De Broglie wavelength

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

Definition
In 1924 a French physicist Louis de Broglie assumed that for particles the same relations are valid as for the photon:
 * $$~ E= h \nu, \qquad $$ $$~c=  \lambda \nu , \qquad $$   $$~ E= \frac {hc} {\lambda} = pc, \qquad $$

where $$~ E$$ and $$~ p$$ are the energy and momentum of the photon, $$~ \nu $$ and $$~ \lambda $$ are the frequency and wavelength of the photon, $$~  h $$ is the Planck constant, $$~  c $$ 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:
 * $$~\lambda_B = \frac {h} {p}. \qquad \qquad (1) $$

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 $$~ m $$ and velocity $$~  v $$ by the formula:
 * $$~ p = \frac {mv} {\sqrt {1-v^2/c^2} }. $$

A simplified equation for the de Broglie wavelength, accurate for speeds much less than c, is so:
 * $$~ \lambda = \frac {h} {mv}. $$

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 the experiment or during the 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, as well as the propagation speed of the de Broglie wavelength:


 * $$~ c_B = \frac { \lambda_B} {T_B }= \frac {c^2}{v}, $$

where $$~ T_B $$  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 $$~ E_e $$ of excited particles is less than the rest energy $$~ mc^2 $$,  then the wavelength is given by the formula:


 * $$~\lambda_2 = \frac {h c^2 \sqrt {1-v^2/c^2} } {E_e v}= \frac {h} {p_ e } \geqslant \lambda_B, \qquad \qquad (2) $$

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

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 $$~\lambda_2 $$ 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 $$~\lambda_2 $$ will be to the $$~\lambda_ B $$. 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:
 * $$~\lambda_1 = \frac {h c \sqrt {1-v^2/c^2} } {E_e }= \frac {h v } { c p_ e }= \frac {\lambda_2 v}{c} = \lambda' \sqrt {1-v^2/c^2},$$
 * $$~ T_1 = \frac {\lambda_1} {v}.$$

This wavelength is subject to Lorentz contraction as compared to the wavelength $$~\lambda' $$ 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, $$~ E_e = mc^2 $$, for the wavelength we have the following:


 * $$~\lambda_{1f} = \frac {h \sqrt {1-v^2/c^2} } { mc }.$$

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 the 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 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.

Derivation of the de Broglie phase wave through the standing (Doppler shifted) electromagnetic waves inside the particles is described in the article. In addition, in the article is assumed that inside a particle there is a rotary electromagnetic wave. According to the conclusion in the article, 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 modeled by a rotating disk, the inner edge of which has the radius $$~ \frac {1}{2} r_B $$ and the outer edge has the radius $$~ \frac {3}{2} r_B $$, where $$~ r_B $$ is the Bohr radius.

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


 * $$~ 2 \pi r = n \lambda_B, \qquad L= r p= \frac { n h }{2 \pi }, \qquad \lambda_B = \frac {h}{p}.\qquad (3) $$

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 $$~ n $$ 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.

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 $$~ \Delta W$$ and the angular momentum $$~ \Delta L $$ away from the atom. For a photon the wave-particle duality is reduced to the direct relation between these quantities, and their ratio $$~\Delta W / \Delta L $$ is equal to the average angular frequency of the photon wave and at the same time to the average angular velocity of the electron $$~ \omega$$, which under corresponding conditions emits the photon in the atom during its rotation. If we assume that for each photon $$~ \Delta L =\frac { h }{2 \pi }= \hbar$$, where $$~ \hbar$$ is the Planck constant, then for the photon energy we obtain: $$~ W = \hbar \omega $$. In this case, during the atomic transitions the electron’s angular momentum also changes with $$~ \Delta L =\hbar $$, 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 $$~ \hbar $$. From (3) it seems that on the electron orbit $$~ n $$ 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.

If any type of ray has the rest mass as zero it will not have de broglie wavelength as de broglie wavelength is associated with the mass of particles