Bab I Mekanika Kuantum
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Transcript of Bab I Mekanika Kuantum
7/30/2019 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
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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 T
R
.0)( T
R
max
d R R T T )(
0
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Chapter 1 Thermal radiation and Planck’s postulate
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
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Chapter 1 Thermal radiation and Planck’s postulate
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
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Chapter 1 Thermal radiation and Planck’s postulate
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
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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
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
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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.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
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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 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
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Chapter 1 Thermal radiation and Planck’s postulate
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
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Chapter 1 Thermal radiation and Planck’s postulate
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
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Chapter 1 Thermal radiation and Planck’s postulate
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
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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
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
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Chapter 1 Thermal radiation and Planck’s postulate
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
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Chapter 1 Thermal radiation and Planck’s postulate
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
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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 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
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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.
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.