Chapter 8. The Uncertainty Principle.   

Heissenberg’s uncertainty principle has by many been interpreted to represent some mystical, inherent uncertainty of nature. By now it should be evident that in our model the uncertainty principle will boil down to statistical fluctuations of the K flux.

In quantum physics, Heisenberg’s uncertainty principle states that the values of certain pairs of conjugate variables cannot both be known with arbitrary precision.

Our theory adds that the K flux varies randomly and follows normal rules of statistical fluctuations. The momentum transferred to a particle from the K flux will vary accordingly. This causes a fermion to randomly shift its position and momentum. Hence there is no such thing as an inherent uncertainty in matter.

The effect known as quantum tunnelling takes place when K interaction, due to statistical fluctuations for a short duration, gives the particle enough momentum in a favourable direction to overcome its potential barrier.

Fig. 13. A particle in a bound state with 2 possible positions of minimum potential energy. Quantum tunnelling is here performed by the randomly changing flux of Ks. When the net impulse transferred by the Ks supersedes what is necessary to overcome the potential barrier, the particle will jump over to the other side. Tunnelling is ruled by the laws of statistical fluctuations of Ks. 

Let us first look at a nuclear event with a certain potential energy barrier, like in the case of tunnelling, then:

• A particle must gather its momentum within a critical reaction time, Δtc, to cross the barrier. 
• The number of K interactions available for the particle is N_K = f_K·Δt _c, 
• The total impact on the particle will be the sum of all vector momentums from Ks during          Δt _c: Σ K _i 
• The standard deviation of the K-momentums is proportional to constant·p_K·√N_K where p_K is the scalar momentum.  

The scalar value of the net momentum for each direction is

p _Kx² + p_Ky² + p_Kz² = p_K² 

And since there is no preferred direction, the variation is the same in all 3 directions

p _Kx² = p_Ky² = p_Kz² = 1/3 p_K² 

If, for instance, the tunnelling requires a push in the +x direction, the number of hits in the relevant direction is N_K/3, and half of the time it goes in the wrong direction, where after it may or may not bounce in the correct direction. But not only the probability for a push in the right direction limits the chance of doing a tunnelling. The particle must also be in a favourable position. If it is temporarily pushed off in the sideways direction, it may not be eligible for a push across the barrier, or at least the chance for such a push is greatly diminished. So the geometry of the tunnelling is also important for the frequency of the tunnelling, not only the potential energy barrier per se.

This is a good starting point for making an estimate for E_K (or m_K). The larger the mass of K, the larger barriers can be overcome, since the number of available K interactions (hits) for a specific event is the same, independent of the mass of K, provided that E = hf tells us that there is 1 or 2 or at least a constant number of Ks emitted at each wavelength of a photon.

For a proton, neutron, electron etc. with mass M we have that the total number of K interactions N_K from which a statistical deviation can be calculated, is:

f_K = N_K/Δt_c = 2f = 2Mc²/h  

N = 2M·Δt_c·c ²/h  

We suppose we can determine a fair estimate for the critical time Δt_c for a given event, then  

N _K = M·constant

Standard deviation of the force which act upon a particle with mass M will then be proportional with √N_K and the momentum carried by each K:

F = constant·pK·√M

(when writing “constant” in different equations, they are usually not the same).

Standard deviation for the acceleration for an event with a known reaction time Δt_c, will then be on the form:  

a = F/M = constant·p_K/√M

where p_K is the average momentum of a K particle, and M is the mass of the particle in question.

This is a good starting point for finding a 1-dimensional representation for the uncertainty principle, even though the constant is both hiding Δt_c and one must decide how much of the total space angle is available for the event to take place. And for practical purposes, the limitation in deciding Δx and Δp will of course be the same. But we know why this limitation exists, and we can break it down for analytical purposes. It is no longer a mystical intrinsic property of nature that position and momentum cannot be decided.  

Ks are omnipresent and very numerous, almost equally numerous in places within the range of strong gravity from planets and stars as in gravitationally neutral zones. Therefore all elementary particles will interact and be pushed in arbitrary directions by the Ks all the time, depending on the natural statistical variations of the K-flux. This constitutes the uncertainty principle.

Consequence 19:

Heisenberg’s uncertainty principle, Δp · Δx ≥ ħ/2, for the least momentum determination, Δp, within a certain distance, Δx, corresponds to the random variation of the net flux of Ks, when the K-flux is seen as having statistical variations as a function of the number of interactions made, and of the momentum or energy of each interaction.  

We have claimed that the reactive cross section (probability of K interaction) is proportional to the mass or energy of an EP. Hence the number of times (N_K) a particle is hit by Ks will be proportional to its mass (M).

Standard deviation for the momentum of the sum of Ks will be proportional to √NK and therefore proportional to √M The uncertainty principle states:

Δp·  Δx ≥ ħ/2  

We have that

F = constant·p_K·√M

To illuminate the connection between the uncertainty principle and standard deviation, let us rewrite the uncertainty principle using

Δp = MΔv: √M·Δv·√M·  Δx ≥ ħ/2  

Heisenberg’s uncertainty principle for the determination in 2 variables is proportional to the standard deviation of each variable for hits from a flux of Ks on a particle with mass M.

Tunnelling.

The tunnelling phenomenon is simply a number of K interactions which statistically occur in surplus, in a direction favourable for enabling the particle in question to overcome a given potential barrier. Tunnelling boils down to simple K-flux variation at the particle level, the rest depends on the geometry of the potential barrier relative to the tunnelling particle.

Consequence 20:

Tunnelling takes place when K interaction due to statistical fluctuations for a short duration has a large enough resulting momentum in a direction favourable for the particle to overcome the potential barrier of the tunnelling.

But a tunnelling is also sort of a pendulum, so quite often the tunnelling particle jumps to the one side, goes so far out that it has enough energy to bounce back over the barrier, and if it does not meet an unfavourable K-flux, it jumps straight back again. Most of the time rapid tunnelling will have a considerable element of this re-bounce effect.

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