On the quantisation of the frequency of electromagnetic radiation

List of Contents

Introduction

Quantisation of the frequency of electromagnetic radiation

A frequency-quantised Fidler diagram

The Fidler diagram as the two-dimensional projection of a three-dimensional surface

The extended Fidler diagram

References

Abstract

It is posited that the frequency of electromagnetic radiation may be quantised and two methods are derived which permit the calculation of the magnitude, of the quantisation.

The work does not establish the existence of frequency quantisation and the two methods derived may be described by the oxymoron, systematically arbitrary.

Both methods rely upon the Fidler diagram [3].

The first method employs the superposition of a fractal path on the diagram and shows that the magnitude is given by: , where (V) is the Planck frequency and n is an odd, disposable integer.

The second method involves the tiling of the Fidler diagram by a succession of self-similar tiles which are progressively-reducing versions of the diagram itself. For this case, the magnitude of the frequency-quantisation is given by: , where n is any disposable integer.

It is shown that the Fidler diagram is the two-dimensional projection of a three-dimensional surface and that the specific energy, specific energy intensity and specific energy density of a photon may, by the procedure outlined at the end of the work, be calculated from the diagram without additions thereto.

Introduction

It is considered axiomatic that the spectrum of electromagnetic radiation, and, in particular, the frequency, is continuous. As is well known, continuity may be considered as a manifestation of the macroscopic viewing of processes which are discontinuous on the microscopic scale; an example of this is the macroscopic treatment of a gas in classical thermodynamics. Here, the gas is considered to be a continuous substance and its behaviour is described by few parameters. We hence propose that the notion of continuity be placed in the same conceptual class as Platonic solids. On a philosophical basis, it may be implied that any phenomenon which is analysed using differential calculus admits of discontinuity, in that differential quantities are never zero; ordering of combinations of such quantities is done on the basis of the mathematical phrase ‘tends towards zero’.

The Planck-Einstein-Schrdinger equation is , where the symbols have their usual connotation. Now, one of the quantum rules stipulates that the quantity , called the photon, may only be emitted or absorbed in integer multiples. The so-called electromagnetic spectrum is the range of all possible frequencies of this photon, and, given that the magnitude of Planck’s constant of action, h, is it is probably true to say that the properties of a single photon have never been measured to sufficient accuracy to resolve the question of whether or not the frequency is quantised.

At the very low frequencies (i.e. a few Hertz), characteristic of the geophysical phenomenon called Schumann waves, a measurable signal could only be obtained from the combined effect of literally trillions of trillions of photons and hence quantisation of the frequency might be masked by the nature of the sheer number of photons, for there is no reason why they should all act in concert. Even the charge-coupled devices associated with that branch of Astronomy which investigates exceptionally-long wavelength radiation, whilst registering the presence of such, are probably incapable of resolving the question of frequency-quantisation. At the other end of the spectrum where measurements may be made of the properties of a relatively small (relatively speaking) number of gamma ray photons, it is doubtful that the frequency could be determined with sufficient accuracy to establish, or otherwise, frequency quantisation, given that the frequency of such photons is of the order of Hz. Given that quantisation of frequency, even at lower frequencies has never been reported it is concluded that if the phenomenon does exist, then it is very small throughout the range of frequency of the spectrum of electromagnetic radiation.

It is known that electronic circuitry incorporating the Josephson effect has been produced that can measure voltages of the order of a few picovolts and it is suggested that this may provide instrumentation which will facilitate the investigation of the posited quantisation of frequency.

If the reality of frequency quantisation can be established then this will extend the quantum rule discussed earlier.

Further, ever since the redefinition in 1967 (together with later amendments), the second has, in a sense, become a quantum-mechanical quantity for it is defined, [1] as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the Caesium-133 atom, and hence, frequency through association with the second may be argued to possess aspects of a quantum-mechanical character. Indeed, it follows that this notion may apply to any quantity so associated.

It is in the spirit of rejection of continuity that the following work examines the quantisation of the frequency of electromagnetic radiation.

We invoke the Jesuit credo,’ it is more blessed to ask for forgiveness than permission’, as articulated by Wilczek [2].

Quantisation of the frequency of electromagnetic radiation

We posit that the frequency,, of electromagnetic radiation is given by the simple linear equation:

Abbildung in dieser Leseprobe nicht enthalten

Where , is a pure number and which must have units of Hertz, is a non-zero increment of frequency. We call the frequency number. Further, it is posited that for the purpose of this work, is a constant, has the character of a finite difference, and hence, is very small.

If the spectrum of electromagnetic radiation is continuous then all are members of the set of real numbers. Conversely, if the frequency is quantised then all are members of the set of natural numbers.

In the case of a continuous spectrum of radiation there is, at bottom, no utility in equation (1) if used in other than the manner prescribed, for the combination of free choices for in conjunction with any number in the range of real numbers renders the equation hyperbolic, and an infinite set of rectangular hyperbolae, with frequency as parameter may be generated, which, in the context of the present work, is devoid of any physical meaning.

In the case of continuous electromagnetic radiation the abscissa, ordinate and hypotenuse of the Fidler diagram [3] are continuous lines and the space enclosed by these lines is filled completely. The magnitude of the division of the bounding lines into discrete intervals is entirely arbitrary and is hence not unique; any such discretisation is made on the basis of convenience and is done by fiat. Further, unlimited interpolation within any interval of discretisation implies continuity.

In keeping with the notation in [3], Planck ‘quantities’ are denoted by upper case letters enclosed in round brackets. Further, we call the point in the Fidler diagram with coordinates (1,1) , the Planck point.

A frequency-quantised Fidler diagram

A discretised, or, ‘discontinuous ‘ Fidler diagram is shown in Fig1.

Abbildung in dieser Leseprobe nicht enthalten

The sloping lines are lines of constant radiation Strouhal number, [2], and represent the spectrum of electromagnetic radiation in different substances. In this representation each spectrum only has meaning at the intersection with a horizontal line; which is isotonic and is the locus of an infinite number of intersections. In order to emphasise this we have omitted the ‘luminal’ line which, in the case of a continuous spectrum in vacuo would pass through the numbered points (and all intermediate points).

The scale of the diagram shown here seems to imply that the very low values of the radiation Strouhal number (and hence very high values of the index of refraction) reported by Hau et al [4], [5] and Schmidt et al [6], could not be accommodated on the diagram. However, the figure depicted is solely for the purposes of illustration and, as will be shown later, can be rendered sufficiently fine-grained to encompass virtually all the above-mentioned experimental data.

We now proceed to devise methods, by means of which the magnitude of the quantisation of frequency may be established. At the outset it must be made clear that the results of these procedures do not establish the existence of frequency quantisation and further, the means by which the results are obtained may be described by the oxymoron ‘systematically arbitrary’.

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