12 量子物理
Sections
1.
2.
3.
4.
5.
6.
Photon and Matter Waves
Compton Effect
Light as a Probability Wave
Electrons and Matter Waves
Schrodinger’s Equation
Waves on Strings and Matter
Waves
7. Trapping an Electron
8. Three Electron Traps
9. The Hydrogen Atom
12-1.1 Photon and Matter
Waves (光子和物質波)
•
Light Waves and Photons
c  f
E  hf ( photon energy )
h  6 . 63  10
 34
J s
The Photoelectric Effect
光電效應
The experiment
• First Experiment (adjusting
V)–
the stopping potential Vstop
K max  eV stop
光電子的最大動能
與光強度無關
• Second Experiment (adjusting
f)–
低於截止頻率時即使光再
強也不會有光電效應
the cutoff
frequency f0
The plot of Vstop against
f
The Photoelectric Equation
hf  K max  

h
V stop  ( ) f 
e
e
h  6 . 6  10
 34
J s
Work
function
12-1.2 Compton Effect
p
hf
c

h

(photon
momentum)
康
普
吞
效
應
實
驗
圖
表
康普吞效應圖示
Energy and momentum
conservation
hf  h f   K
K  mc (   1)
2
2

hf  h f  mc (   1)
h


h

 mc (   1)
pX  h / 
p e   mv
Frequency shift
h


0 
h

h

 
cos    mv cos 
sin    mv sin 
h
(1  cos  )
mc
Compton
wavelength
12-1.3 Light as a Probability
Wave
The
standard
version
The Single-Photon Version
First by Taylor in 1909
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,
Wide-Angle Version
(1992)
The postulate
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

12-1.4 Electrons and
Matter Waves
 
h
• The de Broglie wave length
p
• Experimental verification in 1927
• Iodine molecule beam in 1994
1989 double-slit experiment
7,100,3000, 20,000 and 70,000 electrons
Experimental Verifications
Xray
Electro
n beam
苯
環
的
中
子
繞
射
12-1.5 Schrodinger’s
Equation
• Matter waves and the wave
function
 i t
 ( x, y , z , t )   ( x, y , z )e
• The2probability (per unit time) is 
 ie.  * 
Complex
conjugate
The Schrodinger Equation
from A Simple Wave
Function
 ( x, y , z , t )   ( x, y , z )e
 i t
  A sin( kx )  B cos( kx )
p  h /   k
E  p / 2m   k / 2m
2
2
2
1D Time-independent SE
  A sin( kx )  B cos( kx )
1 d 
2
d  / dx
2
E 


2
2
 k  k  
1 
2
2
d 
2
 2 m dx
d 
2
2
2
2 m dx
2
 E
 dx
2
3D Time-dependent SE


2
d 
2
2 m dx


2


d 
2
(
2m
dx

2
d 
2

dy
2
   i
2
2m

 E
2
2

t

dz
2m
2

2
   V   i
2
d 
2

t

)  E
12-1.6 Waves on Strings
and Matter Waves
駐波與量子化
Quantization
駐波：
 =
2L
n
f 
v

n
v
n = 0 ,1 ,2 ,   
2L
Confinement of a Wave leads to
Quantization – discrete states
and discrete energies
12-1.7 Trapping an
Electron
For a string：
L 
n
n  1, 2 , 3 , 
2
y n  A sin(
n : quantum
n
) x,
L
number
n  1, 2 ,3 , 
Finding the Quantized Energies
of an infinitely deep potential
energy well
  h/ p  h/
2 mE , L  n  / 2
E n  n h / 8 mL ,
2
2
2
n  1, 2 ,3 , 
The Energy Levels 能階
 The
ground state
and excited states
 The Zero-Point
Energy
n can’t be 0
The Wave Function and
Probability Density
For a string
y n  A sin(
 n  A sin(

2
n
n
n  1, 2 , 3 , 
) x,
n  1, 2 ,3 , 
L
n
L
 A sin (
2
) x,
2
n
L
) x , n  1, 2 , 3 , 
The Probability Density
Correspondence principle
(對應原理)
At large enough quantum numbers,
the predictions of quantum
mechanics merge smoothly with
those of classical physics
• Normalization (歸一化)



 ( x ) dx  1
2
n
A Finite Well 有限位能井
d 
2
dx
2
8 m
2

h
2
[ E  E pot ( x )]  0
The probability densities
and energy levels
Barrier Tunneling
穿隧效應
STM 掃描式穿隧顯微鏡
Piezoelectricity
of quartz
12-1.8 Three Electron
• Nanocrystallites 硒化鎘奈米晶
Traps
粒
那種顏色的顆粒比較小
2
En 
t 
n h
2
8mL
c
ft

2
ch
Et
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
U 
1
q1 q 2
4 0
r
En  
me
4
8 h
2
0
 
1
2
n
2

1
e
2
4 0 r
13 . 6 ev
n
2
,
n  1, 2 , 3 , 
氫
原
子
能
階
與
光
譜
線
Bohr’s Theory of the
Hydrogen Atom
The Ground State Wave
Function
1
 (r ) 
a
h 0
3/2
e
r /a
2
a 
 me
2
 5 . 29 pm
(Bohr radius)
Quantum Numbers for the
Hydrogen Atom
The Ground State Dot
Plot
氫原子的量子數
N=2, l=0, ml=0
N=2, l=1