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# De Broglie wavelength

**De Broglie wavelength** is a wavelength, which is manifested in
all the particles in quantum mechanics, according to wave-particle duality, and
it determines the probability density of finding the object at a given point of
the configuration space. The de Broglie wavelength is inversely proportional to
the particle momentum.

## Contents

- 1 Definition
- 2 Derivation of the formula for
the de Broglie wavelength

- 2.1 Waves inside
the particles
- 2.2 Other models

- 3 References
- 4 External links

## Definition

In 1924 a French physicist Louis
de Broglie assumed that for particles the same relations are valid as for the
photon: ^{[1]}

where and are the energy and momentum of the photon, and are the frequency and wavelength of the
photon, is the Planck constant, is
the speed of light.

From this we obtain the
definition of the de Broglie wavelength through the Planck constant and the
relativistic momentum of the particle:

Unlike photons, which always move
at the same velocity, which is equal to the speed of light, the momenta of the
particles according to the special relativity depend on the mass and velocity by the formula:

## Derivation of the formula for the de Broglie wavelength

There are several explanations
for the fact that in experiments with particles de Broglie wavelength is
manifested. However, not all these explanations can be represented in
mathematical form, or they do not provide a physical mechanism, justifying
formula (1).

### Waves
inside the particles

When particles are excited by
other particles in the course of experiment or during collision of particles
with measuring instruments, internal standing waves can occur in the particles.
They can be electromagnetic waves or waves associated with the strong
interaction of particles, with strong
gravitation in the gravitational
model of strong interaction, etc. With the help of Lorentz transformations
we can translate the wavelength of these internal oscillations into the
wavelength detected by an external observer, conducting the experiment with
moving particles. The calculation provides the formula for the de Broglie
wavelength, ^{[2]} ^{[3]}
^{[4]} as well as the propagation speed of the de
Broglie wavelength:

where is
the period of oscillation of the de Broglie wavelength.

Thus, we determine the main
features associated with the wave-particle duality – if the energy of internal
standing waves in the particles reaches the rest energy of these particles,
then the de Broglie wavelength is calculated in the same way as the wavelength
of photons at a corresponding momentum. If the energy of
excited particles is less than the rest energy , then the wavelength is given by the
formula:

where is
the momentum of the mass-energy, which is associated with the internal standing
waves and moves with the particle at velocity .

It is obvious that in the
experiments the de Broglie wavelength (1) is mainly manifested as the boundary
and the lowest value for the wavelength (2). At the same time, experiments with
a set of particles cannot give an unambiguous value of the wavelength according to formula (2) – if excitation
energies of the particles are not controlled and vary for different particles,
the range of values will be too large. The higher the
energies of interactions and of particles’ excitation are, the closer they will
be to the rest energy, and the closer the wavelength will be to the . Light particles, like electrons, achieve
more rapidly the velocity of the order of the speed of light, become
relativistic and at low energies demonstrate quantum and wave properties.

Besides the de Broglie
wavelength, Lorentz transformations give another wavelength and its period:

This wavelength is subject to
Lorentz contraction as compared to the wavelength in
the reference frame associated with the particle. In addition, this wave has a
propagation speed equal to the velocity of the particle. In the limiting case,
when the excitation energy of the particle is equal to the rest energy, , for the wavelength we have the
following:

The obtained wavelength is
nothing but the Compton wavelength in the Compton effect
with correction for the Lorentz factor.

In the described picture the
appearance of a de Broglie wave and the wave-particle duality are interpreted
as a purely relativistic effect, arising as a consequence of the Lorentz
transformation of the standing wave moving with the particle. Moreover, since
the de Broglie wavelength behaves like the photon wavelength with corresponding
momentum, which unites particles and waves, de Broglie wavelengths are
considered probability waves associated with the wave function. In quantum
mechanics it is assumed that the squared amplitude of the wave function at a
given point in the coordinate representation determines the probability density
of finding the particle at this point.

The electromagnetic potential of
particles decreases in inverse proportion to the square of the distance from
the particle to the observation point, the potential of strong interaction in
the gravitational model of strong
interaction behaves the same way. When internal oscillations start in the
particle, the field potential around the particle starts oscillating too, and
consequently the amplitude of the de Broglie wavelength is growing rapidly
while approaching the particle. This corresponds precisely to the fact that the
particle most likely is at the place, where the amplitude of its wave function is the greatest. This is true for a pure state, for
example, for a single particle. But in a mixed state, when the wave functions
of several interacting particles are taken into consideration, the
interpretation that connects the wave functions and probabilities becomes less
accurate. In this case, the wave function would more likely reflect the total
amplitude of the combined de Broglie wave, associated with the total amplitude
of the combined wave field of the particles’ potentials.

Lorentz transformations to determine the de Broglie wavelength were used
also in the article. ^{[5]}

Explanation of the de Broglie wave through the standing waves inside the
particles is also described in the article. ^{[6]} In addition, in the article ^{[7]} is assumed that
inside a particle there is a rotary electromagnetic wave. According to
conclusion in the article, ^{[8]} outside the moving particle should be the De
Broglie wave with amplitude modulation.

### Electrons
in atoms

The motion of electrons in atoms
occurs by means of rotation around the atomic nuclei. In the substantial model the electrons have the form of
disk-shaped clouds. This is the result of the action of four approximately
equal by magnitude forces, which arise from: 1) attraction of the electron to
the nucleus due to strong gravitation and Coulomb attraction of the charges
of electron and nucleus, 2) repulsion of the charged electron matter from
itself, and 3) runaway of the electron matter from the nucleus due to rotation,
which is described by the centripetal force.

In the hydrogen atom the electron in the state with
the minimum energy can be modelled by a rotating disk, the inner edge of which
has the radius and the outer edge has the radius , where is the Bohr radius. ^{[3]}

If we assume that the electron’s
orbit in the atom includes of de Broglie
wavelengths, then in case of a circular orbit with the radius
, for the circle
perimeter and the angular momentum of the electron we will obtain the following:

This corresponds to the postulate
of the Bohr model, according to
which the angular momentum of the hydrogen atom is quantized and proportional
to the number of the orbit and the Planck constant.

However, the excitation energy in
the matter of electrons in atoms on the stationary orbits normally does not
equal the rest energy of the electrons as such, and therefore the spatial
quantization of the de Broglie wave along the orbit in the form (3) should be
explained in some other way. In particular it was shown that on the stationary
orbits in the electron matter distributed over the space the equality holds of
the kinetic matter energy flux and the sum of energy fluxes from the
electromagnetic field and field of the strong gravitation. ^{[3]}

In this case the field energy
fluxes do not slow down or rotate the electron matter. This causes the
equilibrium circular and elliptical orbits of the electron in the atom. It
turns out that the angular momenta are quantized proportionally to the Planck
constant, which leads in the first approximation to relation (3).

Besides, in transitions from one
orbit to another, which is closer to the nucleus, the electrons emit photons,
which carry the energy and the angular momentum away from the atom.

For a photon the wave-particle duality is reduced to
the direct relation between these quantities, and their ratio is equal to the average angular frequency of the photon wave and at the
same time to the average angular velocity of the electron , which under
corresponding conditions emits the photon in the atom during its rotation.

If we assume that for each photon , where is the Dirac constant,
then for the photon energy we obtain: . In this case, during the atomic transitions
the electron’s angular momentum also changes with , and the formula (3)
should hold for the angular momentum quantization in the hydrogen atom.

In the electron’s transition from
one stationary state to another, the annular flux of the kinetic energy and the
internal field fluxes change inside its matter, as well as their momenta and
energies. At the same time, the electron energy in the nuclear field changes,
the photon energy is emitted, the electron momentum increases and the de
Broglie wavelength decreases in (3). Thus, emission of the photon as the
electromagnetic field quantum from the atom is accompanied by changing of the
field energy fluxes in the electron matter, both processes are associated with
the field energies and with the change of the electron’s angular momentum,
which is proportional to . From (3) it seems
that on the electron orbit de Broglie wavelengths can be located. But at the same time the
electron’s excitation energy does not reach its rest energy, as it is required
to describe the de Broglie wavelength in the forward motion of the particles.
Instead, we obtain the relationship between the angular momentum and energy
fluxes in the electron matter in stationary states and the change of these
angular momenta and fluxes during emission of photons.

### Other models

## References

- L. 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). - Fedosin S.G. Fizika i filosofiia
podobiia ot
preonov do metagalaktik,
Perm, pages 544, 1999. ISBN 5-8131-0012-1.
- Sergey Fedosin, The
physical theories and infinite hierarchical nesting of matter, Volume 1, LAP LAMBERT Academic
Publishing, pages: 580, ISBN-13: 978-3-659-57301-9.
- Fedosin S.G. The
radius of the proton in the self-consistent model. Hadronic Journal,
2012, Vol. 35, No. 4, P. 349 – 363.
- Masanori Sato and
Hiroki Sato. Interpretation of De Broglie Waves: At What Time
Does a Massive Particle Obtain the Properties of a Wave?
Physics Essays. 2012, Vol. 25, P. 150-156.
- J. X. Zheng-Johansson
and Per-Ivar Johansson. Developing
de Broglie Wave . Progress in physics. 2006, Vol. 4, P.32-35.
- Malik Mohammad Asif. de Broglie wave and electromagnetic travelling wave
model of electron and other charged particles. Physics Essays.
2014, Vol. 27, P. 146-164.
- J. Domínguez-Montes and
E. L. Eisman, Representative
model of particle–wave duality and entanglement based on De Broglie's
interpretation. Physics Essays. 2012, Vol. 25, P. 215-220.

## External links

**Lessons**

Princ**i**ples of Quantum Mechanics**·** Postulates
of Quantum Mechanics**·** Quantum Mechanics I

Source: http://sergf.ru/wlen.htm