In 1922, Louis de Broglie postulated that any particle or object has an associated wave. This postulate is one of the principles of quantum mechanics.
 
About the observed seemingly continuous path of electron in a cloud chamber Heisenberg states “perhaps we merely see a series of discrete and ill-defined spots through which the electron has passed.”⁸ He meant that in larger scale continuity of the path disappears.

 

This model supposes that along the pilot wave, objects decelerate as they leave the x-axis and reach their linear velocity at the peak. Then they accelerate on their way back to x-axis, where they reach the speed of light and disappear. Please note that in this model we are denying the presence of mass in a portion of particle’s wavelength. However, while the object is present, the speed variance dictates the magnitude of the probability wave at any particular location. The diagram 13 shows the changes in location and momentum seen in this model.

 

Please note that momentum in this model is not fixed. Heisenberg uncertainty about position and momentum necessitates a variable momentum. In this scheme, at the peak, displacement along the x-axis is maximal. When a particle moves down the slope from the peak during its wave function, the changes in position along x-axis (Dx) are diminished. However, the change in its speed, and therefore its momentum (Dp), increases.  Around the x-axis, the particle’s displacement is close to zero. . At this moment the changes of the momentum is maximal.   There will be a time that spatial displacement of the particle is zero (pass the light speed c).

 

On the other hand, close to the peak of spatial displacement, the particle has the least momentum and changes of momentum per unit of time. At the peak, there is a moment when the deflecting force disappears and the attracting force is about to start. At this moment the particle does not possess movement and therefore no changes of momentum. This is kind of realization of Heisenberg location/momentum uncertainty principle.
 
In the section “Mass and Gravity,” I offer a physical description of the Planck constant (h) in this model. According to our calculations, the Planck constant can be taken as the energy that is delivered by the particle to space-time during each wavelength. When the particle appears in space-time, its kinetic energy equals (h). Dirac constant ħ is a reduced Planck constant (ħ = h/2p). At the peak, potential energy equals (h). This description of Planck constant confirms the close match between uncertainty principle and particle-wave behavior in this model.

 

Energy-Time Uncertainty

 

According to the Heisenberg Uncertainty Principle, the relation between energy (E) of a particle and time (t) is obtained as follows:

 

D E D t ≥ ħ/2

D E ≥ ħ / 2Dt

 

We pinpoint time zero as the moment when the particle-wave crosses the x-axis and initiates its wavelength in space-time. Around this point, changes in time are minimal. By the same token, according to our previous assumption, at the entering point changes of mass and therefore energy is at its peak. 

 

The Heisenberg equation indicates that when the change in time is zero, the change in the particle’s energy reaches its peak. At the peak of the curve Dt is maximal and changes in energy are minimal (see diagram 13).

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8 ) Jonathan Allday, Quantum Reality, CRC Press, Boca Raton/ London/ New York, 2009

The above diagram shows were the objects meet the singularity during their wave function.