bab i mekanika kuantum

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Chapter 1 Thermal radiation and Planck’s postulate

FUNDAMENTAL 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.)(T R :)( d R

T represents the emitted energy from a unit area per unit time

between and at absolute temperature T. d

1899 by Lummer and

Pringsheim




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 T

R

.0)( T

R

max

d R R T T )(

0




Stefan’s law (1879): 4284

/1067.5, K m W T R o

T

Stefan-Boltzmann constant

Wien’s displacement (1894): T max

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()

2

sin(),( 0 t

x

E t x E




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

ancnanaa x

x

2//2/2

0

standing wave

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

d c a dn d N

d c a dn c a n

)/4(2)(

)/2()/2(

two polarization states

n 0

))(/2( d c a d

)/2( c a d




for three-dimensional cavity

d c a dr c a r )/2()/2(

the volume of concentric shell dr r r

d c

V d

c

a dr r d N

d c

a d

c

a v

c

a dr r

2

3

2

3

32

23222

8848

12)(

)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

t z E t z E

t y E t y E

t x E t x E

z z

y y

x x

z

y

x

propagation

direction

λ/2

λ/2




for nodes:

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

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

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

z z z

y y y

x x x

n n z a z

n n y a y

n n x a x

222

2222222

/2

)coscos(cos)/2(

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

z y x

z y x

z y x

n n n a

n n n a

n a n a n a

d c a dr c a n n n r

r a c n n n a c c

z y x

z y x

)/2()/2(

)2/()2/(/

222

222

d c a d c a d N

d N dr r dr r dr r N

2323

22

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

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

considering two polarization state

d c V d N 23)/1(42/)(

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




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.K joule k o /1038.1

23

average total energy of each standing wave :KT KT

2/2 the energy density between ν and ν+dν:

kTd c

d T 3

28

)( Rayleigh-Jeans blackbody radiation

ultraviolet catastrophe




1.4 Planck’s theory of cavity radiation

),( T Planck’s assumption: and 0,0

kT

the origin of equipartition of energy:

Boltzmann distribution kT e P kT /)(

/

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

kT

kT e kT e kT kT

d kT

e d P

e kT kT

d kT

e d P

d P

d P

kT kT

kT

kT

kT

])(|)([1

)(

1|)(1

)(

)(

)(

0

/

0

/

0 0

/

0

/

0

/

0

0

0




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

kT kT ,

kT kT ,

kT 0(1) small ν

(2) large large ν 0

s joul h

h

34

1063.6

Planck constant

Using Planck’s discrete energy to find

kT h

e

e n kT

e kT

e kT

nh

P

p

n nh

n

n

n

n

n

kT nh

n

kT nh

n

n

/

1)(

)(

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

0

0

0

/

0

/

0

0




0

0

0

0

0

0

0

ln

n

n

n

n

n

n

n

n

n

n

n

n

n

n

e

e n

e

e

d

d

e

e

d

d

e d d

00

ln]ln[n

n

n

n e

d

d h e

d

d kT

1132

32

0

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

.....1

e X X X X

e e e e

e X

n

n

11)

1

1(

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

/

1

kT h e

h

e

h e

e h

e d

d h e

d

d h

01

/1

/

/

h e kT h

kT kT h e kT h

kT h

kT h




energy density between ν and ν+dν:1

8)(

/3

2

kT h T

e

h

c

1

18)()()(

)()(

/52

kT hc T T T

T T

e

hc c

d

d

d d

Ex: Show )()/4()( T T

R c

dA

dV

r

224

cos

4

ˆ

r

dA

r

r Ad

solid angle expanded by dA is

spectral radiancy:

)(4

sin4

cos)(

)/()4cos()()(

22

20

2/

0

2

0

2

T

t c

T

T T

c

dr r t r

d d

t dAr

dAdV R




Ex: Use the relation between spectral radiancy

and energy density, together with Planck’s radiation law, to derive

Stefan’s law

d c d R T T )()/4()(

3245415/2, h c k T R

T

4

4

3

4

2

0

3

3

4

2

0 /

3

200

15

)(2

1

)(21

2)(

4

)(

T h

kT

c

dx e

x

h

kT

c

d

e

h

c

d c

d R R

x

kT h T T T

15/)1/(

/

4

0

3

dx e x

kT h x

x

32

45

152

h c

k




Ex: Show that 15/)1(41

0

3

dx e x x

dy e y n dx e x dx e e x I

e e e e

dx e e x dx e x I

y

n n

x n

n

nx x

n

nx x x x

x x x

0

3

04

00

)1(3

00

3

0

21

1

0

31

0

3

)1(

1

.....1)1(

)1()1(

Sety x n

e e n y x n dy dx x n y )1(33

,)1/()1/()1(

1 40 4

0

3

16

)1(

16

6

n n

y

n n I

dy e y by consecutive partial integration

?1

14

n n

90

1148

18

5)(

6

1)(

4

1

4

1

4

1

2

2

4

44

2

1

2

2

n n n

x

n

x

n n n x F

n x F :F Fourier series expansion




Ex: Derive the Wien displacement law ( ),T max ./2014.0max k hc T

15

0)1(

50)(

1

8)(

2/

/

/

/5

x

kT hc

kT hc

kT hc

T

kT hc T

e x

e

e

kT

hc

e d

d

e

hc

kT hc x /

x e y

x y

21 ,51

Solve by plotting: find the intersection point for two functions

5/11 x y

x e y

2

T max

5

Y

X

intersection points:

965.4,0 x x

k hc T /2014.0max




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 byK T o

1 K T o

2

1

1

2

1

/

/

kT hc

kT hc

e

e

:

:

2

1

T

T standard temperature ( Au )

unknown temperature

C T o

melting 1068

(2) blackbody radiation supports the big-bang theory. K o3

optical pyrometer




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.

29

5

33

3334

5

102105

10)(106.11063.6

)(105)1.0cos1(1.08.901.0)cos1(

sec)/1(6.11.08.9

21

21

E

E J h E

J mg mgh

l g

The discreteness in the energy is not so valid.

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