Download Chap. 3. Elementary Quantum Physics

Chap. 3. Elementary Quantum Physics
3.1 Photons
- Light: e.m "waves" - interference, diffraction, refraction, reflection

             with     

Ey
y
Velocity = c
Direction
of Propagation
x
z
x
Bz
Fig. 3.1: The classical view of light as an electromagnetic wave. An
electromagnetic wave is a travelling wave which has time varying
electric and magnetic fields which are perpendicular to each other
and to the direction of propagation.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
The instantaneous intensity (energy flow per unit area per second)

    →     

*Light: a stream of discrete energy packets (photons: "particles" of zero rest-mass),
each carrying energy  and momentum  .
- Young's interference experiment:
Path difference    =  for constructive interference

=       for destructive interference

- Bragg's law: x-ray beam from "a single crystal" (diffracted patterns of "spots") or
"a polycrystalline material, powered crystal" (diffracted patterns of bright "rings" no unique orientation of crystal axes)
Existence of a diffracted beam:     
     
Photographic film
Detector
Photographic film
X-rays
1
1
2
2
θ
Scattered X-rays
Single crystal
X-rays with all
wavelengths
(a)
Scattered X-rays
Powdered crystal or
polycrystalline material
X-rays with single
wavelength
d
d
A
θ
dsinθ
dsinθ
B
(c)
(b)
Fig. 3.3: Diffraction patterns obtained by passing X-rays through crystals
can only be explained by using ideas based on the interference of waves.
(a) Diffraction of X-rays from a single crystal gives a diffraction pattern
of bright spots on a photographic film. (b) Diffraction of X-rays from a
powdered crystalline material or a polycrystalline material gives a
diffraction pattern of bright rings on a photographic film. (c) X-ray
diffraction involves constructive interference of waves being "reflected"
by various atomic planes in the crystal.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Atomic planes
Crystal
3.1.1. The Photoelectric Effect
For an incident light with         onto a metal surface, the electrons will be
emitted (the current I is generated).

       

where Vo = the negative anode voltage at which the current I extinguishes.
     where h = Plank's constant.
- The work function:      →   
KEm
Cs
K
slope = h
υ0 3
0
W
υ
υ 02
υ 01
-Φ 3
-Φ 2
-Φ 1
F ig. 3 .6: T h e effect of varyin g th e freq uency of light an d th e
cath od e m aterial in th e p h otoelectric experim en t. T h e lin es for the
d ifferent m aterials ha ve th e sam e slop e of h bu t different in tercep ts.
F ro m P rin cip le s o f E le ctro n ic M a te ria ls a n d D e v ic e s , S e co n d E d itio n , S .O . K a sa p (© M c G ra w -H ill, 2 0 0 2 )
htt p :/ /M a te ria ls.U s a s k .C a
3.1.2. Compton Scattering
- X-ray scattering by an electron:     →  ′ ′ 
KE of the elelctron =   ′
momentum of the photon    
Recoiling electron
X-ray photon
c
Electron
φ
θ
υ, λ
y
Scattered photon
x
υ', λ'
c
Fig. 3.9: Scattering of an x-ray photon by a "free" electron in a
conductor.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
The scattered x-rays are detected at various angles  with respect to the original direction
(    ), their wavelength ′ is measured.
Photon energy:
    
Photon momentum:       where    
X-ray spectrometer
Source of
monochromatic
X-rays
Collimator
λ'
λ0
λ0
θ
X-ray beam
Unscattered xrays
Path of the spectrometer
(a) A schematic diagram of the Compton experiment.
Intensity of
X-rays
θ = 0°
Intensity of
X-rays
Primary beam
λ0
λ
λ0
θ = 90°
λ'
Intensity of
x-rays
λ
λ0
θ = 135°
λ'
(b) Results from the Compton experiment
Fig. 3.10. T he C om pton experim ent and its results.
From P rinciples of E lectronic M aterials and D evices, S econd E dition, S .O . K asap (© M cG raw -H ill, 2002)
http://M ate ria ls.U sask .C a
λ
3.1.3, Blackbody Radiation
- Rayleigh-Jeans law (classical): ∝ and ∝ 
where the spectral irradiance  = the emitted radiation intensity (power per unit area)
per unit wavelength, so that   = the intensity in a small range of wavelength 
- UV catastrophe in the short wavelength range
Escaping black body
radiation
Hot body
Small hole acts as a black body
Spectral irradiance
Iλ
3000 K
Classical theory
Planck's radiation law
2500 K
0
1
λ ( µm)
2
3
4
5
Fig. 3.11. Schem atic illustration of black body radiation and its
characteristics. Spectral irradiance vs w avelength at tw o tem peratures
(3000K is about the tem perature of the incandescent tungsten
filam ent in a light bulb).
From P rinciples of E lectronic M aterials and D evices, S econd E dition, S .O . Kasap (© M cG raw -H ill, 2002)
http://M a te ria ls.U sask .C a
- Plank's blackbody radiation formula:  →    
"Classical" → "Quantum mechanically": light quanta = photon

 



   





3.2. The Electron as a Wave
3.2.1. De Broglie Relationship
- Electron: a wave of wavelength  (wave-like & particle-like)
- Electron diffraction experiments:


 where    ("real" particle)
 

3.2.2. Time-Independent Schroedinger Equation
- A travelling em wave:                      
where          = the spatial dependence.
- The average intensity
     
- 1926, Max Born : a probability wave interpretation for
"the wave-like behavior of the electron"
             : a plane traveling wavefunction for an electric field
    →    
- The wave property of the electron described by         :
1)         = the probability of finding the electron per unit vol. at (x,y,z) at time t.

or          = the probability in a small vol. dxdydz.
2)    has physical meaning, not  itself.
3)       : single-valued (See Fig. 3.14)
     
4)        : continuous (See Fig. 3.14)

ψ(x)
ψ(x)
ψ(x) not continuous
dψ not continuous
dx
x
ψ(x)
x
ψ(x) not single valued
x
Fig. 3.14: Unacceptable forms of ψ (x)
From Principles of Electronic M aterials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://M aterials.U sask.Ca


 
- Total wavefunction               where   , the angular frequency.

- Time-independent Schroedinger equation for     = the spatial dependence
 

       ;














        in 3-dim. space.

  





3.3. Infinite Potential Well: A Confined Electron
- For a certain region,     , an electron is confined.
 
  
  


→
             
Using the b.c. of           ,


            


The energy of the electron:    →


     
 
 
    




V(x)
Electron
∞
V=∞
V=0
V=∞
x
0
0
a
Energy levels in the well ψ(x) ∝ sin(nπx/a) Probability density ∝ |ψ(x)|2
ψ4
Energy of electron
E4
n=4
ψ3
E3
E2
E1
0
x=0
n=3
ψ2
n=2
n=1
ψ1
x=a
0
a0
a
x
Fig. 3.15: Electron in a one-dimensional infinite PE well. The energy
of the electron is quantized. Possible wavefunctions and the
probability distributions for the electron are shown.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Normalization condition (A) :




     


        



 

The normalized wavefunction              


      
- Quantized energy levels:         
 
→

→
(free electron case: continuous)
3.4. Heisenberg's Uncertainty Principle
- For an electron trapped in a 1-dim infinite PE well in the region     ,
The uncertainty in position = a, the uncertainty in momentum =       
For n = 1 (ground state),    →      

            

- Heisenberg's uncertainty principle:  ≥  and  ≥ 
3.5. Tunneling Phenomena: Quantum Leak: Finite PE Well
D
Start here from rest
E
C
A
B
(a)
V(x)
E < Vo
Vo
ψΙ(x)
A1
A2
ψΙΙ(x)
Incident
)
Reflected
Transmitted
I
III
II
x=0
Fig. 3.16
ψΙΙΙ (x
(b)
x=a
x
(a) The roller coaster released from A can at most make to C, but not to E. Its PE at A is
less than the PE at D. When the car is the bottom its energy is totally KE. CD is the energy
barrier which prevents the car making to E. In quantum theory, on the other hand, there is
a chance that the car could tunnel (leak) through the potential energy barrier between C
and E and emerge on the other side of the hill at E .
(b) The wavefunction of the electron incident on a potential energy barrier (V o ). The
incident and reflected waves interfere to give ψ I (x). There is no reflected wave in region
III. In region II the wavefunction decays with x because E < V o .
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.U sask.Ca
- Three regions: I, II, and III (boundary conditions, C2 = 0, normalization)
      



              where   






             



             
   
   


  

    

   
   
                   


          
- Transmission coefficient T : the relative probability that the electron will tunnel from I to III.
   

 

      
where   

      
  → 

      
- For a "wide" or "high" barrier, using  ≫        ≈       
      
 ≈     where   
 
Metal
Metal
Vacuum
ψ(x)
ψ(x)
Vo
V(x)
Vacuum
Second Metal
Vo
V(x)
E < Vo
x
(a)
Probe
x
Itunnel
(b)
Itunnel
Scan
x
Material
surface
Image of surface (schematic sketch)
(c)
Fig. 3.17: (a) The wavefunction decays exponentially as we move away from the
surface because the PE outside the metal is Vo and the energy of the electron, E < Vo..
(b) If we bring a second metal close to the first metal, then the wavefunction can
penetrate into the second metal. The electron can tunnel from the first metal to the
second. (c) The principle of the Scanning Tunneling Microscope. The tunneling current
depends on exp(-αa) where a is the distance of the probe from the surface of the
material and α is a constant.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
3.6. Potential Box (3-Dim. Quantum Numbers)






        with V= 0 in 0<x<a, 0<y<b, and 0<z<c.
       





Let                     , then
                    







Using b.c,       with  = the quantum numbers.



- The eigenfunctions of the electron:



                      



- The energy eigenvalues:
 
           

For a=b=c,      
→



      


        


 


3.7. Hydrogen Atom
3.7.1. Electron Wavefunctions
 
- Potential:       ; the wavefunction:                            

- Principle quantum number:
n = 1,2,3,....
Orbital angular momentum quantum number: l = 0,1,2,3......(n-1) < n
Magnetic quantum number:
ml = -l, -(l-1),.....0,.....(l-1), l
- Labeling of various n l possibilities : n=1 (K), L, M, N ; l=0 (s), l=1 (p), l=2 (d), l=3 (f),...
- The probability that the electron is in the spherical shell of thickness :
          
3.7.2. Quantized Electron Energy
 
 
 
- The electron energy:   
  
 


 




×


where   
 

 
3.7.3. Orbital Angular Momentum and Space Quantization

- Orbital angular momentum:           where         
    where  ≤        states
- Selection rules for EM radiation:
 ± 
    ± 
Energy
0
5
4
3
2
-13.6eV
l=0
l=1
l=2
l=3
5s
5p
5d
5f
4s
4p
4d
4f
3s
3p
3d
2s
2p
n
1
l
Photon
1s
Fig. 3.27: An illustration of the allowed photon emission processes.
Photon emission involves ∆ l = ±1.
From Principles of Electronic M aterials and D evices, S econd E dition, S.O . K asap (© McG raw-H ill, 2002)
http://M a teria ls.U sask.C a
3.7.4. Electron Spin and Intrinsic Angular Momentum S

  
- Spin:          
        ±   

- Magnetic dipole moment of the electron:          


Since       ,    

µorbital
B
N
=
ω
=
A
=
i
-e
B
(a)
S
S
Spin direction
S
=
N
Equivalent current
µspin
Magnetic moment
(b)
Fig. 3.29: (a) The orbitting electron is equivalent to a current loop
which behaves like a bar of magnet.
(b) The spinning electron ican be imagined to be equivalent to a
current loop as shown. This current loop behaves like a bar of
magnet just as in orbital case.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.U sask.Ca
3.7.5. Total Angular Momentum J
- Total angular momentum: J = L + S
        
    
B
J
z
Jz=mjh
S
J
L
S
L
(a)
(b)
Fig. 3.31 (a) The angular momentum vectors L and S precess
around their resultant total angular momentum vector J.
(b) The total angular momentum vector is space quantized.
Vector J precesses about the z-axis along which its component
must be m j h .
[Reading Assignment]
3.8. The He Atom and The Periodic Table
3.9. Stimulated Emission and Lasers
3.10. Time-Dependent Schroedinger Equation
[Homework]
Prob. #3.5