Louis deBroglie’s famous equation is λ=h/(mv), where the wavelength λ is the distance a particle λ-hops.

Satisfying Zeno and deBroglie with the concept of λ-hopping.

(Nothing moves and yet everything changes)

Velocity is Δx/Δt. The notion in Calculus of velocity at a point (dx/dt) is elegant  mathematics and screwy physics. (also see Section 12)

Velocity is a property we calculate for a particle not a property that the particle has.  The logic of this is as follows:

1. Particles appear at a point of space-time then disappear and reappear “nearby” at another point     in space-time.  Wavelength (λ) is the distance between these two points in space-time.

2. A particle moves by going from a point in space-time, to another point in space-time. This is called λ-hopping. See Section 13 and Section 30 for a more complete model of λ-hopping.

3. During the hop the particle cannot be found in space-time. However, the particle can appear before or after its scheduled landing if it either emits or absorbs a photon (accelerates or decelerates) which it can only do during the hop. This action resets the hop after which the particle continues hopping at its new wavelength. It has a new wavelength because it has new calculated velocity. I do not have an reason for λ-hopping. It is just a way of interpreting the deBroglie equation that makes it meaningful. Thus it is a starting axiom. All quantum mechanical particles move by λ-hopping.  Note that Schrodinger interpreted the deBroglie equation as a continuous particle motion describable as a complex linear differential equation, the Schrodinger equation. This is somewhat workable but distorts the underlying reality of discontinuity (IMHO).

A case will also be made that photons (energy) also moves by λ-hopping even though they do not have mass like particles (see Section 30). In addition a case will also be made that gravitons (gravity particles similar to photons) also move by λ-hopping. These gravitons are basically Compton wavelengths and do carry a small amount of mass. Graviton mass will be shown to account for the dark stuff in the universe (see Section 17, Section 19 and Section 20).

4. It takes two position measurements to calculate a wavelength λ. The measurements need to be made with care because the measurement will effect the λ and the velocity of the particle.

5. Once the wavelength λ is known by subtracting the second position from the first position (x2 - x1) we can then use it in the above deBroglie equation (if we know the mass of the particle) to calculate v the velocity of the particle.

6. Alternatively if we have a clock we can calculate the velocity of the particle as:  v = (x2-x1)/(t2-t1).
7. Interestingly, Chemists have been using the concept of λ-hopping when they consider the motion of electrons in atoms, They think of them as hopping from energy level to energy level. The loss of continuity does not seem to bother them too much. It is not much of a stretch from this viewpoint to having electron movement in general to consists of λ-hops.

A particle can propagate but it is never seen moving

a. At the instant a particle appears it has neither velocity nor acceleration; it just is. We infer motion by the history of the particle’s appearances. Note that this type of propagation is not true for ordinary objects like golf balls. The golf ball has so many random λ-hopping particles that they fill in all the space-time and leave no gaps for the golf ball to clear its own dimension, and the concept of a single λ-hop for the golf ball is not possible even though the golf ball is composed of many particles that do λ-hop (see section 29).

b. An observer and a memory (or persistence) of the past are needed to calculate the propagation speed of a particle. In many instances our senses and instruments do this automatically.

With the concept of λ-hopping both Zeno and deBroglie are satisfied.
Zeno is satisfied because a particle is never seen moving, it just makes appearances. Said another way, particles do not have the property of velocity, velocity is given to them by their history.

Louis deBroglie is satisfied because all particles have a movement (velocity) that can be calculated (but is never perceived directly).

A more detailed investigation into λ-hopping is found in section 13 (Dual Slit Experiment) and section 30 (A tale of two wavelengths). It is very similar to what is developed here initially but goes on to speculate that light (and gravitons) are characterized by a single wavelength, while particles are characterized by two wavelengths (a deBroglie wavelength and a Compton wavelength).  Pure fun, check it out.

Where will a particle hop?
a.
If conditions are simple, the particle will maintain its λ-hopping mode unless it is acted upon by other particles and/or energies (photons).  This is similar to Newton’s law of motion, saying that a body will maintain a steady motion unless it is acted upon by an outside force.

b.  I believe that Feynman’s sum over histories technique is also an answer to the question of where will the particle hop, but without the probability interpretation. Feynman needed probably because he considered the electron as having a continuous existence. A sum over histories technique taking into account λ-hopping will predict exactly where the electron will hop. Thus, where a particle will land depends upon the history of all other particles. This is very similar to Newton’s observation that a particle in motion will remain in motion unless acted upon by an outside force.

c.  In Section 16 the mechanism of Compton scattering is introduced.  This mechanism is also a way of understanding how a particle moves when it is “hit” with photons.

d.  In Section 29 the mechanism of Spin is introduced.  Not quite the spinning top model.

The problem in visualizing λ-hopping:

When the λ-hops are short, we can visualize the movement of photons and particles without much trouble. When the λ-hops are light years in length, our notions of a continuous particle become nebulous. This is because we do not have any experimental feel of this type of stuff. And when it comes to a minuscule graviton mass (see section 17) that spans hundreds of light years we are very much at a loss. How do we visualize and measure a very long string of near nothingness? In section 19 I postulate that this string of nothingness (the graviton) has a very small mass and depending upon its geometry is either dark matter and dark energy.

Authors Note:   Some if not most of my readers will find my models of physical reality quite strange.    This is because my starting position is that reality (at the quantum mechanical level) is non-continuous. There is no proving (or disproving) this as Bell tried to do.  There is no finding a missing local variable as Einstein tried to do.  As soon as a scientist has a starting position that quantum mechanical objects must exist continuously, they are in a soup of confusion with many  contradictions and arguments that turn out to be silly when it is realized that their starting axiom of continuity is not the way that nature operates on the quantum level.

The principles of superposition and uncertainty patch up this misreading of nature and have made it    possible to develop the standard model of particle physics. The Standard model of particle physics    is a marvelous, workable structure, but it is at its limit. It is time to begin developing our models of    nature starting with the postulate that quantum mechanical particles do not move (they change    positions).  The λ-hopping motion of particles provides the particle nature and wave nature in one package, giving a better understanding of our world than those models of reality that say our measurement techniques determine whether something is a particle or a wave. Keep on reading, it gets much more interesting !

Albert Einstein on λ-hopping:
Albert Einstein (1879-1955) Germany-Switzerland

"On a Heuristic Viewpoint Concerning the Production and Transformation of Light."

"Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt." Annalen der Physik. Leipzig 17 (1905) 132;

From the last two paragraphs of the introduction:

It seems to me that the observation associated with black body radiation, fluorescence, the photoelectric effect, and other related phenomena associated with the emission or transformation of light are more readily understood if one assumes that the energy of light is discontinuously distributed in space. In accordance with the assumption to be considered here, the energy of a light ray spreading out from a point is not continuously distributed over an increasing space, but consists of a finite number of energy quanta which are localized at points in space, which move without dividing, and which can only be produced and absorbed as complete units.