**9. Meaning and derivation of the De Broglie equations** [Ragazas 2011a]

Consider 0 00 ( , ) ( ,) *x t xt* . We can write 0 = %-change of = 'cycle of change'. For corresponding *x* and *t* we can write, 0 *x* = "distance per cycle of change" and

$$\nu = \frac{\Delta \eta}{\Delta t} = \text{"cycle of change per time". We can rewrite these as } \quad \lambda = \frac{\eta\_0}{\Delta \eta} \quad \text{and} \quad \nu = \frac{\Delta \eta}{\eta\_0} \dots$$

Taking limits and letting 0 *h* (Planck's constant being the minimal that can be measured) we get the *de Broglie equations:*

$$\mathcal{A} = \frac{\eta\_0}{\Delta \eta \bigvee\_{\Delta \Upsilon}} \rightarrow \frac{h}{\hat{\copyrightarrow} \eta \bigvee\_{\hat{\copyrightarrow} \Upsilon}} = \frac{h}{p} \quad \text{and} \quad \nu = \frac{\Delta \eta \bigvee\_{\Delta \Upsilon}}{\eta\_0} \rightarrow \frac{\hat{\copyrightarrow} \eta \bigvee\_{\hat{\copyrightarrow} \Upsilon}}{h} = \frac{E}{h}$$

The Thermodynamics *in* Planck's Law 709

came in direct conflict, however, with Einstein's *Photon Hypothesis* explanation of the Photoelectric Effect which establishes the particle nature of light. Reconciling these logically antithetical views has been a major challenge for physicists. The double-slit experiment

There are many variations and strained explanations of this simple experiment and new methods to prove or disprove its implications to Physics. But the 1989 Tonomura 'single electron emissions' experiment provides the clearest expression of this wave-particle enigma. In this experiment single emissions of electrons go through a simulated double-slit barrier and are recorded at a detection screen as 'points of light' that over time randomly fill in an interference pattern. The picture frames in Fig. 6 illustrate these experimental results.

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

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

embodies this quintessential mystery of Quantum Mechanics.

We will use these results in explaining the *double-slit experiment.*

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

how energy is absorbed in our measurements of it.

electron 'emitted' is not the same electron 'detected'.

Fig. 6.

*Note: Since %-change in can be both positive or negative, and can be both positive or negative.* 

#### **10. The 'exponential of energy'** <sup>0</sup> ( ) *<sup>t</sup> Et Ee*[Ragazas 2010i, 2011a]

From **Section 9.0** above we have that equals "%-change of per unit of time". If we consider *continuous change*, we can express this as 0 *t e* . Differentiating with respect to *t* we have, 0 ( ) *<sup>t</sup> Et e t* and *E*0 0 . Thus, 0 ( ) *<sup>t</sup> Et Ee*
