This document discusses the key principles of quantum physics including: (1) The wave-particle duality of microparticles like electrons described by de Broglie's equation. (2) Energy quantization described by Planck's equation. (3) Heisenberg's uncertainty principle. It describes how Schrodinger's equation is used to model the wave-like behavior of electrons in solids. The energy and behavior of electrons is quantized based on solutions to Schrodinger's equation under different boundary conditions, such as electrons confined in a potential well or interacting with a potential barrier. Quantum theory was needed to fully explain properties of electrons in solids and failures of classical free electron theory

1. Quantum Theory
• Macro particle- Classical Physics based on Newton’s laws
• Micro particle – Quantum Physics; ex- electrons and high
frequency electromagnetic waves
• Basic principles in quantum physics:
(I) energy quanta :- E=hν
(II) Wave particle duality :- λ = h/p
(III) Uncertainty principle :- Δx. Δp ≥ ħ = h/2π & other two
[ħ = 1.054 x 10 -34 J-s is very small; so significant only in the
subatomic level]
• Consequence: we cannot determine the exact position of
an electron, but only determine the probability of finding
an electron at a particular position

2. • Failures of classical free electron theory gave rise to Quantum free
electron theory by Sommerfield.
• Particles of micro dimension like the electrons are studied under
quantum physics
• moving electrons inside a solid material can be associated with
waves with a wave function ψ(x) in one dimension (ψ(r) in 3D)
• Hence its behaviors can be studied with the Schrödinger's equation
time-dependant
&
ψ, is used to describe the behavior of the particle with wave nature &
is a complex quantity

3. • Probability density given by
•
=
With
Boundary conditions used for solution of the Sch. Eqn.

4. Application of Sch. Eqn.
Free Particle: continuous energy
m
k
E
2
22



5. Sommerfield’s model
• As the freely moving electrons can not escape the surface of the
material, they may be treated as particles confined (trapped)in a
box
• Hence, V(x) =0, for 0<x<L . e-
= ∞, for x=0 & x=L V(x)
0 L x
Sch’s equation ,
Solution of the equation can be obtained as
ψ(x) = A sin kx + B cos kx
From boundary conditions , at x=o & x=L , ψ(x)=0,
we can get B=0 and k= ± nπ / L ,
Putting the normalization condition we get A=
0)(
2)(
22
2



x
mE
x
x



P= 0
Ψ= 0
P= 0
Ψ= 0
L
2

6. Energy of the electrons inside the material is quantized and hence is
discrete
ψ3 n=3 E3
ψ2 n=2 E2
ψ1 n=1 E1

7.  Particle in Potential Well : energy is quantized and discrete
 Particle in Potential step: finite probability of particle moving to
step region when E < V0
 Particle in Potential barrier: Tunneling
 Wave theory may be extended to the atoms:
 Electron distribution in the atom obey Pauli’s exclusion principle

8. • Electrons inside the atoms of a solid have wave nature
• Free electrons move throughout the crystal, but restricted
within the surfaces; hence are treated as particle trapped in a
box and studied with help of Sch. Eqn.
• Electrons in a solid can take up discrete energy values & obey
Pauli's principle for their distribution
• To determine the electrical properties in a semiconductor
crystal and develop the current- voltage characteristic for
device application, it is important to understand the properties
of the electrons in the crystal lattice and the statistical
characteristics of the large no of electrons in the crystal.
• Behavior of free electron is different from electron in the
potential field of the crystal
• As current is due to flow of charge, electron behavior in
external electric field is important

9. Failures of quantum free electron theory:
1. It could not explain the negative temperature co-efficient of
resistivity for certain solids
2. It could not classify a solid on the basis of their conductivity
property
3. It failed to explain the sign of Hall co-efficient.
Quantum free electron theory could successfully explain
The electronic sp. heat,
Superconducting phenomena,
Correct value of Lorentz no. & PE effect, Compton Effect, blackbody
radiation.
In classical theory L = Lorenz no. = 3kβ
2 / 2q2 (1.11X 10 -8 WΩ / K2)
In quantum theory L = π2kβ
2/ 3q2 = 2.45X 10 -8 WΩ / K2