The Wave Momentum

Let’s observe in an ideal experiment an electron, considered as isolated in a space without fields, giving out some spherical elementary waves having the characteristic wavelenght for the le rest electron. In Fig.1 the energetic-wave distribution has all around a spherical symmetry.

Fig. 4

• From the infinite a plane wave train-energy photon E_i = h l _i. approaches. The energy of the photon in the A zone is superimposed to the energy of the wave field of the electron (a part of the spherical waves can be considered as plane and parallel to the plane waves of the photon).

• In that zone A two energies coexist at the same time: E_e + E_i.

• The energetic distribution loses its symmetry all around the electron, when an anisotropic variation of energy happens around it.

Let’s consider that a mechanism tending to restore the isotropy of the energetic variation starts at this moment.

• The electron-source of waves reacts moving on the opposite direction to the one where the previous variation of energy had happened.

• The motion of the electron varies the wavelenght given out in the direction of the l _e1,motion.

• It makes it shorter, while it lengthens the wavelenght l _e2, of the waves given out in the opposite direction to the  motion, in compliance with the relativistic Doppler effect.

• When the particle is speed enough to allow the wave density, which is in front, to be equal to the density, which is at the back, the symmetry of the wave variation is reestablished and the wave state remains unchanged, until the arrival of another energetic variation.

It is possible to describe the wave-numbers situation of wave-source-particle and the wave-train-photon.

12)

Here is represented the symmetry to which the disposition of the wave numbers around the particle has to tend:

13)

Replacing the wave numbers of the particle with the equivalent values according to the relativistic Doppler effect:

14)

In order to express the energetic content of the waves, we need to multiply both terms by h, (the energy quantum) and we obtain the relative relativistic momentum:

15)

The photon transfers half of its momentum to the particle (in the following pages, in the “Wave analysis of the
Compton effect”, you will see this is only the first part of the diffraction phenomenon between photon and particle):

16)

This is the real wave mechanism to transfer causally the energy of a photon to a particle.