1. Condensed Matter Systems
Hard Matter
Soft Matter
Colloidal
Dispersions
Liquid
Polymer
Melts and
Solutions
Crystalline Solids
(Metals, Insulators,
Semiconductors)
Non-Crystalline
Solids
Crystals
Crystalline
Solids +
Defects
(point,
Quasicrystals
Amorphous
Solids (Glass)
Biomatter
(proteins,
membranes,
dislocations,
surfaces and
interfaces)
Polymer
Solids (Glass
and Rubber)
nucleic acids)
PHYS 624: Introduction to Solid State Physics

2. ∑ ∑=i 1 2M ∑ji≠ =1 | r − r | Quantum Hamiltonian of
Condensed Matter Physics
“”The general theory of quantum mechanics is now almost complete. The
underlying physical laws necessary for the mathematical theory of a
large part of physics and the whole of chemistry are thus completely
known, and the difficulty is only that the exact application of these laws
leads to equations much too complicated to be soluble.”
P. A. M. Dirac NN
n=1 2Mn
TN : motion of nuclei
Pn2
e2 NN Zn Zm
2 n≠m=1 | Rn − Rm |
VN −N : interaction between nuclei
Hˆ = ∑ ∑
Quantization
∂ ∂
∂r ∂R +
p ≡ −iℏ
, P ≡ −iℏ Ne
pi2
e2 Ne
i j
NN Ne
∑=i 1 2M
∑ji≠ =1 | r − r |
e ∑∑ Zn + + − 2
n=1 i=1 | Rn − r | i i
Te : motion of electrons
Ve−e :int eraction between electrons
Ve−N :interaction between electrons and nuclei
PHYS 624: Introduction to Solid State Physics

3. N N ∼ Ne ∼ 10 Complexity in Solid State Hamiltonian
Even for chemist, the task of solving the
Schrödinger equation for modest multi-
electron atoms proves insurmountable
without bold approximations.
The problem facing condensed matter
physicist is qualitatively more severe:
N N ∼ Ne ∼ 10 23 −
CM Theorist is entrapped in the “thermodynamic limit”
PHYS 624: Introduction to Solid State Physics

4. Time: Separate Length and Energy Scales Energy: ω , T < 1 e V
The Way Out:
Separate Length and Energy Scales
Energy: ω , T < 1 e V
ℏ ℏ
1eV − 19 J
∆ τ ∼ ∼ ∼ 1 0 − 15 s
Time:
Length: | x i − x j |, q − 1 ≫ 1 Å
forget about atom formation + forget about crystal formation me mN
+ Born-Oppenheimer approximation for
≪ 1 ⇓
Helectronic =T +V −I +V −e, V −I (r +rn ) =V −I (r), r = na1 +n2a2 +n3a3
PHYS 624: Introduction to Solid State Physics
ˆ ˆe ˆe ˆe ˆe ˆe n

5. ∑ “Non-Interacting” Electrons in Solids: Band Structure Calculations
In band structure calculations electron-electron interaction is
approximated in such a way that the resulting problem becomes an
effective single-particle quantum mechanical problem …
H IE,H,LDA (r )φk E,H,LDA (r ) = ε IE,H,LDA (k , b )φ IE,H,LDA (r )
ˆ Ib kb
H electronic = Te + Ve − I + Ve − e N
ˆ ˆ ˆ ˆ
IE IEˆ ˆ ˆ ( )e e I iH T H−= = ∑ r e V+
i =1
n(r′)
Hartree e e IH T e d− ′= ∫ r
ˆ ˆ ˆV+
, ( )n =r ∑
H| )b nφk
2( ) | (n= r r r
| r − r′ | ε H (k ,b)≤ EF
LDA[ ]( ) eE nn δ′r
LDA LDALDA IE ( ) , [ ] ( ) ( ( ))eH H e d E n d n e n′= =∫ ∫r r r r r ˆ 2
| r −r′ | δ n(r)
PHYS 624: Introduction to Solid State Physics

6. From Many-Body Problem to Density Functional
Theory (and its LDA approximation)
“Classical” Schrödinger equation approach:
SE Ψ ... Ψ
V (r) ⇒ Ψ(r1 ,…, rNe ) ⇒ average of observables
Example: Particle density
n(r) = N ∫ dr2 ∫ dr3 ⋯∫ drN Ψ* (r,… , rNe )Ψ (r,… , rNe )
DFT approach (Kohn-Hohenberg):
n0 (r) ⇒ Ψ 0 (r1 ,… , rN e ) ⇒ V (r )
Ψ 0 (r1 ,… , rN e ) ≡ Ψ 0 [n0 (r )]
PHYS 624: Introduction to Solid State Physics

7. Pauli Exclusion Principle for Atoms
Without the exclusion principle all electrons would occupy the same atomic orbital.
There would be no chemistry, no life!
PHYS 624: Introduction to Solid State Physics

8. Many-Body Wave Function of Fermions
“It is with a heavy heart, I have decided that Fermi-Dirac, and not
Einstein is the correct statistics, and I have decided to write a short
note on paramagnetism.” W. Pauli in a letter to Schrödinger (1925).
All electrons in the Universe are identical
two physical situations that
differ only by interchange of identical particles are indistinguishable!
Pij Ψ ( x1 ,… , xi ,… , x j ,… , xn ) = Pij Ψ ( x1 ,… , x j ,… , xi ,… , xn )
ˆ ˆ † ˆ ˆ ↑
Ψ P ΨS =ΨS , P ΨA = 
P A Ψ = A Ψ , A = 1 ∑εα P
N ! α
PHYS 624: Introduction to Solid State Physics
+ ΨA , Pˆα is even ΨS
A ˆα ˆα
− ΨA , Pˆα is odd
ˆα ˆ ˆ ˆ ˆα

9. Pauli Exclusion Principle for Hartree-
Fock or Kohn-Scham Quasiparticles
“There is no one fact in the physical world which has greater impact
on the way things are, than the Pauli exclusion principle.” I. Duck and
E. C. G. Sudarshan, “Pauli and the Spin-Statistics Theorem” (World
Scientific, Singapore, 1998).
Two identical fermions cannot occupy the same quantum-mechanical state:
H = h1 + h2 +… + hN
h φi = ei φi ⇒ H Φ1,2,…, Ne = E1,2,…, Ne Φ1,2,…, Ne
E1,2,…, Ne = e1 + e2 +… + eNe
ˆ ˆ ˆ ˆ
ˆi ˆ
φ1 (1) … φNe (1)
⋮ ⋱ ⋮
φ1 ( Ne ) ⋯ φNe ( Ne ) 1
N ! 1
e(εkσ −µ )/ kBT +1
Φ1,2,…, Ne =
⇒ nkσ =
PHYS 624: Introduction to Solid State Physics