**2. Mathematical results**

We list below the main mathematical derivations that are the basis for the results in physics in this Chapter. The proofs can be found in the Appendix at the end. These mathematical results, of course, do not depend on Physics and are not limited to Physics. In *Stocks and Planck's Law* [Ragazas 2010l] we show how the same 'Planck-like' formula we derive here also describes a simple comparison model for stocks.

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

 *tts* is an interval of *t E Et Es* () () is the change of *E* ( ) *<sup>t</sup> s P E u du* is the accumulation of *<sup>E</sup>* <sup>1</sup> ( ) *<sup>t</sup> av <sup>s</sup> E E E u du t s* is the average of *<sup>E</sup>*

Measurement [Ragazas, 2010h] we argue with mathematical certainty that we cannot know

Since we are limited by our measurements of *'what is'*, we should consider these as the beginning and end of our knowledge of *'what is'*. Everything else is just *'theory'*. There is nothing real about theory! As the ancient Greeks knew and as the very word 'theory' implies. In Planck's Law is an Exact Mathematical Identity [Ragazas 2010f] we show *Planck's Law* is a mathematical truism that describes the *interaction of measurement*. We show that *Planck's Formula* can be *continuously* derived. But also we are able to explain *discrete* 'energy quanta'. In our view, *energy propagates continuously but interacts discretely*. Before there is *discrete manifestation* we argue there is *continuous accumulation* of energy. And this is based

Mathematics is a tool. It is a language of objective reasoning. But mathematical 'truths' are always 'conditional'. They depend on our presuppositions and our premises. They also depend, in my opinion, on the mental images we use to think. We phrase our explanations the same as we frame our experiments. In the single electron emission double-slit experiment, for example, it is assumed that the electron emitted at the source is the same electron detected at the screen. Our explanation of this experiment considers that these two electrons may be separate events. Not directly connected by some trajectory from *source* to

We can have beautiful mathematics based on *any* view of the Universe we have. Consider the Ptolemy with their epicycles! But if the view leads to physical explanations which are counter-intuitive and defy common sense, or become too abstract and too removed from life and so not support life, than we must not confuse mathematical deductions with *physical realism*. Rather, we should change our view! And just as we can write bad literature using good English, we can also write bad physics using good math. In either case we do not fault

The failure of Modern Physics, in my humble opinion, is in not providing us with *physical explanations* that make sense; a *physical view* that is consistent with our experiences. A *view* that will not put us at odds with ourselves, with our understanding of our world and our

We list below the main mathematical derivations that are the basis for the results in physics in this Chapter. The proofs can be found in the Appendix at the end. These mathematical results, of course, do not depend on Physics and are not limited to Physics. In *Stocks and Planck's Law* [Ragazas 2010l] we show how the same 'Planck-like' formula we derive here

*P E u du* is the accumulation of *<sup>E</sup>*

*t s* is the average of *<sup>E</sup>*

the language for the story. We can't fault Math for the failings of Physics.

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

*s*

*av <sup>s</sup> E E E u du*

*tts* is an interval of *t*

lives. Math may not be adequate. Sense may be a better guide.

also describes a simple comparison model for stocks.

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

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

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

through direct measurements what a physical quantity *E(t)* is as a function of time.

on the *interaction of measurement*.

*sensor*. [Ragazas 2010j]

**2. Mathematical results** 


#### **2.1 'Planck-like' characterizations** [Ragazas 2010a]

Note that *Eav* T. We can re-write *Characterization 2a* above as,

$$E(t) = E\_0 e^{\nu t} \quad \text{if and only if} \quad E\_0 = \frac{\eta \nu}{e^{\eta \nu f \kappa \mathcal{T}} - 1} \tag{1}$$

*Planck's Law* for blackbody radiation states that, 0 <sup>1</sup> *h kT <sup>h</sup> <sup>E</sup> e* (2)

where *E*0 is the intensity of radiation, is the frequency of radiation and *T* is the (Kelvin) temperature of the blackbody, while *h* is Planck's constant and *k* is Boltzmann's constant. [Planck 1901, *Eqn 11*]. Clearly (1) and (2) have the exact same mathematical form, including the type of quantities that appear in each of these equations. We state the main results of this section as,

*Result I: A 'Planck-like' characterization of simple exponential functions* 

$$E(t) = E\_0 e^{\nu t} \quad \text{if } and \text{ only if } \quad E\_0 = \frac{\eta \nu}{e^{\eta \nu / \kappa \mathcal{F}} - 1}$$

Using *Theorem 2* above we can drop the condition that 0 ( ) *<sup>t</sup> Et Ee* and get, *Result II: A 'Planck-like' limit of any integrable function* 

$$\text{For any } integrable \text{ function } E(t) \, \, \, E\_0 = \lim\_{t \to 0} \frac{\eta \nu}{e^{\eta \nu \{ k \cdot \mathcal{T} \}} - 1}$$

We list below for reference some helpful variations of these mathematical results that will be used in this Chapter.

The Thermodynamics *in* Planck's Law 699

*source and the sensor are in equilibrium. The average energy of the source is equal to the average energy at the sensor. Thus, E kT . 3) Planck's constant h is the minimal 'accumulation of energy'* 

1

*Planck's Formula is a mathematical truism that describes the interaction of energy*. That is to say, it gives a mathematical relationship between the energy locally at the *sensor*, the energy absorbed by the *sensor*, and the average energy at the *sensor* during measurement. Note further that when an amount of energy *E* is absorbed by the *sensor*, *E t*( )resets to *E*<sup>0</sup> .

Note: Our derivation, showing that *Planck's Law* is a mathematical truism, can now clearly explain why the experimental blackbody spectrum is so indistinguishable from the

1. Planck's Formula is an *exact mathematical truism* that describes the interaction of energy. 2. Energy propagates continuously but interacts discretely. The absorption or

3. Before *manifestation of energy* (when an amount *E* is absorbed or emitted) there is an

7. The time *t* required for an *accumulation of energy h* to occur at temperature *T* is given

is the frequency of radiation. [Ragazas 2011a]

measurement of energy is made in discrete 'equal size sips'(energy quanta).

(a) (b)

theoretical curve. **(http://en.wikipedia.org/wiki/File:Firas\_spectrum.jpg)**

*accumulation of energy* that occurs over a duration of time *t* . 4. The absorption of energy is proportional to frequency, *E h*

6. The energy measured *E* vs. *t* is linear with slope

5. There exists a *time-dependent local representation of energy,* <sup>0</sup> ( ) *<sup>t</sup> Et Ee*

 *is the frequency of radiation. 2) When measurement is made, the* 

*.*  1

*kT* for constant temperature *T .* 

*(The Quantization of* 

*,* where *E*0 is the

*h kT*

*<sup>h</sup> <sup>E</sup> e* 

*, where E*0 *is* 

*Assumptions: 1) Energy locally at the sensor at t s can be represented by* <sup>0</sup> ( ) *<sup>s</sup> Es Ee*

Using the above *Mathematical Identity* (6) and *Assumptions* we have *Planck's Formula*,

and so, 0

0

*<sup>h</sup> <sup>h</sup> kT <sup>u</sup> kT <sup>E</sup> h E e du e*

<sup>0</sup> <sup>0</sup>

*at the sensor that can be manifested or measured. Thus we have h*

*the intensity of radiation and* 

Fig. 2.

*Conclusions:* 

*Energy Hypothesis).* 

*kT .* 

by *<sup>h</sup> <sup>t</sup>*

intensity of radiation and

$$E\_0 = \frac{\Delta E}{e^{\Delta \notin \mathcal{E}\_w} - 1} = \frac{\eta \,\nu}{e^{\eta \nu \int \kappa \mathcal{T}\_\eta} - 1} \qquad\qquad\text{(if}\quad E(t) = E\_0 e^{\nu t}\text{)}\tag{3}$$

$$E\_0 \approx \frac{\Lambda E}{e^{\Lambda E \dagger E\_w} - 1} \approx \frac{\eta \nu}{e^{\eta \nu / \kappa \mathcal{T}\_\eta} - 1} \qquad \qquad \text{(if } E(t) \text{ is integrable)} \tag{4}$$

$$E\_0 = \frac{\eta \nu}{e^{\eta \nu / \kappa \mathcal{T}\_\eta} - 1} \text{ is exact if and only if } \begin{array}{c} \eta \nu \\ e^{\eta \nu / \kappa \mathcal{T}\_\eta} - 1 \end{array} \text{ is independent of } \eta \tag{5}$$

Note that in order to avoid using limit approximations in (4) above, by (3) we will assume an *exponential of energy* throughout this Chapter. This will allow us to explore the underlying ideas more freely and simply. Furthermore in **Section 10.0** of this Chapter, we will be able to justify such an exponential time-dependent local representation of energy [Ragazas 2010i]. Otherwise, all our results (with the exception of **Section 8.0**) can be thought as pertaining to a blackbody with perfect emission, absorption and transmission of energy.

### **3. Derivation of Planck's law without** *energy quanta* [Ragazas 2010f]

*Planck's Formula* as originally derived describes what physically happens at the *source*. We consider instead what happens at the *sensor* making the measurement. Or, equivalently, what happens at the *site of interaction* where energy exchanges take place. We assume we have a blackbody medium, with perfect emission, absorption and transmission of energy. We consider that measurement involves an *interaction* between the *source* and the *sensor* that results in energy exchange. This interaction can be mathematically described as a functional relationship between *E s*( ) , the energy locally at the *sensor* at time *s*; *E* , the energy absorbed by the *sensor* making the measurement; and *E* , the average energy at the *sensor* during measurement. Note that *Planck's Formula* (2) has the exact same mathematical form as the mathematical equivalence (3) and as the limit (4) above. By letting *E s*( ) be an *exponential*, however, from (3) we get an *exact* formula, rather than the limit (4) if we assume that *E s*( ) is only an integrable function. The argument below is one of several that can be made. The *Assumptions* we will use in this very simple and elegant derivation of *Planck's Formula* will themselves be justified in later **Sections 5.0, 6.0 and 10.0** of this Chapter.

*Mathematical Identity. For any integrable function E t*( ), ( ) *av s E <sup>s</sup> E u du* (6)

*Proof: (see* Fig. 1*)* 

<sup>T</sup> (if 0 ( ) *<sup>t</sup> Et Ee*

Note that in order to avoid using limit approximations in (4) above, by (3) we will assume an *exponential of energy* throughout this Chapter. This will allow us to explore the underlying ideas more freely and simply. Furthermore in **Section 10.0** of this Chapter, we will be able to justify such an exponential time-dependent local representation of energy [Ragazas 2010i]. Otherwise, all our results (with the exception of **Section 8.0**) can be thought as pertaining to

*Planck's Formula* as originally derived describes what physically happens at the *source*. We consider instead what happens at the *sensor* making the measurement. Or, equivalently, what happens at the *site of interaction* where energy exchanges take place. We assume we have a blackbody medium, with perfect emission, absorption and transmission of energy. We consider that measurement involves an *interaction* between the *source* and the *sensor* that results in energy exchange. This interaction can be mathematically described as a functional relationship between *E s*( ) , the energy locally at the *sensor* at time *s*; *E* , the energy absorbed by the *sensor* making the measurement; and *E* , the average energy at the *sensor* during measurement. Note that *Planck's Formula* (2) has the exact same mathematical form as the mathematical equivalence (3) and as the limit (4) above. By letting *E s*( ) be an *exponential*, however, from (3) we get an *exact* formula, rather than the limit (4) if we assume that *E s*( ) is only an integrable function. The argument below is one of several that can be made. The *Assumptions* we will use in this very simple and elegant derivation of *Planck's* 

<sup>T</sup> (if *E t*( ) is integrable) (4)

*e* 1 

<sup>T</sup> is *independent* of

*<sup>s</sup> E u du* 

(6)

 ) (3)

(5)

<sup>0</sup> <sup>1</sup> <sup>1</sup> *E Eav*

 

a blackbody with perfect emission, absorption and transmission of energy.

**3. Derivation of Planck's law without** *energy quanta* [Ragazas 2010f]

*Formula* will themselves be justified in later **Sections 5.0, 6.0 and 10.0** of this Chapter.

*Mathematical Identity. For any integrable function E t*( ), ( ) *av s E*

 

*e e*

<sup>0</sup> <sup>1</sup> <sup>1</sup> *E Eav*

<sup>T</sup> is *exact* if and only if

*e e*

*<sup>E</sup> <sup>E</sup>*

*<sup>E</sup> <sup>E</sup>*

0

*e* 

*E*

*Proof: (see* Fig. 1*)* 

Fig. 1.

1

*Assumptions: 1) Energy locally at the sensor at t s can be represented by* <sup>0</sup> ( ) *<sup>s</sup> Es Ee , where E*0 *is the intensity of radiation and is the frequency of radiation. 2) When measurement is made, the source and the sensor are in equilibrium. The average energy of the source is equal to the average energy at the sensor. Thus, E kT . 3) Planck's constant h is the minimal 'accumulation of energy' at the sensor that can be manifested or measured. Thus we have h .* 

Using the above *Mathematical Identity* (6) and *Assumptions* we have *Planck's Formula*,

$$h = \int\_0^{\frac{h}{kT}} E\_0 e^{\nu u} du = \frac{E\_0}{\nu} \left[ e^{\frac{h\nu}{kT}} - 1 \right] \text{ and so, } E\_0 = \frac{h\nu}{e^{\frac{h\nu}{kT}} - 1}$$

*Planck's Formula is a mathematical truism that describes the interaction of energy*. That is to say, it gives a mathematical relationship between the energy locally at the *sensor*, the energy absorbed by the *sensor*, and the average energy at the *sensor* during measurement. Note further that when an amount of energy *E* is absorbed by the *sensor*, *E t*( )resets to *E*<sup>0</sup> .

#### Fig. 2.

Note: Our derivation, showing that *Planck's Law* is a mathematical truism, can now clearly explain why the experimental blackbody spectrum is so indistinguishable from the theoretical curve. **(http://en.wikipedia.org/wiki/File:Firas\_spectrum.jpg)** *Conclusions:* 


$$\text{by } \Delta t = \frac{\cdots}{kT} \dots$$

The Thermodynamics *in* Planck's Law 701

0

*t* 

 given above is for a *fixed* **x0** ,*t*<sup>0</sup> . Comparing these we see that whereas our definition of energy is for *fixed* **x0** ,*t*<sup>0</sup> , Schroedinger equation is for *any* **x**,*t* . But otherwise the two equations have the same form and so express the same underlying

viewed as *accumulation of energy* or *action*) and so we can view Schroedinger Equation as in essence *defining* the energy of the system at *any* **x**,*t* while the *wave function*

be understood to express the *accumulation of energy* at *any* **x**,*t* . This suggests that the

The wave function gives the distribution of the accumulation of energy of the system.

*,* for <sup>1</sup> *<sup>t</sup>*

*av E E* 

*<sup>E</sup> <sup>E</sup> e* 

*E u du <sup>p</sup> u du* . Differentiating with respect to *t*, we obtain,

*dx Et p x dt* or more simply, *E pv <sup>x</sup> (energy-momentum equivalence)*

*Schroedinger Equation:* Once the extraneous constants are striped from Schroedinger's

*x x x* 

( )

, where

*<sup>p</sup> u du* .

*t* 

is the energy at *any* **x**,*t* . The definition (7) of energy

in these is one and the same.

. We have the following interesting

(a 'wavelength') we have

(*Characteristic 5)*, we

. Using the definition of

, we again have *Et h*

in (3) above and re-writing this

*av <sup>E</sup> <sup>t</sup>*

0

ln

*E E E E* 

*av*

*E*

. Since *E kT av* and entropy is defined as *<sup>E</sup> <sup>S</sup>*

and of momentum

is the *wave function* ,

(which can be

can

, since

*T* ,

*Energy-momentum Equivalence.* From the definition of energy *E*

( ) *t*

*E u du* and

0

*t* 

Using the *Identity of Eta Principle*, the quantity

*x*

equation, this in essence can be written as *H*

idea. Now (7) *defines* energy in terms of the more primary quantity

is the same as the quantity

*x p*

*x* 

Therefore,

*E t* 

*wave function*

have 1 *av E E* 

we have that

equivalently we have,

we have that,

() ( ) *<sup>x</sup>*

0 0

*t x*

*H* is the *energy operator*, and *H*

*Uncertainty Principle:* Since *E*

 

. Or equivalently, for 1

*if and only if* <sup>1</sup> *<sup>t</sup>*

*h* is the *minimal eta that can be manifested.* Note that since

*Planck's Law and Boltzmann's Entropy Equation Equivalence:* 

<sup>1</sup> *E Eav E E*

Starting with our *Planck's Law* formulation, 0 <sup>1</sup> *E Eav*

*e*

0 0

and so,

*E E*

<sup>1</sup> *Et h* 

*Eth* if and only if *S k* .

interpretation of the wave function.

*t x*

() ()
