Chapter 1 Thermal radiation and Planck’s postulateFUNDAMENTAL CONCEPTS OF QUANTUM PHYSICS Thermal radiation: The radiation emitted by a body as a result of

temperature. Blackbody : A body that surface absorbs all the thermal radiation incident

on them. Spectral radiancy : The spectral distribution of blackbody radiation.)(TR

:)( dRT represents the emitted energy from a unit area per unit time between and at absolute temperature T. d

1899 by Lummer and Pringsheim

Chapter 1 Thermal radiation and Planck’s postulate

The spectral radiancy of blackbody radiation shows that:

(1) little power radiation at very low frequency(2) the power radiation increases rapidly as ν increases from very small value.(3) the power radiation is most intense at certain for particular temperature.(4) drops slowly, but continuously as ν increases , and (5) increases linearly with increasing temperature. (6) the total radiation for all ν ( radiancy ) increases less rapidly than linearly with increasing temperature.

max

)(,max TR

.0)( TR

max

dRR TT )(0

Chapter 1 Thermal radiation and Planck’s postulate

Stefan’s law (1879): 4284 /1067.5, KmWTR oT

Stefan-Boltzmann constant

Wien’s displacement (1894):

Tmax

1.3 Classical theory of cavity radiation

Rayleigh and Jeans (1900): (1) standing wave with nodes at the metallic surface (2) geometrical arguments count the number of standing waves (3) average total energy depends only on the temperature

one-dimensional cavity: one-dimensional electromagnetic standing wave

)2sin()2sin(),( 0 tx

EtxE

Chapter 1 Thermal radiation and Planck’s postulate

for all time t, nodes at .......3,2,1,0,/2 nnx

ancnanaaxx

2//2/20

standing wave

:)( dN the number of allowed standing wave between ν and ν+dν

dcadndNdcadncan

)/4(2)()/2()/2(

two polarization states

n0

))(/2( dcad

)/2( cad

Chapter 1 Thermal radiation and Planck’s postulate

for three-dimensional cavity

dcadrcar )/2()/2(

the volume of concentric shell drrr

dcV

dca

drrdN

dca

dca

vca

drr

23

23

32

23222

884812)(

)2(4)2()2(44

The number of allowed electromagnetic standing wave in 3D

Proof:

nodal planes

)2sin()/2sin(),(

)2sin()/2sin(),()2sin()/2sin(),(

2/cos)2/(

2/cos)2/(2/cos)2/(

0

0

0

tzEtzE

tyEtyEtxEtxE

zz

yy

xx

z

y

x

propagation direction

λ/2

λ/2

Chapter 1 Thermal radiation and Planck’s postulate

for nodes:

.....3,2,1,/2,,0

.....3,2,1,/2,,0

.....3,2,1,/2,,0

zzz

yyy

xxx

nnzaz

nnyaynnxax

222

2222222

/2

)coscos(cos)/2(

cos)/2(,cos)/2(,cos)/2(

zyx

zyx

zyx

nnna

nnna

nanana

dcadrcannnr

racnnnacc

zyx

zyx

)/2()/2(

)2/()2/(/222

222

dcadcadN

dNdrrdrrdrrN2323

22

)/(4)/2)(2/()()(2/4)8/1()(

considering two polarization state

dcVdN 23)/1(42/)(

:/8)( 32 cN Density of states per unit volume per unit frequency

Chapter 1 Thermal radiation and Planck’s postulate

the law of equipartition energy: For a system of gas molecules in thermal equilibrium at temperature T, the average kinetic energy of a molecules per degree of freedom is kT/2, is Boltzmann constant.

Kjoulek o/1038.1 23

average total energy of each standing wave : KTKT 2/2

the energy density between ν and ν+dν:

kTdc

dT 3

28)( Rayleigh-Jeans blackbody radiation

ultraviolet catastrophe

Chapter 1 Thermal radiation and Planck’s postulate

1.4 Planck’s theory of cavity radiation

),( T Planck’s assumption: and 0,0

kT the origin of equipartition of energy: Boltzmann distribution kTeP kT /)( /

:)( dP probability of finding a system with energy between ε and ε+dε

kT

kTekTekTkT

dkTe

dP

ekTkT

dkTe

dP

dP

dP

kTkT

kT

kTkT

])(|)([1

)(

1|)(1)(

)(

)(

0

/0

/

0 0

/

0/

0

/

0

0

0

Chapter 1 Thermal radiation and Planck’s postulate

Planck’s assumption: ..............4,3,2,,0 kTkT ,

kTkT ,

kTkT ,

kT0 (1) small ν

(2) large large ν0

sjoulh

h

341063.6

Planck constant

Using Planck’s discrete energy to find

kTh

e

enkT

ekT

ekTnh

P

p

nnh

n

n

n

n

n

kTnh

n

kTnh

n

n

/

1)(

)(

......3,2,1,0,

0

0

0

/

0

/

0

0

Chapter 1 Thermal radiation and Planck’s postulate

0

0

0

0

0

0

0ln

n

n

n

n

n

n

n

n

n

n

n

n

n

n

e

en

e

edd

e

edd

edd

00

ln]ln[n

n

n

n edd

hedd

kT

1132

32

0

)1()1(.......1

.....1

eXXXX

eeee

eX

n

n

11)

11(

)]1ln([)()1ln(

/

1

kTheh

ehe

eh

edd

hedd

h

01/1

/

/

hekTh

kTkThekThkTh

kTh

Chapter 1 Thermal radiation and Planck’s postulate

energy density between ν and ν+dν: 1

8)( /3

2

kThT e

hc

118)()()(

)()(

/52

kThcTTT

TT

ehcc

dddd

Ex: Show )()/4()( TT Rc

dA

dV

r22 4

cos4

ˆr

dArrAd

solid angle expanded by dA isspectral radiancy:

)(4

sin4cos)(

)/()4

cos()()(

2220

2/

0

2

0

2

T

tc

T

TT

c

drrtr

dd

tdAr

dAdVR

Chapter 1 Thermal radiation and Planck’s postulate

Ex: Use the relation between spectral radiancy and energy density, together with Planck’s radiation law, to derive Stefan’s law

dcdR TT )()/4()(

32454 15/2, hckTRT

44

3

4

2

0

3

3

4

2

0 /

3

200

15)(2

1)(2

12)(

4)(

ThkT

c

dxex

hkT

c

deh

cd

cdRR

x

kThTTT

15/)1/(

/4

0

3

dxex

kThxx

32

45

152

hck

Chapter 1 Thermal radiation and Planck’s postulate

Ex: Show that 15/)1( 41

0

3

dxex x

dyeyn

dxexdxeexI

eeee

dxeexdxexI

y

nn

xn

n

nxx

n

nxxxx

xxx

0

3

04

00

)1(3

00

3

0

21

1

0

31

0

3

)1(1

.....1)1(

)1()1(

Set yxn eenyxndydxxny )1(33 ,)1/()1/()1(

14

04

0

3

16)1(

16

6

nn

y

nnI

dyey by consecutive partial integration

?11

4

n n

90114818

5)(

61)(

4

14

14

12

24

44

2

12

2

nnn

x

n

x

nnnxF

nxF :F Fourier series expansion

Chapter 1 Thermal radiation and Planck’s postulate

Ex: Derive the Wien displacement law ( ), Tmax ./2014.0max khcT

15

0)1(

50)(1

8)(

2/

/

/

/5

x

kThc

kThc

kThcT

kThcT

ex

ee

kThc

edd

ehc

kThcx /

xeyx

y 21 ,5

1

Solve by plotting: find the intersection point for two functions

5/11 xy

xey 2

Tmax

5

Y

X

intersection points:965.4,0 xx

khcT /2014.0max

Chapter 1 Thermal radiation and Planck’s postulate

1.5 The use of Planck’s radiation law in thermometry

(1) For monochromatic radiation of wave length λ the ratio of the spectral intensities emitted by sources at and is given byKT o

1 KT o2

11

2

1

/

/

kThc

kThc

ee

::

2

1

TT standard temperature

( Au ) unknown temperature

CT omelting 1068

(2) blackbody radiation supports the big-bang theory. Ko3

optical pyrometer

Chapter 1 Thermal radiation and Planck’s postulate

1.6 Planck’s Postulate and its implication

Planck’s postulate: Any physical entity with one degree of freedom whose “coordinate” is a sinusoidal function of time (i.e., simple harmonic oscillation can posses only total energy nh

Ex: Find the discrete energy for a pendulum of mass 0.01 Kg suspended by a string 0.01 m in length and extreme position at an angle 0.1 rad.

295

333334

5

102105

10)(106.11063.6

)(105)1.0cos1(1.08.901.0)cos1(

sec)/1(6.11.08.9

21

21

EE

JhE

Jmgmghlg

The discreteness in the energy is not so valid.