The true nature of matter waves or de Broglie waves. Unsolved and misinterpreted. Now the nature is found and verified mathematically.

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MATERIAL PARTICLES
Wave-Particle Nature of
new-physics.com

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Wave Particle Duality
of Light
After fighting over hundreds of
years, the issue of weather light is
a particle or a wave is finally
settled, or rather, compromised. It
is now generally accepted that
light is a particle as well as a
wave. It is a hybrid called a wave-
particle.
Some clever people coined the
word ‘wavicle’ for a particle that
is wave and at the same time as
particle. They are like the mythical
hybrids of ancient Greece.

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Light imitates material
particles and become
wavicles. But what about
the material particles
themselves?
There are many micro-
particles such as electrons,
neutrons, and protons and
even quarks and gluons. Do
they behave like waves as
well?
Material Particles?

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Matter Waves
Photons are different from
matter particles in that they
are singularly characterized.
Matter particles are
traditionally believed to be
solid, localized and unwavy
– even when they are in
motion.
The idea of matter particles
that waves like a photon
was not known until 1924.
Photon = Particle + Wave
Matter particles and objects

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de Broglie Hypothesis
In 1924, a young French
nobleman named Louis de
Broglie submitted his doctoral
thesis to the University of Paris
under the title “Research in the
Quantum Theory”[1].
It contained a new message that
seemed to have the potential to
completely revolutionize the
classical view of matter.
[1] Louis de Broglie: ‘Recherches sur la théorie des
quanta (Researches on the quantum theory)’ Thesis
(Paris), 1924; L. de Broglie, Ann. Phys. (Paris) 3, 22
(1925).
Louis de Broglie (1892-1987) ⦊

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Particle Waves
Based on the work of Max
Planck (1858-1947)[2] and
Albert Einstein (1879-1955)[3],
de Broglie proposed in his
thesis that material particles
such as electrons should have
both wave and particle
properties, just like photons.
[2] Max Planck: Energy of radiation [Link]
[3] Albert Einstein: Relativistic energy [Link]
Photon = Particle + Wave
Matter particles

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Early Days of the Theory
His thesis was so original that it was
easily recognized. But there was not
any experimental evidence of such a
kind of wave at the time. So this new
idea of matter-wave was not
considered to have any physical
reality.
However, Paul Langevin (1872-1946)
drew the attention of Albert Einstein
(1879-1955) to the matter.
Einstein immediately recognized its
significance and promoted it to the
attention of other physicists.
Good
stuff!
[Because it
came from my
formula!]

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Nobel Prize Award
Three years later in 1927, de
Broglie’s idea was confirmed
by experiments and he
received the Nobel Prize for
his discovery of the wave
nature of electrons in 1929.
This made him the first
person to receive a Nobel
Prize on a PhD thesis.
Nobel Prize Medal

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Debut of Matter Waves
de Broglie said in 1929:
“We thus find that in order to
describe the properties of Matter,
as well as those of Light, we must
employ waves and corpuscles
simultaneously. We can no longer
imagine the electron as being just
a minute corpuscle of electricity:
we must associate a wave with
it.” [4].
[4] Louis de Broglie: Nobel prize speech 1929.
Classical picture of
an electron in
motion
Matter wave picture of
an electron particle in
motion

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Significance
With this new discovery, de
Broglie opened the gateway to
the development of wave and
quantum mechanics in the early
19th century. Although in the
later stage of development, the
precision descriptions of a
particle in motion no longer
prevailed and gave way to
probabilistic interpretations, the
discovery of de Broglie
remained a historic landmark in
physics.
Picture source: Wikipedia

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de Broglie’s
WAVE EQUATION

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Determining the Matter
Wave Equation
To determine the wavelength
of the wavy electron, de
Broglie made use of the
relations between the energy
𝐸𝐸, the velocity of light 𝑐𝑐, the
momentum 𝑝𝑝 and the
frequency 𝑓𝑓 of a photon or
particle established by Planck
and Einstein at the time.
To start with, de Broglie first
employed Einstein’s
relativistic energy equation.
Light 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = 𝑐𝑐
Light f𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 = 𝑓𝑓
𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 = 𝑚𝑚
𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑜𝑜𝑜𝑜 𝑎𝑎 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

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Classical Momentum
In classical mechanics, the momentum 𝑝𝑝𝑝𝑝 of a particle is equal to the product of its
mass 𝑚𝑚𝑝𝑝 and velocity 𝑣𝑣𝑝𝑝, or 𝑝𝑝𝑝𝑝 = 𝑚𝑚𝑝𝑝 𝑣𝑣𝑝𝑝. If the speed is so high as close to the
speed of light 𝑐𝑐 (relativistic speed), its momentum will be governed by Einstein’s
relativistic equation.
𝑣𝑣𝑝𝑝 ≪ 𝑐𝑐 𝑣𝑣𝑝𝑝 ≈ 𝑐𝑐
Classical Newtonian Einsteinan
Your need to use
my equations
Velocity of particle Velocity of light

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Einstein’s Energy Equation
Einstein’s equation for the energy
𝐸𝐸𝑟𝑟𝑟𝑟𝑟𝑟 of a particle at high speed is
written as:
𝐸𝐸𝑟𝑟𝑟𝑟𝑟𝑟
2
= 𝑝𝑝2
𝑐𝑐2
+ (𝑚𝑚𝑜𝑜 𝑐𝑐2
)2
Taking the square roots on both
sides, we have:
𝐸𝐸𝑟𝑟𝑟𝑟𝑟𝑟 = 𝑝𝑝2 𝑐𝑐2 + (𝑚𝑚𝑜𝑜 𝑐𝑐2)2
At the same time, Einstein's theory of
relativity pointed out that for a particle
like a photon of zero rest mass 𝑚𝑚𝑜𝑜 = 0.
So we can neglect the (𝑚𝑚𝑜𝑜 𝑐𝑐2
)2
term
and the relativistic energy becomes:
𝐸𝐸𝑟𝑟𝑟𝑟𝑟𝑟 = 𝑝𝑝2 𝑐𝑐2 + (𝑚𝑚𝑜𝑜 𝑐𝑐2)2
= 𝑝𝑝2 𝑐𝑐2 = 𝑝𝑝𝑝𝑝

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Planck’s Equation
On the other hand, according to Planck,
the energy 𝐸𝐸γ of a photon is related to
its frequency 𝑓𝑓𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 and Planck’s
constant ℎ by the famous Planck’s
equation:
𝐸𝐸γ = ℎ𝑓𝑓γ
where ℎ is Planck's constant; 𝑓𝑓𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 is
the frequency of the radiation or
photon.
𝑓𝑓γ
Photon
frequency
Gamma - symbol
for photon h – Planck’s constant

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Speed & Wavelength
In radiation (light), the
frequency 𝑓𝑓𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 of a
photon is related to its
velocity 𝑐𝑐 and wave length 𝜆𝜆
by:
𝑓𝑓𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 =
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤
=
𝑐𝑐
λ
So in terms of λ, the Planck’s
energy relationship can be
written as:
𝐸𝐸𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = ℎ𝑓𝑓 = ℎ 𝑐𝑐/λ
Or:
λ𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 𝑐𝑐/𝑓𝑓
𝐸𝐸𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = ℎ𝑐𝑐/λ
λ
c
λ𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 𝑐𝑐/𝑓𝑓𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

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Planck + Einstein
Linking up Planck’s formulae with
Einstein’s energy equation, de
Broglie had:
𝐸𝐸 = ℎ𝑓𝑓 = 𝑝𝑝𝑝𝑝
ℎ𝑓𝑓 = 𝑝𝑝𝑝𝑝
or:
𝑝𝑝𝑝𝑝 = ℎ𝑓𝑓
That is: Planck’s frequency energy
= Einstein’s relativistic energy
Kinetic energy
of photon
Frequency
energy of photon

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Wavelength and
Momentum
By manipulating the equation
a little bit in moving the
terms on both sides, we have
a new equation which finally
becomes:
𝜆𝜆 = ℎ/𝑝𝑝
As seen in previous page
𝑐𝑐/𝑓𝑓 = 𝜆𝜆.
𝑝𝑝 𝑐𝑐 = ℎ𝑓𝑓
𝑐𝑐/𝑓𝑓 = ℎ/𝑝𝑝
𝜆𝜆 = ℎ/𝑝𝑝
Swap side
Swap side

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De Broglie Hypothesis
At this point, de Broglie made an
ingenious intuitive guess that if the
electron is also a wave particle, its
formulae should also be like that of a
photon wave. That is, the same formula
works also for the electron:
𝜆𝜆𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 =
ℎ
𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
𝜆𝜆𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 =
ℎ
𝑝𝑝𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
Photon
wave
Electron
wave

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de Broglie equation
This relation between the wavelength
and the momentum of the electron
later became known as the famous de
Broglie equation. 𝜆𝜆𝑒𝑒 is called the de
Broglie wavelength of the electron:
𝜆𝜆𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 =
ℎ
𝑝𝑝𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
So the particle bursts open and
becomes a wave-particle. It is an
assumption that if an electron is free, it
would behave like a photon.

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Exercise 01 - The Wavelength of an Electron
Find the de Broglie wavelength of an
electron (𝑚𝑚 = 9.11 × 10−31 𝑘𝑘𝑘𝑘)
moving at 2 × 106 m/s.
The de Broglie wave equation is:
𝜆𝜆 =
ℎ
𝑚𝑚𝑚𝑚
𝜆𝜆 =
6.63 × 10−34 𝐽𝐽 � 𝑠𝑠
9.11 × 10−31 𝑘𝑘𝑘𝑘 × 2 × 106 𝑚𝑚/𝑠𝑠
= 3.639 × 10−10 𝑚𝑚
Compared with the classical electron
radius which is about 2.8179×10−15 m,
this is a relatively large wave length.

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Exercise 02 - The Wavelength of a Baseball
A baseball with a mass of 0.15 kg is
pitched at 45 m/s What is its De
Broglie wavelength?
𝜆𝜆 =
ℎ
𝑚𝑚𝑚𝑚
=
6.63 × 10−34
𝐽𝐽 � 𝑠𝑠
0.15𝑘𝑘𝑘𝑘 × 45𝑚𝑚/𝑠𝑠
= 9.8 × 10−35
Diffraction effects of a baseball are
negligible.
This is an incredibly small figure
compare with the size of the ball.
However this is a wrong example,
as we shall see later.