The nature of particle waves or de Broglie waves. Long forgotten and misinterpreted. Now true nature found and mathematically verified. More details: new-physics.com

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WHAT IS THERE WAVING?
If an electron is a wave
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Beginning of Matter
Waves
When de Broglie first came up with the
idea of matter waves, he was not able
to pin point what actually waves. The
idea came to him when he saw the
probability of as an analogy between
electrons and photons. Photon as a
wave-particle was quite well
established; but matter wave at that
moment was more a mathematical
construct than a reality since electron
wave was not yet known. However, it
happened that the idea turned out
unexpectedly to be very helpful and so
he carried on with it.
Photon wave
Electron wave

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Wave Nature not Considered in Beginning
When de Broglie was working on his thesis, he was not overly concerned with the
nature of the wave. At the beginning, he was only trying to find a theory to
compromise their coexistence of both wave and particle properties in the photon.
He regarded the coexistences as a curious kind of dualism that may be intrinsic in
the nature of things:
“When in 1922-1923, I had my first ideas about wave mechanics, I was guided by
the vision of constructing a true physical synthesis, resting upon precise concepts,
of the coexistence of waves and particles. I never questioned then the nature of the
physical reality of waves and particles.”*
When the particle wave concept became a celebrated theory, de Broglie began to
feel the need to investigate into its physical reality.

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Electromagnetic Nature of Light
The classical electromagnetic theory of Maxwell provided a physical basis to the
nature of light. It is simply the oscillations of the electric and magnetic fields.
Although no further explanation was given to the nature of the fields, the classical
model of the photon envisages a wave propagating in the 𝒛𝒛 direction and the
electromagnetic fields (𝐸𝐸 and 𝐵𝐵)* waving in the direction transverse to the
propagation. They are simply the oscillations of the electric and magnetic fields,
although no further explanation was given to the nature of the fields. Matter
wave presents a more mysterious shroud over its nature.
B
E
P
Magnetic field Electric field
Poynting vector
Direction of
Poynting vectorMagnetic field

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Plane Matter Waves
Matter wave presents a more mysterious shroud over its nature.
At first, de Broglie thought that these waves were sinusoidal and plane in
nature with their fronts perpendicular to the particle's direction of
propagation, just like plane electromagnetic waves. However, Broglie later
realized that a plane monochromatic wave is but an idealization which is not
physically viable.

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Pilot Waves
According to de Broglie, all particles were accompanied by actual physical waves
which acted like a pilot guiding the particle along its trajectories. The wave is
physically real and occupies a certain region in space while the particle is a
material point having a certain position in the wave. He called them the pilot
waves. He believed that these distinctive assignments to both wave and particle
are in closest accord with classical concepts of waves and particles.

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Probability Waves
de Broglie also incorporated the probability
element advocated by Born into this
interpretation in that the probability of
finding the particle is proportional to the
intensity of the wave at the point.
In the classical picture, when the particle
wave incidents on a boundary between two
media, it splits into a reflected wave and a
refracted wave. The probabilities of the
particle in these two opposite waves are
determined by the amplitude of these
waves. Thus the difficulty of having one
particle partially reflected and partially
refracted is lifted.
Incident wave Reflected wave
Refracted wave
%
%

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Probability Wave too fast
However, such waves at
times will be travelling at
speed greater than that of
light. This is taboo in the
theory of relativity. At the
same time, prediction of
particle energy in
bichromatic waves basing
upon this hypothesis did not
agree with experiment. As a
result, de Broglie had to give
up the interpretation.

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Mathematical Analysis
de Broglie also tried to break
down a wave into complex
waves represented by Fourier
integrals - forming a wave by
the superposition of a number
of component waves.
In essence, the wave was a
physical wave of very weak
amplitude whose essential
role was to guide the motion
of the particle. This
interpretation was untenable
and was subsequently
discarded as well.
𝑓𝑓 𝑥𝑥 =
𝑎𝑎𝑜𝑜
2
+ �
𝑛𝑛=1
∞
𝑎𝑎𝑛𝑛 𝑐𝑐𝑐𝑐𝑐𝑐 𝑛𝑛𝑛𝑛 + �
𝑛𝑛=1
∞
𝑏𝑏𝑛𝑛 𝑠𝑠𝑠𝑠𝑠𝑠 𝑛𝑛𝑛𝑛
Fourier analysis

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Mathematical Wave
At a certain stage, de Broglie thought that
matter was purely made of waves which
were the only reality in nature.
In order to explain such a reality, he came
up with a vague theory of mathematical
structure. According to him, a particle is a
localized concentration of energy in the
form of waves with extremely short
wavelengths. In mathematical terms, a
particle is represented by a point-
singularity in the wave field. This kind of
singularity was non-physical in nature. But
no further light was shed on the term
except some mathematical manipulations.

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Wave function
In modern quantum mechanics,
the reality of the de Broglie
wave has undergone
tremendous changes.
The wave is no longer real. A
“wave” isn't what is normally
imagined as something that
moves up and down and moves
in one direction, like ripples in
water. It's just a function that
evolves with time and has a
different value at different point
in space.

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Wave function 𝜓𝜓
The familiar wave is replaced by a
mathematical function called the wave
function 𝜓𝜓 (psi).
This wave does not "exist" per se in
physical space. It can be drawn
(superimposed) on physical space, but
that just means that it has a value at
every point there. The absolute value of
the function is the squared |𝜓𝜓(𝑥𝑥)|2 of the
wave function. It gives the probability
density of finding the particle in a given
location. Here, it is the wave function is
waving and what it waves is probability,
not a physical entity.

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Is the matter-wave an
extended object?
Some scientists tried to think of the electron as
an extended object. An electron may be
considered as a collection of millions of fragments
instead of a single integrated particle. It spreads
out as a hump and there is the powder of an
electron at every point.
In such a picture there is no electron-particle.
What one observes is only the fraction
corresponds to the probability of finding the
electron there. The denser are the powdery parts,
the more likely is the electron found. The
fractions behaves like an electron because they
clump together the minute one tries to make an
observation. So it is meaningless in asking what is
it that is waving in the electron. An electron is an
extended object. In the field of an atom, the
orbital electrons extended smoothly like clouds
round the nucleus.

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Probability Density
Some other scientists would
support the idea by saying
that the product of the
charge −𝑒𝑒 and the
probability density |𝜓𝜓(𝑥𝑥)|2
can be interpreted as a
charge density.
This is due to the motion of
the electron in an atom. It
moves so fast that the forces
they exert on other charges
are essentially equivalent to
the forces exerted by a
charge distribution
prescribed by −𝑒𝑒|𝜓𝜓(𝑥𝑥)|2
.

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Is the matter-wave an extended object?
The idea of the smeared out electron is but a murky
transition of a single particle to a collection of fractional
particles. Though it is an intuitive attempt to explain the
nature of the quantum wave, the idea of an electron as a
smeared object or a charge distribution was met with
much objections. Firstly because this form of the electron
is different from the traditional form. Secondly the Charge
density is only valid in the presence of large number of
charged particles. An electron is an electron, not a
collection of smaller particles.
The renowned physicist Richard Feynman strongly
protested: “the wave function of an electron in an atom
does not, then, describe a smeared-out electron with a
smooth charge density. The electron is either here, or
there, or somewhere else, but wherever it is, it is a point
charge”.

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Wave Packets
Erwin Schrӧdinger (1877-1961) also worked on the idea that
the de Broglie wave was formed by the superposition of
several waves. His adeptness in mathematics enabled him to
put his findings in complicated and abstract mathematical
forms, among which the famous Schrӧdinger’s equation was
one of the sublime examples.
He came up with the notion that these waves worked well
with the fictitious wave function 𝜓𝜓 which propagated in a
fictitious space.
Schroëdinger suggested that a particle was only a wave
packet (Wellenpaket) of de Broglie waves. The wave packet
assumed a well-defined locality in space and time. It is
therefore an ideal candidate to represent highly localized
matter. What is more, its group velocity coincided perfectly
with the trajectory of the particle. This eliminated the
dilemma that the individual waves may travel faster than the
particle itself.
𝝍𝝍

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Quantum Mechanics
Some physicists found the reality of the wave
packet unacceptable. For one thing, such a
group would be destroyed by dispersion during
diffraction experiments, so that the particle
would no longer be found in the scattered
beams. A typical example is found in the
refraction and reflection of a matter wave
incident on a boundary between two media. It
is extremely hard to accept that both the
refracted and reflected wave group still
represent the one and only original electron.
For another, the wave group spreads out in
time. It cannot therefore represent a particle in
the aspect of stable existence.

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End of de Broglie Wave
In quantum mechanics (QM), the de Broglie wave has become a mathematical construct. It is
probability and not anything physical that is waving. So it can be said that the quest for the
nature of de Broglie wave meets its end here. It is no longer of any physical meaning to ask
the question: “What is it waving?” As a consequence, the original matter waves gradually lost
much of their physical attributes and became grossly fictitious. The new wave idea turned out
to be an abstract theory constructed over a purely mathematical substructure. The corpuscle
itself becomes a term represented by symbols and abstract notions, representing a quantum
world that is so contrary to conventional perception.

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What is waving there?
de Broglie spent a lot of time much time after
his formulation of matter-waves. His efforts
went without much success and this dilemma
stayed unsolved ever since. So in spite of all
the successful experimental verifications of
the existence of the de Broglie waves and its
applications, the question remains unsolved
from 1929 to the present time:
What exactly is in existence in a matter
wave?
- a mathematical symbolism?
- a particle?
- a wave?

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de Broglie’s wave
WHERE DID IT GO WRONG?

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de Broglie equation
The relation between the wavelength
and the momentum of a particle is
the core essence of the ground
breaking de Broglie equation.
In the equation, 𝜆𝜆 is called the de
Broglie wavelength of the particle, 𝑝𝑝
is its momentum, and ℎ is Planck’s
constant:
𝜆𝜆 =
ℎ
𝑝𝑝
Wavelength
of particle
Planck’s
constant
Momentum of
the particle
𝜆𝜆 =
ℎ
𝑝𝑝

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de Broglie’s Problem
However in spite of all the efforts in
the rest of his life, de Broglie could
not figure out what is there waving.
Instead, it ends up in the hands of
other physicists as probability wave
without any physical substance.
Probability became the
fundamental essence of modern
quantum mechanics. The earlier
particle-wave picture is no longer
seriously considered.
This is why de Broglie was so upset
with the later versions of quantum
mechanics.

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Where did it go
wrong?
In the earlier days of the
discovery, electron was taken
as the typical particle that
exhibited the particle-wave
phenomenon. So in our
discussion, we concentrate
on the case of the electron:
𝜆𝜆 =
ℎ
𝑝𝑝
𝜆𝜆𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 =
ℎ
𝑝𝑝𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
Wavelength
of electron
Planck’s
constant
Momentum of
the electron
Electron
𝒆𝒆−