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
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
function
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 Electron
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 01 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 Ub 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 f
c
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
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 an2, 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
1 2
2 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
2.180 10 J 13.60 eV
= , for 1, 2,3,
En n
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 2
4 0
U 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 ml =0
The 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 P r for n 45, n 1 44