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The Use and Abuse of the “photon” in Nanomechanics
David P. Norwood
Southeastern Louisiana University, Hammond LA 70402, USA [email protected]
Abstract
In this paper, I decry misuse of the word “photon”, primarily because of the misleading notions it creates
in students and physicists alike. Like Dr. Lamb [Lamb, 1995] I blame the “particle-like” nature of the
word “photon” for planting in the minds of impressionable students a host of “particle-like” ideas about
EM radiation: ideas that are misleading or wrong, and persist into the young physicist’s old age. I begin
by reviewing examples of particularly egregious misuse of the photon. I review the “semi-classical” and
“neo-classical” approximations, and the conventional quantum theory of radiation, upon which the word
“photon” is based. I discuss applications of classical and quantum theories of radiation to experiments
involving the interaction of light with matter. I call attention to cases where commonly held opinions
regarding the QTR are misleading or wrong. I finally discuss the relevance of this to the field of
nanomechanics.
Keywords: QED, semiclassical, radiation-reaction,
1. Introduction – What’s wrong with ‘photon’?
Misuse of word photon falls into two classes: pretentious and wrong. An egregious example of the
former would be the phrase “stream of visible photons”• instead of “light”. This is not factually wrong,
but it is unfortunate that physics, a field that seeks to describe the world in the simplest terms, should
produce such a tortured reference. Such examples abound, and no more need be said, except perhaps to
paraphrase Albert Einstein and say that the language used to describe a physical effect should be as
simple as possible, but no simpler. I move on to discuss various statements about photons that the author
feels are simply wrong (or at best misleading).
1.1 Photons are small and localized
This statement can be traced at least as far back as Einstein’s discussion of the production and
transformation of light [Einstein, 1905], in which he describes light as consisting of “independently
moving points” and “energy quanta localized at points of space”. More recent authors have gone further
and characterized photons as “point particles”. In fact, neither the classical or quantum theories of
radiation provide a means to even discuss the size or “position” or an electromagnetic wave. It is almost
unfortunate that the symbol typically used as an index or label for the electromagnetic field, the symbol r,
is the same as that used to refer to the position of an actual material particle. Unfortunate since they play
such completely different roles in reference to the two different entities.
But in fact, this notion (of small corpuscles of light) might not have survived had the Nobel
committee not chosen to mention particularly Einstein’s description of the photoelectric effect when
awarding him the Nobel Prize in 1921. (Consider that his description of the Stokes shift, mentioned
immediately before the photoelectric effect in the same paper, and explained on the same basis, is seldom
mentioned).
1.2 Photons are required to understand the photoelectric effect

In this introduction, I include quotes without references in order to protect the guilty.
This is simply one example of a great many physical effects that are taken as proof that the
electromagnetic field must be quantized. To be sure, there are physical effects that must be described by
a quantum theory of radiation. But as is discussed in the section on applications, the photoelectric effect
and many others can be described at least qualitatively, and in many cases, quantitatively by a classical
theory of electromagnetism coupled with a quantum theory of matter. This is the so-called semiclassical
approximation and is described later.
Particularly entertaining is a statement in the introduction of a quantum mechanics textbook that “the
frequency dependence of the [photoelectric] effect is not comprehensible within that framework [i.e., in a
classical theory of electromagnetic waves]”, a textbook that goes on to present a full description of the
photoelectric effect using a classical theory of electromagnetism.
1.3 There is no delay in photoelectron emission with decrease in intensity of radiation
This statement is simply untrue. While for large intensities, the onset of photoelectron flux begins
immediately and the flux scales with the intensity, for sufficiently low intensities, the time between the
emission of photoelectrons becomes longer and longer. These are the “photon-counting” experiments.
And this shows that there IS a delay in order for an electron to accumulate enough energy to be liberated
from the bound state.
1.4 A photon wave function collapses to transfer energy
This point is sometimes raised when discussing the absorption of electromagnetic energy by an electron.
As the electron is so much smaller than the “photon” (its size taken to be it’s wavelength), there must be a
“collapse” in order for the energy to be transferred. Again, this simply fails to understand classical
electrodynamics. When subjected to an incident EM wave, an electron will generate a dipole field that,
when added to the incident field will “channel” the electromagnetic energy to the dipole. An electron
absorbing energy from visible light is no more mysterious than a miniature radio with a 10 cm antenna
absorbing energy from a radio wave with a 1000 m wavelength.
1.5 Photons are required if light is to carry and transfer momentum
This very common assertion simply reflects a poor understanding of classical electrodynamics. It is well
understood that a classical electromagnetic wave can carry and transfer energy, momentum and angular
momentum. And the energy and momentum of classical fields (in a unit volume) even satisfies the
relation often touted as quantum mechanical: E = pc.
1.6 Photons are required to explain light scattering anisotropically
This claim is typically mentioned in the context of the Compton effect, in which light incident upon a
solid or molecule is scattered in a well-defined direction, rather than isotropically. The claim is made that
a classical theory cannot explain the Compton effect because a classical theory predicts that light will be
scattered isotropically. This again reflects a poor understanding of classical electrodynamics. While it is
true that in the limit (v/c)~0, classical electrodynamics predicts isotropic scattering, for a relativistic
electron (which condition obtains in the Compton effect), radiation is highly directional.
These examples are typical of the sorts of claims found in introductory textbooks in quantum
mechanics and modern physics. And they likely reflect less upon the writer’s expertise as a physicist than
upon the “cookie cutter” approach to textbook writing. If fact, the very best textbooks begin by simply
talking about how to do QM, not what it means. (Landau and Lifshitz 1958, Griffiths, 1994)
In fact, the physical effects that prove that the electromagnetic field is quantized are subtle and
generally hard to find (as opposed to being forced upon us in the way that quantum effects in atoms and
molecules are). This is particularly relevant to general problems in the interaction of light with matter,
including in the field of nanomechanics. Here, the primary interest lies NOT in proving that the
electromagnetic field is quantized but rather in solving problems in the interaction of electromagnetic
fields with matter (including some cases in which the field is not deliberately applied – e.g., the Casimir
effect, to be discussed below).
2. Modeling the Interaction of Light with Matter
2.1 Semi-classical approximation
While it has different meanings in different contexts, here semiclassical approximation refers to a
treatment of the interaction of light and matter in which the material particles are described quantum
mechanically and the electromagnetic fields are described classically, using Maxwell’s equations. The
first mention the author has found of this sort of calculation is by Erwin Schroedinger, presented in four
lectures given in March of 1921 [Schroedinger, 1928].
In Shroedinger’s presentation, he uses this semiclassical approximation to show how electromagnetic
radiation induces transitions between electronic states of energy Ek and El, so long as the frequency, f, of
the field (i.e., the perturbation) satisfies f = ∆E/h (termed the “frequency-rule” of Bohr), where ∆E is the
energy difference between the two electronic states and h is Planck’s constant. Here we see energy being
transferred in discrete amounts, not because a photon is a “point particle that cannot be divided” but
rather because the electron can only accept energy in discrete amounts as it transitions between discrete
states. And the presence of Planck’s constant, h, is not due to the inherently quantum properties of the
photon, but rather due to the inherently quantum properties of the electron.
In these lectures, Schroedinger also described some other aspects of the interaction of light with
matter: the absence of certain transitions that might be naively expected (selection rules), dispersion
(including anomalous dispersion near a resonance), and the polarization and intensity relations for the
Stark and Zeeman effects.
In summary, the typical implementation of the semiclassical approximation follows Schroedinger’s
approach and treats electromagnetic radiation as a perturbation to a system described by Schroedinger’s
equation. Transitions to and/or from a continuum are usually described using Fermi’s Golden Rule. This
type of calculation is well described by recent introductory quantum mechanics texts, and will not be
described further here [Cohen-Tannoudji 1977, Griffiths 1994, Merzbacher 1998]. In the section on
Applications, I describe some successful uses of this approach (as well as some for which it is
inadequate).
2.2 Neo-classical approximation
While he did not develop it, Schroedinger provides a hint of what is called the “neoclassical”
approximation in those same lectures of March, 1921. In it, he interprets the wave function as
representing an electron charge density by writing: ρ = -e Ψ∗Ψ (i.e, as the elementary charge times the
magnitude squared of the electron wavefunction).
But the neoclassical approximation was extensively developed by Jaynes and Cummings and others
[see Jaynes and Cummings, 1963 and references therein]. In this approximation, the material system is
again described by quantum mechanics and the electromagnetic fields classically (using Maxwell’s
equations), the difference being that the matter wavefunctions are used as source terms in the EM field
equations. That is, the charge density is taken to be that shown above (by Schroedinger) and the current
density by the standard probability flux equation from quantum mechanics (multiplied by the elementary
charge):
J ( r, t ) = − e[ψ * ∇ ψ − ψ ∇ ψ *]
(1)
(This expression for the probability flux is found in every introductory quantum mechanics book with
which the author is familiar and will not be justified here). This “neoclassical” approximation is often in
the form of a self-consistent solution of the quantum mechanical equations describing matter and
Maxwell’s equation describing the electromagnetic fields.
This method extends the range of applicability beyond the set of problems successfully attacked by
the previously described semiclassical approximation, but the important conclusions from above remain –
in most cases described by this formalism, the “quantum” effects observed come from the inherent
quantum properties of matter rather than those of the electromagnetic field.
2.3 Radiation Reaction
This is an effect well known from classical electrodynamics [Jackson, 1975], frequently scorned (by those
who are even familiar with it) because of its counterintuitive and “unrealistic” predictions. (One wonders
how quantum mechanics in general, and quantum electrodynamics in particular, might have fared had the
same sort of skepticism prevailed during the development of those theories). In fact, this appears in both
classical and quantum theories of electrodynamics, and plays a significant role in both.
To review briefly, this is the effect of the field created by charge upon that very charge. One of two
effects that produce comparatively modest discomfort is the modification of the mass of the electron,
essentially modifying the “bare” electron mass by adding the Coulomb energy of the electron divided by
c2 (and suitably “cut off” to avoid the divergence of the Coulomb energy). The other effect that seems
less disturbing to physicists is the damping of the motion of an oscillating charge due to the interaction of
the charge with the field it radiates. That the motion of the charged particle must be damped as energy is
radiated away seems clear, and so this effect also leads to comparatively little discomfort (and in fact, this
is the effect for which radiation reaction is named).
On the other hand, two other effects that lead physicists to such scorn for radiation reaction are
preacceleration and “runaway” solutions. In the simplest case, at least, one can choose which of these
effects obtains. “Runaway” solutions are solutions in which the motion of the charge tends to infinity
with time (an obviously unsatisfactory state of affairs). Preacceleration is an acausal effect in which a
charge begins to accelerate under an influence that has not yet reached the location of the charge. While
the author shares the concern over exponentially divergent speeds and accelerations, the nonlocal nature
of preacceleration seems less distasteful, particularly in view of experiments in quantum mechanics for
which nonlocal effects remain a conceivable explanation. (In short, while it seems clear that nonlocal
effects are forbidden macroscopically by the unequivocal success of relativity – a classical theory – it
does NOT seem resolved that such effects are forbidden to quantum objects).
In any event, some authors claim to have resolved these questions in both a classical [Valentini, 1988]
and quantum mechanical [Moniz, 1977] context, and term preacceleration and “runaway” solutions as
artifacts of an improper development.
3. Outline of Quantum Electrodynamics
The generally accepted (and thus most commonly used) quantum theory of radiation is based upon the
Hamiltonian formulation of electrodynamics. Quantization of the electromagnetic field proceeds much as
the quantization of matter. That is, one begins with a Hamiltonian, recasts the dynamical variables from
real numbers to operators and imposes upon those operators the uncertainty relation (in the form of a
nonzero commutation relation). But as many practicing physicists are not well versed in the Hamiltonian
formulation of even classical electrodynamics•, it may be no accident that they are likewise uncertain of
the actual predictions of quantum electrodynamics (which then leads them to make unsupportable
comments regarding the position and size of a photon). I will briefly and superficially sketch out the

In which category, the author, alas, qualified for most of his career. Which perhaps explains his missionary zeal
on this topic.
typical development of the quantum theory of radiation.[ Cohen-Tannoudji, 1989; Gerry and Knight,
2005]
I begin by noting that in the Lagrangian development of electrodynamics (which leads directly to the
Hamiltonian formulation), one does not construct a Lagrangian, but rather a Lagrangian density (related
to the energy density of the electromagnetic field). One begins with the classical energy density of an
electromagnetic field, which can be transformed to be formally the same as a harmonic oscillator:
H=
(
)
(
1
1 2
ε 0 E2 + c 2B =
pˆ + ω 2 qˆ 2
2
2
)
(2)
where the carets denote that the classical fields are now considered operators. (Note that I have
included no term which allows the field to interact with matter – while this certainly makes this
Hamiltonian useless from a practical point of view, it suffices to make the points necessary here).
The solution of this problem is well known and results in a quantized energy in a unit volume of
(n+1/2) hf, where n is the number of photons of mode frequency, f, and h is Planck’s constant.
Note that even when there are zero photons in this mode, there is a nonzero energy. And while
the mean field in such a state is zero (reflecting the fact that the operators for the number of
photons and electric field do not commute), the field has a nonzero second moment. These are
the so-called vacuum fluctuations, which are invoked as the sole explanation for the effects of
spontaneous emission, the Lamb shift and the Casimir effect (this latter effect of particular
interest to those in the field of nanomechanics). But what is not often acknowledged are the
cosmological effects of this non-zero vacuum energy density. In fact, the energy density is
divergent unless some high frequency cutoff is invoked to prevent the divergence. I discuss this
in the section on Applications.
4. Applications
In this section I discuss a more or less random selection of physical phenomena for which it commonly
held that there is no classical or semiclassical explanation. That is, phenomena that are taken to establish
the need for a quantum theory of radiation, with particular attention to those taken to establish the reality
of the vacuum electric field. It is my desire NOT to deny the need for a quantum theory of radiation (that
need I feel is well established) but rather to establish some guidelines as to where and when such a
quantum theory is truly necessary and point out the effectiveness of some alternatives. And if, in the
process, I can clean up some sloppy and unsupported language in reference to the photon, then my time
has not been ill spent.
4.1 Spontaneous emission
Frequently taken as definitive proof that the vacuum electric field has observable physical effects, in fact
there are other approaches that can predict spontaneous emission.
The basic issue is that an electron in an excited, but stationary (i.e., time independent) state should
remain there. In order to decay, there must be a perturbation that disturbs the electron in its state, causing
a transition to a lower state. The conventional description has it that the omnipresent vacuum electric
field provides just this perturbation. But semiclassical descriptions that include radiation reaction [Crisp
and Jaynes, 1969; Stroud, Jr. and Jaynes, 1970] produce spontaneous emission with the correct Einstein
coefficient, although other authors [Nesbet, 1971] claim that this SC description cannot give the correct
thermal (i.e., Planck) distribution of radiation, while other authors compare a QED and SC theory
(including radiation reaction) of spontaneous emission from harmonic oscillator states and find no
difference between the theories [Doslic and Bosanac, 1995]. More recent papers [Camparo, 2001]
compare a SC theory of spontaneous decay of atoms in a microcavity to measurements and find good
agreement.
Perhaps more significant, it has been shown that radiation reaction (in QED) and vacuum fluctuations
are complementary explanations for spontaneous emission [Ackerhalt, et al., 1973, Milonni and Smith,
1975], and in fact, whether spontaneous emission, and the widths and shifts of the lines, is attributable to
vacuum fluctuations or to a theory of QED including radiation reaction is shown to depend upon the
ordering of commuting atomic and field operators [Milonni, et al., 1973].
4.2 Lamb shift
The Lamb shift has much in common with spontaneous emission. Not in the essential effect (spontaneous
emission is a downward electronic transition accompanied by the emission of radiation, while the Lamb
shift is a measured difference between two electronic states), but in their physical descriptions. The
Lamb shift is another effect taken to prove the existence of the vacuum electric field, and in fact, most of
the references above address both spontaneous emission and the Lamb shift and much of that discussion
would be replicated here. Additionally, there is a SC description of a “two photon” Lamb dip, which
agrees with experiment, but only qualitatively [Shimizu, 1974].
4.3 “Photon counting”
As mentioned in the Introduction, “photon counting” is a phenomenon in which extremely low light
intensities are seen to delay the absorption of light by an electron (the lower the light intensity, the longer
the delay). Also as mentioned, this effect is taken as evidence of a “photon wave function” collapse (to
enable the transfer of the energy from the extended electromagnetic wave to the compact electron), thus
demonstrating the quantum nature of electromagnetic radiation. (And perversely, the delay produced as
the light intensity decreases is taken as further evidence of a quantum nature, despite the fact that no delay
at higher intensities is likewise “proof” of light’s quantum nature).
Low light levels (i.e., small number of photons) is generally taken as a line between classical and
quantum descriptions (similar to, and perhaps because of, a similar criterion in the quantum theory of
matter). In fact, there are SC theories of “photon counting” [Mandel and Wolf, 1966] or other low light
level experiments [Ugaz, 1994; Koshino and Ishihara, 2004].
4.4 Nonlinear optics
Under this general heading, I consider the interaction of matter with intense electromagnetic radiation,
giving rise to effects such as two- and multi-photon absorption, nonlinear refraction, multiphoton
ionization of atoms and intensity dependent index effects. These effects, again, have a long history of SC
and QED explanations, again with varying levels of success. It was found early on that the SC and QED
calculations of two-photon absorption give the same results [Carasutto, et al., 1967]. Other authors
[Chang and Stehle, 1971; Gordov and Tvorgov, 1973] claimed that a SC explanation is inadequate to
describe the behavior atoms interacting with high intensity fields (those giving rise to nonlinear effects).
Note what is being claimed: that the SC approximation is untenable at high intensities – i.e., in the limit
of large numbers of photons. This is refuted by other authors who continue to present SC theories of twophoton [Burrows and Salzman, 1977] and multi-photon [Salzman, 1977] absorption, and multiphoton
ionization of atoms in strong fields [Akberg, et al., 1991]. Other nonlinear effects successfully treated by
the SC approximation are bleaching by excitons in polymers [Li, et al., 1992] and two photon Doppler
free spectroscopy [Cesar and Kleppner, 1999]. Comparisons of QED and SC theories of high order
harmonic generation conclude that the two descriptions agree at low harmonic orders but disagree at high
orders (and that the QED calculations better describe the experiments) [Gao, et al., 2000]. Relevant to
nanomechanics is a SC description of frequency conversion in a microcavity [Martinez, 2005] and of a
two-photon nonlinearity of an exciton in a quantum dot [Koshino and Ishihara, 2005]. Also, QED and SC
descriptions of nonlinear optics in one-dimensional photonic crystals [Centini, et al., 2005] are found to
agree with experiment and one another.
4.5 Compton effect
Again, this was described in the introduction, and there exist SC and QED descriptions of the effect. A
SC description is said to be equivalent to the QED description (and exactly equivelant if the coherent state
is used in the quantum description [Frantz, 1965]). A later paper corrects previous SC and QED
descriptions, but comes to the same conclusions [Kormendi and Farkas, 1999].
4.6 Mixed states and entanglement
Under this heading, I consider experiments designed to be sensitive to the entanglement of the
wavefunction of two photons that were created in such an entangled state. These experiments generally
are inspired by Bell’s consideration [Bell, 1964] of the Einstein-Podolsky-Rosen paradox [Einstein, et al.,
1935] or are otherwise related to it. Generally, an inequality is calculated that distinguishes between a SC
and QED description, such as that by Franson [Franson, 1991A; Franson, 1991B]. This does in fact,
seem to be an intrinsically quantum effect that has no SC description. But it gave rise to the unfortunate
statement that these experiments provide "graphic illustration of the lack of objective realism of the
electric field". Other scientists who obtained the same experimental results made less dramatic
statements [Pryde, et al., 2005]. There are also cavity measurements that seem to require a QED
description due to interference effects [Jyotsna and Agarwal, 1994] although other cavity measurements
can be described by a SC approximation for “large” fields (number of photons greater than 2) [Thompson,
et al., 1998].
A superficially related effect is that of photon “antibunching” in which a photon is “forced” to
“decide” whether to (for example) reflect or transit through a beam splitter. Experimental results from
these sorts of experiments [Beveratos, et al., 2001] seem to require a quantum theory of radiation.
4.7 Cosmological effects
I mention two effects of cosmological significance that do not have a SC explanation, but do have a QED
description that incorporates radiation reaction, and thus do not call for vacuum electric fields. These are
the well-known Hawking radiation and the related (but less well known) Unruh effect [Barut and
Dowling, 1990], in which an accelerated charge experiences a thermal bath of radiation.
But by far a more profound cosmological effect is that produced by the immense energy density
provided by the vacuum electric field. If the vacuum field is to be strong enough to explain, for example,
the observed Lamb shift, then it produces an energy density that will not only perturb the Kepler relation
between orbital periods and orbital semimajor axes, but will destabilize the orbits of the planets [Jaynes,
1978, Jaynes, 1990]. There have been attempts to resolve this problem [e.g., Hawton, 1994] but the
author remains unsatisfied.
5. Relevance to Nanomechanics
The field of nanomechanics is inextricably tied to the interaction of light with matter. This might be by
necessity, if the materials or devices in question are intended to generate, detect or guide electromagnetic
radiation. Or if the structures are to be constructed using light, as in the technique of photolithography.
Many of the papers discussed above have direct relevance to the field of nanomechanics, and an attempt
was made to point these out as they arose. In addition, there are specific structures for which SC and
QED calculations have been made and compared [Altewischer, et al., 2005] in which the SC
approximation is successful. Other cases present only a QED calculation [Savasta, et al., 2002, ; Kiraz, et
al., 2004].
But even in those cases in which there is no intention of using or interacting with light, the
construction and use of nanostructures will invariably result in the intrusion of effects that are ignorable
for a macroscopic structure. An example is the Casimir effect [see, Lamoreaux, 2005, and references
therein], a force between two neutral conductors that has its origin in the quantum theory of radiation.
The Casimir force scales as d-4, where d is the distance between the conductors and it is clear that for d
sufficiently small, the Casimir force can play a significant and even dominating role in the construction,
manipulation and operation of nanostructured devices [Klimchitskaya and Mostepanenko, 2006]. Other
effects will no doubt come into play as the practice of nanomechanics develops.
Conclusion
After all that, can the basic question posed be answered: is there a hard and fast rule that can guide the
physicist in deciding whether to choose a quantum or classical description of the interaction of light with
matter. Several answers are typically posed.
There are authors who have attempted to provide such a roadmap, some unreservedly and specifically
criticizing the SC approximation [Clauser, 1972; Lee and Stehle, 1976; Raimond, et al., 2001; Braun, et
al., 2004; Dente, 1975].
One typical response is that a QED prescription is to be used for small fields and a SC prescription for
large (i.e., intense) fields [Oliver, 1977; Lin and Chuu, 2001]. But there exist calculations and
experiments (references above) that refute both claims.
There are indications (again, references above) that suggest that coherence and wavefunction
entanglement require a fully quantum mechanical treatment – both matter and electromagnetic field. But
many of the experiments described were painstakingly contrived to produce these entanglement effects.
It is not clear that they will arise in a natural way. In fact, the author’s experience in the “original
nanotechnology” (i.e., the creation and characterization of polymers) has led him to the opinion that a
fully quantum mechanical description of the interaction of light with matter is seldom necessary. It may
be that the nanotechnologist must make do with the same guide that every other physicist has – perform
the calculation and see how it compares to experiment.
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