1. Photon and Matter Waves 2. Compton Effect

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

• Second Experiment (adjusting f)–

the cutoff frequency f

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 _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

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 c  f

 

2 2 2

8mL

h

E

 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 

, 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

n

n n



   

44

Energy Changes

39-

Substituting f=c/ and using the energies E_n 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  

4

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

m

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}