Download 物理(十三)

12 量子物理
Sections
1.
2.
3.
4.
5.
6.
7.
8.
9.
Photon and Matter Waves
Compton Effect
Light as a Probability Wave
Electrons and Matter Waves
Schrodinger’s Equation
Waves on Strings and Matter Waves
Trapping an Electron
Three Electron Traps
The Hydrogen Atom
12-1 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  eVstop
光電子的最大動能與
光強度無關
•Second Experiment (adjusting f)–
the cutoff frequency f0
低於截止頻率時即使光再
強也不會有光電效應
The plot of Vstop against f
The Photoelectric Equation
hf  K max  
h

Vstop  ( ) f 
e
e
34
h  6.6 10 J  s
Work
function
12-2 Compton Effect
hf h
p

c 
(photon momentum)
康
普
吞
效
應
實
驗
圖
表
康普吞效應圖示
Energy and momentum
conservation
hf  hf   K
K  mc (  1)
2
2

hf  hf  mc (  1)
h
h

 mc(  1)
 
p X  h /  pe  mv
Frequency shift
h
h

cos   mv cos 
 
h
0  sin   mv sin 

h
 
(1  cos  )
mc
Compton wavelength
12-3 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, WideAngle Version (1992)
50μm
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

12-4 Electrons and
Matter Waves
•The de Broglie wave length
•Experimental verification in 1927
•Iodine molecule beam in 1994
h

p
1989 double-slit experiment
7,100,3000, 20,000 and 70,000 electrons
Experimental Verifications
X-ray
Electron
beam
苯
環
的
中
子
繞
射
12-5 Schrodinger’s Equation
• Matter waves and the wave function
( x, y, z, t )   ( x, y, z )e
 i t
•The probability (per unit time) is 

2
ie.  *
Complex conjugate
共軛複數
The Schrodinger Equation from A
Simple Wave Function
 i t
 ( x, y , z , t )   ( x, y , z ) e
  A sin( kx)  B cos( kx) (1D)
p  h /   k
E  p / 2m   k / 2m
2
2
2
1D Time-independent SE
  A sin( kx)  B cos( kx)
1d
d  / dx  k  k  
2
 dx
2
2
2
2
1  d
E
2
 2m dx
2
2
 d

 E
2
2m dx
2
2
2
3D Time-dependent SE
d 2

 E
2
2m dx
2
2
2
2
  

( 2  2  2 )  E
2m x
y
z
2


 i

2m
t
2
2


   V  i

2m
t
2
2
12-6 Waves on Strings and
Matter Waves
駐波與量子化 Quantization
駐波：
2L
v
v
=
f  n
n = 0,1,2,   
n

2L
Confinement of a Wave leads to
Quantization – discrete states and
discrete energies
12-7 Trapping an Electron
For a string：
n
L
2
n  1,2,3,
n
yn  A sin(
) x, n  1,2,3, 
L
n : quantum number
Finding the Quantized Energies of an infinitely
deep potential energy well
  h / p  h / 2mE , L  n / 2
En  n h / 8mL ,
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
n
n  1,2,3,
) x,
yn  A sin(
L
n
n  1,2,3,
 n  A sin( ) x,
L
2 n
2
2
 n  A sin ( ) x, n  1,2,3,
L
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  A  2 / L

2
n
A Finite Well 有限位能井
d  8 m
 2 [ E  E pot ( x)]  0
2
dx
h
2
2
The probability densities and
energy levels
Barrier Tunneling
穿隧效應
•Transmission
coefficient
T e
2 kL
k
8 2m( E pot ( x)  E )
h2
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
2
2
nh
En 
8mL2
c ch
t  
f t 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
2
1
q1q2
1 e
U

40 r
40 r
4
me 1
13.6ev
En   2 2 2  
,
2
8 0 h n
n
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
e
 (r ) 
3/ 2
a
2
h 0
 5.29pm
a
2
me
(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