**12.1 Plausible explanation of the double-slit experiment**

The basic logical components of this *double-slit experiment* are the 'emission of an electron at the source' and the subsequent 'detection of an electron at the screen'. It is commonly assumed that these two events are directly connected. The electron emitted at the source is assumed to be the same electron as the electron detected at the screen. We take the view that this may not be so. Though the two events (emission and detection) are related, they may not be directly connected. That is to say, there may not be a 'trajectory' that directly connects the electron emitted with the electron detected. And though many explanations in Quantum Mechanics do not seek to trace out a trajectory, nonetheless in these interpretations the detected electron is tacitly assumed to be the same as the emitted electron. This we believe is the source of the dilemma. We further adapt the view that while energy propagates continuously as a wave, the measurement and manifestation of energy is made in discrete units (*equal size sips*). This view is supported by all our results presented in this Chapter. And just as we would never characterize the nature of a vast ocean as consisting of discrete 'bucketfuls of water' because that's how we draw the water from the ocean, similarly we should not conclude that energy consists of discrete energy quanta simply because that's how energy is absorbed in our measurements of it.

The 'light burst' at the detection screen in the Tonomura *double-slit experiment* may not signify the arrival of "the" electron emitted from the source and going through one or the other of the two slits as a particle strikes the screen as a 'point of light'. The 'firing of an electron' at the source and the 'detection of an electron' at the screen are two separate events. What we have at the detection screen is a separate event of a light burst at some point on the screen, having absorbed enough energy to cause it to 'pop' (like popcorn at seemingly random manner once a seed has absorbed enough heat energy). The parts of the detection screen that over time are illuminated more by energy will of course show more 'popping'. The emission of an electron at the source is a separate event from the detection of a light burst at the screen. Though these events are connected they are not directly connected. There is no trajectory that connects these two electrons as being one and the same. The electron 'emitted' is not the same electron 'detected'.

The Thermodynamics *in* Planck's Law 711

Using our derivation of *Planck's Law* as a *Rosetta Stone* (linking Mechanics, Quantum Mechanics and Thermodynamics) we considered the *quantity eta* that naturally appears in our derivation as *prime physis.* Planck's constant *h* is such a quantity. Energy can be defined as the time-rate of *eta* while momentum as the space-rate of *eta*. Other physical quantities can likewise be defined in terms of *eta*. Laws of Physics can and must be mathematically derived and not physically posited as Universal Laws chiseled into cosmic dust by the hand

We postulated the *Identity of Eta Principle*, derived the Conservation of Energy and Momentum, derived Newton's Second Law of Motion, established the intimate connection between entropy and time, interpreted Schoedinger's equation and suggested that the *wave-function ψ* is in fact *prime physis η*. We showed that The Second Law of Thermodynamics pertains to *time* (and not entropy, which can be both positive and negative) and should be reworded to state that *'all physical processes take some positive duration of time to occur'*. We also showed the unexpected mathematical equivalence between *Planck's Law and Boltzmann's Entropy Equation* and proved that "*if the speed of light is a constant, then* 

The proofs to many of the derivations below are too simple and are omitted for brevity. But

the propositions are listed for purposes of reference and completeness of exposition. *Notation.* We will consistently use the following notation throughout this APPENDIX:

We will use the following characterization of exponential functions without proof:

*rt rs E Et Es Ee Ee* ,

*E t*( ) is a real-valued function of the real-variable *t*

of God.

*light is a wave".* 

**14. Appendix: Mathematical derivations** 

*tts* is an 'interval of t'

<sup>1</sup> ( ) *t*

*s E E E u du*

( ) *t*

**14.1 Part I: Exponential functions** 

while 0 00 1 *<sup>t</sup>*

*s*

*s*

*av*

*E Et Es* () () is the 'change of *E'* 

*P E u du* is the 'accumulation of *E'* 

*Basic Characterization:* <sup>0</sup> ( ) *rt Et Ee if and only if D E rE <sup>t</sup> Characterization 1:* <sup>0</sup> ( ) *rt Et Ee if and only if E Pr*

*r r*

Therefore by the *Basic Characterization,* <sup>0</sup> ( ) *rt Et Ee* . *q.e.d* 

*ru rt rs*

*<sup>E</sup> P E e du E e E e*

*t s* is the 'average of *E* '

*Dx* indicates 'differentiation with respect to *x* ' *r* is a constant, often an 'exponential rate of growth'

*Proof:* Assume that <sup>0</sup> ( ) *rt Et Ee* . We have that 0 0

. Therefore *E Pr* .

Assume next that *E Pr* . Differentiating with respect to *t*, *D E rD P rE t t* .

What is emitted as an electron is a burst of energy which propagates continuously as a wave and going through both slits illuminates the detection screen in the typical interference pattern. This interference pattern is clearly visible when a large beam of energy illuminates the detection screen all at once. If we systematically lower the intensity of such electron beam the intensity of the illuminated interference pattern also correspondingly fades. For small bursts of energy, the interference pattern illuminated on the screen may be undetectable as a whole. However, when at a point on the screen *local equilibrium* occurs, we get a 'light burst' that in effect discharges the screen of an amount of energy equal to the energy burst that illuminated the screen. These points of discharge will be more likely to occur at those areas on the screen where the illumination is greatest. Over time we would get these dots of light filling the screen in the interference pattern.

We have a 'reciprocal relation' between 'energy' and 'time'. Thus, 'lowering energy intensity' while 'increasing time duration' is equivalent to 'increasing energy intensity' and 'lowering time duration'. But the resulting phenomenon is the same: the interference pattern we observe. This explanation of the *double-slit* experiment is logically consistent with the 'probability distribution' interpretation of Quantum Mechanics. The view we have of energy propagating continuously as a wave while manifesting locally in discrete units (*equal size sips)* when *local equilibrium* occurs*,* helps resolve the *wave-particle dilemma*.

## **12.2 Explanation** *summary*

The argument presented above rests on the following ideas. These are consistent with all our results presented in this Chapter.


Our thinking and reasoning are also guided by the following attitude of *physical realism*:


Thus, using *physical realism* we argue that if we keep the experimental apparatus constant but only replace our 'detection devices' and as a consequence we detect something contradictory, the physics of the double slit experiment does not change. The experimental behavior has not changed, just the display of this behavior by our detection device has changed. The 'source' of the beam has not changed. The effect of the double slit barrier on that beam has not changed. So if our detector is now telling us that we are detecting 'particles' whereas before using other detector devices we were detecting 'waves', *physical realism* should tell us that this is entirely due to the change in our methods of detection. For the same input, our instruments may be so designed to produce different outputs.
