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Lecture contents
• Quantum mechanics and solid state physics review
– Interaction of two wells
– Many wells: formation of bands
NNSE 618 Lecture #2
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Secondary quantization: photons
• Velocity
• Momentum
• Photon energy:
• Energy of optical field
(ensemble of harmonic
oscillators)
c = ln
p  k  
2
l

h
l
E    hn
1

E k ,     nk ,    
2

NNSE 618 Lecture #2
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Two wells
V(x)
0
V1(x)
L
V2(x)
x
E

First approximation of perturbation theory:

E   1*V2 ( x) 1dx   1 | V2 |  1

What will happen if the wells are resonant ?
Hˆ  (r )  (T  V1  V2 )  E
Solution in the form:
Negative
  C1 1  C2 2
NNSE 618 Lecture #2
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Two wells: Solution
NNSE 618 Lecture #2
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Two wells: energy level splitting
V(x)
x
0

0
L
anti-bonding

E
E
bonding
Negative
E    1 | V2 |  1   1 | V1 |  2
 
1
2
 1  2 
•
Resonant levels split
•
Splitting increases at small distances
NNSE 618 Lecture #2
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Orbital structure and energy levels of Si atom
3p
3s
Orbital structure
Energy levels of
electrons in silicon
Silicon: 1s22s22p63s23p2
NNSE 618 Lecture #2
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Bonding and antibonding
When two atoms brought together,
wave functions interact each other
and energy levels split.
Wave function
Probability density function
Antibonding
Bonding
NNSE 618 Lecture #2
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Periodic Potential Well
V(x)
R
x
E  2  1 | V1 |  2
• Formation of energy band
• Width depends mainly on interaction with the closest
neighbors
• Wave-functions are renormalized
NNSE 618 Lecture #2
Energy levels of Si as a function of interatomic spacing
Low
Repulsion
Attraction
9
When N Si atoms
brought together
3p2
Potential
energy
3s2
High
Interatomic spacing
Close
Far
NNSE 618 Lecture #2