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Considering Photons As “Spatially Confined Wave-Packets”

Dimitris J. Panagopoulos National Center for Scientific Research "Demokritos", Athens, Greece. Department of Biology, University of Athens, Greece Radiation and Environmental Biophysics Research Centre, Athens, Greece

Abstract(2)

The present theoretical study examines a possible revision in the current corpuscular photon model, to account for a physical quantum explanation of the light interference facts as originally described in Thomas Young’s double-slit experiments as well as in subsequent experiments. The light interference effect still constitutes a “mystery” for quantum mechanics, in spite of theoretical efforts to explain it (in terms of the current photon model) that lack physical meaning. We re-examine the facts that led to the establishment of the current corpuscular photon model. We accept that - at least - natural electromagnetic radiation does not consist of continuous waves as was considered in classical electromagnetic theory, but consists of energy quanta (photons), emitted as a whole and carrying energy hν. However, we suggest that each individual photon should not be considered as (indivisible) “particle” transmitted in a single direction and absorbed only as a whole, but rather as a wave-packet. Further we suggest that this wave-packet consists of a (divisible) bundle of identical adjacent, radial waves/pulses of certain length and distinct frequency, polarization, and phase, transmitted independently from each other on a part of a spherical surface in multiple directions within a solid angle. The radius of the spherical surface increases with the velocity of light. The photon’s divergence angle is determined by the screening during its emission by the electron clouds of its atomic/molecular source. Such a wave-packet can spread discontinuously over ever larger volumes by increasing the distance between adjacent radial pulses, or can be collimated and confined within narrow beams as in the case of lasers or directional antennas. According to this new photon model which we name “Spatially Confined Wave Packet” (SCWP), each photon may be generated as a whole but not necessarily absorbed or transmitted as a whole. In this way the light interference facts can be given a quantum mechanical interpretation with physical meaning, as the result of superposition between two parts of the same divisible bundle (photon) of identical pulses. We show that - in spite of widespread opposite information from text books - actually there is no condition in Planck’s, Einstein’s, or Compton’s original works that force us to accept that photons are transmitted in a single direction or absorbed only as a whole as “particles”, and neither any subsequent experimental fact force us to do so. Considering photons as SCWP, we present an alternative explanation of the photoelectric effect, as well as an alternative proof of Compton’s equation, and suggest a natural quantum interpretation of the light interference facts.

1. Introduction

2. Revisiting Important Data on Photon Nature

2.1. “Black-body” Radiation and the Postulation of Light Quanta

The concept of quanta or photons as the energy elements of a radiating body was first introduced by Max Planck in 1901 during his study on the “black-body” radiation2. He postulated that the emission of electromagnetic energy from a “black-body” is quantized and that (in his own words) “the energy element should be proportional to the number of vibrations” (frequency) (Planck 1901):

${\displaystyle E=h\cdot v}$ (1)

(h = 6.62510-34 J s the Planck’s constant,  the frequency of radiation). The derivation of his famous law of “black body” radiation was based on this crucial assumption. More specifically, Planck assumed that: a) The molecules on the walls in the interior of the cavity oscillate producing multiply reflected waves within the cavity, and these waves are then emitted outside once they are emitted/reflected towards the cavity’s hole. b) There is an energy element in the molecular oscillations (and the corresponding produced waves) at each individual frequency ν. This energy element is given by hν, and the individual energy En of each oscillation (and corresponding produced wave) at frequency ν within the cavity is an integer multiple of this energy element: En = nE = nhν, (n an integer number). The cavity is thus a system of innumerous resonators (oscillating molecules), producing multiply reflected waves (considered as standing waves), and at each individual frequency ν the average energy radiated by each resonator is U. Based on the above postulations and in combination with mathematical/statistical operations (Planck 1901; Trachanas 1981), Planck found that this average radiating energy U of each resonator at a given frequency ν is given by the equation:

${\displaystyle U={\frac {hv}{e{\frac {hv}{kT}}-1}}}$ (2)

(k =1.38110-23 JK-1 the Boltzmann constant, T the absolute temperature in K). Multiplying U by the number of produced standing waves (supposedly equal to the number of resonators) per unit volume per unit frequency () (Trachanas 1981) we get the spectral energy density u(ν) of the emitted radiation (energy per unit volume per unit frequency) which is the famous Planck’s law:

u(ν) = (3)

(c = λν, is the velocity of light in the vacuum or in the air, and λ the wavelength of produced waves). Since Planck’s law (Eq. 3) explained the prior experimental observations for the spectral distribution of the “black-body” radiation, and since its derivation was based on the assumption that the energy of each individual resonator at a given frequency ν is an integer multiple of the elementary amount hν, his assumption was considered to be correct. Then Planck calculated the value of the constant h based on earlier experimental measurements on the “black-body” radiation. Thus, Planck’s law was derived from experimental facts (the already known spectrum of “black-body” radiation) and his successful attempt (by combination of specific postulations and mathematical operations) to find a specific formula which can describe these facts. Planck’s law implies that the emission of electromagnetic energy by individual vibrating molecules is quantized in integer multiples of the quantity hν at each individual frequency3. This does not necessarily apply for the absorption or transmission of electromagnetic energy, since - theoretically - the cavity’s molecules may absorb EMR in quotients of hν, store it, and emit it in integer multiples of hν. Moreover, each emitted quantum of energy hν is not necessarily indivisible as a “particle”. Planck himself denied that the absorption or the transmission of energy is quantized and suggested that the resonator in general (not only at a specific frequency) possesses a non-integer number of quanta (Kuhn 1978; Brush 2007).

2.2. The Photoelectric Effect and the Wave-Particle Duality Principle

Planck’s hypothesis for the existence of elementary energy quanta was subsequently extrapolated by Einstein in his interpretation of the photo-electric effect (Einstein 1905a). Einstein postulated that the quantization does not only apply for the emission, but in addition for the transmission and absorption of light. Most importantly, he postulated that the light quanta are localized at points in space and transmitted in a single direction as (indivisible) “particles”, and thus emitted or absorbed only as a whole. This postulation was called “the Light Quantum Hypothesis” (LQH). The photoelectric effect was already discovered by Heinrich Hertz in 1887 and studied extensively by Philipp Lenard4: When a metal was illuminated by ultraviolet or even visible light, free electron emission occurred from the metal surface which could be transformed into electric current. The current intensity was proportional to the light intensity, but the energy/velocity of the emitted electrons was proportional to the frequency and not to the intensity of light (Lenard 1902). Einstein’s explanation of the photoelectric effect based on Lenard’s descriptions was summarized in his famous “photoelectric equation”5. In Einstein’s own words: “The simplest picture is one where the light quantum gives its entire energy to a single electron; we assume that this will occur….The kinetic energy of such electrons is…” (and wrote his famous equation):

hν = W + me u2 (4)

(W called “work function” is the necessary work - characteristic for the metal - for the release of a free electron from the metal, me u2/2 is the electron’s kinetic energy when leaving the metal surface, me the electron’s mass, and u the electron’s velocity at that moment). But then Einstein added: “However, it must not be excluded that electrons accept the energy of light quanta only partially” (Einstein 1905a). Thus, in the more general case, Equation (4) - as stated by Einstein himself - should be written as a condition:

hν ≥ W + me u2 (5)

recognizing the possibility that the electron may only partially absorb the energy hν of the incident photon. This condition is - in our opinion - extremely important, underestimated by physicists, and totally ignored in modern physics text books. In combination with Planck’s law, Einstein’s Condition means to us that, although the emitted energy is an integer multiple of h, its absorption does not necessarily follow the same quantization, and thus it can also occur in quotients of h. Although Einstein recognized the possibility that light quanta may be absorbed only partially as his Condition 5 states, in the same study (Einstein 1905a) he also stated his LQH asserting that Planck’s light quanta with energy E = hν behave as “particles” transmitted in a single direction and are “absorbed or generated only as a whole”. This hypothesis - of course - contradicts Condition 5 and this contradiction implies that Einstein himself was not sure on whether photons are absorbed partially or only as a whole (in other words he was not sure whether photons are divisible or indivisible quantities). However, strangely the subsequent evolution of quantum mechanics was based entirely on his postulation that photons are emitted, transmitted, and absorbed only as a whole, and totally ignored his Condition 5 as well as his related statement for partial photon absorption. Trying to reconcile the conflicts between the double-slit interference effects and his quantum theory which considered photons as “particles”, Einstein introduced in 1909 the “particle-wave duality principle” for light (Einstein 1909a; 1909b; Trachanas 1981) postulating that photons behave either as waves or as particles depending on the specific conditions. In our opinion, such a dual nature is unphysical and complicated.

2.3. An Alternative Explanation for the Photoelectric Effect

Robert A. Millikan in 1916 published a long paper with his experimental results on the photoelectric effect. Millikan’s experiment measured the maximum kinetic energy of the emitted photo-electrons by a metal surface. Knowing the frequency of the light that exposed the metal, he estimated the work function for the free electrons according to Einstein’s Eq. 3 for typical metals to be  2-5 eV (Millikan 1916) 6. However, he did not accept Einstein’s postulation for the “particle” nature of the energy quanta. In his own words: “Despite then the apparent complete success of the Einstein equation, the physical theory of which it was designed to be the symbolic expression is found so untenable that Einstein himself, I believe, no longer holds to it” (Millikan 1916; Wroblewski 2006). Millikan’s statement seems contradicting when declaring the “apparent complete success of the Einstein equation” and at the same time characterizing his theory “untenable”… But here there is another detail missing from modern physics text books: Millikan’s experimental setup was such that he was measuring not the kinetic energy of the photoelectrons in general, but their maximum kinetic energy. A brief description of Millikan’s experiment is as follows: A photocathode is irradiated by ultraviolet light which causes the emission of electrons and generates an electric current between the cathode and the electron collector (anode). The intensity of electric current is measured by a galvanometer. In addition, a voltage V is applied in the circuit such that the electrons are hindered in advancing towards the collector. When the voltage transcends a critical value, the energy of the applied field becomes higher than the kinetic energy of the emitted electrons (qe V > me u2) and no electrons arrive at the collector (qe, me, u are the photoelectron’s charge, mass, and velocity respectively). If however the voltage is regulated to a value Vo such that the current (as measured by the galvanometer) tends to zero, then qe Vo equals the maximum kinetic energy of the emitted photoelectrons (Millikan 1916). Thus, the kinetic energy measured by Millikan’s experiment was the maximum kinetic energy:

qe Vo = me umax2 = hν – W

Thus, Einstein’s photoelectric equation 4 must be re-written as:

hν = W + me umax 2 (6)

Considering that the ejected electrons are the free electrons on the metal surface and not inner electrons which require a greater work function, and that these electrons are ejected having a range of velocities u  umax, the general formula for the photoelectric effect must be written as Condition 5:

hν ≥ W + me u 2

Considering all these, it is not surprising at all that in 1912 Owen W. Richardson derived Eq. 4 for the maximum kinetic energy of the photoelectrons without any postulation for “particle” nature of light quanta, based purely on statistical and thermodynamic principles combined with the laws of classical electrodynamics. He treated electron emission as evaporation due to heat absorption. He applied Planck’s law for the spectral distribution of absorbed electromagnetic radiation, but he did not assume any corpuscular nature of the light, neither that each light quantum interacts with a single electron and transfers all of its energy to that electron. In Richardson’s own words: “It appears therefore that the confirmation of the above equation by experiment would not necessarily involve the acceptance of the unitary theory of light” (Richardson 1912; Wroblewski 2006; Jagielski 2009). We conclude that Einstein’s formula was actually found to be correct as an equation for the maximum electron velocity and as a condition in general, and thus, the postulation of light “particles” was totally unnecessary.

2.4. Photon Momentum, “Matter Waves”, and Complementarity Principle

According to Einstein’s special theory of relativity, only massless entities (e.g. photons) can travel at the speed of light. Photons are thus considered to have zero rest mass but they do have energy hν. Einstein’s relativistic formula relating the energy of a particle of rest mass mo with its momentum p (Einstein 1905b) is:

E = (7)

For the photon (mo = 0) the equation becomes:

E = cp (8)

In 1909, Johannes Stark explicitly attributed to the light quanta a momentum value p (Stark 1909) as this comes naturally from Eq. 8:

p ==  p = (9)

(For a particle at rest (u = 0, p = 0) with mass mo, we get from Eq. 7 Einstein’s famous formula for the energy corresponding to its (rest) mass: E = moc2). Then, Louis de Broglie in his doctoral thesis in 1924 based on Einstein’s wave-particle duality of light, assumed that just as light exhibits both corpuscular and wave behaviour, subatomic particles such as electrons possess in addition a wave nature (De Broglie 1924). De Broglie ascribed a wavelength on elementary particles in accordance with Eq. 9 for the photon momentum:

λ =  	(10)


(m, u the particle’s mass and velocity respectively).

His hypothesis was very soon confirmed experimentally when it was shown that electrons produce diffraction patterns just like x-rays (Davisson and Germer 1927). (In our opinion, it is easier to comprehend a (sub)atomic particle -moving naturally at speeds approaching the speed of light- exhibiting a wave-like behaviour, than a photon being a “particle”). Finally, Niels Bohr7, after the discovery of the Compton effect, abandoned his strong opposition to Einstein’s corpuscular photon model and introduced the “Complementarity principle” according to which: “Wave and particle are two aspects of describing physical phenomena, which are complementary to each other. Depending on the measuring instrument used, either waves or particles are observed, but never both at the same time, i.e. wave- and particle-nature are not simultaneously observable” (Bohr 1928). Thus Bohr’s Complementarity is very close to Einstein’s Duality.

2.5. The Compton Effect

What is considered as the principal experimental “proof” that light quanta behave as indivisible “particles” (and as such transmitted in a single direction and not only emitted but also absorbed only as a whole, and carrying momentum in addition to energy), was provided in 1923 by Arthur H. Compton’s experiments with scattering of x-rays by electrons on a material (graphite) (Compton 1923a; 1923b). In those studies, Compton adopted Einstein’s postulation that the quanta behave as “particles” with total energy hν and have a momentum given by Stark’s Eq. 9. Moreover, Compton postulated (in his own words), “that each quantum of x-rays is scattered by an individual electron”, and that “it spends all of its energy and momentum” during its interaction with this individual electron. The electron due to this interaction recoils and leaves the material with a velocity u (Compton 1923a; 1923b). Compton considered the interaction at a reference coordinate system/frame where the electron is still before the interaction. At such a system the velocity of the photon is again c according to Einstein’s Special Theory of Relativity (Einstein 1905b). Then, applying the energy and momentum conservation principles, he showed that the increase in the wavelength of the scattered ray should be given by the equation:

λ - λo = (1-cosθ) (11)

(where λο, λ the wavelengths of the primary and scattered rays respectively, mo the electron’s rest mass, and θ the angle between the scattered and the primary rays). The corresponding equation in terms of frequency modification is:

-  = (1-cosθ) 	(12)


(where νο, ν the frequencies of the primary and secondary photons), and in terms of photon energy modification:

hν = (13)

Let us see how Compton reached Eq. 11. Assuming the photon to be traveling along X-axis before collision, Compton applied the energy and momentum conservation laws as follows:

hνο + moc2 = hν + mc2 (14) (Energy conservation)

= cosθ + mu cosφ                          (15) (Momentum conservation along X-axis)


0 = sinθ – mu sinφ (16) (Momentum conservation along Y-axis)

In addition he applied Einstein’s relativistic equation for the electron’s mass (Einstein 1905b):

m = (17)

(νο, ν the frequencies of the primary and secondary photons, mo, m the electron’s mass before and after interaction with the photon, u the electron’s recoil velocity after interaction with the photon, and θ, φ the angles of the secondary ray and electron recoil respectively, with the direction of the primary ray). Squaring both sides of Eq. 14 and multiplying by c2, we get

m2c4 = h2(νο2 + ν2 – 2 νο ν) + 2mοhc2 (νο – ν) + mo2c2 (18)

Squaring both sides of Eq. 15 and 16, adding together and multiplying by c2, we get

m2u2c4 = h2c2(νo2 + ν2 – 2 νoν cosφ) (19)

Then, combining Eq. 17, 18, 19 we get Eq. 11 (Compton’s law). The experimentally detected increase in wavelength of the “modified” ray in regard to the primary ray was found to be in satisfying although not complete agreement with Compton’s Eq. 11 (Mo-K x-rays scattered by graphite at π/2 angle from the primary direction, had an observed wavelength increase λπ/2 – λο = 2.2×10-8 m, while the corresponding increase computed by Eq. 11, is 2.4×10-8 m). Overall, for scattering angles 0-180 the wavelength increase predicted by Compton’s theory was in satisfying agreement with the experimental results, and the recoiled electrons although had not been observed but only predicted by Compton (Compton 1923a) were soon detected by Charles Wilson with use of his “cloud chamber” (Wilson 1923) 8. Since Compton assumed that an electron absorbs completely one primary photon (as if the photon were a particle), the verification of his equation by the experiment in combination with the detection of recoiled electrons by Wilson, was interpreted as a verification of the photon’s particle nature. However, we should note that Compton’s postulation that each photon interacts with a single electron only and spends all of its energy on that electron only, is a simplification, not to say an oversimplification. Moreover, there are some additional data in Compton’s experiments which are usually ignored or underestimated in physics text books: a) The acquired spectra of the scattered ray always contain not only the scattered ray with the longer wavelength (“modified” scattered ray) but in addition a scattered ray with the same wavelength as the primary ray (“unmodified” scattered ray). The energy and momentum of this unmodified ray was not taken into account in Compton’s calculations although the intensity of this “unmodified” ray was even stronger than that of the “modified” one at small angles (up to 45). It was considered that the “unmodified” photons are scattered without any loss of energy by inner electrons - which possess a greater work function - as was suggested by classical physics (Thomson scattering) (Compton 1923a). At each different scattering angle the two scattered rays (“unmodified” and “modified”) were both detected, meaning that there was a whole spectrum of scattered rays, not just two rays. b) Eq.11 does not give information regarding the total amount of energy distributed in both “modified” and “unmodified” scattered rays neither the minimum amount of photon energy required for the effect, but only partial information regarding the change in wavelength of a single photon in the “modified” part of the scattered radiation. Moreover, it is noteworthy that Eq. 11 states that this change in wavelength is independent of the incident photon’s energy/frequency/wavelength, and depends only upon the scattering angle. [Thomson scattering refers to excitation of an atomic electron by an incident electromagnetic wave. The wave induces a forced-vibration on the electron but due to unstableness the electron immediately re-emits the absorbed wave in a random direction with the same frequency (i.e. without loss of energy: hν ≈ hνo). According to Eq. 13, this phenomenon predominates in relation to Compton scattering when: hνo  mo c2, or at very small scattering angles (for which cosθ  1) (Klimov 1975)].

2.6. An Alternative Explanation for the Compton Effect

In contrast to widespread information through most physics text books, Compton’s Eq. 11 does not necessarily prove that a photon is totally absorbed by an electron. To show this, instead of postulating that incident photons are “particles” and each one is totally absorbed by an electron, let us postulate that the primary incident photons - being wave-packets - are partially absorbed by single electrons. More specifically, let us postulate that only 1/n part of a primary photon energy and momentum interacts and is scattered by an individual electron (1/n of the part of spherical surface corresponding to the whole photon’s wave-front according to our suggested model). Moreover, we postulate that the scattered photon is not a different photon than the incident one, but the same incident photon (the same part the spherical surface) due to “collision” with an electron is scattered with a different frequency. Applying again the energy and momentum conservation principles exactly the same way as Compton did, we come to Eq. 11, without the postulation that each photon is a particle and absorbed as a whole by an electron. [The fact that we apply the momentum conservation principle ascribing momentum to the photon does not mean that it has to be a particle. As explained already (Eq. 9) its “mass” is due to its energy (m = )]. Let us examine this in detail. Assuming that only 1/n part of a primary photon hνo (e.g. 1/n of the whole part of spherical surface corresponding to the whole wave-packet) interacts with a single electron and gets scattered with an altered frequency ν, Eq. 14, 15, 16 become respectively:

+ moc2 =  + mc2 	(20)

= cosθ + mu cosφ 	(21)


0 = sinθ – mu sinφ (22)

Applying also Eq. 17 and performing exactly the same operations as described before (section 2.5), we reach again Eq. 11 without postulating the photon to be a particle that transfers all of its energy into a single electron. The above considerations make clear that Compton’s hypothesis that one photon is totally absorbed by a single electron, is not only arbitrary but also unnecessary, and that Eq. 11 is only restricted in the energy exchange of a “modified” photon scattered at a certain angle and does not refer to the energy exchange during the whole phenomenon in all scattering angles. By modifying his hypothesis and not considering the photons as particles, we reached the same equation for the increase in the wavelength of the “modified” scattered rays. We showed that the interaction of a part of the incident photon () instead of a whole photon (hνο) results in the same change in wavelength of the modified ray. This is in agreement with our remark that the change in wavelength in Compton’s Eq. 11 is independent of the incident photon’s energy, and depends only upon the scattering angle. Moreover, even though hνo is not negligible compared to moc2 for x-rays, and thus the probability for Thomson scattering is small, there is abundant co-existence of the “unmodified” secondary spectrum in the Compton effect, not only at small scattering angles (θ 45) where the unmodified ray is expected to predominate, but also at large scattering angles (90-180) where this ray still has a significant intensity according to experimental results presented by Compton himself (Compton 1923b). However, if Wilson had shown that all the recoiled electrons have energies corresponding to the absorption of a whole x-ray photon, that would have been a proof of Compton’s and Einstein’s postulations. But this is not the case either. A close look at Wilson’s studies shows that Wilson detected a whole range of “long” and “short” electron paths in his cloud chamber which do not undoubtedly correspond to whole x-ray photon absorption. Thus, while according to Compton’s calculations the maximum kinetic energy of a recoiled electron having absorbed a whole x-ray photon is ~300 keV (corresponding to a maximum electron velocity, umax = 0.82c), Wilson detected electrons the vast majority of which had energies significantly below 20 keV (Compton 1923b; Wilson 1923; 1927). This can be interpreted that a few electrons absorb a whole x-ray photon while the vast majority of recoiled electrons absorbed quotients of a photon. This is in agreement with our suggested photon model, and the fact that x-rays (and consequently photons) are significantly more localized (confined in space) than infrared, visible, or ultraviolet. Thus, in reality Wilson’s experiments indeed verified the ejection of electrons by the x-rays, but not necessarily Compton’s postulation that photons are absorbed “only as a whole” by each recoiled electron. Thus, the Compton effect is erroneously regarded as the “proof” for the photon’s “particle” nature. Compton discovered and explained the change in the wavelength/frequency/energy of the modified ray, as well as predicted the ejection of electrons from a material by x-rays (which were extremely important findings anyway), but nothing more than that. The existence of a whole spectrum of scattered photons with unmodified wavelength as classic physics would predict (Thomson scattering) shows that classical interpretation is also partly valid, since about half the number of the primary photons interact with electrons and get scattered without any loss of energy. Even if we consider that all the “unmodified” photons interact with inner electrons for which a greater work is necessary to be offered in order to be extracted from the atom, the x-ray photons used in Compton’s experiments are produced by de-excitation of inner electrons and thus they carry sufficient energy to expel inner electrons. Moreover, they certainly carry abundant energy to expel outer electrons. Why should we accept that they do not interact with outer electrons but in contrast, they leave the outer electrons unaffected, penetrate and interact with inner electrons? Isn’t this a less likely scenario? But according to our suggested SCWP model the reasoning gets significantly simpler: A fraction of the x-ray photon energy may well be not sufficient for expelling inner (or even outer) electrons, so it is simply reflected without energy exchange. Thus our suggested photon model may possibly provide a more realistic interpretation for the abundant “unmodified” spectrum in the Compton effect. A possible explanation for the whole spectrum (“modified” and “unmodified”) acquired in Compton’s experiments in accordance with our suggested photon model (SCWP) can be that, one part of an incident photon which carries sufficient energy gets scattered by an electron (which recoils) and gets modified (), while a smaller part of the same (or of another) photon - corresponding to insufficient amount of energy (n2 > n1) - is simply reflected by another electron without energy exchange. By the above alternative derivation of Compton’s equation, we do not prove that photons are indeed partly absorbed by individual electrons, but we do show that this is possible, and in addition, this hypothesis possibly offers a more realistic explanation for the existence of the “unmodified” spectrum of the scattered rays. In other words, we show that Compton’s equation does not necessarily prove that photons behave like “particles” as was erroneously interpreted and spread widely in text books. Thus, it is clear that neither the photoelectric nor the Compton effect necessarily indicate or even more prove the “particle” nature of photons, as dogmatically claimed in modern physics text books. This conclusion has been independently stated by other investigators as well, following very different approaches than ours (Milonni 1997; Brush 2007; Jagielski 2009).

2.7. Wave-Function and Schroedinger Equation

In 1926 the Austrian physicist Erwin Schroedinger in an attempt to describe mathematically the matter waves associated (according to De Broglie) with an elementary particle (e.g. an electron in an atom), introduced the quantum mechanical “wave-function”

(r, t) = ei(pr-Et)/ ħ (23)

(where: ħ = h/2π) as this is directly deduced from the classical wave-function

ξ(r,t) = ei(kr-ωt)

that describes a plane wave of circular frequency ω=2πν (ν the frequency) and wave-number k = 2π/λ (λ the wavelength), at distance r from its source along the direction of propagation (Alonso and Finn 1967). [i is the imaginary unit (i2 = -1)]


The square of the wave-function 2 supposedly describes the probability for the particle to be localized at the position r at a certain instant t (e.g. the probability for an electron to be found at distance r from the nucleus at a given instant). That was arbitrarily accepted in analogy with classical wave physics (again) where the square of the oscillating quantity is proportional to the energy density of the wave. Equation 23 comes directly from the classical wave function, by substituting ω and k with their corresponding quantum mechanical expressions (derived directly from Planck’s and De Broglie’s Equations 1 and 10 respectively):

ω = E/ħ and k = p/ ħ Differentiating Eq. 23 with respect to r and t, we get correspondingly:

-i ħ(/r) = p (24)

and i ħ(/ t) = E (25)

Thus, the operator –iħ( / r) corresponds to the momentum of the particle, and the operator iħ( / t) corresponds to its energy. In classic physics the total (conserved) energy value E of a particle with mass m and momentum p moving in a potential V(r) is given by the Hamiltonian function

H(r, p) = p2/2m + V(r) 	(26)


and it is the sum of kinetic and potential energy. Thus, the equation

E = p2/2m + V(r)


expresses the energy conservation law. Since the wave-function (r, t) was introduced to represent the wave associated with the particle under study, Schroedinger demanded a-priori that it must satisfy the equation:

E = (p2/2m) + V(r) 	(27)


Substituting (a-priori again) in the last equation the energy and momentum by the corresponding operators from Equations 24, 25, we get:

i ħ( / t) = - (ħ2/2m)( 2/ r2) + V(r) (28)

Thus, Equation 28 describes the energy conservation law for a “matter-wave” at position r and it is called the Schroedinger equation in one dimension. For a particle/”matter-wave” moving in three dimensions the Schroedinger equation becomes:

i ħ= -2 + V(r) 	(29)


(where: 2 =  2/ x2 +  2/ y2 +  2/ z2 the Laplace operator) In a space free from any fields, V(r) = 0, and the Schroedinger equation becomes:

i ħ= -2 	(30)


According to Eq. 29, the quantum mechanical wave-function cannot be a Real function since in such a case the first part of Eq. 29 would be imaginary and the second part real which is impossible. Thus,  does not represent a real wave that can be physically observed (measured/detected) as with classical waves in which the wave-functions are Real functions. It is a complex function possessing a real part and an imaginary part, and represents a “probability wave”. Its square supposedly represents the probability density for the particle to be found at a certain location. The Schroedinger equation9 (29) describes the energy conservation for the transmission of a “matter-wave” associated with an elementary particle. It was arbitrarily introduced without proof by Schroedinger based on De Broglie’s assumption that elementary particles posses a wave nature. As described, Schroedinger took the classical wave-function, substituted ω and k by their quantum mechanical expressions given by Planck and De Broglie respectively, and combined (multiplied) his quantum mechanical wave-function with the energy conservation law for a particle. Surprisingly, his equation was successfully applied to describe the movement of elementary particles, such as the electrons in an atom. Its solutions are certain expressions for the wave-function the square of which represents the probability of finding the electron at a specific location around the nucleus at any moment. For the simplest case of the hydrogen atom with a single electron, Schroedinger equation gives the energy values for the electron predicted by Bohr’s theory. The quantum numbers introduced by Bohr and Sommerfeld empirically to describe the energy states of the electron in the hydrogen atom (shape of the electron “cloud”, or shape of the different orbitals) can be deduced by solving the Schroedinger equation for this atom. For multi-electron atoms the equation becomes very complicated to be solved and we accept that the electrons occupy successive electronic states of the hydrogen atom.

2.8. Uncertainty Principle

Let us now consider a number of plane waves transmitted in a certain direction (e.g. along the X-axis). These waves are superimposed to each-other forming a resultant wave-packet. This wave-packet includes a range of circular frequencies Δω and a range of wave-numbers ΔkX, and thus includes a temporal and spatial dispersion Δt and Δx respectively which can be shown classically (Alonso and Finn 1967; Tarasov 1980) that satisfy the conditions:

Δkx Δx ≥ 1 	(31)


and Δω Δt ≥ 1 (32)

These relations are well-known in classical wave-physics. Based on these relations and substituting ω and k by Planck’s and De Broglie’s equations (ω = E/ħ and k = p/ ħ), the German physicist Werner Heisenberg10 stated in 1927 his famous “Uncertainty Principle” according to which, the simultaneous accurate measurement of the position and momentum (or velocity) of a particle, is not possible. Equivalently, the simultaneous accurate measurement of a particle’s energy and time is similarly not possible:

Δpx Δx ≥ ħ 	(33)


and ΔE Δt ≥ ħ (34)

(where: Δx, Δpx the uncertainties in the measurement of the particle’s position, and momentum respectively, and ΔE, Δt the uncertainties in the measurement of energy and time for the particle). As we saw, Conditions 33, 34 are direct transformations of the classical conditions 31, 32 respectively. Position and velocity (or momentum) of a point particle are conjugate independent quantities from each other and their combined notion determines the kinetic condition of the particle. Similarly energy and time are also conjugate independent quantities. Condition 33 can also be written as:

Δux Δx ≥ ħ /m 	(35)


(where Δux is the uncertainty in the measurement of a particle’s velocity, and m the particle’s mass). Condition 35 implies that for large masses (macroscopic objects) the uncertainty is negligible, but for elementary particles it becomes significant. What actually the uncertainty principle declares is that when we refer to elementary particles such as electrons (and to their corresponding wave-functions) there is no meaning to talk about a specific position or velocity or energy of such a particle, but we can only talk about probabilities for specific values of these quantities. The Schroedinger equation and the uncertainty principle form the basis of modern quantum mechanics. It is certainly noteworthy that both were derived directly from corresponding classical relations by simply substituting ω and k by their quantum mechanical expressions given by Planck and De Broglie.

2.9. The Spreading of the Wave-Packets according to Quantum Mechanics

Based on Schroedinger’s equation for a wave-packet (associated with an elementary particle of mass m) transmitted in the free space (V(r) = 0), it is shown that the uncertainty in the position increases with time (Tarasov 1980; Trachanas 1981). Thus, if the initial uncertainty is (Δx)o, after time t the uncertainty in position (Δx)t is shown to be given by the expression:

(Δx)t = (Δx)o  	(36)


Equation 36 shows that the uncertainty in the particle’s position constantly increases with time, in other words the wave-packet - describing the probability for the particle to be detected at a specific point in space - constantly spreads. Moreover it shows that the spreading increases proportionally with time, in other words the wave-packet spreads within a constant divergence angle. This is in agreement with our suggested photon model even though our model does not refer to a particle but directly to a wave-packet which is considered to be the photon itself.

2.10. More recent Experiments on Photon Nature

Recent considerations supporting the “particle” nature of photons are based on experiments with light-detectors (photomultipliers) giving an electric pulse whenever monochromatic light shines on them, and when the light gets dimmer the pulses become fewer but remain just as strong (Feynman 1985). But these results do not necessarily constitute a proof for a particle nature either, because: 1. Light sources used in laboratories usually are directional, sending light in a specific direction just like a directional antenna or waveguide confines its electromagnetic wave emission within a narrow beam. In such a case, similar quantities of energy from the directional source reach the detector which is usually directional as well. The result is then pulses of equal intensity. Indeed, electromagnetic waves are not elastic/mechanical waves which spread their energy in every direction within a material medium because of the elastic forces between adjacent molecules in all material media. In contrast, electromagnetic waves may be emitted in all directions from a point source or a wire antenna, but they can also be confined within narrow beams in certain directions without significant spreading in other directions (e.g. lasers, hard x-rays, gamma rays, polarized signals from directional antennas). 2. Photodetectors respond by giving electric pulses of equal strength for every incident photon with energy hν between a lower and an upper limit:

ΔEmin = hνmin ≤ hν ≤ hνmax = ΔEmax 	(37)


The min and max values correspond to light-induced electronic transitions from the valence band of the photomultiplier’s semiconductor p-n junction to any level between the lower and upper limits of the conduction band (Roychoudhuri and Tirfessa 2008). Thus, equal pulses of photoelectron emission do not necessarily correspond to photons of equal energy/frequency, even if we accept that photons are indivisible packets of energy or “particles”. This is further supported by the fact that linearity of photomultipliers’ response at very low light intensities (below 104 photons per sec, corresponding theoretically to single photon detection) was not carefully investigated before the early eighties, and when investigated it was found that the response is not linear (Panarella 1987; 2008). This very simply means again, that in “single-photon” detection, higher photon energy does not necessarily produce a more intense pulse. Other investigators have also questioned the one-to-one correspondence for photoelectron emission (Lamb and Scully 1969; Lamb 1995). Actually what can really be considered as experimental proof for the particle behavior of photons - except for the Compton effect which finally as we showed does not constitute such a proof - is relatively very recent and comes from the work of Grangier et al (1986). This study employed “single photons” emitted by a non-classical source (radiative cascade originating from excited calcium atoms), and either reflected by or transmitted through a beam-splitter. Then the reflected and the transmitted photons were detected by two separate photomultipliers (one for the reflected and one for the transmitted parts of the initial beam)11. The first experiment of the study showed a particle-like behavior. More specifically, it showed that single photons do behave like particles since their detection after the beam splitter by the two photo-detectors presents anticoincidence. The second experiment of the same study involved interference test and showed that the reflected and the transmitted beams produce interference with each other, which is a wave-like behavior (Grangier et al 1986). This result contradicts the observed anticoincidence since it implies that single photons may split in two parts by the beam-splitter which then produce an interference pattern with each other. It may simply mean that photons are actually divisible wave-packets, and that electromagnetic energy can exist in quotients of a photon, and that would support our hypothesis that a photon may be partially absorbed. The authors’ interpretation of the above results was based on the assumption that the beam-splitter (a half-silvered mirror) splits equally the beam in a 50:50 manner which is a simplification (if not an oversimplification) especially for the short durations and low intensities applied in the experiments in order to have “single-photon” beams. This in combination with the limited efficiency and noise level of the detectors and the non-linearity of photomultipliers’ response at the low intensities of “single photons” which were not taken into account (but they are of extreme importance) could favor the anticoincidence result. Finally, if the experiments involved a classical light source, such as e.g. a laser, perhaps the results would be different (Panarella 1987; 2008; Jagielski 2009). Indeed, if the “half”-silvered mirror (“beam-splitter”) does not split accurately each incident wave-packet into two equal parts (which is the most likely, especially for such low intensities corresponding to “single photons”), this in combination with the limited photodetectors’ efficiencies (which may also differ between each other) will most likely result in anticoincidence. A probable scenario considering the photon as a wave-packet, is that the beam-splitter splits each photon into a bigger and a smaller part, or simply does not split it at all but simply some photons are reflected and some are transmitted resulting - of course - in complete anticoincidence. In the case of splitting of the wave-packet, if the smaller part is below the photodetector’s threshold it will not be detected while the bigger part will be. Such an event would be counted as an “anticoincidence” result, while in reality it is not. Even when the splitter splits each photon into two (unequal) parts which are both detected, the two separate detectors must have identical temporal response and identical efficiency in order to show coincidence. Finally, any detection of stray photons by the photomultipliers will further enhance the “anticoincidence” result. The degree in which these parameters are controlled in “coincidence photon experiments” determines the results. A recent attempt to reproduce the results of the Grangier et al (1986) experiments with a “classical” light source (coherent He-Ne laser beam) did not verify the anticoincidence effect) (Jagielski 2009). Other experiments of the Grangier et al group with faint light (attenuated light pulses from light emitting diodes - LED) at the level of 0.01 photon per pulse did not indicate a particle-like behavior (Aspect and Grangier 1991). In other more recent experimental attempts to produce interference pattern on a screen by “single photon” emission the interference pattern was observed. Covering the one slit, the interference bands disappeared (Fox 2006; Grimes and Grimes 2005; Ficek and Swain 2005; Loudon 2003). When a detector (photomultiplier) was added in one or in both slits to detect the incident photons, the interference pattern on the screen disappeared again, in other words, “detecting the photon” at the slit removes the effect (Fox 2006; Ficek and Swain 2005; Loudon 2003). All these experimental facts support our hypothesis that any single photon is not transmitted in a single direction as a “particle” but transmitted as a (divisible) wave-packet in multiple directions within a solid angle, and thus different parts of the same photon go simultaneously at the two slits. In regard to “single photon emission” our opinion is that this cannot be determined simply by decreasing the intensity of light emitted by a source after or during the emission. All light sources emit large numbers of photons at any instant. Decreasing the rate of emission on a LED or a laser, or decreasing the intensity after the emission by filters at a level corresponding theoretically to single photons, does not prove that the source indeed emits single photons at any given instant, neither that the detectors actually receive discrete single photons. They certainly receive light of intensities corresponding to single photons (or even less). These detected intensities may be the sum of quotients from several photons, which is in agreement with a divisible (wave-packet) photon character. Supporting to this opinion are other experiments suggesting that the emission of a single photoelectron actually requires the simultaneous presence of several (at least four) photons (Panarella 1987; 2008) Thereby, even the more recent experimental results allegedly supporting the corpuscular photon model are not totally persuading.

2.11. Interference Patterns by Interferometers can be Explained Postulating a Photon Wave-Packet Character

3. A Suggested New Photon Model

4. Other Suggested Wave-Packet Photon Models

It seems that not everybody in the quantum physics community accepts the established corpuscular photon model. A wave-packet photon model is offered by Hunter and colleagues (Hunter and Wadlinger 1989; Hunter et al 2008). According to this suggested model which originates as a monochromatic solution of Maxwell’s equations, the photons are monochromatic wave-packets traveling rectilinearly at the speed of light, having the shape of a circular ellipsoid in the case of “circularly polarized states” with its long axis parallel to its propagation and equal to its wavelength λ, and its diameter (small axis) equal to λ/π. This ellipsoid wave-packet called “soliton” carries energy hν, and is surrounded/accompanied by an “evanescent” wave of zero energy which according to the authors is responsible for the production of the interference phenomena. According to this model, while the “soliton” is spatially confined within the ellipsoid and thus it passes through the one slit in a particle-like manner, its “evanescent” wave extends over and interacts with both slits. In spite of the offered explanation, our opinion is that this model is too complicated to be real. Moreover, how a single photon can ever be circularly polarized? Circular polarization is the result of superposition of two fields/waves (photons in this case) of identical energy and linear polarization with a phase difference 90 between them, or the result of superposition of three fields/waves of identical energy and linear polarization, with a phase difference 120 between each two of them (Alonso and Finn 1967). Moreover, the linearly polarized superposed fields must be in specific geometrical arrangement preserved during transmission. Another wave-packet photon model which is simpler and perhaps more relevant than the “soliton” model was recently proposed (Roychoudhuri and Tirfessa 2008). According to this model, the photon is also a time-finite and space-finite wave-packet of energy hν, which constantly expands during its transmission but kept also within an angle that remains constant and has “the physical shape of a Gaussian spatial wave-front and semi-exponential temporal envelope”. The authors combine their suggestion with recent direct measurement of kinetic energy of electrons detached from neon atoms in the presence of an intense laser field. The measured energy was in the form of Gaussian-shaped sinusoidal pulses allegedly corresponding to single photons, and each one containing approximately five periodical oscillations (Goulielmakis et al 2008). In our opinion it is not clear whether the detected undulations are indeed related with single photons. One objection to both of the above wave-packet photon models is that, if the photon consists of a single wave/undulation, how can the amplitude of this undulation increase infinitely as the wave-packet expands, while its energy hν remains constant? Our suggested SCWP photon model offers a solution in this problem by considering not a single wave/pulse but a bundle of adjacent identical waves. The bundle/photon expands by increasing the distances between adjacent waves. One similarity between our SCWP model and the Gaussian-shaped pulse model (Roychoudhuri and Tirfessa 2008) is that both consider propagation within a divergence angle that remains constant. Further, the interference patterns acquired by the Michelson and Mach-Zehnder interferometers cannot be explained by the established corpuscular light model, but neither by the “soliton” model (Hunter and Wadlinger 1989; Hunter et al 2008) or the Gaussian-shaped pulse model (Roychoudhuri and Tirfessa 2008). In this case we do not simply have an interaction between the photon and two slits which may be possible for an “evanescent” wave or a single sinusoidal Gaussian-shaped pulse, but we have two very different paths (which may be separated by great distances) to be followed by the two parts of light before rejoining again on the screen. It is unlikely for an “evanescent” wave with zero energy to split from its “soliton”, or a single Gaussian pulse to split in two parts which will travel these two different/separated paths simultaneously without dissociating from each-other. This is also admitted by Hunter et al (2008). Finally, the observed “coherence length” in Michelson interferometer cannot be explained with the “soliton” model, since this model suggests that the photon (“soliton”) length is equal to its wavelength (which e.g. for visible light is λ  3.8-7.6×10-5 cm), and thus about 105-106 times smaller than the observed “coherence length”. Similarly cannot be explained with the Gaussian-shaped pulse model which suggests a length five times as the wavelength.

Discussion and Conclusion

“hν ”

All black body radiation All the spectrum variations All atomic oscillations Vary as “hν”

. . .


There would be a mighty clearance, We should all be Planck’s adherents, Were it not that interference

Still defies “hν”


I take the freedom to revise slightly the poem in order to adjust it with the ideas of this chapter, hoping that Dr Shearer would have no objection:

“Particle Light”

All black body radiation(s) All the spectrum variations All atomic oscillations Vary as “particle light”

. . .


There would be a mighty clearance, We should be Albert’s adherents, Were it not that interference Still defies “particle light”

Again, these are the thoughts of a radiation biophysicist, not a quantum physicist. Maybe most quantum physicists will find these thoughts “strange” or even wrong. But sometimes someone coming from “outside” may have a more clear view than someone who is deep “inside” a difficult physical (and philosophical) problem among countless details. I would be happy if I offered a relevant alternative view on an unsolved paradox in addition to that of most experts in the particular field. Or if I simply offered some pleasant time to the reader of the present essay even in case that the points raised are indeed irrelevant.

“Dr Panagopoulos’ study is definitely significant, even though there are only a couple of spots where I do not completely agree with him. His analysis related to Einstein’s photoelectric equation, Compton Effect and the brief representation of Schroedinger’s equation and the related comments are excellent for the next generation researchers. His general approach is quite objective and shows his deeper thinking is anchored in exploring the real physical interaction processes taking place in nature; in contrast to simply rushing to validate the measurable data by simple-minded mathematical equations. Our thinking diverges on the details of the physical propagation mechanism of the “Spatially Confined Wave-Packets” after they have been emitted by atoms and molecules. QM is correct that any spontaneous emission starts with a discrete amount of energy ΔE = hν. My current understanding converges with that of Dr. Panagopoulos that this energy packet evolves as a wave packet with a classical time-finite amplitude envelope containing a carrier frequency ν. We differ after this point. I think this wave packet propagates diffractively following Huygens-Fresnel’s diffraction integral since the integral is mathematically a solution of the Maxwell’s wave equation and it is continuing to be the bedrock behind the progress of optical science and engineering from the beginning of its mathematical formulation by Fresnel. In his original book, Huygens clearly stated that his postulated secondary wavelets do not interfere (interact) with each other; which I have generalized in my recent book, “Causal Physics”. The book details how all optical phenomena are strictly causal when we simply eliminate many unnecessary self-contradictory and non-causal postulates that are still prevalent in modern books. Divergences and diversities in our thinking are healthy. Science is always a work in progress. We never have a final theory, unlike religions.”

Professor Dr. Chandrasekhar Roychoudhuri,
(Photonics Laboratory, Physics Department,
University of Connecticut, USA)


Acknowledgments

Many Thanks to Dr C. Roychoudhuri, and Dr G. Fanourakis for critical reading of the manuscript and valuable discussions.

References