**11. Proposition:** *"If the speed of light is constant, then light is a wave"*  [Ragazas 2011b]

*Proof:* We have that *<sup>h</sup> p* , *<sup>E</sup> h* and *c* . Since *p x* and *E t* , we have that *t* . Differentiating, we get

$$D\_t(\mathcal{A}\boldsymbol{\nu}) = \frac{\frac{\partial^2 \boldsymbol{\eta}}{\partial t^2} \cdot \frac{\partial \boldsymbol{\eta}}{\partial \mathbf{x}} - \frac{\partial \boldsymbol{\eta}}{\partial t} \cdot \frac{\partial^2 \boldsymbol{\eta}}{\partial t \partial \mathbf{x}}}{\left(\frac{\partial \boldsymbol{\eta}}{\partial \mathbf{x}}\right)^2} \quad \text{and} \quad D\_\mathbf{x}(\mathcal{A}\boldsymbol{\nu}) = \frac{\frac{\partial^2 \boldsymbol{\eta}}{\partial \mathbf{x} \partial \mathbf{t}} \cdot \frac{\partial \boldsymbol{\eta}}{\partial \mathbf{x}} - \frac{\partial \boldsymbol{\eta}}{\partial t} \cdot \frac{\partial^2 \boldsymbol{\eta}}{\partial \mathbf{x}^2}}{\left(\frac{\partial \boldsymbol{\eta}}{\partial \mathbf{x}}\right)^2}$$

Since *c* , we have that 0 *Dt* and 0 *Dx* . Therefore,

$$
\frac{
\partial^2 \eta
}{
\partial t^2
} \cdot \frac{
\partial \eta
}{
\partial \mathbf{x}
} - \frac{
\partial \eta
}{
\partial t
} \cdot \frac{
\partial^2 \eta
}{
\partial t \partial \mathbf{x}
} = 0 \quad \text{and} \quad \frac{
\partial^2 \eta
}{
\partial \mathbf{x} \partial t
} \cdot \frac{
\partial \eta
}{
\partial \mathbf{x}
} - \frac{
\partial \eta
}{
\partial t
} \cdot \frac{
\partial^2 \eta
}{
\partial \mathbf{x}^2
} = 0
$$

Using *t x* and *c* , these can be written as,

$$\frac{\partial^2 \eta}{\partial t^2} = c \cdot \frac{\partial^2 \eta}{\partial t \partial \mathbf{x}} \quad \text{and} \quad \frac{\partial^2 \eta}{\partial \mathbf{x} \partial t} = c \frac{\partial^2 \eta}{\partial \mathbf{x}^2}$$

Since 'mixed partials are equal', these equations combine to give us,

$$\frac{\partial^2 \eta}{\partial t^2} = c^2 \cdot \frac{\partial^2 \eta}{\partial \mathbf{x}^2} \quad , \text{ the wave equation in one dimension}$$

Thus, for the speed of light to be constant the 'propagation of light' must be a solution to the wave equation. *q.e.d*

### **12. The double-slit experiment** [Ragazas 2011a]

The 'double-slit experiment' (where a beam of light passes through two narrow parallel slits and projects onto a screen an interference pattern) was originally used by Thomas Young in 1803, and latter by others, to demonstrate the wave nature of light. This experiment later

*x*

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 embodies this quintessential mystery of Quantum Mechanics.

Fig. 6.

708 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

 *and* 

equals "%-change of

 

. Thus, 0 ( ) *<sup>t</sup> Et Ee*

*c* . Since *p*

and

[Ragazas 2010i, 2011a]

*t e* 

> *x*

*x* 2 *xt x t <sup>x</sup> <sup>D</sup>*

. Therefore,

2 2

 

*xt x t x* 

2 2 <sup>2</sup> *c x t x*

 

and *E*

2 2

 

*x*

<sup>2</sup> 0

must be a solution to

*can be both positive or negative.* 

*t* 

2

, we have that

. Differentiating with respect to *t*

per unit of time". If we

*can be both positive or negative,*

**11. Proposition:** *"If the speed of light is constant, then light is a wave"* 

and 0 *Dx*

and

and

, *the wave equation in one dimension*

The 'double-slit experiment' (where a beam of light passes through two narrow parallel slits and projects onto a screen an interference pattern) was originally used by Thomas Young in 1803, and latter by others, to demonstrate the wave nature of light. This experiment later

*Note: Since %-change in* 

we have, 0 ( ) *<sup>t</sup> Et e*

*Proof:* We have that *<sup>h</sup>*

[Ragazas 2011b]

*t x*

 

Using *t*

Since  **10. The 'exponential of energy'** <sup>0</sup> ( ) *<sup>t</sup> Et Ee*

consider *continuous change*, we can express this as 0

 and *E*0 0 

> *h*

2 2

 

*x*

 

 

2 2 <sup>2</sup> *c t t x* 

Since 'mixed partials are equal', these equations combine to give us,

Thus, for the speed of light to be constant the 'propagation of light'

2 2 2 2 2 *c t x*

**12. The double-slit experiment** [Ragazas 2011a]

 

*c* , these can be written as,

2 2 <sup>2</sup> 0 *t x t tx* 

and

 

*p* 

2 *t* 2 *<sup>t</sup> x t tx <sup>D</sup>*

. Differentiating, we get

*c* , we have that 0 *Dt*

*x*

and

 

the wave equation. *q.e.d*

, *<sup>E</sup>*

From **Section 9.0** above we have that

*t* 

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