Table Of Content7 The conception of photons
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n Urjit A. Yajnik
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keywords : Photons, light quanta, birth of the quantum, Planck spectrum, photoelectric effect, quantum
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h indistinguishability, Bose-Einsteinstatistics, Glauber-Sudarshanrepresentation,principle of linear superpo-
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Contents
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1 Prologue 3
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p 1.1 From an embarrassment to a paradigm shift . . . . . . . . . . . . . . . . . . . . . . . 3
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2 The antecedents 4
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2.1 The quiet before the storm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
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6 2.2 The chinks in the armour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
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3 The stage is set 6
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3.1 A challenge to theorists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
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v 3.2 Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
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3.3 An act of desperation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
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4 The light quantum proposal 13
4.1 A heuristic viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Entropy as a hint to discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.3 Photoelectric effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
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5 Towards the birth of the quantum 18
5.1 Seeds of the dreaded rules of the Quantum? . . . . . . . . . . . . . . . . . . . . . . . 18
5.2 Opportunism of theorists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.3 A curious case of inadequate diffusion of scientific knowledge? . . . . . . . . . . . . . 20
5.4 Confirmation from far away, far later . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6 Quantum Mechanics 23
6.1 Ideas whose time had come . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6.2 States and quanta : the essence of quantum physics . . . . . . . . . . . . . . . . . . 24
7 Conclusion and outlook 26
7.1 Final story of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
7.2 The enigmatic Quantum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2
Part I
of its own only if the contemporary people ac-
cept it and propagate it as interesting. There
Planck, Einstein, and key events in the early
are many ideas for which it is said their time
history
had come, and in this case several different peo-
pleindependentlythinkofthesamethingwithin
Abstract : In the year 1900 Max Planck was a short period of time. Einstein’s was an excep-
led by experimental observations to propose a tional case to have come up with not one but
strange formula for the intensity as a function of several differentprofoundideas withinjustafew
frequencyforlight emitted byacavity madeina years’ time and to have articulated them within
hot substance such as a metal. lPlanck provided a single year, 1905, and to have received a quick
a derivation based on peculiar properties to be acceptance for them. There is some evidence
obeyed bytheemitters andabsorbersinthecav- that Special Relativity was being contemplated
ity. I attempt to point out some nuts and bolts by several others, some of them stalwarts, at the
reasoning that could have provided a clue to the same time. But the conviction and clarity with
physical reasoning.
which Einstein expounded its fundamental na-
ture probably got it instant fame.
In 1905, Einstein made the bold hypothe-
sis that under certain circumstances, radiation However another of Einstein’s profound ideas
could be absorbed and emitted as packets of en- from the same year, contained in a paper “On
ergy and also propagated without spreading out a heuristic point of view concerning the produc-
like waves. Einstein was able to predict the for-
tion and transformation of light” had a rather
mulaforthephotoelectric effect basedon hishy- different fate. This is the paper that introduced
pothesis. While the formula was experimentally the idea of what we now call a photon. It was
verified by 1913, his peers seem to have rejected the only paper to take forward the rather het-
its interpretation in terms of light quanta. Ein- erodox ideas put forward by Max Planck about
stein himself was aware of its inherentcontradic- the behaviour of light in his early papers in the
tions. The first part of this article goes over this year 1900. This too was written with the same
period of struggle with the photon concept, and clarity and conviction that characterised Ein-
sets the stage for the entry of S N Bose’s critical stein’s papers. Not only was this idea not ac-
contribution in 1923. cepted, it was in fact considered to be an em-
barrassment by his contemporaries. And even
while he became reputed for his other papers,
1 Prologue
which earned him a full professorship at Berlin,
there was pressure on him to withdraw this par-
1.1 From an embarrassment to a ticular paper. In practical terms, the patently
paradigm shift ludicrous nature of the idea delayed his admis-
sion intothePrussianAcademy byseveral years,
and therefore also probably delayed his Nobel
If ideas could speak they would tell us very
prize. It was not until S. N. Bose from India
strange tales. Firstly how the idea arose is it-
provided him with a “missing link” derivation in
self an interesting question. And it has any life
3
1924 that Einstein himself fully, and the rest of Newtonproposed,orawave, asperHuygens’so-
the world for the first time, became convinced phisticated constructions, finds a culmination in
of the correctness of the idea. It is ironic in- this modern description. It does not endorse ei-
deed that Einstein received his Nobel Prize for ther side as “true”, but shows both descriptions
none of his other astounding ideas but precisely as merely two facets of a multifaceted, subtle
this one which was causing so much embarrass- phenomenon!
ment. However, the Prize was not given for the
While the new Mechanics openedup thegates
profoundness of its core concept, but merely for
tonewphenomena,newmaterialsandnewforces
it being a correct phenomenological prediction.
of nature, its originator seems to have remained
And the award of the Prize, however circum-
unconvinced ofits additional conceptualfounda-
spect, happened in 1921, before the clinching
tions. In this sense, this is the story of an idea
proof provided by Bose.
which was arduously protected by its proposer
Thegoalofthisarticleistoputtheseeventsin for decades under attack from the contempo-
perspective, while also discussing the science in- raries, but whose subsequent implications were
volved, along with the evidence it is based upon. rejectedbythesameoriginatorjustaftertheidea
Webeginwithadiscussionofwhatthechallenge received a resoundingconfirmation by numerous
ofBlack Bodyradiationwas,followed byapossi- experiments and an enthusiastic acceptance by
ble reasoning that may make plausible Planck’s the rest of the world.
path to the leap into the unknown. Next I dis-
cuss Einstein’s paper, his argument for the core
new concept it advanced and the possible rea-
sons for the conviction Einstein carried for an
2 The antecedents
apparently irreconcilable stance which came to
be eschewed by all his colleagues. I also try to
conjecture why it fell to S. N. Bose in the dis- 2.1 The quiet before the storm
tantcolonialIndianuniversityofDhakaandwith
a gap of two decades, to provide the systematic
The end of the nineteenth century appeared to
derivation. Inthelater partsofthearticleItake
mark an epoch of triumphs in the science of
uptheaftermathoftherevolutionstartedbythis
Physics. Heat, light and electricity, indepen-
paper, viz., the emergence of Quantum Mechan-
dent subjects under Physics in any school text-
ics. We shall be concerned mainly with Ein-
book, were suddenly coming closer. The science
stein’s own response to the later developments,
of Thermodynamics had been put on sound and
in that he came to disapprove of the schema of
consistent footing. Maxwell had put together
the new Mechanics that his path breaking paper
a comprehensive mathematical and conceptual
had helped to unravel. I also briefly give the fol-
framework of Electromagnetism. The several
low up story of the circumstances that led to the
laws duetoCoulomb, Amp`ere, Faraday andoth-
development of the complete definitive theory of
ers painstakingly put together over a century
light in the hands of Glauber and Sudarshan.
were beautifully united in a common conceptual
Thegreat debate of whetherlight is a particle as
framework. An added wonder of this accom-
4
plishment was that light for the first time could dard conditions, was putting an atomistic view
be clearly understood to be an Electromagnetic of the world on a sound footing.
phenomenon. Finally,radiatedheatcouldbeun-
A dichotomy immediately becomes apparent,
derstood and a form of light.
since the continuum mechanics of solids and flu-
It may be these developments that led Lord ids assumed substances to be ideal continua,
Kelvin to announce that all the major discov- whileallthematerialswereslowlybeingrevealed
eries of Physics had already been made and fu- as an agglomeration of only a fixed basic list
ture explorations will only help to improve the of “atoms”. To be sure, atomistic view of the
preciseness of the value of “this or that con- worldhadlongbeenproposedinmanyschoolsof
stant”. Tobesure,Newton’s schemaof Mechan- thought around the world. The reason is some-
ics was the conceptual and mathematical frame- what obvious, with hindsight. Iron or copper
work towhichall motion shouldconform. Euler, from any mine in the world had the same prop-
Bernoulli, Poisson and others had extended the erties. Wood or oil always burned; water in any
Newtonian framework to deal with continuum water bodywasmoreor lessthesamesubstance,
systems like fluids and solids. Heat had been with some differences. Thus it was easy to guess
recognised as a form of energy and interconvert- that there were some primary substances, with
ible with mechanical energy. Energy was also varying manifestations. Further, since each had
getting a place in the scheme of Maxwell’s Elec- a characteristic property differentiating it from
tromagnetism which in turn was also unifying the others, there had to be a basic unit that car-
thephenomenonoflightwiththerestofPhysics. ried these special qualities. If all materials were
These developments comprehensively embraced indefinitely divisible, it would be difficult to dis-
all the domains of phenomena that Physics as tinguish between them in the ultimate limit of
a mathematical science could embrace, placing subdivision. On the other hand the presence of
them on a platform of universal concepts and a basic unit, the so called atom could be a can-
frameworks. didate for encoding the few basic properties like
colour, smell taste etc that are characteristic of
thesubstance. Thuscontinuummechanics could
2.2 The chinks in the armour be understood as a good effective description on
a larger scale, while atomic picture as the more
fundamental one.
Yet, there were developments that needed reck-
oning, some happening soon after Lord Kelvin’s There was another direction where new phe-
declaration. The biggest development that was nomena demanded understanding. Kirchhoff
to eventually challenge all of Physics were brew- and Bunsen had been carrying out experiments
ing slowly in Chemistry. Dalton’s theory of with metals heated to very high temperatures.
atoms and valence were very crucial develop- This was made possible by that humble equip-
ments towards atomistic and unified theory of ment available to every school laboratory to-
the world. Combined with Avogadro’s observa- day, the Bunsen burner. But the efficiency with
tion about a standardised number of atoms that which it burned gas to produce complete com-
wouldoccurinanymaterialunderspecifiedstan-
5
3 The stage is set
bustion and a blue flame made possible experi-
ments that were previously difficult to perform.
One of the great observations of Kirchhoff and
3.1 A challenge to theorists
Bunsen was that the spectra observed in light
from heavenly bodies like stars were exactly the
same as those that could be obtained from sub-
KirchhoffandBunsenmadeasalientobservation
stances found on the Earth, such as Hydrogen,
other than the universality of substances men-
Calcium, Sodium etc. We may consider this a
tioned above. It is a common observation that
next step in unification since Newton. Newton
all metals when hot begin to glow. They glow
had shown that curved paths of planetary mo-
in a specific colour that is characteristic of the
tion were just a generalisation of the fall of the
metal that is being heated. However, Kirchhoff
apple on the earth. The heavenly bodies obeyed
and Bunsen observed that as the temperature
exactly the same law as the bodies on Earth.
becomes really high, the characteristics specific
Now Kirchhoff and Bunsen were showing that
to the substance grew gradually less important,
the constitution of the substances in the distant
and all the metals glowed in a universal manner.
stars was the same as that of the substances on
They had made measurements of the intensity
the Earth. Together with the atomistic theory,
of the glow in different wavelength ranges, i.e.,
this was showing that the Earth was just one
at different colours of the spectrum. Kirchhoff
among a large number of bodies made up of the
came to conclude that the rate of energy emis-
same materials obeying the same laws.
sion in a given frequency range depends only on
Onemoredevelopmentthatbegantounfoldin the temperature of the emitting metal, and in-
the 1890’s was radioactivity. Fluorescence was a dependentofthespecificpropertiesofthemetal.
known phenomenon and could be understood as As early as 1859 Kirchhoff wrote this as a paper
an excited state ofan atom oramolecule. When challenging theorists to find an explanation.
HenriBecquerel firstobserved aradioactive sub-
In what follows we are not going to adhere to
stance he mistook it for a more peculiar kind of
the original chain of development of ideas. We
fluorescent substance. Henri Poincar´e was the
shall adopt an ahistoric vain and pose ourselves
one to point out that the properties that Bec-
“what if” questions such that if we could be se-
querel had observed suggested that this peculiar
lective about the sequence of discoveries, what
radiation arose spontaneously from deep inside
would be a good sequence in which to explore
the atom, and not merely from the excitation
the facts so that a coherent picture of the physi-
and de-excitation of the same atom. As we now
cal laws emerges inductively. The history of the
know, these were the first clues to new funda-
personalities and of the main events is neverthe-
mental force laws of nature, the Weak and the
less interesting andweshallusethatas themain
Strong nuclear forces. The stupendously high
framework for exposition, but in the interest of
energy outputs and the longevity of stars could
keepingtheemergingphysicalprinciplesclearwe
be understood only after the nuclear forces were
may skip some false starts and topical interme-
understood.
diatedevelopments. Inthisvainwemayproceed
by recapitulating Kirchhoff’s observation as
6
a) the spectrum of emitted energy is indepen- sentials, without interference from other inci-
dent of the substance emitting it, and dental effects. As is usually done in theoretical
physics, to get to such a situation some idealisa-
b) the emitted total intensity depends only on
tion andsimplification wereintroduced. Itcould
temperature.
be shown that the property Kirchhoff was high-
lighting could beobserved withoutother encum-
Considering the fact that individual substances
brance if one dug a small hole in the surface of
are known to have their characteristic frequen-
a metal. Then the cavity acted like a “perfect
cies in which they absorb or emit, we may in-
emitter”, as well as a “perfect absorber”, also
terpret a) to mean that the spectrum was some-
called a “Black Body”. Kirchhoff’s proposal of
how a property of light itself. Whereas b)means
the frequency distribution being completely de-
that the only characteristic that could change
termined by the temperature applied accurately
from one situation to another was the temper-
to Black Body radiation. The stated problem
ature. This in turn can be meant to read that
thus also came to be known as the Black Body
light must be subject to thermodynamic laws in
radiation problem.
a manner similar to atoms and molecules.
To keep the physics issues clear we now jump
From the ordinary substances, standard gas
aheadheretotheanswertothischallenge, whose
laws were already known to exist. Further, a
history will be dealt with later in Sec. 3.3.
microscopic picture existed from which to derive
The distribution of electromagnetic energy into
thebulk laws. Thedistributionof kinetic energy
frequencies as determined by experiment, and
amongtheatomsormoleculesinanidealgaswas
matching Kirchhoff’s hypothesis was discovered
known in the form of the Maxwell-Boltzmann
correctly by Max Planck in the year 1900. It
formula,
looks like this
f(v) = m 34πv2e−mv2/2kT (1) 8πhν3 e−hν/kT
r(cid:16)2πkT(cid:17) ρ(ν)= c3 1−e−hν/kT (3)
Here v is a possible speed for the molecule, m
Here ν (greek letter nu, distinct from the letter
is the mass of a molecule, k is Boltzmann’s con-
v used above for velocity) is the frequency of the
stant appearing in all of Thermodynamics, and
light, ρ(greek letter rho)is theintensity of emit-
T is the temperature expressed in Kelvin’s abso-
ted radiation perunitvolume perunitfrequency
lute scale. The expression mv2/2 is the kinetic
interval, c is the speed of light and finally, h is
energy of an atom or molecule. For the purpose
a new constant of nature, called Planck’s con-
of comparison with what follows we can write
stant. We have written out the formula in the
down the kinetic energy density in an interval of
modern notation. While Planck was quick to
speeds v to v+dv as
realise that he had identified a new constant of
1
ρ (v) = mv2f(v) (2) nature, many aspects of the formula were going
M−B 2 to remain rather unclear for two decades.
In view of point a) of Kirchhoff’s challenge, Formulas (2) and (3) are complicated even for
the effect needed to be understood in its es- a college student. The main point to focus on is
7
that the factor same amounts of energy, in other words, no en-
ergy dependence in the form of the Boltzmann
e−Energy/kT (4) factor would appear in the energy distribution
function. This observation was made by Lord
occurring in formula (2), an exponenetial, is Rayleigh in 1998 but seems to have escaped at-
known as the Boltzmann factor, and a similar tention. Further, Lord Rayleigh observed that
factor occurs in Eq. (3). The Boltzmann factor fromclassicalpointofview,afactor ν2 wastobe
is the intrinsic probability that a particle will be expected for modes of oscillation in three space
found to have that particular value of energy in dimensions. He also noted that at sufficiently
thehotmediumattemperatureT. Theproduct small frequencies, the exponential in Wien’s for-
kT which has the dimension of energy is deter- mula approaches unity, and then the classical
mined from the bulk measurement of temper- assumption of a distribution independent of fre-
ature by say a thermometer, while “energy” in quency would becomereliable. Based on this, he
thenumeratorisintrinsicenergyofamicroscopic wentontoproposethatatlowfrequencieswhere
member of the gas. Hence, if we associated hν the usual thermodynamics begins to be appli-
with energy of light, then by analogy with for- cable, the front factor should be kTν2 instead
mula(2), onecould thinkof formula(3)as relat- of ν3. This is exactly what Rubens found two
ing to the “gas of light” or “light gas”. In fact years later and communicated to Planck prior
before Planck’s complete formula there was an to publication, and which spurred Planck to ar-
“almost” correct formula of Wien, which read riveatthecorrectformula. Planck’s1900papers
however do not refer to Lord Rayleigh’s remarks
8πhν3
ρ (ν) = e−hν/kT (5) which hadappeared inthe British journalPhilo-
Wien c3 sophical Magazine in 1898. It is an interesting
question whether Planck’s fortuitous foray could
Wien’s formula had found considerable suc-
have been made two years earlier had he known
cess in the short wavelength or high frequency
of and relied upon Lord Rayleigh’s remark. The
regime, buthad begun to show deviation at long
question whether it was the language barrier or
wavelengths andlowfrequencies. Butfromtheo-
a cultural difference in professional circles that
retical point of view, it was a completely unfath-
kept the efforts across the English channel se-
omable formula at that stage of knowledge. Nei-
questered from each other is also an interesting
ther the Boltzmann exponential, nor the front
one, because as we shall later argue, this may
factor of ν3 could be understood.
also have been the reason why the discovery of
Based on Maxwell theory, light was known the full law concerning photons awaited S. N.
to be waves. The energy of a wave was deter- Bose for two whole decades, and who also was
mined by its amplitude and not its frequency, far away from the flourishing centres of science.
i.e., by the extent of vibration, not by the speed
If we were very bold, (and willing to be os-
of vibration. Thus if a fundamental principle of
tracised by the community of Physicists of that
Thermodynamicscalledthe“equipartitionofen-
time), we could argue something like this. Since
ergy” was applied, all the modes of light would
lightwasknowntobewaves, thishadtobeagas
oscillate with the same amplitude to absorb the
8
of waves, and Wien’s formula should betaken to Planck’s caution inavoiding application ofthe
mean that the energy of individual microscopic thermodynamic argument to light gas was justi-
unit of this gas has energy hν as suggested by fied. There was a fundamental difference in con-
theBoltzmannfactor,anassociationnevermade ception between the two substances. Maxwell’s
before for waves. Further, combined with Lord theory relied on the fact that the Electromag-
Rayleigh’s argumentthat thefrontfactor should netic field phenomena were continuum phenom-
have a ν2 just from the counting of modes, the ena in the ideal sense. There was no limit to
additional ν in the ν3 factor could be correctly how much you subdivide the space to be ob-
understood as being proportional to the energy. served, there would be newer degrees of freedom
Now in analogy with mv2/2 as the energy of a of the Electromagnetic field. This was unlike
molecule in formula (2), and comparing Eq. (5), other substances, where the atomic properties
wecouldbegintoidentify hν in formula(3)with would become observable beyond some extent of
the energy of light. Thus the spectrum would subdivision,andattemptstosubdividethespace
correctly display theBoltzmann likedistribution into regions smaller than the atomic dimensions
of energy. There would still be a major discrep- would not bring in any new degrees of freedom.
ancy in the denominator, but we would be set
Despite the difficulties of understanding for-
substantially on the right track.
mula (3), it was good news. If we deviate from
As it turned out, the correct formula (3) was Planck and adopt the view that some kind of
arrivedatafterfurtherexperimentaldetailofthe thermodynamic argument could in fact be ap-
spectrum was known, and Planck would have plied to light then thinking along the lines of
been greatly assisted in the process of trying the bold hypothesis would befruitful. Light had
to explain it, had he adopted such a bold hy- been understood as a wave phenomenon, but
pothesis. Planck however was a very conserva- here its collective system was cast in a form sim-
tive physicist, and in any case, bold hypotheses ilar to a gas of particles. Heat had been clearly
without foundation are not the way of science. understoodas a form of energy of molecular mo-
He therefore first made sure the formula fitted tion only half a century earlier, and the far infra
the experimental data, and in a later paper, to red light was understood as heat waves. Now
explain the origin of the formula, adopted what it was being found that the energy contained in
seemed like a thermodynamic argument applied light, when confined to a cavity, also had the
toabsorbersandemitters inthewalls ofthecav- properties of heat, and had a temperature char-
ity. He did not focus attention on light itself, acterising its thermodynamics. Diversephenom-
because that would have led him immediately to ena were coming closer and appearing to obey
the dead end suggested by Equipartition Princi- the same framework of laws. We may think of
ple mentioned above. Planck’s thermodynamic this as the meta-principle that may have guided
argument was circuitous and puzzling to many Einstein in making the hypothesis even bolder
physicists, but that was closest to anything like than ours, to be discussed in Sec. 4. However,
anexplanationonecouldadvancewiththatstate a further mysterious difference was the modified
of knowledge. denominator of formula (3). In subsection 3.3,
we shall see how Planck could deduce the pres-
9
ence of the extra −e−hν/kT in the denominator, amountof,therespectivequantitiesS orQ. This
but its full derivation awaited S. N. Bose. profound and very subtle identification balanced
the Thermodynamic equations and accounted
for all observations. However a microscopic ex-
3.2 Statistical Mechanics planation of entropy at molecular level was lack-
ing.
To further understand and appreciate Kirch-
This gap in the understanding was sought
hoff’s challenge we need a diversion into Ther-
to be fulfilled by Ludwig Boltzmann. He in-
modynamics and its basis in Mechanics, as pro-
troduced the concept of “ensembles”, to mean
vided by Statistical Mechanics. Thermodynam-
the set of all possible states the mechanical sys-
ics had made great strides, with the recognition
tem could assume in principle. He then set
that heat was a form of energy and that pres-
about proposing the rules of computation that
sure could be understood as the collective aver-
wouldexplainhowthe observedThermodynam-
ageforceexertedonthewalls ofthecontainer by
icswouldemergefromthisfundamentalcounting
the motion of myriads molecules. A major intel-
of all possible configurations. He could obtain
lectual challenge to full understanding of Ther-
both the distribution law (1), and the weightage
modynamics as collective mechanical behaviour
factor(4)asauniversalfeature,andalsoidentify
of molecules was thefactthat heatcouldonly be
a fundamental quantity associated with ensem-
dissipated, and if channelised to produce usable
bles as the entropy of gases as quantified in eq.
mechanical energy, there were theoretical limits
(6) from his fundamental postulates of Statisti-
on what fraction of it could be so converted.
cal Mechanics. His famous formula reads
A formula had been found by Nicolas Sadi
Carnot, relating heat considered as energy and S = klogW (7)
its capacity to do work. This formula when ex-
amined showed that therewould always besome where k as before is the Boltzmann constant,
heat which will be wasted and cannot be con- and W is the number of all the possible states
vertedtousablework. Basedonthisfact,Rudolf that the system can attain consistent with gen-
Clausiusdevelopedtheconceptof“entropy”sug- eral conservation laws. In a sense then, the chal-
gesting the sense of “wasted” form of energy ( lenge thrown up by Kirchhoff’s observation was
trope meaning turned away). For an amount to identify and enumerate thelist of possiblemi-
of heat ∆Q being exchanged with a heat bath croscopic states of light treated as some kind of
at temperature T, the associated entropy ∆S, substance. Everybody took the Maxwell theory
quantifyingtheirrecoverablepartofthetotalen- as the basis of their computation and enumer-
ergy, is given as ated the fundamental microscopic states accord-
ingly. And the results were a disaster. They
∆Q obtained an answer that suggested indefinitely
∆S = (6)
T large contribution to energy at shorter wave-
lengths1. This classical expectation elaborated
Here ∆, (upper case of Greek letter delta) is
meant to suggest a small change in, and a small 1This is true for example if we take Lord Rayleigh’s
10
