12 量子物理
Sections
1. Photon and Matter Waves 2. Compton Effect
3. Light as a Probability Wave 4. Electrons and Matter Waves 5. Schrodinger’s Equation
6. Waves on Strings and Matter Waves
7. Trapping an Electron 8. Three Electron Traps 9. The Hydrogen Atom
12-1 Photon and Matter Waves
(
光子和物質波 )• Light Waves and Photons
s J 10
63 .
6
) energy photon
(
34
h
hf E
f c
The Photoelectric Effect
光電效應
• First Experiment (adjusting V)–
the stopping potential Vstop
• Second Experiment (adjusting f)–
the cutoff frequency f0
stop
max
eV
K
The experiment
光電子的最大動能與 光強度無關
低於截止頻率時即使光再 強也不會有光電效應
The plot of V
stopagainst f
The Photoelectric Equation
s J 10
6 . 6
) (
34 stop
max
h
f e e
V h
K
hf Work
functio n
12-2 Compton Effect
momentum) (photon
h c
p hf
康普 吞效 應實 驗圖 表
康普吞效應圖示
mv p
h p
h mc h
mc f
h hf
mc K
K f
h hf
e
X
/
) 1 (
) 1 (
) 1 (
2
2
Energy and momentum
conservation
) cos
1 (
sin sin
0
cos cos
mc h h mv
h mv h
Frequency shift
Compton wavelength
12-3 Light as a
Probability Wave
The
standard version
The single-photon, double- slit experiment is a
phenomenon which is impossible, absolutely
impossible to explain in any classical way, and which has in it the heart of quantum
mechanics - Richard Feynman
The Single-Photon Version
First by Taylor in 1909
The Single-Photon, Wide-Angle Version (1992)
50μm
Light is generated in the source as photons
Light is absorbed in the detector as photons
Light travels between source and detector as a probability wave
The postulate
12-4 Electrons and Matter Waves
p
h
• The de Broglie wave length
• Experimental verification in 1927
• Iodine molecule beam in 1994
1989 double-slit experiment
7,100,3000, 20,000 and 70,000 electrons
Experimental Verifications
X- ray
Electro n
beam
苯環 的中 子繞 射
12-5 Schrodinger’s Equation
• Matter waves and the wave function
• The probability (per unit time) is
t
e
iz y x
t z y
x , , , ) ( , , )
(
2ie. *
Complex conjugate
共軛複數
The Schrodinger Equation from A Simple Wave
Function
m k
m p
E
k h
p
kx B
kx A
e z
y x
t z y
x
i t2 / 2
/ /
) cos(
) sin(
) ,
, (
) ,
, ,
(
2 2
2
(1D)
dx E d m
dx d E m
dx k d
k dx
d
kx B
kx A
2 2 2
2 2 2
2 2 2
2 2
2
2
2 1
1 /
) cos(
) sin(
1D Time-independent
SE
2 2 2
2 2 2 2
2 2 2
2
2
2 2
2
( )
2
2 2
d E
m dx
m x y z E
m i t
V i
m t
3D Time-dependent SE
12-6 Waves on
Strings and Matter
Waves
Confinement of a Wave leads to Quantization –
discrete states and discrete energies
駐波與量子化
Quantization
n = , , ,
L n v
f v n
L 0 1 2
2
= 2
駐波：
number
quantum :
, 3 , 2 , 1
, ) sin(
, 3 , 2 , 1 2
n
n L x
A n y
n n L
n
12-7 Trapping an Electron
For a string ：
, 3 , 2 , 1
, 8
/
2 / ,
2 /
/
2 2
2
n mL
h n E
n L
mE h
p h
n
Finding the Quantized Energies of an infinitely deep potential
energy well
The ground state and
excited states
The Zero- Point
Energy
n can’t be 0
The Energy Levels
能階
, 3 , 2 , 1
, ) (
sin
, 3 , 2 , 1
, ) sin(
, 3 , 2 , 1
, ) sin(
2 2
2
n L x
A n
n L x
A n
n L x
A n y
n n n
The Wave Function and Probability
Density
For astring
The Probability
Density
•
Normalization ( 歸一 化 ) 2( ) 1 2 /
n
x dx A L
Correspondence principle
(
對應原理 )At large enough quantum numbers, the predictions of quantum mechanics
merge smoothly with those of classical physics
0 )]
( 8 [
2 2 2
2
x E
h E
m dx
d
pot
A Finite Well
有限位能井The probability
densities and energy
levels
Barrier Tunneling
穿隧效應2 2
2
8 ( ( ) )
pot
kL m E x E
T e k
h
• Transmission coefficient
STM
掃描式穿隧顯微鏡Piezoelectricity of quartz
12-8 Three Electron Traps
• Nanocrystallites 硒化鎘奈米晶粒 那種顏色的顆粒比較小
t t
t E
ch c f
2 2 2
8mL h En n
A Quantum Dot
An Artificial Atom
The number of electrons can be controlled
Quantum Corral
量子圍欄
12-1.9 The Hydrogen Atom
• The Energies
, 3
, 2 , 1
ev , 6
. 13 1
8
4 1 4
1
2 2
2 2
0 4
2
0 2
1 0
n n n
h E me
r e r
q U q
n
氫原 子能 階與 光譜 線
Bohr’s Theory of the
Hydrogen Atom
radius) (Bohr
pm 29
. 5 ) 1
(
2 0 2
/ 2
/ 3
