Thermodynamic s, Thsnr.pl Radiation and the Beginning*
or
luantum Phvsics,
P. 3.0. Bruskiewich
Mathematical-Physics, University ©f British Columbia, Vancouver B.C.
The beginnings of Quantum Theory is introduced in such
a manner as to present the continuity of effort between the
"classical" and "modern" physics. Emphasis will be on the
specific problem of the energy spectrum of a blackbody source
in thermal equilibrium, and the transfer of energy on the atom-
ic scale.
In the year 1900»Max Planck
presented a paper to the Grerras.n
Physical Society in Berlin(l) that
helped to change the intrinsicJchara-
cter of science, Eis concern m this
paper was with the' mechanical ,thermo
dynamic and elect rodynamic processes
specifically associated with " ideal"
resonators inside an "ideal" black-
body enclosure. In a sense, his work
represented the consistent extension
of the endeavours of many other"clasa
ical" physicists like Kirchoff ,Clausius >
Maxwell, St ©fan ,Wi en and Bolt smarm, but
it also represented a personal daring
(encompassed in the introduction of the
concept of discrete action ) that was
later to blossom into what we now call
quantum physics, Needless to say, at the
time his approach did not follow that
of others ( James Jeans, for instance),
but the conclusions, though originally
heuristic and highly theoretical, that
can be drawn from his work have shown,
beyond any doubt, that Planck's concept
of discrete action is very fundamental
to the intrinsic understanding of the
physical universe.
There are many ways one could
approach Planck's achievement, however,
to appreciate the fundamental importance
of his work it would be best to follow
Planck' s own formulation of his blackbody
spectral law.
Max Planck' s career as a physic
ist began with over fifteen years ©f
study in the field of thermodynamics, mere
specifically the second law, thermodynamic
potentiality, and irreversibility, Because
of his beginnings (as a diadple of Rudolf
Olauaius, the formulator of the
second law ) , Planck considered
the second lav; to be universally
valid, namely processes in which
the total entrop«> f a system decreg
sed over extended time were to be
considered srictly impossible. '.
originally did not consider Ludwig
Boltsmann' s reformulation of the
second lav/ (into a statistical law)
as being acceptable because Boltzmann's
statistical mechanics did not make -
the increase of entropy absolutely
certain, only highly probable. Planck
often expressed the deep, esoteric
conviction that the loftiest goal
of any if not all science was the
search for the absolute.This convict
ion is clearly reflected in his
appreciation of the general physical
laws, but beyond the formulation of
fundamental laws, he attempted, as
others attempted before him (e.g.
Ludwig Boltzmann ) , to relate the
unexplained physical processes and
paradoxes of his time, to fundament
ally universal laws.Flanck believed
that the Principle of Increasing
Entropy could bo preserved intack as
a rigorous theorem in a comprehensive
world view .When it came to the unexp
lained phenomenon of blackbody radia
tion, it was this belief that acted
as the touchstone to Planck's attempt
at expl aination*
In March I895 Planck presented
a paper(2) in which he discussed the
problem of resonant scattering by an
oscillating dlpole(of dimension comp
atible with the wavelength) , of a
pi
"
plane elctromagnetic wave(5) .'What this
scattering process offered to him was a
way of relating the equilibrium ctate(4)
radiation in a blackbody enclosure to
the states of hypothetically ideal re son
at or s } which Planck introduced as making
up the walls of an ideal blackbody enclo
sure(5) • Planck studied the mechanisms
of emi s si on/ab sorption end of damping of
these hypothetical resonators ;He realised
that by disregarding mechanical damping
by concerning himself uniquely with radi
ation damping, and by taking the mechanism
for the system's irreversibility as being
the conversion of the incident plane waves
into spherical waves, you could more fully
begin to understand the dynamics of the
radiation equilibrium in the enclosure.
However, a very fundamental flaw in his
approach was brought up by Boltzmann(it
was a flaw that forced Planck to change
his attitude towards Boltzmann 1 s statist!
cal mechanics ) • The equations of electro
dynamics do not lead to a monotonic conver
gence to a state of thermal equilibrium,
Nothing in the laws of electrodynamics
prevent the inverse of the scattering
processes from occuring^This important
fact forced Planck to admit that statist!
cal descriptions were necessary(6).It also
forced him to consider energy exchange by
.discrete means as opposed to continuous
energy exchange.
Backtracking alittle bit, we can
now begin to approach the mathematics of
Planck's concept. Experiments by Josef
Stefan in 1879 j s -nd theoretical work by
Ludwig Boltzmann in 1884, established that
JD(1T.-
law. Wien* a spectral law was
be a reasonable representation of the
blackbody spectral distribution known
up to 1899* In 1899 > Planck, continuing
hi 3 work begun in 1895 on ^n e analogy
of the" ideal" resonators, proved (7)
that the distribution function must be
of the form
where 3(v,T)
represents the average energy of the
"ideal" resonators at frequency v, at
cavity temperature. Though this equati
on applies to a hypothetical wall comp
osed of "ideal" resonators, because of
Kirchoff ' s 1&59 proof regarding the
constituents of the walls of a black*
body enclosure, once a valid frequency-
temperature relation is found for the
average resonator energy E(v»T), it can
be applied to a cavity formed by any
real material. Because of" his belief
in the universality of the second law*
Planck attempted to discern the form
of 3(v,T) by looking at the fundamental
relationship between the energy and
entropy of a system in a state of thermal
equilibrium. He began with
dfc
T
, and by
differentiating again he came up with
the equally important equation
J —
„' proved rigorously that the distribution
function p(v,T) must be of the form
The derivative dS/dT
could be evaluated by combining Wien 1 s
spectral law with Planck's distribution
In 189 2 ) Wilhelm Wien function so that « „,
<trt* '
p(at,t) - \r S-iy/r)
In I896, ";;ien, using
a somewhat questionable theoretical present
ation, proposed that f(v,T) was of the form
U«/t) " A £
«>r
1
.
anc
The entropy - energy derivative
then becomes
ent constants, giving
p(v,T} ■= Av^e, *
where A & B rep res
Rv
|5irE
The simplicity of this entropy-
energy derivative greatly impressed
Planck, but spectral measurements taken
Wien 1 s spectral (at the same time that Planck did his work
(2)
in 1899) &t high temperatures and low
frequencies mads it apparent that T >fien' s
spectral lav; had serious empirical limit
ations. In 1900 , working with quite new
data ^Planck derived a simple dependence,
which agreed with the high temperature
data (i.e. p(v,T)c*iT, 2(v,T><r, dE/dT is
constant ) .Planck realized that these two
\* <js cases were limiting cases, one for relati
VS^y?" ^ high energy and the other for low.
' $-* ' Almost within the same few days he formul
o, ' ■ atc-a a single equation which sufficed to
"' X 6 «- exoress both limiting cases, namely
« -. - \e (a +& )}
-I
this single equation yields
Integration of
41
CA-tG)] 4-i
with d serving
as the constant of integrstion.Replacing
dS/dE by l/T, the equation becomes
is\
e
A
a — rr;
"valuation of
the constant of integration(i.e. looking
at the equation as l/T •> ) gives d - 0.
Solving the last equation yielded an appr
opriate equation for the resonant energy,
namely
(l%~
giving the expression
A
r
3
(*>~0
and p(v,T) the form
Q-v
Ji
P<W7 = 7—^
which can be evaluated
for B and thenceforth G, giving
" ~F
av»
/4 ^ - SjLvfk
h being a new universal constant now
known as Planck's constant (h-6. 6?,6 \10"^ T
J-sec).
Planck, himself a theoretician,
was not satisfied with just fitting a
formulation to empirical data and began
to put his concepts into a rigorous
form. As I pointed out before, criticism
By Ludwig Boltzmann forced Planck to
turn towards a statistic;! interpretation
of entropy .This statistical approach
yielded dramatic results.
Planck began by looking at the N
ideal" resonators making up the walls of
his hypothetical blackbody enclosure .Each
resonator has an average energy<E>and an
average entropy^S^ giving a total energy
E t of 11(E) and a total entropy S^of l$>.It
is here that Flanck introduced Boltzmann' s
famous expression pfL ^i^-wv-a^wtl ^<J\j^>Jn{(J3\
n«Hw»l ,and follow&%
the^Bolztmann devised previously to deal
with the entropy divided amongst many
oscillators(8) . By proposing that the total
energy E. be made up of n discrete units
of energy, each of amount e,he was able
to equate 1^2) to ne and open up the approach
to the fundamental problem of W; How to
calculate the number of ways n discrete
units of energy can-
distributed amongst
for the distribu
tion function. Returning to the limiting
case encompassed in Wien' s spectral law,
Planck showed that A, a (supposive) constant*
really was a function depending on v, so
that A = F' v% with P representing a(real }
constant .Thus f(v,T) takes on the form
N resonators. More fundamentally, Planck
realized that previous limitations in his
reasoning, as well as in ioltzmann's work,
could be removed by considering the energy
transfer to actually be occuring by discrete
means.
Flanck now had a way of relating
W to N and n. Beginning with
\r
I
-+K -0
lAt (ti-i
and making the
(5)
approximation { on the assumption that
both N and. n are largo )
expression
.•■„
■v\)l
M-
and by using
Stirling" s ftyyiitU - l a a-j^^/je/Ci ha <*wk_
y\
-ti\ ~ ~t{v\-L--t
he wag able to
express klnW as
jUfi-fhe entropy of
single "ideal" resonator is
a
s -
c'
He transformed this
entropy relation into an energy
relation ££r the taking jj^the
derivative
X> -
d£ \e
•\
l ? A
E
and by (again) equating
this to 1/T, so then
j(/\
e, v
G
\
e
Solving for 2(v,T), he
ended up with his previously derived
expression
eKt
q, " x
By returning to Wien's
distribution function
PKtV nrHOf
and his own
he - was ablo- te reasoned
fl^fe- that e«w or (again) returning to Planck 1 ' a
prf ri i OTn fnrrmi 3 m ti l <m J thfiti" 5 hv. Thus
he was able t
function
1 1 . * ' ■*■
his distribution
(a 5 * -i;
quite rigorously.
TMsenew radiation formulae
brought a great deal of pleasure^' rSei.
they were in agreement with the then new
spectral data of Kurlbsum and Rubens, who
at the time were doing precise spectral
measurements at high temperature and low
frequencies ) , but also a great deal of
distress in that, if one accepte d his
formulation^ th on"^he.rgy„tran sfer appear ed^
to be occuring b y discrete steps as opposed
to the cTxLssicaliBSEIc] continuous transfer ,
Planck, even though he was
the formulator of this important discovery,
refused for many years, to accept fully
the consequences of discrete energy transfers
It was left to someone as bold and ira-gina
tive as Albert Sinstien to expand Planck's
original work -IwtJfW luiU maim^ JL ^Mx/SW^ ^
************ pLfiiw,
(1) Zur Theorie des Gesetzes der 3ner<tie~
vertoilung in Normalspecrtu m, M.Planck, Verb.
d.Deutsch.Phys.Ges. 2, 202,(1900) , ( english
translation! Planck's Original Papers ,Taylor
and Francis, 1972) .,
(2)M.Planck,Ann.Hrys.(5) 57,1,(1396) .
(5) The intensity of the plane polarised light
is independent of the orientation of the plane
of polarization as well as the position and
direction of the light. Inside the enclosure
the radiation is isotropic and homogeneous.
(4) The state of thermodynamic equilibrium
is that state in which the entropy of the
system has the maximum value compatible with
the total energy as fixed by the initial
conditions of the system,
(5) Planck was able to use the analogy of
discrete "ideal" resonators as constituting
the enclosure's walls because Gustav Kirchoff
had previously provelft (in 1©?9) that the
energy spectrum of a large blackbody enclosure
is entirely independent of the constituents,
shapes and sizes of the materials making up
its walls.
(4)
(6) In thia resolution lay dormant the
complete theoretical description of the
mechanism of radiation.
(7) How Planck aid this is best seen in
the englieh translation of his book
Waermestrahlung (191?) » by M.Masiua,
Blakiston's Son & Qo, 19l4 f titled
Planck's Heat P.adiation «
(8) Boltzmann originally introduced this
statistical description as a computational
aid, however ; later he became quite satisfied
to consider its application as being more
iAott~ w Mew
general.
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