Question on the nature of the particle wave. de Broglie and other scientists struggled to find out but failed. Ended in the probability interpretation.

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WHAT IS THERE WAVING?
Matter-Waves [003] : If an electron is a wave

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Notice
This section of slides is for
continuation only.
It is boring and brings no
positive results.
It is here only to show how
the pioneer scientists
struggled with the search
for answers which eluded
them.
You may skip this section
and go to the next one.

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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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THE CARRIER THEORY
To be continued on : Matter-Waves [004]
ABCC