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 V
stop• Second Experiment (adjusting f)–
the cutoff frequency f
0stop
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 stop against f
stop
( ) h
V f
e e
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 i
z y x
t z y
x , , , ) ( , , )
(
2 ie. *
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 t
2 / 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 a
string
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 ( )
bL
m U
bE
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
E
n 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
41
The Bohr Model of the Hydrogen Atom
39-
Fig. 39-16
Balmer’s empirical (based only on observation) formula on
absorption/emission of visible light for H
2 2
1 1 1
, for 3, 4,5, and 6
R 2 n
n
Bohr’s assumptions to explain Balmer formula 1) Electron orbits nucleus
2) The magnitude of the electron’s angular momentum L is quantized
, for 1, 2,3,
L n n
42
Coulomb force attracting electron toward nucleus
Orbital Radius is Quantized in the Bohr Model
39- 1 2
2
F k q q
r
2 2
2 0
1 4
e v
F ma m
r r
Quantize angular momentum
l : sin n
rmv rmv n v
rm
Substitute v into force equation
:
2 0 2
2
, for 1, 2,3,
r h n n
me
r an
2, for n 1, 2,3,
Where the smallest possible orbital radius (n=1) is called the Bohr radius a:
2 0 10
2
5.291772 10 m 52.92 pm a h
me
43
The total mechanical energy of the electron in H is:
Orbital Energy is Quantized
39- 2 2
12 2
0
1 4 E K U mv e
r
Solving the F=ma equation for mv2 and substituting into the energy equation above:
2
0
1 8 E e
r
Substituting the quantized form for r:
4
2 2 2 0
1 for 1, 2,3,
n
8
E me n
h n
18
2 2
2 . 1 8 0 1 0 J 1 3 . 6 0 e V
= , f o r 1 , 2 , 3 ,
E
nn
n n
44
Energy Changes
39-
Substituting f=c/ and using the energies En allowed for H:
This is precisely the formula Balmer used to model experimental emission and absorption measurements in hydrogen! However, the premise that the electron orbits the nucleus is incorrect!
Must treat electron as matter wave.
2 2
low high
1 1 1
R n n
high low
hf E E E
4
2 3 2 2
0 high low
1 1 1
8 me
h c n n
Where the Rydberg constant
4 7 -1
2 3 0
1.097373 10 m 8
R me
h c
氫 原
子 能
階 與
光 譜
線
Schrödinger’s
Equation and the Hydrogen Atom
Fig. 39-17
24
0U r e
r
radius) (Bohr
pm 29
. 5 ) 1
(
2 0 2
/ 2
/ 3
me a h
a e
r r a
The Ground State Wave
Function
Quantum Numbers for the Hydrogen Atom
For ground state, since
n
=1 l=0 and →m
l =0The Ground State Dot
Plot
50
Fig. 39-21
Wave Function of the Hydrogen Atom’s Ground State
Fig. 39-20 39-
2
r
Probability of finding electron within a small volume at a given position
Probability of finding electron within a within a small distance from a given radius
P r
氫原子的量子數
N=2, l=0, m l =0
N=2, l=1
54
As the principal quantum number increases,
electronic states appear more like classical orbits.
Hydrogen Atom States with n>>1
39-
Fig. 39-25