GENERAL ⎜ ARTICLE
Black-Body Radiation
G S Ranganath
T o w a r d s th e e n d o f th e n in e te e n th c e n tu ry , it
b e c a m e c le a r th a t w ith in th e fra m e w o rk o f c la ssic a l p h y s ic s, it w a s n o t p o ss ib le to u n d e r sta n d
th e p h e n o m e n o n o f B la c k -B o d y R a d ia tio n . In
1 9 0 0 M a x P la n c k c a m e u p w ith a th e o r y th a t a t
o n e str o k e a c c o u n te d su c c e ssfu lly fo r a ll its o b se r v e d fe a tu r e s. T h is th e o r y h a s b e e n p r e se n te d
a g a in st a b a c k d r o p o f th e m a jo r d isc o v e r ie s th a t
le d to it.
In tr o d u c tio n
O n 2 7 th A p ril 1 9 0 0 L o rd K elv in g av e a lectu re a t th e
R oy a l In stitu tio n o f G rea t B rita in o n `N in eteen th-C en tu ry C lou ds over the D yn am icalT heory of H eat an d L ight'.
T h e tw o clo u d s th a t h e referred to w ere th e B la ck -B o d y
S p ectru m a n d th e resu lt o f th e M ich elso n { M o rley ex p erim en t. H e sa id th a t th e `b ea u ty a n d clea rn ess o f th eo ry '
w a s ov ersh a d ow ed b y th ese `tw o clo u d s'. It is n ow a
p a rt o f h isto ry h ow th e stu d y o f th e ¯ rst led to th e b irth
o f Q u a n tu m M ech a n ics a n d th a t o f th e seco n d resu lted
in th e d ev elo p m en t o f th e T h eo ry o f R ela tiv ity. T h u s
B la ck -B o d y R a d ia tio n o ccu p ies a cen tra l p o sitio n in th e
h isto ry o f m o d ern p h y sics. W e sh a ll lo o k a t th e ea rly
a ttem p ts to u n d ersta n d it a n d a lso M a x P la n ck 's n o n cla ssica l th eo ry o f th e b la ck -b o d y ra d ia tio n . A n a ttem p t
h a s b een m a d e to p resen t th e su b ject in its h isto rica l
p ersp ectiv e.
E a r ly S tu d ie s o n R a d ia tio n
N a tu re o f T h e r m a l R a d ia tio n
W e k n ow th a t h ea t tra n sfer ta k es p la ce th ro u g h th e
p ro cesses o f co n d u ctio n , co n v ectio n a n d ra d ia tio n . T h e
RESONANCE ⎜ February 2008
G S Ranganath was
formerly at the Raman
Research Institute,
Bangalore. He is fan of
Alexandre Dumas, Conan
Doyle and Bernard Shaw.
He is interested in classical
Karnatak Music.
Keywords
Black-body radiation, thermal
radiation, heat, electromagnetic
radiation, Stefan’s Law, Stefan–
Boltzmann Law, Wien’s Law,
Rayleigh–Jeans Law, blackbody spectrum, ultraviolet catastrophe, zero point energy,
photon.
115
GENERAL ⎜ ARTICLE
The famous British
astronomer William
Herschel, the
discoverer of the
planet Uranus, was
probably the first to
demonstrate that heat
is a form of light
beyond the red end of
the visible spectrum.
Heat is an
electromagnetic
radiation with a
wavelength ranging
from 1 to 100 micro
meters or microns i.e.
larger than 0.7
microns
corresponding to the
red end of the visible
spectrum.
116
¯ rst tw o p ro cesses ca n ta k e p la ce o n ly in a m ed iu m .
T h e p ro cess o f ra d ia tio n d o es n o t n eed a m ed iu m . It
is th ro u g h th is p ro cess th a t h ea t fro m th e S u n rea ch es
th e E a rth . T h e fa m o u s B ritish a stro n o m er W illia m H ersch el, th e d iscov erer o f th e p la n et U ra n u s, w a s p ro b a b ly
th e ¯ rst to d em o n stra te th a t h ea t is a fo rm o f lig h t b ey o n d th e red en d o f th e v isib le sp ectru m . In a n ex p erim en t ca rried o u t in 1 8 0 0 , h e sta ck ed u p a n a rray o f
th erm o m eters b eh in d a p rism w h en th e su n lig h t w a s in cid en t o n th e o th er sid e o f it. T h ere w ere th erm o m eters
cov erin g n o t o n ly th e v isib le p a rt o f th e sp ectru m b u t
a lso b ey o n d . A fter a few h o u rs o f ex p o su re ea ch th erm o m eter a ro u n d a n d b ey o n d th e red reg io n in d ica ted
a rise in tem p era tu re. T o h is su rp rise h e fo u n d th a t
a s h e m ov ed aw ay fro m th e red en d in to th e in v isib le
(in fra red ) reg io n , th e tem p era tu re g ra d u a lly in crea sed .
H e p u b lish ed th ree p a p ers o n th is su b ject in th e P hilosophical T ran saction s of the R oyal S ociety o f L o n d o n .
In h is seco n d p a p er h e co n clu d ed th a t h ea t a n d lig h t
w ere a p a rt o f th e sa m e sp ectru m . It b eca m e clea r fro m
la ter stu d ies th a t h ea t is a n electro m a g n etic ra d ia tio n
w ith a w av elen g th ra n g in g fro m 1 to 1 0 0 m icro m eters
o r m icro n s i.e., la rg er th a n 0 .7 m icro n s co rresp o n d in g
to th e red en d o f th e v isib le sp ectru m . L a ter in v estig a tio n s esta b lish ed th a t th e S u n em its ra d ia tio n n o t o n ly
in th e v isib le a n d th e in fra red reg io n s b u t a lso in th e
u ltrav io let reg io n .
B la c k -B o d y R a d ia tio n
T o q u a n tify su ch a sp ectru m , w e h av e to m ea su re th e
a m o u n t o f ra d ia tio n en erg y a p p ea rin g a t d i® eren t w av elen g th s. F o r th is w e n eed a d etectin g m a teria l th a t a b so rb s th e ra d ia tio n a t a ll w av elen g th s co m p letely. G u stav R o b ert K irch h o ® , a G erm a n p h y sicist a n d m a th em a ticia n , in terested h im self in th is p ro b lem . H is stu d ies in 1 8 6 0 led h im to lo o k a t th e p ro p erties o f su ch
a m a teria l w h ich h e term ed a s a p erfect a b so rb er. H e
p o in ted o u t th a t w h en su ch a m a teria l is h ea ted , it w ill
RESONANCE ⎜ February 2008
GENERAL ⎜ ARTICLE
em it ra d ia tio n s o f a ll w av elen g th s i.e. it w ill b e a p erfect
em itter. S u ch a b o d y is referred to a s a b la ck b o d y. A
m a teria l th a t a p p ea rs b la ck b eh av es v ery n ea rly lik e a
b la ck b o d y. B u t h ow to g et a tru e b la ck b o d y ? T h e
a n sw er to th is q u estio n w a s p rov id ed b y W ilh elm W ien
a n d O tto L u m m er in 1 8 9 5 . T h ey sh ow ed th a t a h o llow
b o d y w h o se w a lls a re a t th e sa m e tem p era tu re b eh av es
lik e a b la ck b o d y. T h ey p o in ted o u t th a t it w ill b e a
p erfect em itter em ittin g th ro u g h a tin y h o le in its su rfa ce, ra d ia tio n s o f a ll w a v elen gth s. F u rth er, a n y ra d ia tio n en terin g su ch a cav ity w ill u n d erg o in ¯ n itely m a n y
re° ectio n s in sid e a n d w ill lo se en erg y a t ev ery re° ectio n .
T h u s n o in cid en t ra d ia tio n w ill em erg e o u t o f th e tin y
h o le. In o th er w o rd s, it is a lso a p erfect a b so rb er. T h u s
su ch a cav ity, w h ich ca n b e ea sily co n stru cted , is a lm o st
a p erfect b la ck b o d y. T h e a m o u n t o f ra d ia tio n em itted
b y a b la ck b o d y ca n b e m ea su red u sin g a b o lo m eter, a
th erm o p ile o r a ra d io m eter.
How to get a true
black body? The
answer to this
question was
provided by Wilhelm
Wien and Otto
Lummer in 1895.
S te fa n 's L a w
S cien tists h a d stu d ied ra d ia tion ev en b efo re th e co n cep t
o f a b la ck b o d y h a d em erg ed . F o r ex a m p le, th e F ren ch
p h y sicist P ierre L o u is D u lo n g a n d th e F ren ch m a th em a ticia n A l¶e x is T h ¶e rμe se P etit h a d p u b lish ed w ay b a ck
in 1 8 1 7 resu lts fro m w h a t th ey co n sid ered to b e p u rely
ra d ia tio n h ea t tra n sfer b etw een a sp h erica l b u lb a n d a
sp h erica l ch a m b er. V a rio u s g a ses ¯ lled th e g a p b etw een
th e tw o a n d th ey h a d m ea su red th e ra te o f ch a n g e o f
tem p era tu re o f th e b u lb ov er a ra n g e o f p ressu res. T h ey
ex tra p o la ted th eir resu lts to zero p ressu re h o p in g th a t
o n ly th e ra d ia tiv e p ro cess rem a in ed . T h ey co n clu d ed
fro m th eir stu d ies th a t th e to ta l en erg y E em itted p er
u n it a rea p er seco n d is rela ted to a b so lu te tem p era tu re
T o f th e b o d y. T h ey ev en g o t a n em p irica l rela tio n sh ip
b etw een E a n d T .
In 1 8 7 9 , th e A u stria n p h y sicist J o sef S tefa n rea ssessed
th is w o rk o f D u lo n g a n d P etit. S tefa n fro m h is ex p eri-
RESONANCE ⎜ February 2008
Scientists had
studied radiation
even before the
concept of a black
body had
emerged.
117
GENERAL ⎜ ARTICLE
Using his own law
Josef Stefan arrived
at the important result
that Sun is at a
temperature of about
6,000 K.
m en ts o n g a ses h a d sh ow n th a t th e th erm a l co n d u ctiv ity o f g a ses w a s n o t p ressu re d ep en d en t. H en ce, D u lo n g
a n d P etit, b y th eir ex tra p o latio n to zero p ressu re h a d
elim in a ted co n v ectio n b u t n ot co n d u ctio n . T h erefo re,
S tefa n co rrected fo r th is a n d ca lcu la ted th e p u re ra d ia tio n co m p o n en t o f h ea t tra n sfer b etw een th e b u lb a n d
th e ch a m b er. H e d iscov ered th a t a rela tio n o f th e ty p e
E = ¾T
4
m a tch ed b etter th e ex p erim en ta l resu lts o f D u lo n g a n d
P etit. T h u s w a s b o rn h is fa m o u s law . T h e co n sta n t o f
p ro p o rtio n a lity ¾ is ca lled th e S tefa n C o n sta n t. T h is
law w a s in d eed a v ery im p o rta n t d iscov ery. U n fo rtu n a tely it d id n o t a ttra ct th e a tten tio n o f m a n y p h y sicists
till L u m m er to o k n o tice o f it. In 1 8 8 9 , L u m m er, a lo n g
w ith E rn st P rin g sh eim a n d F erd in a n d K u rlb a u m ex p erim en ta lly m ea su red th e a m o u n t o f ra d ia tio n em itted b y
a cav ity a n d su cceed ed in su b sta n tia tin g S tefa n 's law .
T h ey g o t th e fo llow in g va lu e fo r S tefa n 's co n sta n t:
¾ = 5 :7 0 £ 1 0 ¡ 5 erg cm
We can also apply
Stefan’s law to work
out the temperature
of the Earth.
118
¡2
sec ¡1 d eg ¡4 (1 0 ¡8 W m
¡2
K
¡4
):
S tefa n h im self in d ica ted th e im p o rta n ce o f h is law . B y
th a t tim e, th e a m o u n t o f so la r ra d ia tio n th a t rea ch es th e
E a rth , h a d b een m ea su red a n d w a s fo u n d to b e 1 3 6 0
W / sq m . F ro m a stro n o m ica l o b serva tio n s, th e E a rth {
S u n d ista n ce a n d th e S u n 's d ia m eter w ere k n ow n . W ith
a ll th is d a ta ava ila b le to h im , a n d a ssu m in g th e S u n
to b e a b la ck b o d y, S tefa n co u ld ea sily ca lcu la te its E .
T h en u sin g h is ow n law h e a rriv ed a t th e im p o rta n t resu lt th a t S u n is a t a tem p era tu re o f a b o u t 6 ,0 0 0 K . T h is
w a s in d eed a sig n i¯ ca n t resu lt sin ce u n til th en n o o n e
w a s su re o f th e S u n 's tem p era tu re. T em p era tu res a n y w h ere b etw een 2 ,0 0 0 K a n d 1 0 ,0 0 0 K w ere b ein g q u o ted
b y p h y sicists. W e ca n a lso a p p ly S tefa n 's law to w o rk
o u t th e tem p era tu re o f th e E a rth . W e k n ow th e a m o u n t
o f so la r ra d ia tio n rea ch in g th e E a rth . A b o u t 3 0 % o f th is
in co m in g ra d ia tio n is re° ected b a ck in to sp a ce b y th e
RESONANCE ⎜ February 2008
GENERAL  ARTICLE
E a rth a n d th e rest h ea ts u p th e E a rth . S o o n th e E a rth
sta rts em ittin g ra d ia tio n a s a b la ck b o d y. It rea ch es a
stea d y sta te w h en it sta rts em ittin g a s m u ch a s it receiv es. It is ea sy to w o rk o u t th is p ro b lem a n d a rriv e
a t th e E a rth 's tem p era tu re. It tu rn s o u t to b e a b o u t
2 5 5 K o r ¡ 1 8 ±C . In fa ct th is is clo se to th e tem p era tu re th a t sa tellites h av e m ea su red o u tsid e th e E a rth 's
a tm o sp h ere.
We can also apply
Stefan’s law to
work out the
temperature of the
Earth.
B la c k -B o d y S p ec tru m
W e h av e so fa r d iscu ssed th e to ta l ra d ia tio n em itted b y
th e b la ck b o d y. It is im p o rta n t to k n ow h ow th e ra d ia n t
en erg y is d istrib u ted a m o n g st th e d i® eren t w av elen g th s
in its sp ectru m . L u m m er a n d P rin g sh eim a d d ressed th is
p ro b lem . A s a resu lt o f th eir la b o rio u s a n d p a in sta k in g
w o rk , th e g en era l n a tu re o f th e b la ck -b o d y sp ectru m b eca m e clea r. F igu re 1 sh ow s so m e ty p ica l cu rv es th a t rep resen t th e sp ectru m a t d i® eren t tem p era tu res T o f th e
b la ck b o d y. E a ch cu rv e g iv es u = u (¸ ), th e en erg y p er
u n it v o lu m e (en erg y d en sity ), a t d i® eren t w av elen g th s ¸ ,
in u n it w av elen g th in terva l (d ¸ = u n ity ). N o tice th ree
im p o rta n t fea tu res:
Figure 1. Black-body spectrum.
RESONANCE  February 2008
119
GENERAL ⎜ ARTICLE
Boltzmann took a
bold step in
applying
thermodynamics to
radiation.
1 . T h e sp ectru m is co n tin u o u s. It h a s a ll th e w av elen g th s ¸ , fro m 0 to 1 .
2 . It h a s a p ea k a t a p a rticu la r w av elen g th .
3 . T h e p ea k h eig h t in crea ses a n d sh ifts to sh o rter w av elen g th s a s th e tem p era tu re o f th e b la ck b o d y in crea ses.
N o te: T h e v isib le p a rt o f th e sp ectru m ex ten d s fro m 4 0 0
n m to 7 0 0 n m .
T h e r m o d y n a m ic D e sc r ip tio n o f R a d ia tio n
S te fa n { B o ltz m a n n L a w
It is g rea t to k n ow th a t th e b la ck -b o d y ra d ia tio n o b ey s
S tefa n 's law . B u t to g et th is law fro m th e scien ce o f
h ea t o r th erm a l p h y sics is g rea ter still. T h is b eca m e th e
n ex t b ig ch a llen g e fo r p h y sicists. In th is g a m e, L u d w ig
B o ltzm a n n , a stu d en t o f S tefa n , b ro k e n ew g ro u n d in
1 8 8 4 . T ill th en p h y sicists h a d a p p lied th erm o d y n a m ics
o n ly to m a teria l o b jects. B o ltzm a n n to o k a b o ld step
in a p p ly in g th erm o d y n a m ics to ra d ia tio n . T h e p ressu re
P o f th e electro m a g n etic ra d ia tio n in term s o f its to ta l
en erg y d en sity U ca n b e ca lcu la ted . It tu rn s o u t th a t
P =
Boltzmann conceived
a Carnot ‘Ether
Engine’ with radiation
as the working
substance and driven
purely by pressure of
radiation.
120
μ ¶
1
U:
3
S o , B o ltzm a n n co n ceiv ed a C a rn o t `E th er E n g in e' w ith
ra d ia tio n a s th e w o rk in g su b sta n ce a n d d riv en p u rely
b y p ressu re o f ra d ia tio n . T h is en g in e w a s a ssu m ed to
h av e p erfectly re° ectin g w a lls. T h e so u rce a n d receiv er
w ere ta k en to b e b la ck b o d ies. D u rin g th e iso th erm a l
ex p a n sio n , th e h ea t (en erg y ) a b so rb ed is d Q = (U d V +
P d V ). W o rk d o n e in th e C a rn o t cy cle is d W = d P d V .
T h u s th e e± cien cy o f th is en g in e is
´ =
dW
dU
=
:
dQ
4U
F o r a C a rn o t en g in e th is sh o u ld b e eq u a l to d T = T . It
is a n ea sy m a tter th en to sh ow th a t th e to ta l en erg y
RESONANCE ⎜ February 2008
GENERAL ⎜ ARTICLE
d en sity U o f th e th erm a l ra d ia tio n d ep en d s o n a b so lu te
tem p era tu re T a cco rd in g to th e rela tio n :
U = bT 4 ;
w h ere b is a co n sta n t. F ro m th is, B o ltzm a n n ea sily a rriv ed a t th e S tefa n 's law o f ra d ia tio n w ith ¾ = (bc= 4 );c
b ein g th e v elo city o f lig h t. T h a t is h ow in la ter litera tu re, S tefa n 's law ca m e to b e k n ow n a s th e S tefa n {
B o ltzm a n n L aw .
Wien was the first
one to account for
some of the features
of the black body
spectrum. He, like
Boltzmann, applied
thermodynamics to
radiation.
W ie n 's L a w
T h e fea tu res o f th e b la ck -b o d y sp ectru m rem a in ed a
m y stery till W ien a p p ea red o n th e scen e. H e w a s th e
¯ rst o n e to a cco u n t fo r so m e o f th e fea tu res o f th e b la ck b o d y sp ectru m . In 1 8 9 3 , h e, lik e B o ltzm a n n , a p p lied
th erm o d y n a m ics to ra d ia tio n . H e co n sid ered th e p ro b lem o f ra d ia tio n co n ¯ n ed in sid e a ch a m b er o f v o lu m e
V , w h o se w a lls a re u n d er a d ia b a tic ex p a n sio n . B eca u se
th e w a lls o f th e cav ity a re m ov in g , rela tiv ely, th e w av elen g th s o f th e w av es rea ch in g th e su rfa ce g et D o p p ler
sh ifted in w av elen g th . C a lcu la tio n s led W ien to th e su rp risin g resu lt th a t in th is p ro cess (V = ¸ 3 ) rem a in s a co n sta n t. F ro m th is, h e ea sily ca lcu la ted th e en erg y u (¸ ;T )
em itted p er u n it v o lu m e p er sec in u n it w av elen g th in terva l. H e fo u n d th a t:
u (¸ ;T ) =
f (¸ T )
;
¸5
w h ere th e n a tu re o f th e fu n ctio n `f ' rem a in ed u n d eterm in ed [1 ]. In cid en ta lly in teg ra tio n o f u ov er ¸ g iv es U ,
th e to ta l en erg y d en sity o f a b la ck b o d y. In th e litera tu re th is rela tio n is o ften referred to a s W ien 's law .
W ien sh ow ed fro m th is rela tio n th a t th e p ea k va lu e u p o f
u a n d th e w av elen g th ¸ p a t w h ich th is p ea k o ccu rs, a re
rela ted to th e tem p era tu re T o f th e b la ck b o d y th ro u g h
th e rela tio n s:
T ¸p = A ;
RESONANCE ⎜ February 2008
121
GENERAL ⎜ ARTICLE
It is not generally
appreciated that water
vapour is a
greenhouse gas.
Water molecules
absorb
electromagnetic
radiations of
wavelength more than
10 microns.
u p = B T 5;
w h ere A a n d B a re co n sta n ts. T h ese ex tra o rd in a ry rela tio n s b rin g o u t tw o im p o rta n t a sp ects o f th e b la ck -b o d y
sp ectru m , o n e p erta in in g to th e p o sitio n o f its p ea k a n d
a n o th er to th e h eig h t o f its p ea k . T h o u g h W ien d id th is
¯ n e p iece o f w o rk in 1 8 9 3 , h e p u b lish ed it o n ly in 1 8 9 6 .
In 1 9 0 0 L u m m er a n d P rin g sh eim fro m th eir ex p erim en ts
o n cav ity ra d ia tio n sh ow ed th a t b o th th ese rela tio n s o f
W ien a g reed w ell w ith th eir resu lts. T h ey fo u n d th e
co n sta n t A to b e 2 :8 9 8 £ 1 0 ¡3 m K . T h e ¯ rst rela tio n is
k n ow n in th e litera tu re a s W ien 's D isp la cem en t L aw . It
m ay b e m en tio n ed h ere th a t W ien g o t th e 1 9 1 1 N o b el
P rize in P h y sics fo r th e d iscov ery o f th is law .
T w o im p lica tio n s o f W ien 's d isp la cem en t law a re w o rth
m en tio n in g h ere. O u r ¯ rst ex a m p le refers to a tu n g sten ¯ la m en t la m p . It em its ra d ia tio n a t a b o u t 2 0 0 0 K .
T h u s it em its m o stly a t a w av elen g th o f 1 .5 m icro n s i.e.
h ea t a n d n o t lig h t. It is a v ery in e± cien t la m p . O u r
seco n d ex a m p le h a s to d o w ith th e tem p era tu re o f th e
E a rth . A s a lrea d y m en tio n ed a b ov e, th e tem p era tu re
o f th e E a rth d u e to so la r h ea tin g a lo n e is a b o u t 2 5 5 K
o r ¡ 1 8 ±C . S u ch a b la ck b o d y em its p ea k ra d ia tio n a t
a w av elen g th o f a b o u t 1 2 m icro n s. W e k n ow th a t th e
E a rth is su rro u n d ed b y a b la n k et o f w a ter va p o u r. It is
n o t g en era lly a p p recia ted th a t w a ter va p o u r is a g reen h o u se g a s. W a ter m o lecu les a b so rb electro m a g n etic ra d ia tio n s o f w av elen g th m o re th a n 1 0 m icro n s. W a ter
m o lecu les in th e a tm o sp h ere, a fter a b so rb in g th ese lo n g
w av elen g th ra d ia tio n s, ra d ia te th em b a ck to th e E a rth
th u s ra isin g its tem p era tu re. A n ew ra d ia tio n eq u ilib riu m is rea ch ed a n d th e E a rth 's tem p era tu re rises to
a b o u t 3 0 0 K i.e 2 7 ±C .
M ic ro sc o p ic D e sc r ip tio n o f R a d ia tio n
W ie n 's F o r m u la
W ien w a s a lso th e ¯ rst to su g g est a m icro sco p ic v iew o f
122
RESONANCE ⎜ February 2008
GENERAL ⎜ ARTICLE
th e b la ck -b o d y ra d ia tio n . In sp ired b y th e k in etic th eo ry
o f g a ses, h e co o k ed u p th ro u g h `lo o se' a rg u m en ts a n ex p o n en tia l fo rm fo r th e fu n ctio n `f ' a n d p ro p o sed in th e
sa m e p a p er o f 1 8 9 6 :
f (¸ T ) = C ex p
μ
¡ D
¸T
¶
;
Lord Rayleigh and
James Jeans looked
at the black-body
radiation from the
point of view of
statistical mechanics.
th u s g ettin g fo r u th e fo llo w in g fo rm :
u (¸ ;T ) =
C ex p
h³
¡D
¸T
¸5
´i
;
w h ere C a n d D a re co n sta n ts. T h is is o ften referred to a s
`W ien 's fo rm u la '. L u m m er a n d P rin g sh eim sh ow ed th a t
it a g rees v ery w ell w ith ex p erim en ta l resu lts n ea r th e
low w av elen g th en d o f th e b la ck -b o d y sp ectru m . B u t
th ey fo u n d th a t it d ev ia tes, g iv in g lesser va lu es, a t th e
h ig h er w av elen g th s w ith la rg e d ev ia tio n s n ea r th e lo n g
w av elen g th ta il o f th e sp ectru m .
R a y le ig h { J ea n s L a w
L o rd R ay leig h a n d J a m es J ea n s lo o k ed a t th e b la ck b o d y ra d ia tio n fro m th e p o in t o f v iew o f sta tistica l m ech a n ics. T h ey w ere p ro b a b ly th e ¯ rst to d ev elo p a selfco n sisten t cla ssica l m icro sco p ic p ictu re o f b la ck -b o d y ra d ia tio n . T h ey co n sid ered th e b la ck -b o d y ra d ia tio n co n ¯ n ed in a cav ity. S in ce ra d ia tio n s a re n o th in g b u t electro m a g n etic w av es, th ese w av es w ill h av e to `¯ t' in to th e
cav ity sp a ce. H en ce, a n y su ch w av e w ill h av e, lik e a v ib ra tin g strin g , a n in teg ra l n u m b er o f h a lf w av elen g th s
ov er th e len g th it cov ers in sid e th e cav ity. E a ch su ch
p erm itted w av e is ca lled a `m o d e'. R ay leig h a n d J ea n s
w o rk ed o u t th e n u m b er Z o f th e p o ssib le electro m a g n etic m o d es p er u n it v o lu m e th a t ca n ex ist in sid e th e
cav ity, a t a w av elen g th ¸ fo r u n it w av elen g th in terva l.
T h ey g o t a fter ta k in g in to co n sid era tio n th e p o la riza tio n
o f th e electro m a g n etic w av es:
Z (¸ ) =
RESONANCE ⎜ February 2008
8¼
:
¸4
123
GENERAL ⎜ ARTICLE
The famous
Rayleigh–Jeans
Law completely
fails near the low
wavelength end.
N ex t th ey in v o k ed th e eq u ip a rtitio n th eo rem o f M a x w ell's cla ssica l sta tistics. A cco rd in g to th is, th e th erm a l
en erg y o f a sy stem is eq u a lly d istrib u ted a m o n g st th e
d i® eren t d eg rees o f freed o m o f th e sy stem . E a ch d eg ree
o f freed o m a cco m m o d a tes a n en erg y o f (k T = 2 ), k b ein g
th e B o ltzm a n n co n sta n t eq u a l to 1 :3 8 £ 1 0 ¡ 1 6 erg d eg ¡1
(£ 1 0 ¡ 2 3 J / K ). R ay leig h a n d J ea n s a p p lied th is th eo rem
to ra d ia tio n in sid e th e cav ity. T h ey reco g n ized th a t th e
en erg y o f ra d ia tio n is d istrib u ted a m o n g st its d i® eren t
m o d es in sid e th e cav ity. N ow ea ch m o d e is lik e a h a rm o n ic o scilla to r a n d h en ce h a s tw o d eg rees o f freed o m .
A s a co n seq u en ce ea ch m o d e in sid e th e cav ity h a s a n en erg y o f " = k T . A n d th ere a re Z m o d es p er u n it v o lu m e
in sid e th e cav ity. T h erefo re th e to ta l ra d ia tio n en erg y
p er u n it v o lu m e a t ¸ in u n it w av elen g th in terva l is:
u (¸ ;T ) = " Z =
(8 ¼ k )T
:
¸4
T h is is th e fa m o u s R ay leig h { J ea n s L aw . R ay leig h g o t
th is resu lt in 1 9 0 0 w ith a n u n d eterm in ed co n sta n t in
p la ce o f th e n u m erica l co e± cien t (8 ¼ ). L a ter in 1 9 0 5
J ea n s w o rk ed o u t th is co n sta n t. F o r a ll its b ea u ty, th is
fo rm u la d o es n o t a g ree w ell w ith th e ex p erim en ta l b la ck b o d y sp ectru m ex cep tin g a t h ig h w av elen g th s n ea r th e
ta il o f th e sp ectru m . F u rth er it co m p letely fa ils n ea r
th e low w a v elen g th en d w h ere it d iv erg es to a n in ¯ n ite
va lu e fo r u (¸ ;T ). T h is is o ften referred to a s th e U ltra v io let C a ta stro p h e. It is in terestin g to n o te th a t w e ca n
g et R ay leig h { J ea n s law fro m W ien 's law if w e ta k e th e
fu n ctio n f (¸ T ) = 8 ¼ k T ¸ .
P la n c k 's F o r m u la
S o , w e h a v e tw o fo rm s fo r u (¸ ;T ): th e o n e d u e to W ien
th a t w o rk s w ell a t th e low w av elen g th en d a n d th e o th er
o n e d u e to R ay leig h a n d J ea n s th a t w o rk s w ell a t th e
h ig h w av elen g th ta il o f th e b la ck -b o d y sp ectru m . B o th
th e fo rm u la e fa il in th e in term ed ia te reg io n o f th e sp ectru m . H av in g g o t fo rm u la e a t th e tw o w av elen g th lim its,
124
RESONANCE ⎜ February 2008
GENERAL ⎜ ARTICLE
it seem s a rea so n a b le step to g et a fu ll fo rm u la b y in terp o la tin g b etw een th ese tw o . T h is im p o rta n t step w a s
ta k en b y M a x P la n ck a ro u n d 19 0 0 . P la n ck su g g ested a n
in terp o la tio n fo rm u la b y ta k in g fo r W ien 's fu n ctio n `f '
th e fo rm
(8¼ k ¯ )
o;
f (¸ T ) = n ³ ¯ ´
ex p ¸ T ¡ 1
w ith k a s th e B o ltzm a n n co n sta n t a n d ¯ a s a n a d ju sta b le
p a ra m eter. H en ce fo r th e b la ck -b o d y sp ectru m h e g o t
Planck presented
the discovery of
Planck’s formula
at the German
Physical Society
meeting that took
place on October
9, 1900.
(8¼ k ¯ )
³ ´
o i:
u (¸ ;T ) = hn
ex p ¸¯T ¡ 1 ¸ 5
T h is is th e fa m o u s P la n ck fo rm u la . W e ca n q u ick ly v erify th a t th is g o es ov er to th e R ay leig h { J ea n s law a t h ig h
w av elen g th s a n d to th e W ien 's fo rm u la a t low w av elen g th s w ith th e co n sta n ts C = 8 ¼ k¯ a n d D = ¯ . In
v iew o f w h a t h a s b een sa id o f th e W ien a n d R ay leig h {
J ea n s fo rm u la e, w e co n clu d e th a t th e P la n ck fo rm u la
a g rees w ith ex p erim en ta l resu lts a t b o th th e sh o rt a n d
lo n g w av elen g th lim its. H ow is it in th e rest o f th e sp ectru m ? It is o f h isto rica l in terest in th is co n tex t to k n ow
th a t o n S u n d ay ev en in g o f O cto b er 7 , 1 9 0 0 , P la n ck sen t
h is fo rm u la o n a p o stca rd , to h is frien d H ein rich R u b en s
w h ich h e receiv ed th e fo llow in g m o rn in g . A co u p le o f
d ay s la ter, R u b en s in fo rm ed P la n ck th a t th e fo rm u la
w o rk ed p erfectly. A ty p ica l d ata sh ow n in th e F igu re 2
illu stra tes th is fa ct.
Figure 2. Experimental
black body spectrum and
Planck’s formula.
P la n ck p resen ted th e d iscov ery o f th is fo rm u la a t th e
G erm a n P h y sica l S o ciety m eetin g th a t to o k p la ce o n
O cto b er 9 , 1 9 0 0 . L a ter h e com m u n ica ted th e sa m e to
th e B erlin A ca d em y o f S cien ces.
P la n c k 's T h e o r y
T h e ex tra o rd in a ry su ccess o f h is fo rm u la w o u ld b y itself
h av e g iv en P la n ck a p erm a n en t p la ce in th e a n n a ls o f
p h y sics. T h e g rea tn ess o f P la n ck lies in try in g to m a k e
RESONANCE ⎜ February 2008
125
GENERAL ⎜ ARTICLE
Planck quantized
the permitted
energies of an
oscillator.
sen se o u t o f h is fo rm u la . Its v ery su ccess m u st h av e set
h im th in k in g d eep ly a b o u t th e u n d erly in g p h y sics. T ill
th en o n ly th e R ay leig h { J ea n s fo rm u la h a d b een d eriv ed
o n `so u n d ' p h y sics. S in ce th e cla ssica l d escrip tio n d id
n o t lea d to th e rig h t `fo rm u la ' P la n ck h a d to th in k o f a
n o n -cla ssica l a p p ro a ch .
H e k n ew th a t th e ra d ia tio n th a t w a s in sid e th e cav ity
w a s b ein g co n tin u o u sly em itted a n d a b so rb ed b y th e
`m a teria l' o f th e cav ity. A t th a t tim e, th e p h y sicists
th o u g h t th a t th e a to m s o f a m a teria l w h ich a b so rb ed
o r em itted ra d ia tio n a s h a rm o n ic o scilla to rs. H en ce,
o scilla to rs w ere in eq u ilib riu m w ith ra d ia tio n ex ch a n g in g en erg y w ith it. H e k n ew th a t ra d ia tio n s a re electro m a g n etic w av es. H en ce th e cav ity w a s ¯ lled w ith
th ese w av es. T h ese w av es w ere a b so rb ed a n d em itted b y
th e o scilla to rs th a t b eh av ed lik e cla ssica l `p en d u lu m s'.
E n o rm o u s ex p erim en ta l ev id en ce h a d b a ck ed th e w av e
n a tu re o f electro m a g n etic w av es. P la n ck co u ld n o t ig n o re th ese a n d th u s reta in ed th e w av e n a tu re o f ra d ia tio n a s su ch . B u t th e o scilla to r m ech a n ism o f a b so rp tio n
a n d em issio n o f electro m a g n etic w av es w a s m o re a th eo retica l m o d el. H e th erefo re, v en tu red to a m en d th e law s
g ov ern in g its b eh av io u r. H e m u st h av e tried m a n y a ltern a tiv es a n d fa iled , to a rriv e a t h is ow n fo rm u la fo r th e
b la ck -b o d y sp ectru m . T h en ¯ n a lly, in d esp era tio n a s h e
h a s co n fessed , h e su g g ested th a t a h a rm o n ic o scilla to r
ca n n o t h av e a n y en erg y b u t o n ly in in teg ra l m u ltip les
o f a q u a n tu m o f en erg y " 0 = h º , w h ere º is th e n a tu ra l
freq u en cy o f th e o scilla to r a n d h a co n sta n t to b e d eterm in ed . In o th er w o rd s, P la n ck q u a n tized th e p erm itted
en erg ies o f a n o scilla to r. T h u s, th ey w ere strictly n o n cla ssica l in n a tu re w ith en erg ies " = n h º , n b ein g a n
in teg er. T h ese o scilla to rs a re in th erm a l eq u ilib riu m a t
a n y tem p era tu re. T h erefo re, P la n ck in v o k ed th e B o ltzm a n n d istrib u tio n to d escrib e th em . A cco rd in g ly, th e
n u m b er N o f o scilla to rs o f en erg y " is g iv en b y :
126
RESONANCE ⎜ February 2008
GENERAL ⎜ ARTICLE
μ
Ã
¶
!
¡"
¡ nhº
N = N 0 ex p
= N 0 ex p
kT
kT
w ith N 0 a s th e to ta l n u m b er o f o scilla to rs o f zero en erg y.
T h en th e av era g e o scilla to r en erg y " o f th e sy stem o f
o scilla to rs b eco m es
T o ta l en erg y
" =
T o ta l n u m b er o f o scilla to rs
=
=
(
P
P
N ²)
N
P n
ex p
P n
³
¡n h º
kT
ex p
³
´o
¡n h º
kT
fn h º g
´o
(su m m a tio n is ov er n )
=
n
(h º )
ex p
³
hº
kT
´
¡ 1
o:
N ow ea ch p erm itted m o d e th a t is in sid e th e cav ity is a b so rb ed o r em itted b y a n o scilla to r th a t is in th e m a teria l
o f th e cav ity. T h erefo re P la n ck id en ti¯ ed ea ch o scilla to r
o f n a tu ra l freq u en cy º w ith a cav ity m o d e o f th e sa m e
freq u en cy. H en ce in term s o f th e w av elen g th ¸ (= c= º )
o f th e m o d e, w e g et th e av era g e en erg y o f a m o d e to b e
g iv en b y
³ ´
hc
"= n
ex p
³¸
hc
¸ kT
´
¡ 1
o:
P la n ck rep ea ted th e p ro ced u re o f R ay leig h a n d J ea n s.
T h ere a re Z m o d es p er u n it v o lu m e p er u n it w av elen g th
in terva l. F u rth er, ea ch m o d e h a s a n av era g e en erg y ".
H en ce, th e to ta l en erg y o f ra d ia tio n p er u n it v o lu m e a t
¸ p er u n it w av elen g th in terva l is
u ( ¸ ;T ) = "Z (¸ );
(8 ¼ h c)
³
´
o i:
u (¸ ;T ) = hn
ex p k h¸ cT ¡ 1 ¸ 5
RESONANCE ⎜ February 2008
127
GENERAL ⎜ ARTICLE
T h is is n o th in g b u t th e P la n ck fo rm u la w ith ¯ = (hc/k ).
T h u s, h is q u a n tized o scilla to r m o d el led P la n ck to th e
fo rm u la th a t h e h a d ea rlier d iscov ered . u (¸ ,T ) h a s a
p ea k a t a w av elen g th ¸ p g iv en b y :
T ¸p =
hc
;
4 :9 6 5 1 k
a n d th e to ta l en erg y d en sity U is o b ta in ed b y in teg ra tin g
u (¸ ;T ) ov er ¸ fro m 0 to 1 ,
U =
H en ce,
b=
Ã
8¼ 5k 4
1 5 c3 h 3
Ã
!
8¼ 5 k 4
1 5 c3 h 3
T 4:
!
:
P la n ck u sed th e va lu e o f T ¸ p = 0 :2 9 4 cm d eg a s o b ta in ed
b y L u m m er a n d P rin g sh eim in 1 9 0 0 . F u rth er h e u sed
th e d a ta o n th e to ta l ra d ia tio n em itted b y a b la ck b o d y
a s m ea su red b y K u rlb a u m in 1 8 9 8 . T h is g av e h im b =
7 :0 6 1 £ 1 0 ¡ 1 5 erg cm ¡3 d eg ¡4 . F ro m th ese va lu es
P la n ck g o t
k = 1 :3 4 6 £ 1 0 ¡1 6 erg d eg (= 1 :3 4 6 £ 1 0 ¡ 2 3 J = K );
h = 6 :5 5 £ 1 0 ¡2 7 erg sec (= 6 :5 5 £ 1 0 ¡ 3 4 J sec):
Planck had to wait
for more than a
decade for the
scientific community
to accept his
revolutionary ideas
on quantization.
128
T h e clo se a g reem en t b etw een th e co m p u ted a n d k n ow n
va lu es o f th e B o ltzm a n n co n sta n t k is a testim o n y to
th e su ccess o f h is fo rm u la . T h is p ro b a b ly em b o ld en ed
h im to rep o rt h is ¯ n d in g s in th e G erm a n P h y sica l S o ciety m eetin g th a t to o k p la ce o n D ecem b er 1 4 , 1 9 0 0 .
H ere th e q u a n tiza tio n o f o scilla to r en erg y w a s p resen ted
fo r th e ¯ rst tim e. T h a t is w h y m a n y lo o k u p o n th is a s
th e d a te o f b irth o f Q u a n tu m T h eo ry. P la n ck co m m u n ica ted h is ¯ n d in g s to th e B erlin A ca d em y o f S cien ces in
1 9 0 0 a n d la ter p u b lish ed h is w o rk in A n n alen der P hysik
in 1 9 0 1 . P la n ck h a d to w a it fo r m o re th a n a d eca d e
fo r th e scien ti¯ c co m m u n ity to a ccep t h is rev o lu tio n a ry
RESONANCE ⎜ February 2008
GENERAL ⎜ ARTICLE
id ea s o n q u a n tiza tio n a n d h en ce h is th eo ry o f ra d ia tio n .
In fa ct E in stein w a s a critic o f P la n ck 's w o rk . H e u sed
W ien 's fo rm u la in h is stu d y o f b la ck -b o d y ra d ia tio n th a t
led h im to p ro p o u n d th e q u a n tu m n a tu re o f lig h t in h is
fa m o u s 1 9 0 5 p a p er. G ra d u a lly p h y sicists sta rted a p p recia tin g P la n ck 's th eo ry. T h ese a d m irers w ere in d eed
h a p p y w h en P la n ck g o t th e 1 9 1 8 N o b el P rize in P h y sics
fo r th is g rea t w o rk .
Admirers were
indeed happy
when Planck got
the 1918 Nobel
Prize in Physics
for this great work.
P la n c k a n d th e Z e r o P o in t E n e r g y
It h a s a lrea d y b een sta ted th a t P la n ck 's fo rm u la g o es
ov er to th e R ay leig h { J ea n s fo rm u la in th e lo n g w av elen g th o r eq u iva len tly h ig h tem p era tu re lim it. T h erefo re, it w a s n a tu ra l fo r P la n ck to ex p ect th e av era g e o scilla to r en erg y o f h is q u a n tized o scilla to r to g o ov er, a t
h ig h tem p era tu res, to th e cla ssica l va lu e o f (k T ). It h a s
b een sh ow n a b ov e th a t th e av era g e en erg y o f a P la n ck ia n o scilla to r is
³ ´
hc
¸
³
´
.
"= n
ex p ¸ hk cT ¡ 1 g
It is v ery ea sy to sh ow b y ex p a n d in g th e ex p o n en tia l
term a n d reta in in g o n ly th e ¯ rst p o w er (h c= ¸ k T ), th a t
th is ex p ressio n fo r " red u ces to k T in th e h ig h tem p era tu re lim it. T h is is w h a t P la n ck d id in h is 1 9 0 1 p a p er
o n th e q u a n tu m th eo ry o f b la ck -b o d y ra d ia tio n . L a ter
in 1 9 1 2 , fo r rea so n s th a t a re n o t clea r, h e rea ssessed h is
ea rlier w o rk . H e fo u n d th a t w h en h e w en t to th e n ex t
h ig h er p o w er in (h c= ¸ k T ) a n d ca lcu la ted ", h e w a s in
fo r a su rp rise. H e g o t
" = kT ¡
³ ´
hc
¸
2
= kT ¡
(h º )
:
2
It d id n o t g o ov er to th e cla ssica l va lu e o f k T . T o fo rce
th e cla ssica l lim it o n h is m o d el, h e p ro p o sed in h is 1 9 1 2
p a p er th a t h is q u a n tized o scilla to r h a s its en erg y lev els
RESONANCE ⎜ February 2008
129
GENERAL ⎜ ARTICLE
The first experimental
evidence for the
existence of the Zero
Point Energy came in
1924 from a work of
the famous American
chemist Robert
Mulliken.
sh ifted u p w a rd b y ( 12 h º ). T h a t is, th e o scilla to r en erg ies
a re
μ
¶
1
h º;
"= n +
2
in stea d o f " = n h º . In o th er w o rd s, ev en in th e g ro u n d
sta te (n = 0 ), it h a s a ¯ n ite en erg y o f 12 h º . T h u s,
P la n ck 's o scilla to r w a s n o n -cla ssica l in tw o d i® eren t w ay s.
It h a d n o t o n ly d iscrete en erg ies b u t w o u ld n o t b e a t rest
ev en in th e low est en erg y sta te. In o th er w o rd s, it w o u ld
h av e a ¯ n ite en erg y ev en a t a b so lu te zero o f tem p era tu re. In la ter litera tu re th is w a s referred to a s th e Z ero
P o in t E n erg y o f a q u a n tu m o scilla to r. A s a co n seq u en ce
o f th is, th e b la ck -b o d y ra d ia tio n sp ectru m is d escrib ed
by
u (¸ ;T ) =
h
n
(8 ¼ h c) 1 = ex p
³
hc
k¸T
¸5
´
o
¡ 1 +
1
2
i
:
P la n ck h im self ca lled th is th e S eco n d L aw o f R a d ia tio n .
It is n o t o u t o f co n tex t to p o in t o u t h ere th a t p ro b a b ly th e ¯ rst ex p erim en ta l ev id en ce fo r th e ex isten ce o f
th e Z ero P o in t E n erg y ca m e in 1 9 2 4 fro m a w o rk o f
th e fa m o u s A m erica n ch em ist R o b ert M u llik en . H e w a s
stu d y in g th e sp ectra o f B o ro n M o n ox id es (B O ), w ith
tw o d i® eren t B o ro n iso to p es, in o n e ca se w ith m a ss 1 0 ,
in a n o th er m a ss 1 1 . (A s a co n seq u en ce, th ese m o lecu les
w ill h av e slig h tly d i® eren t red u ced m a sses). B y co m p a rin g th e v ib ra tio n a l a n d th e ro ta tio n a l sp ectra o f th ese
m o lecu les, h e d iscov ered th a t th e o b serv ed sp ectra h a d
a b etter ¯ t w ith th e th eo ry, if th e en erg y lev els w ere
a ssu m ed to sta rt a t (h º )= 2 ra th er th a n 0 .
Q u a n tiz e d O sc illa to r
Q u a n tiza tio n o f th e en erg y o f a n o scilla to r is cen tra l to
P la n ck 's th eo ry o f b la ck -b o d y ra d ia tio n . T h ere is a n im p ressio n th a t th e ju sti¯ ca tio n fo r th is co m es o n ly fro m
fo rm a l q u a n tu m m ech a n ics w h ere th e S ch rÄo d in g er W av e
E q u a tio n is so lv ed fo r a n o scilla to r. T h e m eth o d em -
130
RESONANCE ⎜ February 2008
GENERAL ⎜ ARTICLE
p loy ed to so lv e th e S ch rÄo d in g er eq u a tio n is m a th em a tica lly q u ite co m p lex . It is n o t g en era lly a p p recia ted th a t
th e essen tia l resu lt i.e., " = n h º ca n b e a rriv ed a t in a
m u ch sim p ler w ay b y ex p lo itin g th e M a tter W av e C o n cep t a s en u n cia ted b y d e B ro g lie. In th is d escrip tio n a
p a rticle o f m a ss m m ov in g w ith a v elo city v b eh av es lik e
a w a v e o f w av elen g th
¤ =
h
(m v )
:
A s th e p a rticle tra v els a d ista n ce `d s' in its p a th , th e
n u m b er o f w av elen g th s it cov ers is (d s/ ¤ ). In a p erio d ic
m o tio n a s in a m o tio n o n a circle o r in a h a rm o n ic o scilla to r, th e p a rticle co m es b a ck to th e sa m e sta te o n ce
in a cy cle. In th e w av e d escrip tio n th is m ea n s th a t in
th a t cy cle it m u st h av e trav ersed a n in teg ra l n u m b er o f
w av elen g th s. In o th er w o rd s, ov er a cy cle
ZÃ
ds
¤
!
=
Zμ
m v
h
¶
d s = n (a n in teg er):
W h en th e p a rticle is m ov in g o n a circle w ith a u n ifo rm
v elo city v , th e in teg ra tio n is ea sy a n d lea d s to th e resu lt
m v r = n h = 2 ¼ ; th e w ell-k n ow n B o h r q u a n tu m co n d itio n
w h ich is a t th e h ea rt o f h is th eo ry o f a to m s. W e d escrib e a h a rm o n ic o scilla to r in term s o f its d isp la cem en t
X g iv en b y
X = X 0 sin (2 ¼ º t):
T h e a m p litu d e o f its o scilla tio n s is X 0 a n d its n a tu ra l
freq u en cy is º = (® = m )1 = 2 w ith ® a s th e sp rin g co n sta n t.
N ow th e en erg y o f a h a rm o n ic o scilla to r is
m v2
®X 2
+
:
2
2
"=
A s a lrea d y sa id ov er o n e p erio d w e m u st h av e a n in teg ra l
n u m b er o f w av elen g th s. i.e.
Zμ
m v
h
RESONANCE ⎜ February 2008
¶
dX = n :
131
GENERAL ⎜ ARTICLE
Suggested Reading
[1]
[2]
[3]
[4]
[5]
[6]
J K Roberts and A R
Miller, Heat and Thermodynamics, Blackie and
Sons Ltd., 1966.
J Stefan, Sitzungsberichte
der Kaiserlichen Akademie der Wissenschaften,
Mathematische Naturwissenschaftliche Classe
Abteilung, Vol.279, p.391,
1879.
L Boltzmann, Annalen
der Physik und Chemi.,
Vol.22, p.31, 291, 1884.
W Wien, Ann. Physik.,
Vol.58, p.622, 1896.
J W S Rayleigh, Phil.
Mag., Vol.49, p.539, 1900.
J H Jeans, Phil. Mag.,
Vol.10, p.91, 1905.
[7]
M Planck, Berlin Academy of Sciences, Vol.X,
p.19, 1900.
[8]
M Planck, Berlin Academy of Sciences, Vol.XII,
p.14, 1900.
M Planck, Annalen der
Physik., Vol.l4, p.553,
1901.
[9]
[10]
M Planck, Annalen der
Physik., Vol.37, p.642,
1912.
In teg ra tin g ov er a cy cle, w e g et th e req u ired resu lt,
" = n h º:
E p ilo g u e
P la n ck 's u n o rth o d ox so lu tio n to th e p ro b lem o f th e b la ck b o d y sp ectru m h era ld ed th e era o f M o d ern P h y sics. T h e
stu d y o f b la ck -b o d y ra d ia tio n b y la ter in v estig a to rs b eca m e ex trem ely fru itfu l a n d im p o rta n t. W e list so m e o f
th em h ere:
1 . E in stein a n a ly sed th e low w av elen g th en d o f th e
b la ck -b o d y sp ectru m in th e W ien a p p rox im a tio n a n d
su g g ested th a t `L ig h t' b eh av ed lik e p a rticles w ith ea ch
p a rticle, la ter ch risten ed a s P h o to n s, h av in g a n en erg y
o f h º . T h is led to h is N o b el P rize w in n in g w o rk o n th e
p h o to electric e® ect.
2 . E in stein stu d ied ca refu lly th e th erm a l eq u ilib riu m
b etw een ra d ia tio n a n d a to m s in a cav ity a n d co n clu d ed
th a t ra d ia tio n n o t o n ly b eh av ed lik e p a rticles o f en erg y
h º b u t ea ch p a rticle a lso h a d a m o m en tu m o f h º = c. F u rth er, h e sh ow ed th a t th e a to m s a b so rb ed o r em itted ra d ia tio n o f en erg y h º = j(E 2 ¡ E 1 )j in a tra n sitio n o f a n
a to m fro m sta te o f en erg y E 1 to a n o th er o f en erg y E 2
th u s ju stify in g B o h r's a ssu m p tio n co n cern in g em issio n
a n d a b so rp tio n o f ra d ia tio n . H is la ter stu d y o f th e sa m e
sy stem sh ow ed th a t th ere ca n b e b o th sp o n ta n eo u s a s
w ell a s ra d ia tio n -in d u ced tra n sitio n o f a n a to m fro m a
h ig h er en erg y sta te to a low er en erg y sta te. T h is is a t
th e h ea rt o f L a ser a ctio n .
3 . In d ia n p h y sicist B o se sh ow ed th a t E in stein 's p h o to n
p ictu re o f ra d ia tio n w o u ld lea d to P la n ck 's ra d ia tio n fo rm u la if th e p h o to n s a re a ssu m ed to b e n o t o n ly in d istin g u ish a b le b u t a lso to o b ey a sta tistica l d istrib u tio n to ta lly d i® eren t fro m th e w ell-k n ow n M a x w ell{ B o ltzm a n n
d istrib u tio n . T h is led to th e d ev elo p m en t o f Q u a n tu m
S ta tistics.
132
RESONANCE ⎜ February 2008
GENERAL ⎜ ARTICLE
4 . T h e co sm ic m icro w av e b a ck g ro u n d ra d ia tio n d iscov ered b y W ilso n a n d P en zia s h a s a b la ck -b o d y sp ectru m
w ith a p ea k ra d ia tio n a ro u n d a w av elen g th o f a b o u t 1
m m a n d co rresp o n d s to a tem p era tu re o f a b o u t 3 K .
T h is stro n g ly su p p o rted th e B ig B a n g T h eo ry a cco rd in g to w h ich th is ra d ia tio n is a rem n a n t o f th e B ig B a n g .
T h ey g o t th e 1 9 7 8 N o b el P rize in P h y sics, fo r th is w o rk .
Address for Correspondence
G S Ranganath
422, 10th Cross, 8th Main
Padmanabhanagar
Bangalaore 560 070, India.
Email:
[email protected]
Max Planck in Memoriam
A man to whom it has been given to bless the world with a great creative idea has no need for the praise of
posterity. His very achievement has already conferred a higher boon upon him.
Yet it is good – indeed, it is indispensable – that representatives of all who strive for truth and knowledge should
be gathered here today from the four corners of the globe. They are here to bear witness that even in these times
of ours, when political passion and brute force hang like swords over the anguished and fearful heads of men,
the standard of our ideal search for truth is being held aloft undimmed. This ideal, a bond forever uniting
scientists of all times and in all places, was embodied with rare completeness in Max Planck.
Even the Greeks had already conceived the atomistic nature of matter and the concept was raised to a high degree
of probability by the scientists of the nineteenth century. But it was Planck’s law of radiation that yielded the
first exact determination – independent of other assumptions – of the absolute magnitudes of atoms. More than
that, he showed convincingly that in addition to the atomistic structure of matter there is a kind of atomistic
structure to energy, governed by the universal constant h, which was introduced by Planck.
This discovery became the basis of all twentieth-century research in physics and has almost entirely conditioned
its development ever since. Without this discovery it would not have been possible to establish a workable
theory of molecules and atoms and the energy processes that govern their transformations. Moreover, it has
shattered the whole framework of classical mechanics and electrodynamics and set science a fresh task: that of
finding a new conceptual basis for all physics. Despite remarkable partial gains, the problem is still far from a
satisfactory solution.
In paying homage to this man the American National Academy of Sciences expresses its hope that free research,
for the sake of pure knowledge, may remain unhampered and unimpaired.
– Albert Einstein
Statement read at the Memorial Services for Max Planck, April, 1948.
RESONANCE ⎜ February 2008
133