Download 物理(十五)

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  eVstop
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

Vstop  ( ) f 
e
e
34
h  6.6 10 J  s
Work
function
12-1.2 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
12-1.3 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
( x, y, z, t )   ( x, y, z )e
 i t
• The2probability (per unit time) is 

ie.  *
Complex
conjugate
The Schrodinger Equation
from A Simple Wave
Function
 i t
 ( x, y , z , t )   ( x, y , z ) e
  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)
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
 2 d 2

 E
2
2m dx
2
2
2
2
 d d d

( 2  2  2 )  E
2m dx
dy
dz


2

   i 
2m
t
2


2

   V  i 
2m
t
2
12-1.6 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
12-1.7 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
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
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
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