.
This work is
licensed under the Creative
Commons Attribution International
License (CC BY
4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access
A. Orefice et al.
DOI:
10.4236/jamp.2018.612218 2622 Journal of Applied Mathematics and Physics
of any corpuscle the propagation of a wave. But the homogeneous wave that I
had been led to consider, and that became the wave
ψ
of the usual wave
mechanics, did not seem to me to describe the physical reality (...).
Giving no
particular prerogative to any point in space
,
it was not capable of representing
the position of the corpuscle
: we could suppose at most, as was very shortly
done, that it gave, by its square, the probability of presenceof the corpuscle in
each point (...)
No physicist ignores today (1959) that Wave Mechanics has received for more
than thirty years a purely probabilistic interpretation in which the wave
associated with the corpuscle is no more than a probability representation
dependent on the state of our information, and likely to vary abruptly with it
(Heisenberg’s “reduction of the probability packet), while
the corpuscle is
conceived as having no permanent location in space and
,
consequently
,
as not
describing a well-defined trajectory
. This way of conceiving the wave-particle
dualism has received the name of complementarity, a rather vague notion that
was tentatively extrapolated, in a somewhat perilous way, outside the realm of
physics.
This interpretation of Wave Mechanics, quite different, I will recall, from the
one I had considered at the beginning of my research, is mainly due to Born,
Bohr and Heisenberg, whose brilliant works are worthy, no doubt, of the greatest
admiration. It has been adopted fairly quickly by almost all theorists, despite the
express reserves made by such eminent minds as Einstein and Schrödinger and
despite their objections. Personally, after having proposed a radically different
interpretation, I joined the one that had become orthodox”, and I taught it for
many years. Since 1951, however, in particular after the attempts made at that
time by Bohm and Vigier, I asked myself, once more, if my original orientation
towards the problem posed by the existence of the wave-corpuscule dualism
could be the good one.
A few years have passed, and it seems to me that the time has come for a new
review of the state of the question, taking into account the progress made since
my 1953-54 presentations.
The strongest objections that can be raised against the currently accepted
interpretation of wave mechanics
concern the non-localization of the corpuscle
in this interpretation. It admits, indeed, that, if the state of our knowledge on a
corpuscle is represented by an extended wave-train, the corpuscle is present in
all the points of this wave-train with a probability [density] equal to
2
ψ
: this
presence could be qualified as potential, and it is only at the moment when we
notice the presence of the corpuscle at a point of the wave-train by an
observation that this potentiality is actualized”—in the language of philosophers.
Such a conception encounters difficulties which have been pointed out with
force, and in various ways, by Einstein and Schrödinger; L. de Broglie [1].
As exemplified by [2], Standard Quantum Mechanics (SQM) was generally
presented, since the very beginning, by “
conveniently assuming as fundamental
postulates
the time-dependent
Schrödinger equation and its probabilistic inter-
A. Orefice et al.
DOI:
10.4236/jamp.2018.612218 2623 Journal of Applied Mathematics and Physics
pretation,
since every department of deductive science must necessarily be
founded on certain Primary Assumptions
”.
We show in the present paper that an answer to
de Broglie
s problem con-
cerning the particle localization
may be obtained from quite simpler, and more
evident, Primary Assumptions, suggested by the very foundations of Wave
Mechanics.
Section 2 presents the demonstration that
any
Helmholtz-like equation is as-
sociated with a set of
exact
Hamiltonian ray trajectories”. Section 3 shows that
both the Schrödinger and the Klein-Gordon
energy-dependent
equations, be-
cause of their belonging to the Helmholtz-like family, are associated with mat-
ter-wave trajectories along which, thanks to de Broglie’s Primary Assumption
=pk
, the particle motion is addressed and piloted. These trajectories provide
therefore, both in the non-relativistic and in the relativistic case, the Guidance
Laws allowing to solve
de Broglie
s problem
. The
energy-dependence
of those
equations allows an
exact
dynamic representation of the particle motion, run-
ning as close as possible to the corresponding
classical
description, and basically
distinctas discussed in Sections 4 and 5from the
probabilistic flow lines
and
Guidance Laws of the Bohmian theory, whose hydrodynamic approach is based
on the same logic and Primary Assumptions as SQM.
2. Helmholtz Ray-Tracing
Referring to a stationary medium with refractive index
( )
,nωr
, sustaining an
electromagnetic wave of the form
, (1)
where
( )
,u
ω
r
is assumed to be a solution of the Helmholtz equation
( )
2
2
0
0u nk u+=
, (2)
the standard replacement
( ) ( )
( )
,
, ,e
i
uωRω
ϕω
=
r
rr
, (3)
with real amplitude
( )
,Rωr
and phase
( )
,
ϕω
r
, is easily seen to split Equa-
tion (2), after the separation of real and imaginary parts and after the definition
of the wave-vector
()
,
ϕω
=kr
(4)
and of the function
( ) ( )
( )
2
0
,
,2,
R
c
WkR
ω
ωω
= − r
rr
, (5)
into the equation system
( )
( )
( )
( )
2
2
2
0
0
0 (6)
, , , 0 (7)
2
R
c
D k nk W
k
ϕ
ωω
⋅=

−+ =

rk r
∇∇
A. Orefice et al.
DOI:
10.4236/jamp.2018.612218 2624 Journal of Applied Mathematics and Physics
The differentiation
d d0
DD∂∂
⋅+ ⋅ =
∂∂
rk
rk
of Equation (7) associates to Equa-
tion (2) an exact Hamiltonian ray-tracing system of the form
() ( )
( )
()
0
2
0
2
0
d
d
d,,
d2
0
0
Dc
tk
ck
DnW
t
R
t kc
ωω
ω
= ≡

=−≡ −


⋅=
= = =
rk
k
krr
r
k
k
(8)
whose time-integration provides a
stationary
ray-trajectory system coupled by
the term
( )
,W
ω
r
(which we called
Wave Potential
) acting
perpendicularly
, at
each point, to the relevant ray trajectories, together with
the time-table of the
rays
along these trajectories
.
The ray-trajectory coupling due to the (monochromatic) Wave Potential is the
one and only cause of any diffraction and interference process. When, however,
the space variation length
L
of the wave amplitude
( )
,R
ω
r
turns out to satisfy
the condition
0
1kL
, Equation (7) reduces to the
eikonal equation
( )
( )
2
2
2
0
k nk
ϕ
, (9)
describing the geometrical optics limit, where the rays are seen to propagate in-
dependently from one another, without any diffraction and/or interference
process.
In conclusion
,
any Helmholtz-like equation of the form
(2)
is associated to a
stationary system of exact ray-trajectories
:
a basic information which was not
available until
2009 [3]-[8].
3. Back to de Broglie’s Wave Mechanics
Let us refer now, indifferently, to
non-relativistic
or
relativistic
Dynamics, and
let us consider the simple case of
non-interacting point-particles
of mass
m
,
rest
mass
0
m
and total energy
E
, launched
with an initial momentum
0
p
into an
external force field deriving from a
time-independent
potential energy
( )
Vr
.
The classical
non-relativistic
dynamics of each particle is summarized, as is
well known [9], by the time-independent Hamilton-Jacobi (H-J) equation
( ) ( )
2
2
S mE V= −


r
, (10)
while the classical
relativistic
dynamics is summarized by the time-independent
Hamilton-Jacobi equation
( ) ( )
( )
2
2
2
0
EV
S mc
c

= −


r
. (11)
Both in Equation (10) and Equation (11) the basic property of the H-J func-
tion
( )
,
SEr
is the fact that the particle momentum is given by
()
,SE=pr
. (12)
A. Orefice et al.
DOI:
10.4236/jamp.2018.612218 2625 Journal of Applied Mathematics and Physics
Recalling the
Fermat
and
Maupertuis
variational principles, Louis de Broglie
[10] [11] [12] was induced to associate each
particle
of momentum
p
with a
suitable
matter wave
(with wave-vector
k
) of the form
( ) () ( )
()
,
,, ,e ,e
it
it
tu R
ϕ ωω
ω
ψω ω ω


= ≡
r
rr r
(13)
under Planck’s condition
E
ω
=
, (14)
according to the Primary Assumption (laying the foundations of
Wave Me-
chanics
)
=pk
. (15)
We have therefore, from Equation (4) and Equation (12), the relations
( )
,SE
ϕ
=r
and
( ) ( )
( )
,
, ,e
iSE
uE RE
r
rr
, (16)
The
scalar form
pk=
(whence
2πp
λ
=
) of de Broglie’s Primary As-
sumption was very soon verified by the Davisson-Germer electron diffraction
experiments [13], which established once and for all the physical reality of mat-
ter waves and of the wave-particle duality, and was sufficient by itself to grant a
Nobel Prize to all of them.
The
vector form
of Equation (15) appeared, in its turn, together with Equation
(16), to be quite eloquent: they told that the H-J surfaces
( )
,S E const=r
rep-
resent the phase-fronts of the newly contrived matter waves, and that the particle
momentum
p
is addressed along the wave-vector
k
, orthogonal to the
phase-fronts of the relevant matter waves.
The discovery of a satisfactory Guidance Law of the particles along their dy-
namic trajectories
,
however
,
had not yet been reached
.
An important step in
this direction was performed by Schrödinger [14] [15] [16] [17], assuming that
the laws of Classical Mechanics (represented here by Equation (10) and Equation
(11)) are the eikonal approximation of suitable
Helmholtz-like equations
of the
form (2).
By performing therefore, in the
non-relativistic
case (10), the
replacement
( )
( )
2
22
022
2pm
nk k E V→≡ =

(17)
suggested by the eikonal Equation (9), into Equation (2), we get the
Helm-
holtz-like equation
( ) ( ) ( )
2
2
2
, ,0
m
uE EV uE∇ +− =


r rr
, (18)
which is the so-called time-independent(but
energy-dependent
) Schrödinger
equation [18] [19]: an eigen-value equation admitting in general both continu-
ous and discrete eigen-function spectra and energy eigen-values, which bypass
the heuristic prescriptions of the oldquantum theory.
By performing, similarly, in the
relativistic
case (11), the replacement
( )
( )
22
2
220
02
EV mc
p
nk k c


→≡ =




r

(19)
A. Orefice et al.
DOI:
10.4236/jamp.2018.612218 2626 Journal of Applied Mathematics and Physics
into Equation (2), we get the
Helmholtz-like equation
()()()
22
20
, ,0
EV mc
uE uE
c




∇+ − =






r
rr

, (20)
which is the so-called time-independent(but
energy-dependent
) Klein-Gordon
equation (holding even in the case of particles with
0
0m=
).
In order to perform the final step toward a reliable Guidance Law
in the ab-
sence of further information
, de Broglie considered the idea [20] [21] [22] of a
non-linear double-solutionunderlying the Schrödinger and Klein-Gordon
equations: a theory which did never get, for him, a satisfactory level.
As we know from Section 2, however,
we are nowadays informed
of the prop-
erty of Helmholtz-like equations of being associated with exact kinematic sets of
ray-trajectories.
The reply to de Broglie
s question
:
can a particle have a per-
manent localization in space
?”
appears therefore to be almost immediate
.
Both
in the
non-relativistic
and in the
relativistic
case,
in fact,
we have only to
repeat
the procedure of Section
2,
by replacing the function
( )
,uEr
, given by Equa-
tion (16), into the Helmholtz-like Equation (18) and/or Equation (20), and by
separating, once more, real from imaginary parts.
The long-desired
Guidance Law
is finally reached in the general form
d
d
H
t
=
r
p
(21)
where the energy function
()
,,HE
rp
takes on the form
()()( )
2
,, ,
2
p
H E W EV
m
≡+ +rp r r
(22)
in the
non-relativistic
case, and the form
( ) ( ) ( )
( )
( )
2
22
0
,, 2 ,H E V pc m c EW E≡+ + +rp r r
(23)
in the
relativistic
case. In
both cases
, the particle trajectories and time-tables are
found by time-integrating Equation (21) in parallel with the relevant Hamilto-
nian system of dynamical equations reported in Appendix II. Not by sheer co-
incidence Equation (21) takes on, in the
relativistic
case, the same form as de
Broglie’s Guidance Law of [20] [21] [22], showing that, in spite of his dissatis-
faction, he wasn’t far from his goal. The missing pieces of the puzzle, in his ap-
proach, were the equations accompanying Equation (21) in the Hamiltonian
systems AII-1 and/or AII-2, and determining the relevant Helmholtz trajectories.
In both systems, in particular, a suitable
Wave Potential
function
( )
,WEr
,
acting
orthogonally to the particle motion
and exerting therefore an
energy-conserving
gentle drive
, is seen to be the cause of any diffraction and/or interference
wave-mechanical process.
The time-integration of the wave-mechanical systems AII
-1
and
/
or AII
-2
pro-
vides
,
in conclusion
,
de Broglie
s missing link
, without any further assumption
and without resorting to any kind of probabilistic interpretation. The particle
trajectories and time-tables are simply found, in fact [8], by assigning
( )
,0Et=r
,
A. Orefice et al.
DOI:
10.4236/jamp.2018.612218 2627 Journal of Applied Mathematics and Physics
( )
,0Et=p
and
()
,, 0R Et=
r
over a suitable launching surface, and making
use of the equation (see both AII-1 and AII-2)
( )
20R⋅=p
, (24)
expressing the constancy of the flux of
2
Rp
along any tube formed by the tra-
jectories, in order to obtain the wave amplitude
( )
,REr
and the Wave Poten-
tial function
( )
,WEr
at each time-step. The
energy-dependence
of Equation
(18) and Equation (20) provides, moreover, a crucial analogy with Classical Me-
chanics, allowing to build up exact trajectories unfolding as close as possible to
the relevant
classical
limits, to which they reduce when the Wave Potential term
is neglected,
i
.
e
. in their eikonal approximation.
Let us consider for instance, in Figure 1 and Figure 2, the
non-relativistic
case of a particle beam of the initial Gaussian form
( )
( )
22
0
; 0 expR xz x w=∝−
,
with half-width
0
w
, launched along the z-axis, from the left hand side, into a
potential field
( )
,V xz
representing a lens-like focalizing structure [8]. The
point-like focus
(Figure 1) obtained in the eikonal limit,
i
.
e
. according to
Classical
Figure 1. Lens-like potential: point-like focusing in the absence of Wave Potential.
Figure 2. Lens-like potential: finite-focusing due to Wave Potential.
A. Orefice et al.
DOI:
10.4236/jamp.2018.612218 2628 Journal of Applied Mathematics and Physics
Mechanics
, by dropping from the system AII-1 the Wave Potential term
( )
,WEr
,
is seen to be replaced by a
finite focal waist
(Figure 2) in
Wave Mechanics
, when
the diffractive role of
( )
,WEr
is taken into account.
4. Time-Dependent Equations
We could stop here
,
since the wave-particle duality is already adequately de-
scribed, as we have shown, by the
energy-dependent
(and time-independent)
Helmholtz-like Equation (18) and Equation (20) and by their dynamic trajectory
systems AII-1 and AII-2.
Because, however, of the history itself of Quantum Mechanics, it’s interesting
to remind that two
time-dependent
equations [18] [19] may be obtained, mak-
ing use of Equation (13), from Equation (18), in the form, respectively, of the
usual-looking
wave equation
( )
2
2
22
2mEV
Et
ψ
ψ
∇=
(25)
with a phase velocity
( )
2
E mE V
, and of the
unusual-looking
, and
energy
independent
, equation
( )
2
2
2
iV
tm
ψψψ
=− ∇+
r
. (26)
which is the so-called “time-dependent” Schrödinger equation. Equation (26)
was adopted, as is well known [18] [19], as the most significant generalization of
Equation (18).
Referringin order to fix ideasto a discrete energy spectrum of Equation
(18), and defining both the eigen-frequencies
nn
E
ω
and the eigen-functions
() ( )
,e
n
iE t
nn
tu
ψ
=rr
, (27)
any linear superposition (with arbitrary constant coefficients
n
c
) of the form
( ) ( )
,,
nn
n
tc t
ψψ
=
rr
, (28)
turns out to be a solution of Equation (26), splitting it into a time-evolving su-
perposition of Helmholtz equations, in the form
() () ( )
2
2
2
e0
n
E
it
n n nn
n
m
c u EV u

∇+ − =




r rr
. (29)
The function (28) is a weighted average performed over the whole set of ei-
gen-functions
( )
,
nt
ψ
r
, representing a particular “packet” of wave-trains. As we
know from Section 3, however, each eigen-function
( )
n
ur
has its own trajecto-
ries, leading in general to the progressive space-dispersion of any wave-packet.
The group velocity of a wave-packet takes on, indeed, the suggestive form
( )
( )
( )
2
d2
d
d
dd d
g
pm
E
m
ω
≡= = =
p
vkp p
, (30)
referring to the packet center and
apparently
coinciding with the particle veloci-
ty, but obtained for a progressively diverging range around
p
. In Born’s words
A. Orefice et al.
DOI:
10.4236/jamp.2018.612218 2629 Journal of Applied Mathematics and Physics
[23], 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 interpre-
tation comes up against insurmountable difficulties
,
since a wave-packet of this
kind is in general very soon dissipated
”. Let us recall, by comparison, that in our
classical
Hamiltonian case AII-1, holding for
point-like particles
, we have from
Equation (21) an
exact
particle velocity
d
d
H
tm
= ≡
rp
p
(31)
without any dispersion.
Born’s proposal [24], giving rise to SQM, was to view the function (28)
(which was called “
Born Wave-Function
”) as representing a
unique
physical
quantity
carrying (before observation) the
most complete information
about the
possible state of a particle,
ranging in its full set of eigenstates
, according to the
(duly normalized) probabilities 2
n
c. The continuous evolution of the Born
Wave-Function was assumed, moreover, to (discontinuously) collapse into the
energy eigen-value observed by the experimental set-up.
Born’s Primary Assumption, leaving no room to physical intuition and discus-
sion, was therefore givenas we said in the Introductionby the “time-dependent”
Schrödinger Equation (26) itself, to be viewed in a probabilistic perspective.
In the
Bohmian approach
[25] [26], whose state of art is thoroughly described
in [27]-[32],
Born
s Wave-Function
was written in the form
( ) ( ) ( )
( )
,
, , ,e
iG t
nn
n
t c t Rt
ψψ
≡=
r
r rr
(32)
with real
( )
,Rtr
and
( )
,Gtr
, and
2
2
R
ψ ψψ
≡≡
. The expression (32) was
“shaped” on de Broglie’s Equation (16), with the aim of viewing Born’s
“Wave-Function” as a generalized de Broglie’s wave, participating, possibly, in
Davisson-Germer’s detectability. The role of de Broglie’s Primary Assumption
=pk
(and of de Broglie himself as founding father of Wave Mechanics) was
therefore overshadowed, and somewhat diminished, by this cooptation. Con-
cerning the function (32), it’s easily verified that
( )
**
*
,
2
Gt
m mi
ψ ψ ψψ
ψψ
r
∇∇
. (33)
Since, in SQM [18] [19], the fluid-like
probability current density is given by
the expression
( )
**
2mi
ψ ψ ψψ
≡−J∇∇
, the term
( )
,Gt
m
r
is seen to coincide
with the velocity
( )
,
prob
tvr
at which the
fluid-like probability density
is trans-
ported. That is why, in the Bohmian theory, the particle Guidance Law is
as-
sumed
in the form
( ) ( )
,
d,
d
prob
Gt
t
tm
= r
rvr
, (34)
where the particle is represented by a wave-packet centered at
r
.
This choice gives rise to a
hydrodynamic visualisation
of SQM, utterly differ-
ent both from the
dynamic
Guidance Law” that de Broglie was looking for and
A. Orefice et al.
DOI:
10.4236/jamp.2018.612218 2630 Journal of Applied Mathematics and Physics
from our own Guidance Law (21).
5. Discussion and Conclusions
As we wrote in the Introduction, the direct assumption of Schrödinger’s equa-
tions (together with their probabilistic interpretation) as axiomatic Primary
Assumptions of Quantum Mechanics doesn’t help the intuitive understanding
of its standard interpretation and of its possible alternatives. Starting, on the
contrary, from de Broglie’s Assumption
=pk
(in its complete vectorial form)
is quite helpful both for the physical intuition of Wave Mechanics and for its
subsequent development.
Both in his juvenile years [10] [11] and in his later papers [1] [12] [20] [21]
[22] de Broglie had clear in mind the problem of localizing and addressing the
particles along a classical-looking path, starting from assigned launching condi-
tions and according to a consistent Guidance Law. No forward step could be
performed, however, before the discovery [3] of the Hamiltonian ray-tracing
properties of Helmholtz-like equations.
As we have shown, the desired Guidance Law is given by Equation (21), duly
accompanied by the full Hamiltonian systems AII-1 and/or AII-2. Their
time-integration tells us that an exact and classical-looking point-particle dy-
namics, guided by matter waves, is both possible and easily practicable, contrary
to the assertion of an intrinsically probabilistic and indeterministic nature of
physical reality: an assertion whose extrapolations lead to a host of
quantum
paradoxes
[33] [34], including doubts about the physical reality itself of material
particles.
As far as the natural development of the present study is concerned, we are
presently working on its extension to many-particle applications. Here, too, de
Broglie’s juvenile work appears to provide an essential contribution, in striking
contrast with the SQM route. At the Solvay Conference of 1927 the father of
Wave Mechanics happened to write, in fact [35]: “
It appears to us certain that if
one wants to physically represent the evolution of a system of N corpuscles
,
one
must consider the propagation of N waves in space
,
each of the N propagations
being determined by the action of the N
-1
corpuscles connected to the other
waves
.
(…)
Contrary to what happens for a single material point
,
it does not ap-
pear easy to find a single wave that would define the motion of a system taking
Relativity into account
”.
Conflicts of Interest
The authors declare that there is no conflict of interest concerning the present
paper.
References
[1] de Broglie, L. (1959) L’interprétation de la Mécanique Ondulatoire.
Le Journal de
Physique et le Radium
, 20, 963-979.
https://doi.org/10.1051/jphysrad:019590020012096300
A. Orefice et al.
DOI:
10.4236/jamp.2018.612218 2631 Journal of Applied Mathematics and Physics
[2] Pauling, L. and Wilson Jr., E.B. (1935) Introduction to Quantum Mechanics.
McGraw-Hill Company, Inc., NewYork, London.
[3] Orefice, A., Giovanelli, R. and Ditto, D. (2009) Complete Hamiltonian Description
of Wave-Like Features in Classical and Quantum Physics.
Foundations of Physics
,
39, 256. https://doi.org/10.1007/s10701-009-9280-2
[4] Orefice, A., Giovanelli, R. and Ditto, D. (2012) Beyond the Eikonal Approximation
in Classical Optics and Quantum Physics. In: Oriols, X. and Mompart, J., Eds.,
Ap-
plied Bohmian Mechanics
:
From Nanoscale Systems to Cosmology
, Chapter: 7,
PAN Stanford Publishing, Singapore, 425-453
[5] Orefice, A., Giovanelli, R. and Ditto, D. (2013) A Non-Probabilistic Insight into
Wave Mechanic.
Annales de la Fondation Louis de Broglie
,
38, 7-31.
[6] Orefice, A., Giovanelli, R. and Ditto, D. (2015) From Classical to Wave-Mechanical
Dynamics.
Annales de la Fondation Louis de Broglie
,
40, 95.
[7] Orefice, A., Giovanelli, R. and Ditto, D. (2015) Is Wave Mechanics Consistent with
Classical Logic?
Physics Essays
, 28, 515-521.
https://doi.org/10.4006/0836-1398-28.4.515
[8] Orefice, A., Giovanelli, R. and Ditto, D. (2018) Dynamics of Wave-Particle Duality.
Journal of Applied Mathematics and Physics
, 6, 1840.
[9] Goldstein, H. (1965) Classical Mechanics. Addison-Welsey, Boston.
[10] de Broglie, L. (1924) A Tentative Theory of Light Quanta.
The London
,
Edinburgh
,
and Dublin Philosophical Magazine and Journal of Science
, 47, 446-458.
https://doi.org/10.1080/14786442408634378
[11] de Broglie, L. (1925) Recherches sur la théorie des quanta.
Annales de Physique
, 10,
22.
[12] de Broglie, L. (1965) La nature ondulatoire delélectron. Nobel Lectures in Physics
1922-1941, Elsevier Publ. Co., Amsterdam, 244.
[13] Davisson, C.J. and Germer, L.H. (1927) The Scattering of Electrons by a Single
Crystal of Nickel.
Nature
,
119, 558-560. https://doi.org/10.1038/119558a0
[14] Schrödinger, E. (1926) Quantisierung als Eigenwertproblem I.
Annalen der Physik
,
79, 361. https://doi.org/10.1002/andp.19263840404
[15] Schrödinger, E. (1926) Quantisierung als Eigenwertproblem II.
Annalen der Physik
,
79, 489. https://doi.org/10.1002/andp.19263840602
[16] Schrödinger, E. (1926) Quantisierung als Eigenwertproblem III.
Annalen der
Physik
, 80, 437. https://doi.org/10.1002/andp.19263851302
[17] Schrödinger, E. (1926) Quantisierung als Eigenwertproblem IV.
Annalen der
Physik
, 81, 109. https://doi.org/10.1002/andp.19263861802
[18] Persico, E. (1950) Fundamentals of Quantum Mechanics. Prentice-Hall, Inc., Upper
Saddle River.
[19] Messiah, A. (1959) Mécanique Quantique, Dunod.
[20] de Broglie, L. (1956) Une tentative d’interprétation causale et non-linéaire de la
Mécanique Ondulatoire: La théorie de la double solution. Gauthier-Villars, Paris.
[21] de Broglie, L. (1971) L’interprétation de la Mécanique ondulatoire par la théorie de
la double solution. In: d’Espagnat, B., Ed.,
Foundations of Quantum Mechanics
,
Academic Press, New York, 345-367.
[22] de Broglie, L. (1987) Interpretation of Quantum Mechanics by the Double Solution
Theory.
Annales de la Fondation Louis de Broglie
, 12, 1.
[23] Born, M. (1935) Atomic Physics. Blackie & Son Ltd., London.
A. Orefice et al.
DOI:
10.4236/jamp.2018.612218 2632 Journal of Applied Mathematics and Physics
[24] Born, M. (1926) Quantenmechanik der Stoßvorgänge.
Zeitschrift für Physik
, 38,
803-827. https://doi.org/10.1007/BF01397184
[25] Bohm, D.J. (1952) A Suggested Interpretation of the Quantum Theory in Terms of
“Hidden” Variables I.
Physical Review
, 85, 166.
https://doi.org/10.1103/PhysRev.85.166
[26] Bohm, D.J. (1952) A Suggested Interpretation of the Quantum Theory in Terms of
“Hidden” Variables II.
Physical Review
, 85, 180.
https://doi.org/10.1103/PhysRev.85.180
[27] Holland, P.R. (1992) The Quantum Theory of Motion. Cambridge University Press,
Cambridge.
[28] Wyatt, R.E. (2005) Quantum Dynamics with Trajectories: Introduction to Quantum
Hydrodynamics. Springer, Berlin.
[29] Dürr, D. and Teufel, S. (2009) Bohmian Mechanics. Springer-Verlag, Berlin.
[30] Oriols, X. and Mompart, J. (2012) Applied Bohmian Mechanics: From Nanoscale
Systems to Cosmology. Pan Stanford Publishing.
[31] Sanz, A.S. and Miret-Artès, S. (2012-2014) A Trajectory Description of Quantum
Processes. Vol. I-II, Springer, Berlin.
[32] Benseny, A., Albareda, G., Sanz, A.S., Mompart, J. and Oriols, X. (2014) Applied
Bohmian Mechanics.
The European Physical Journal D
, 68, 286.
https://doi.org/10.1140/epjd/e2014-50222-4
[33] Aharonov, Y. and Rohrlich, D. (2005) Quantum Paradoxes: Quantum Theory for
the Perplexed. Wiley, Hoboken. https://doi.org/10.1002/9783527619115
[34] Hardy, L. (1992) Quantum Mechanics, Local Realistic Theories and Lorentz In-
variant Theories.
Physical Review Letters
, 68, 2981.
https://doi.org/10.1103/PhysRevLett.68.2981
[35] de Broglie, L. (1927) The New Dynamics of Quanta. In: Bacciagaluppi, G. and
Valentini, A., Eds.,
The Book Quantum Theory at the Crossroads
:
Reconsidering
the
1927
Solvay Conference
, Part III, Cambridge University Press, Cambridge, 374.
A. Orefice et al.
DOI:
10.4236/jamp.2018.612218 2633 Journal of Applied Mathematics and Physics
Appendix I
We report here the beginning of de Broglie
s original text of the paper
L
interprétation de la Mécanique Ondulatoire
[1]
written
,
in French
,
in
1959.
Dans mes premiers travaux sur la Mécanique ondulatoire, qui remontent à
1923, javais clairement aperçu quil fallait dune façon générale associer au
mouvement de tout corpuscule la propagation dune onde. Mais londe
homogène que javais été amené à considérer, et qui est devenue londe
ψ
de la
Mécanique ondulatoire usuelle, ne me paraissait pas décrire la réalité physique.
(...)
Ne donnant aucune prérogative particulière à aucun point de l
espace
,
elle
n
était pas susceptible de représenter la position du corpuscule
: tout au plus
pouvait-on supposer, comme on le fit bientôt, quelle donnait par son carré la
“probabilitè de présence” du corpuscule en chaque point (...).
Aucun physicien nignore aujourdhui (1959) que la Mécanique ondulatoire a
reçu depuis plus de trente ans une interprétation “purement probabiliste” dans
laquelle Ionde associée au corpuscule nest plus quune représentation de
probabilité dependant de létat de nos informations à son sujet, et susceptible de
varier brusquement avec elles (réduction du paquet de probabilité au sens de
Heisenberg), tandis
que le corpuscule est conçu comme n
ayant pas de
localization permanente dans l
espace et
,
par suite
,
comme ne décrivant pas une
trajectoire bien définie
. Cette manière de concevoir le dualisme onde-corpuscule
a reçu le nom de “complémentarité, notion assez peu précise que lon a cherché
à extrapoler, dune façon un peu périlleuse, en dehors du domaine propre de la
Physique.
Cette interprétation de la Mécanique ondulatoire, bien différente, je le
rappellerai, de celle que javais envisagée au début de mes recherches, est due
principalement à MM. Born, Bohr et Heisenberg dont les brillants travaux sont
d’ailleurs dignes de la plus grande admiration. Elle a été assez rapidement
adoptée par presque tous les théoriciens malgré les réserves expresses que
faisaient à son sujet des esprits aussi éminents que MM. Einstein et Schrödinger
et les objections quils lui opposaient. Personnellement, après avoir proposé une
interprétation tout à fait différente je me suis rallié à celle qui devenait
“orthodoxe”, et je lai enseignée pendant de longues années. Mais depuis 1951, à
la suite notamment de tentatives faites à cette époque par MM. Bohm et Vigier,
je me suis à nouveau demandé si ma première orientation vis-à-vis du problème
posé par lexistence du dualisme onde-corpuscule nétait pas la bonne. Quelques
années ont passé et il me semble que le moment est venu de faire une nouvelle
mise au point de létat de la question en tenant compte des progrès accomplis
depuis mes exposés de 1953-1954.
Les objections les plus fortes que lon peut élever contre linterprétation
actuellement admise de la Mécanique ondulatoire
sont relatives à la non-localization
du corpuscule
dans cette interprétation. Elle admet, en effet, que, si létat de nos
connaissances sur un corpuscule est représenté par un train donde
ψ
étendu,
le corpuscule est présent dans tous les points de ce train dondes avec une
A. Orefice et al.
DOI:
10.4236/jamp.2018.612218 2634 Journal of Applied Mathematics and Physics
probabilité égale à
2
ψ
: cette présence pourrait être qualifiée de “potentielle” et
c’est seulement au moment où nous constatons la présence du corpuscule en un
point du train dondes par une observation que cette potentialité s’actualise, pour
employer un langage de philosophes. Une telle conception se heurte à des
difficultés qui ont été signalées avec force et de diverses manières par MM. Ein-
stein et Schrödinger.
Appendix II
We report here (from [8]) the Hamiltonian systems of particle trajectories
holding, respectively, in the non-relativistic and in the relativistic case.
A II-1
Non-Relativistic
Hamiltonian Trajectories
AII-2
Relativistic
Hamiltonian Trajectories
( ) ( )
( )
( )
2
0
d
d
d,
d
0
02
H
tm
HV WE
t
R
t p mE
= ≡
=− ≡− +


⋅=
=≡=
rp
p
prr
r
p
p
( )
()( ) ( )
( )
( ) ( ) ( )
2
2
22
00
d
d
d,
d
0
0
Hc
t EV
HE
V WE
t EV
R
t p Ec mc
= ≡
∂−
=− ≡−
∂−
⋅=
=≡= −
rp
pr
prr
rr
p
p
∇∇
( ) ( )
( )
2
2
,
,2,
RE
WE mR E
= − r
rr
( ) ()
( )
2
22
,
,2,
RE
c
WE ER E
= − r
rr
ResearchGate has not been able to resolve any citations for this publication.
Article
Full-text available
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 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. A numerical treatment not substantially less manageable than its classical counterpart shows, in a number of examples, that each particle "dances a wave-mechanical dance” around its classical trajectory.
Article
Contrary to a wide-spread commonplace, an exact, ray-based treatment holding for any kind of monochromatic wave-like features (such as diffraction and interference) is provided by the structure itself of the Helmholtz equation. This observation allows to dispel - in apparent violation of the Uncertainty Principle - another commonplace, forbidding an exact, trajectory-based approach to Wave Mechanics.
Article
The time-independent Schroedinger and Klein-Gordon equations - as well as any other Helmholtz-like equation - turn out to be associated with exact sets of Hamiltonian ray-trajectories (coupled by a "Wave Potential" function, encoded in their structure itself) describing any kind of wave-like features, such as diffraction and interference. This property suggests to view Wave Mechanics as a direct, causal and realistic, extension of Classical Mechanics, based on exact trajectories and motion laws of point-like particles "piloted" by de Broglie's mono-energetic matter waves and avoiding the probabilistic content and the wave-packets both of the standard Copenhagen interpretation and of Bohm's theory. FULL TEXT AVAILABLE AT http://aflb.ensmp.fr/AFLB-401/aflb401m806.pdf
Article
The usual interpretation of the quantum theory is self-consistent, but it involves an assumption that cannot be tested experimentally, viz., that the most complete possible specification of an individual system is in terms of a wave function that determines only probable results of actual measurement processes. The only way of investigating the truth of this assumption is by trying to find some other interpretation of the quantum theory in terms of at present "hidden" variables, which in principle determine the precise behavior of an individual system, but which are in practice averaged over in measurements of the types that can now be carried out. In this paper and in a subsequent paper, an interpretation of the quantum theory in terms of just such "hidden" variables is suggested. It is shown that as long as the mathematical theory retains its present general form, this suggested interpretation leads to precisely the same results for all physical processes as does the usual interpretation. Nevertheless, the suggested interpretation provides a broader conceptual framework than the usual interpretation, because it makes possible a precise and continuous description of all processes, even at the quantum level. This broader conceptual framework allows more general mathematical formulations of the theory than those allowed by the usual interpretation. Now, the usual mathematical formulation seems to lead to insoluble difficulties when it is extrapolated into the domain of distances of the order of 10-13 cm or less. It is therefore entirely possible that the interpretation suggested here may be needed for the resolution of these difficulties. In any case, the mere possibility of such an interpretation proves that it is not necessary for us to give up a precise, rational, and objective description of individual systems at a quantum level of accuracy.