(zenodo.org)

Transverse and longitudinal wave coupling

1. In classical mechanics, transverse waves can only exist in an elastic medium where there are shear stresses (e.g., solids).
2. Longitudinal waves exist in solids as well as in liquids and gases. They transmit disturbance through compression and rarefaction.

If we talk about a particle as some wave structure in space, we can ask the question:

• If the particle has a wave nature, which wave creates the interference pattern — transverse or longitudinal?

What happens in a single-slit experiment?

• When a particle passes through the slit, its wave function envelopes the obstacle and creates an interference pattern.
• This property is characteristic of all waves, whether they are transverse or longitudinal.

But assuming that the de Broglie wave is inherently longitudinal, this might explain:

1. Why the wave function obeys the Schrödinger equation, which is similar to the equation for acoustic waves.
2. Why a particle experiences wave effects even without the presence of a medium (which is strange for ordinary mechanics).

Is a transverse wave always accompanied by a longitudinal wave?

In mechanics, not always, but often:

• For example, when elastic waves propagate in a solid, longitudinal and transverse components can coexist.
• In the case of medium deformation, compression can induce perpendicular displacements, that is, a longitudinal wave can induce a transverse wave.

How can this be verified?

1. See how diffraction behaves when the width of the slit is changed. If there is a critical width at which the interference abruptly disappears, this may hint at a connection with longitudinal effects.
2. Check the behaviour of particles in media with different densities. If the wavelength varies, this may indicate a longitudinal nature.
3. Try a similar experiment with acoustic waves to see the similarities.

Let us assume that the particles are travelling with a longitudinal wave and not just «smeared» as a wave function. If the wave associated with the particle is longitudinal, then the de Broglie wavelength can indeed be a characteristic of its spatial interaction.

Connection with Anderson Transition

Recent studies have shown that the Anderson transition is observed not only in electronic systems but also in optical media. In particular, light waves passing through a disordered medium can transition into a localized state, ceasing to propagate diffusely. This effect has been experimentally observed in waveguides with irregular structures.

If we draw an analogy between a disordered medium and the narrowing of a diffraction slit, we can assume that chaos, in a certain sense, acts as a dynamic barrier that restricts wave propagation. This means that Anderson localization can be considered a spatial constraint on wave propagation, similar to how diffraction changes when the slit width decreases.

What does this imply?

1. If the de Broglie wave is a spatial wave of energy density, then its behavior in disordered media may follow the same laws as light waves during the Anderson transition.
2. If particles in a chaotic medium start behaving as if their «effective space» has decreased, this may confirm that energy density in space plays a key role in quantum effects.
3. This effect can be tested experimentally by studying wave propagation in systems with varying degrees of disorder and comparing the results with classical diffraction through slits of different widths.

Thus, the Anderson transition may serve as evidence that de Broglie waves are not merely mathematical abstractions but real spatial waves of energy interacting with the medium.

Connection to Light Refraction and Transition Radiation

Let us consider how photons behave when transitioning between media with different densities. It is well known that when light moves from one medium to another, its wavelength changes according to the refractive index, while its frequency remains unchanged. This phenomenon confirms that an electromagnetic wave can alter its spatial characteristics depending on the conditions of the medium in which it propagates.

If de Broglie waves are indeed spatial energy waves, then a similar effect could manifest for massive particles. However, unlike photons, charged particles experience another important process when crossing the boundary between two media with different dielectric properties—transition radiation. This effect occurs when a charged particle moves through an interface between two media, emitting electromagnetic radiation as a result of the change in its environment.

This suggests that a particle does not merely change its wavelength but may also lose part of its energy during the transition, much like photons modify their wavelength during refraction. Thus, transition radiation could serve as a mechanism that allows massive particles to adjust their wavelength depending on the medium, supporting the idea that their wave nature is indeed linked to the spatial characteristics of energy.

This fact further emphasizes that the de Broglie wavelength is not just a mathematical description but a real physical property determined by the distribution of energy in space.

What does this have to do with the Schrödinger equation?

Schrödinger equation for a free particle:

This equation is similar to the equation of longitudinal waves in an elastic medium:

where u is the displacement in the medium, v is the velocity of wave propagation.

If we think of a particle as a localised standing wave in space, then:

• Its de Broglie wavelength can be related to the size of the particle‘s spatial structure.
• As a particle moves, velocity can change its spatial distribution, which results in a change in wavelength.

Why the confusion then?

The confusion may be because it is common to think of de Broglie’s wavelength as an analogue of electromagnetic waves. But in fact, if this wavelength is related to a wave in space, then:

1. It does not have to be transverse.
2. It can depend on the energy density in space, not just the velocity of the particle.
3. It may be related to the «size of the particle» in a spatial sense, rather than its internal structure.

What can be done next?

If we want to confirm or disprove a hypothesis, we can:

• Find analogues of the Schrödinger equation in classical mechanics for longitudinal waves.
• To check how the de Broglie wavelength behaves under changing conditions (e.g. in different potentials).
• To try to relate mathematically the wavelength to the size of the particle in space.

Some calculations

If we make calculations of how many half-waves of de Broglie waves will be placed inside a particle at a known size. For non-relativistic velocities we obtain that for the electron this value is much less than unity, but for the proton and neutron it already approaches 1, but it is still not enough if we consider that the particle should be a standing wave.

In the case of the neutron, the picture is similar to that of the proton, and with its mass slightly higher than that of the proton, the result also becomes closer and closer to unity.

What is just as important is that as the speed decreases, the number of half-waves that can be accommodated decreases dramatically.

This may mean that the proton and neutron has a spatial wave structure closer to its size than the electron. This may be due to its larger mass and shorter de Broglie wavelength.

To consider other particles simply does not make sense. If we consider the hypothesis that all particles are standing waves, then short-lived particles simply do not have time to form as stable standing waves, and hence their behaviour will not correspond to the expected one.

Let’s look at the limit at the speed of light

At the speed of light v= c the relativistic mass goes to infinity and the de Broglie wavelength tends to zero. However, if we consider a limiting approximation, such as 0.999999999999999c, we can get a finite value. Let’s do the calculations.

Results for the proton at a velocity of 0.9999999999999c:

• De Broglie wavelength: 1.32×10^-15 м
• Number of half-waves along the diameter (2r): 2.55

What’s interesting:

• The number of half-waves remains finite. Moreover, in my hypothesis the proton is a standing wave with four nodes, which means that the number of half-waves should be equal to 2.5, which is very close to the calculated value.
• It does not tend to infinity despite approaching the speed of light.
• This may indicate a structural feature of the proton related to wave processes.

If the de Broglie wavelength does describe the spatial structure of a standing wave, then:

1. The wavelength is not the electromagnetic wavelength, but the wave structure of space itself.
2. The vibrational nature of mass can be related to this standing wave.

Results for the neutron at a velocity of 0.999999999999999c:

• De Broglie wavelength: 1.32×10^-15 м
• Number of half-waves along diameter 2r: 2.55

Pretty much the same as the proton

This supports the idea that the de Broglie wave is a spatial standing wave, not an electromagnetic wave.

Results for the electron at a velocity of 0.9999999999999c

• De Broglie wavelength: 2.42×10^-12м
• Number of half-waves along diameter 2r: 0.00465

Unlike the proton and neutron, the number of half-waves for the electron is very small!

How can the different results be interpreted? Why for the proton the result is in perfect agreement with the hypothesis, for the neutron it is also close but still not correct, and for the electron it does not agree with the hypothesis at all? For the electron the number of half-waves, based on my hypothesis, should be equal to 1.5. Why can this deviation be observed?

Here I think, it should be considered that the mass (energy density) will be distributed not uniformly inside the whole volume of the particle, but in the form of a wave. I.e. in fact in case of experiments to determine the size of particles they cannot be considered as point objects. The mass or energy density inside the particle is not uniformly distributed. For an electron, all the mass will be concentrated in a small volume, in its centre part, and the surrounding part will create an electric charge. Therefore, when a particle, in this case the electron, is bombarded by other particles, some of them, due to the increased energy, can fly through the outer shell of the electron, which will create an incorrect idea of its size. Consequently, my calculations will be different from the expected ones.

In the case of a proton the case is different. For the proton, the increased energy density will be observed both in the centre of the particle and along its boundary. This is ideal for obtaining the true size of the particle. This is the reason why the calculated result is so close to the theoretical one.

The neutron differs from the proton by one half-wavelength and has an internal rotation, which can distort the resulting size due to centripetal force. Therefore, when determining the radius of the neutron, the centripetal force can introduce distortions. The experimental data may be ambiguous.

Then it turns out that the measured dimensions are not the particles themselves, but their «energy frame»!

So, perhaps the de Broglie wave describes exactly the structure of space that forms these standing waves.

This approach offers a new perspective on the nature of mass and its relationship to electromagnetic processes. A more detailed discussion of this hypothesis and its philosophical implications can be found in the following works:

— (dzen)

— (Zenodo)