28. Quantum Physics28. Quantum Physics28-1. Black-Body Radiation and Plank’s Theory
T
Thermal radiation : The radiation depends on the temperature and properties of objects
Color of a Tungsten filament – Black– Red– Yellow– White
• Classic Point of View
The thermal radiation was considered to be simply due to accelerated charged particles near the surface.⇒ Not right !
• Black-body Radiation
All the light is absorbed. But the radiation depends on the temperature of the inside wall.
KmT ⋅×= −2max 102898.0λ
Wien’s Displacement law
Wavelength
Inte
nsity Experimental
Classical theory
Ultraviolet catastrophe
• Plank --- Explain the black-body radiationwith two assumptions related to the oscillating charges.
1. The radiation energy is Quantized.
nhfEn =
2. The rasonators emit energy, the so-called photon.
hfE =
Plank succeeded in reproducing the black-body radiation curve. But no body including Plank himself did not accept the quantum concept. -- Considered the assumptions unrealistic.
28-2. The Photoelectric Effect
Photoelectric effect (광전효과)
The first discovery by Herz in 1887.
∆Vs : Stopping potential (independent of the radiation intensity)
• Electrons having a kinetic energy K
sVeK ∆=max
• Characteristics in the photoelectric effect
i) Cutoff frequency, fc
⇒ No photoelectronsii) Kmax is independent of the light intensity.iii) iv) Photoelectric effect occurs instantaneously ( ~ 10-15 sec.)
cff <
fK ∝max
• Einstein (1905)
Extend the quantum concept of Plank’sEnergy of the electromagnetic waves⇒ PhotonsEach photon can give its energy to a single electron.
φ−= hfKmax
Work function
φ=chf
Minimum energy bound in the metal (~ 3-6eV)
i) Cutoff frequencyii)iii) iv) The particle theory of light
chf=φφ−= hfKmax
fKmax ∝
Kmax
ffc
φφλ hc
hc
fc
cc ===
/
Cutoff wave length
28-3. The Compton effect
• Einstein hfE =
cE
chfp ==
Energy
Momentum
• Holly Compton and Peter Debye in 1923 carried an experiment to prove Einstein’s point-like particle concept.
cEphfE
==
The photoelectric effect (x-ray scattering): The total momentum of the photon-electron pair
must be conserved.
(a) Classical Model
Doppler effects
(b) Quantum Model
)cos1(0 θλλ −=−′cm
h
e
Compton Shift
Compton Wavelengthnmcm
h
ec 00243.0==λ
( )p,pppp ′<<∆=′−0
Compton’s Experiment
nm071.00 =λ
Example 28.4
°= 45θnm200000.00 =λ
nmcc 000710.0)2/11()cos1( =−⋅=−=∆ λθλλ
nm200710.0=′λ
00354.0/ =∆ EE
28-4. Photons and Electromagnetic Waves
• Light has a dual nature , Wave & Photon.
Low frequency : Long wavelength ⇒ More wave likeHigh frequency : Short wavelength ⇒ More particle like
28-5. The Wave Properties of Particles
Particle also has a dual nature!!
In 1923, Louis Victor de Broglie (Ph. D. dissertation) postulated an electron also has a dual nature.Perhaps all forms of matter have wave as well as particle properties.
• Photon:
The wavelength of photon can be defined by the momentum.
hfE = phh
cEp =⇒== λ
λ
• Electron:
de Broglie wave : de Broglie wavelegnth
frequency of matter
mvp =
mvh
ph ==λ
hEf =
Quantization of Angular Momentum in the Bohr model
de Broglie ; a dual nature of MatterBohr’s theory : Semiclassical theory
A standing wave form
r λ
rn πλ 2= L,3,2,1=n
vmh
ph
e
==λ
rvm
nh
e
π2= hnhnvrm, e =π
=2
Angular momentum
hnvrmL e == Quantization of angular momentum
Standing wave : Discrete frequency
rn πλ 2≠If , no standing wave ⇒ no closed circular orbit.
• De Broglie (1923): All matters have a dual nature. Then an electron must exhibit diffraction and interference effects.
• Davisson-Germer Experiment (1927): Measure the wavelength of electrons.
Crystalized NiO target
Diffraction patterns due to electron beam.
Extended work on many single-crystalline targets
λhp =Conclude
• G. P. Thomson (1928)Electron diffraction patternfrom electrons passing through a gold foil
• Helium atom, Hydrogen atom, Neutron also show the diffraction pattern.⇒ The matter wave is an Universal Nature
28-6. The Double-Slit Experiment
2sin λθ =D
xph
=λ
xDph
D 22sin ==≈
λθθ
Minimum
The number of electrons detected at a certain spot is proportional to the intensity of two interfering matter waves.
How do we understand the wave-character of electrons?
BErr
,Photon ⇒ Electromagnetic Wave
⇒ Interference effects2EI ∝
ψ : Wave function
*2 ψψψ =∝I
21 ψψψ +=
φψψψψ
ψψψ
cos2 212
22
1
22
21
2
++=
+≠=I
• Which slit does the electron pass through? Slit 1 or Slit 2
28-7. The Uncertainty Principle 1927 Werner Heisenberg
• Heisenberg Uncertainty Principle
A measurement of position is made with precision ∆x, anda measurement of momentum is made with precision ∆px.
2h
≥∆⋅∆ xpx
It is fundamentally impossible to make simultaneous measurementsof a particle’s position and momentum with infinite accuracy.
Similarly
2h
≥∆⋅∆ tE ⇒ Life-time of a particle
λhp =
λhpx =∆
Position of electron λ=∆x hxpx =∆⋅∆⇒
hxpx ≥∆⋅∆
28-8. An Interpretation of Quantum Mechanics
de Broglie ; matter waveMax Bohr ; Explain atomic discrete energy levelSchrödinger ; Wave equation
A particle is described with a wave function ψ(x, y, z, t).
The probability density
dxdx)x(P 2ψ=
12 =ψ= ∫∫∞
∞−dxdx)x(P
Expectation valuethe average position x∫
∞
∞−= dxxx 2ψ
xph
=λ
kxAxAx sin2sin)( =
=λπψ
λπ2
=k( )( )ikxAex =ψ
∞=∆⇒=∆ xp 0If k is determined, then
A traveling electron (wave-pocket)
22
222
22
22
2
2
)()(
xx
xxx
xxxx
xxx
−=
+−=
+−=
−=∆
222)( ppp −=∆
28-9. A Particle in a Box
In classical theory
In Quantum Mechanics
from Schrödinger EquationkxAx sin)( =ψ
Lxx== 0
0)(0)(
==
Lx
ψψ
Boundary Conditions
Lnk
nkLπ=
π=
L,3,2,1=n
xLnsinA)x( π
=ψ∴
de Broglie wave
Analogous to the standing wave
2λnL =
nL2
=λ
Lnh
nLhhp
2/2===
λ
22
222
8221 n
mLh
mpmvEn
=== L,3,2,1=n
Energy Quantization!!
The energy of the particle is quantized.
22
2
8n
mLhEn
=
Example. (i) 1g of a ball, L = 1 m m = 10-3 kg , h = 6.63 × 10-34 J·s En = 5.5 × 10-65 n2 J
If v = 10 m/s , J.mvEk 050221 ==
31102.3~ ×n Very large value
(ii) an electron L = 1 nm = 10-9 mm = 9.11 × 10-31 kg
eVEeVEeVE
6.36.14.0
3
2
1
===
eVnJnEn2218 4.0106 ≈×≅ −
JeV 19106.11 −×=
28-10. The Schrödinger Equation
ph=λλ
π= 2kkxAx sin)( =ψ
hppk h == π2
xpsinA)x(h
=ψ
xpApdx
xdhh
cos)(=
ψ
)(sin)( 22
2
2
xpkxApdx
xd ψψ
−=
−=
hh
2
222
dxdp h−=⇒
mpmvEk 22
1 22 ==
)(2
)(2
)( 2
222
xdxd
mhx
mpxEk ψψψ −==
)(2)( 22
2
xEmxdxd
kψψh
−=
A particle in a potential U(x)
UEEk −=
ψψ )(222
2
UEmdxd
−−=h
• Schrödinger equation is originated from a wave equation.
)tkx(iAe)t,x()tkxsin(A)t,x(
ω−=ψ
ω−=ψ
λ
π=
2k kpp
hh=⇒=λ
)t,x(x
)t,x(k)t,x(p ψ∂∂
−=ψ=ψ 2
22222 hh
ω=⋅==π
ωh
2hhfE
),(),(),(),( txt
itxti
txtxE ψψωψψ∂∂
=∂∂
−== hh
h
Um
pUEE k +=+=2
2
),(),(2
),( 2
22
txUtxxm
txt
i ψψψ +∂∂
−=∂∂ h
h
General form
),(2
),( 22
trUm
trt
i rhrh ψψ
+∇−=
∂∂ Time-dependent
Schödinger equation
( )∞→∆tTime-independent ⇒ E is fixed
)(2
)( 22
rUm
rE rhr ψψ
+∇−=
Kinetic energy Potential energyTotal energy
A simple form of the one-dimensional Schödinger equation
)(2
)( 2
22
xUxm
xE ψψ
+
∂∂
−=h
h
mEk 2=
)()(2)( 222
2
xkxEmxdxd ψψψ −≡−=
h
ikxikx eCeCx −+= 21)(ψ
0)0( 21 =+= CCψ
12 CC −=
Boundary condition at x=0
−=⇒
−
ieeiCx
ikxikx
22)( 1ψ
sin kxkxAx sin)( =ψ
Boundary condition at x=L
0sin)( == kLALψ πnkL =⇒
LnmEk π
==h
2
22
22
2
22
82n
mLhn
mLE
==
πh
=
LxnAx πψ sin)(
How to define A
Px =2)(ψ
12
2cos12
sin1)(
2
0
2
0
22
0
2
==
−=
==
∫
∫∫
LA
dxxLnA
dxL
xnAdxx
L
LL
π
πψ0
2/12
=⇒
LA
π
=ψ∴
Lxnsin
L)x(
212
28-11. Tunneling through a Barrier
0 Dd d+L
LeT κ−≅ 2 Transmittance
1<<Tif where
h
)EU(m −=κ
2
1=+ RT
If an electron is in a such potential well, what is the probability that the electron is in each region ?
In region I
)(2
),( 12
22
1 xdxd
mtxE ψψ h
−= kxsinA)x( =ψ⇒ 1
)(2
),( 22
22
2 xUdxd
mtxE ψψ
+−=
h
( ) ( ) ( )
)x(
xEUmxdxd
22
2222
2 2
ψκ=
ψ−=ψh
h
)EU(m −=κ
2
( ) xBex κ−=ψ2
In region II
In region III
)()(2)( 32
3232
2
xkxEmxdxd ψψψ −=−=
hikxikx eCeCx −+= 213 )(ψ
Boundary Conditions
( ) ( )( ) ( )( ) 03
32
21
=ψ+ψ=+ψ
ψ=ψ
LLdLd
dd
Normalized Condition
( ) ( ) ( ) ( )1
23
220
21
2
=
ψ+ψ+ψ=ψ ∫∫∫∫ +
+∞
∞−
D
dL
dL
d
d dxxdxxdxxdxx
Determine the coefficient A, B, C1, and C2