DOI: 10.31038/NAMS.2023611

Abstract

Einstein published his light-quantum hypothesis (LH) 1905, suggesting that a light ray transmits discrete energy quanta h.fi, emitted at speed c from their source, where h refers to Planck’s constant – and fi to the frequency of a ray of light Lx in accordance with the wave equation  c=λi.fi – with λi  indicating the wavelength of Lx. Now, let fv and fr stand for the frequency of violet and red light, respectively, assuming that  fv=2.fr .  Following these premises, a source of light Qv emitting h.fv light-quanta would transmit an energy amounting to (h.fv.fv) per second, whereas a source Qr, releasing h.fr light-quanta would only emit an energy equivalent to (h.fr.fr). In other words: The LH implies that the energy a ray of light transports per time-unit corresponds to the square of its frequency. This stands at odds with the experimentally established view that the energy carried by different light-rays per time-unit is linearly proportional to their (respective) frequency.

Introduction

Einstein introduced 1905 the light-quantum-hypothesis (LH) in his paper «Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt» [On a heuristic point of view concerning the production and transformation of light]. With the help oft he LH he aimed to interpret the photoelectric effect – as well as other phenomena related to it – in a consistent, straightforward way.

In the present paper, I shall

1. Consider the LH as a purely theoretical system, based on a number of explicit and implicit postulates;
2. Check whether the postulates underlying the LH are self-consistent and mutually compatible;
3. Analyse certain implications of the LH, and compare them with the generally accepted interpretation of the experimental findings.

*An earlier version of this paper was published in GALILEAN ELECTRODYNAMICS, Volume 30, Number 5, September/October 2019, p. 95-97.

Explicit and Implicit Postulates Underlying the LH

P1:          Let sources of light Qx emit homogenous, mono-frequent rays of light Lx.

P2: Let the wave-equation

(1) c=λi . fi

be valid for every light-ray Li emitted by a source Qi, with λi referring to the wavelength of a ray of light Li, and fi to the frequency of Li.

P3: Let Lv be of a higher frequency than Lr, i.e. fv > fr, with ‘v` standing for “violet“, ‘r’ for red.

P4: Let every source of light Qi which emits rays of light of a given frequency fi, emit these rays as bundles of light-quanta or photons Phi having the energy h.fi, i.e.

(2) Ei=h.fi,

where ‘h‘ refers to Planck’s constant (elementary quantum of action), and ‘fi’ to the frequency.

P5: Let the light-quanta (=energy quanta, photon) Phi, which are emitted by Qi, move through space as indivisible entities, and let them be absorbed as such by adequate targets.

P6: Let each light-quantum Phi move with speed c  in relation to its source Qi – if mutual conditions of rest are given between source, receiver, and medium (insofar a medium needs to be taken into account).

A short explanation of the foregoing postulates reads as follows:

Postulate P1 is seldom assumed explicitly. Einstein, in his introductory remarks, observed that he would consider the energy transmitted by a light-ray emitted by a «Lichtpunkt» (i.e. a very small source of light). With respect to the following analysis, I’ll assume the existence of « quasi-linear » rays of light, i.e. tiny bundles of light the cross-sections of which are so small that only one light-quantum per unit of time can hit and perforate an adequately placed perpendicular plane. Said in other terms, and in accordance with Einstein: I’ll consider rays of light in which light-quanta move in a linear row.

Postulate P2 is very rarely formulated explicitly. Nevertheless, it is implicitly taken for granted as soon as one:

(a) alludes to the frequency fi of a given light-ray, and distinguishes rays of light of different frequencies;

(b) introduces c as the constant speed of every light-ray Li in relation (i) to its source, (ii) to the medium (insofar a medium has to be considered), (iii) the receptor, and (iv) the « external observers » – albeit on the premises that source, medium, receptor, and « external observers » are at rest with respect to each other;

(c) assumes that the fronts of the rays of light – e.g. of violet and red light – proceed from their respective sources at equal speeds c.

Postulate P3 is generally accepted and needs no further comment.

Postulates P4 and P5 comprise the main, central, statements of Einstein’s LH [1]. Strictly speaking, P4 and P5 contain three separate assertions, concerning:

(i) the emission of light-quanta (see: P4);

(ii) the propagation through space of indivisible light-quanta (see: P5), and

(iii) the absorption of indivisible light-quanta (see: P5).

However, since I am dealing in the present paper only with the emission and the propagation of light-quanta, I have subsumed propagation and absorption to the same postulate P5.

Postulate P6 is seldom stated explicitly. In a configuration stating mutual rest of source, receiver, medium (insofar a medium needs to be considered), and observers – P6 is self-evident. Furthermore, P6 can be derived from P2 in conjunction with P4 and P5 [2-5].

Discussion of the LH

We shall start by considering the postulates P1 – P6 independently of one another: As far as I can see, there are no ambiguous demands being made, and nothing otherwise untenable can be discerned. Thus, every single postulate  P1 – P6, taken on its own, is free from contradictions [6,7].

We must now look at several combinations of these postulates:

(i) It follows from P3 and P4 that Lv-photons consist of a larger amount of energy than Lr-photons. In other terms:

(Ev=h.fv) > (Er=h.fr).

(ii) According to P4 in conjunction with P5, the light-quanta spread from their respective sources as indivisible entities, in their respective frequencies fv and fr – which is a constitutive factor of the light-rays we are considering.

(iii) With regard to P2, it follows that every frequency fi is univocally correlated to a corresponding wavelength λi, so that any kind of event belonging to Li (e.g. its wave-peaks) will be repeated with the corresponding frequency fi at any well-defined point along the path of Li that continues to move forth with speed c.

(iv) According to P4, source Qv emits light-quanta of energy h.fv, whereas a source Qr will release light-quanta of energy h.r.

(v) Now, following P3, ‘fv’ refers to the frequency of violet light-rays and ‘fr’ to the frequency of red light-rays, with fv > fr. On the basis of P2, we will then, reciprocally, have to infer that λv < λr.

This assertion implies that source Qv will not only emit light-quanta Phv which excel in energy the light-quanta Phr emitted by source Qr by the factor (fv – fr), but also that source Qv is due to propagate its light-quanta Phv with a frequency which surpasses the frequency with which source Qr releases its light-quanta Phr by the same factor (fv – fr).

In other words: On the basis of P1 – P6, we are compelled to deduce that if we – for instance – compare a ray of light Lv of wave-length λv=4000 nm with a ray of light Lr of wavelength λr=8000 nm, the former one (i.e. Lv) must transmit four times (and not twice) as much energy per unit of time than the latter (i.e. Lr).

It is hard to see, how this unexpected implication could be avoided: The frequency fi is a firmly bound variable of equation (1) [c=λi.fi]. – As soon as we use the symbol ‘fi‘ to point at the energy of a light-quant h.fi, we are forced to accept that these quanta are linked to equation (1) and are, therefore, emitted fi times per unit of time.

(vi) Einstein believed that the amount of energy contained in a single light-quantum  h.fi is linearly proportional to ist frequency, i.e. to the fi. However, on the basis of his premises – i.e. of his postulates P1 – P6 – we are logically compelled to infer that Einstein’s premises do not support what Einstein believed to be true, but -on the contrary – supported the erroneous assumption that the amount of energy contained in a single light-quantum h.fi is proportional to the square its frequency.

(vii) In 1916 [8]: Millikan pointed out: “The hypothesis [i.e. Einstein’s LH] was apparently made solely because it furnished a ready explanation that when an electron is thrown out of a metal by ultra-violet light or X-rays it is independent of the intensity of the light while it depends on its frequency .“ However, if one correctly argues – based on the experimental findings established by Millikan – that the amount of energy being transported by a ray of light Li per unit of time is linearly proportional to its frequency – one is compelled to dismiss as erroneous the core of Einstein’s theoretical premises and his reasonings based on them. In short: Einstein’s theoretical premises and his reasonings based on them stand at odds with his belief and with the experimental findings established by Millikan.

Conclusion

Einstein’s light quantum hypothesis (LH) prescribes that every ray of light Li transports and transmits discrete energy-light-quanta of magnitude h.fi.If every source of light Qi emitted discrete light-quanta of a specific, frequency-dependent magnitude h.fi, it would also have to release these energy-quanta with the corresponding frequency fi. This would, in turn, imply that the energy Qi emitted per unit of time with a ray of light Li, had to be proportional to the square of its frequency, i.e. the amount of the propagated and transmitted energy would have to be equivalent to fi.(h.fi) per unit of time.However, if we hold to the generally acknowledged – and by Millikan [9] experimentally established – view that every ray of light transports and transmits an amount of energy per unit of time which is linearly proportional to its frequency, Einstein’s premises and his reasonings cannot be maintained.

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