# 物理(十五)

```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

The experiment
the stopping potential Vstop
K max  eV stop

the cutoff frequency f0

The plot of Vstop against f
h

V stop  ( ) f 
e
e
The Photoelectric Equation
hf  K max  

h
V stop  ( ) f 
e
e
h  6 . 6  10
 34
J s
Work
function
12-2 Compton Effect
p
hf
c

h

(photon
momentum)

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
12-3 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

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

The Schrodinger Equation from A
Simple Wave Function
 ( x, y , z , t )   ( x, y , z )e
 i t
  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 )
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


2
(
2m
 E
2
x
2

2
 i

2
2m

2
y
2

t

   V  i

2m
z
2
)  E

2
2

2

t

12-6 Waves on Strings and
Matter Waves

 =
2L
n
f 
v

n
v
n = 0 ,1 ,2 ,   
2L
Confinement of a Wave leads to
Quantization – discrete states and
discrete energies
12-7 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  A 
2
n
2/L
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

•Transmission
coefficient
8 m (U b  E )
2
T e
 2 bL
k 
h
2
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

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

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 , 
The Bohr Model of the Hydrogen Atom
Balmer’s empirical (based only on
observation) formula on
absorption/emission of visible light for H
1 
 1
 R  2  2  , for n  3, 4, 5, and 6

n 
2
1
Bohr’s assumptions to explain Balmer formula
1) Electron orbits nucleus
2) The magnitude of the electron’s angular
momentum L is quantized
Fig. 39-16
L  n , for n  1, 2, 3,
39- 41
Orbital Radius is Quantized in the Bohr Model
Coulomb force attracting electron toward nucleus
F 
e
2
4 0 r
2
1
Quantize angular momentum l :
F k
q1 q 2
r
 v2 
 ma  m  

 r 
2
 rm v sin   rm v  n  v 
n
rm
Substitute v into force equation:
h 0
2
r 
 me
r  an , for n  1, 2, 3,
2
n , for n  1, 2, 3,
2
2
Where the smallest possible orbital radius (n=1) is called the Bohr radius a:
h 0
2
a
 me
2
 5.291772  10
 10
m  52.92 pm
39- 42
Orbital Energy is Quantized
The total mechanical energy of the electron in H is:
2


1
e
2
1
E  K  U  2 mv   
2 
4


r
0


Solving the F=ma equation for mv2 and substituting into the energy
equation above:
E 
1
e
2
8 0 r
Substituting the quantized form for r:
En  
2.180  10
n
2
 18
J
=
En  
13.60 eV
n
2
me
4
1
8 h n
2
0
2
2
for n  1, 2, 3,
, for n  1, 2, 3,
39- 43
Energy Changes
hf   E  E high  E low
Substituting f=c/ and using the energies En allowed for H:
 1
1

 2
2 3 
2

8  0 h c  n high n low
1
 1
1
 R 2  2
n

n high
 low
1
me




4




Where the Rydberg constant
R
me
4
8 0 h c
2
3
 1.097373  10
7
m
-1
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.
39- 44

Schrödinger’s Equation and
the Hydrogen Atom
Fig. 39-17
U r 
e
2
4 0 r
The Ground State Wave Function
1
 (r ) 
a
h 0
3/2
e
r /a
2
a 
 me
2
 5 . 29 pm
Quantum Numbers for the
Hydrogen Atom
For ground state, since n=1→ l=0 and ml =0
The Ground State Dot Plot
Wave Function of the Hydrogen Atom’s Ground State
Probability of finding electron
within a within a small distance
Probability of finding electron
within a small volume at a
given position

P r
2
r
Fig. 39-21
Fig. 39-20
39- 50

N=2, l=0, ml=0
N=2, l=1
Hydrogen Atom States with n>>1
As the principal quantum number increases,
electronic states appear more like classical orbits.
Fig. 39-25
P  r  fo r n  4 5,
 n  1  44
39-54
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