ELECTRONIC PROPERTIES OF MATERIALS
Chapter 3.
Introduction to Quantum
Mechanics
Prof. Jeong Hwan Han
Department of Materials Science and Engineering
Seoul National University
Before we start..
1. 𝐸𝑥𝑝𝑜𝑛𝑒𝑡𝑖𝑎𝑙 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
𝑦
𝑒
→
𝑒
=𝑒 ,
𝑒 𝑑𝑥
𝑒
2. 𝑆𝑖𝑛𝑒, 𝑐𝑜𝑠𝑖𝑛𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
𝑧 𝑎 𝑖𝑏, 𝑥
1
𝑧
𝑎
𝑖𝑏
𝑟 𝑐𝑜𝑠𝜃
3. 𝐸𝑢𝑙𝑒𝑟 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
𝑒
𝑐𝑜𝑠𝜃
𝑖𝑠𝑖𝑛𝜃
𝑟𝑒
 Euler's formula
𝑖𝑠𝑖𝑛𝜃
2
An equation for matter wave
De Broglie postulated that every particles has an associated wave of wavelength:
  h/ p
Wave nature of matter confirmed by electron diffraction studies etc (see earlier).
If matter has wave-like properties then there must be a mathematical function
that is the solution to a differential equation that describes electrons, atoms and
molecules.
The differential equation is called the Schrödinger equation and its solution is
called the wavefunction, .
What is the form of the Schrödinger equation ?
3
Derivation of Schrodinger Equation
Photon Momentum ∶ 𝑝
ℎ
𝜆
ℏ𝑘
·········· 1
Photon Energy ∶ E
ℎ𝜈
ℎ𝑐
𝜆
Ψ 𝑥, 𝑡
𝜔𝑡
𝑖𝐴𝑠𝑖𝑛 𝑘𝑥
Ψ 𝑥, 𝑡
𝐴𝑐𝑜𝑠 𝑘𝑥
𝐴𝑒
·········· 4
ℎ 𝜕Ψ 𝑥, 𝑡
𝑖
2𝜋 𝜕𝑥
𝑖
ℎ 𝜕Ψ 𝑥, 𝑡
2𝜋 𝜕𝑡
ℎ
𝑖
𝑘Ψ 𝑥, 𝑡
2𝜋
𝑖
ℎ
𝜔Ψ 𝑥, 𝑡
2𝜋
ℏ𝜔 ·········· 2
𝜔𝑡 ·········· 3
𝜕Ψ 𝑥, 𝑡
𝜕𝑥
𝜕Ψ 𝑥, 𝑡
𝜕𝑡
𝑖𝑘Ψ 𝑥, 𝑡 ·········· 5
𝑖𝜔Ψ 𝑥, 𝑡 ·········· 6
ℎ 2𝜋
Ψ 𝑥, 𝑡
2𝜋 𝜆
ℎ
2𝜋𝜈Ψ 𝑥, 𝑡
2𝜋
ℎ
𝑖
2𝜋
𝑝Ψ 𝑥, 𝑡 ·········· 7
𝐸Ψ 𝑥, 𝑡 ·········· 8
4
Derivation of Schrodinger Equation
𝑖
ℎ 𝜕Ψ 𝑥, 𝑡
2𝜋 𝜕𝑥
𝑖ℏ
ℎ 𝜕Ψ 𝑥, 𝑡
𝑖
2𝜋 𝜕𝑡
E
𝐸
𝜕
Ψ 𝑥, 𝑡
𝜕𝑥
𝜕
𝑖ℏ Ψ 𝑥, 𝑡
𝜕𝑡
𝑝
2𝑚
𝐸 →𝐸
𝑖ℏ
𝑝Ψ 𝑥, 𝑡 ·········· 9 → 𝑝
𝐸Ψ 𝑥, 𝑡 ·········· 10 → 𝐸
𝑖ℏ
𝜕
·········· 11
𝜕𝑥
𝜕
𝑖ℏ ·········· 12
𝜕𝑡
𝑉 ········· 13
𝑖ℏ
𝑉
ℏ
𝑉 ·········· 14
Η, Hamiltonian : Energy operator
𝑖ℏ
𝛹 𝑥, 𝑡
ℏ
Ψ 𝑥, 𝑡
𝑉Ψ 𝑥, 𝑡 ·········· 15 ↔ 𝐸Ψ 𝑥, 𝑡 = ΗΨ 𝑥, 𝑡
: Time-dependent Schrodinger Wave Equation for 1-D
5
Schrodinger eq. and Newton’s eq.
 ( x, t )
 2  2  ( x, t )

i
 V ( x )  ( x, t )
2
t
2m x
Given suitable initial conditions (Ψ(x,0)) Schrodinger’s equation determines
Ψ(x,t) for all time
d 2x
V ( x)
m 2 
dt
x
Given suitable initial conditions (x(0), v(0)) Newton’s
2nd law determines x(t) for all time
Classical Energy Conservation
Maximum height
and zero speed
Zero speed start
Fastest
• total energy = kinetic energy + potential energy
• In classical mechanics,
• V depends on the system
• e.g., gravitational potential energy, electric potential energy
Schrodinger Equation and Energy Conservation
... The Schrodinger Wave Equation !
𝜕
𝑖ℏ 𝛹
𝜕𝑡
Total E term
ℏ 𝜕
Ψ
2𝑚 𝜕𝑥
K.E. term
𝑉Ψ
P.E. term
... In physics notation and in 3-D this is how it looks:
Time independent Schrodinger Equation
𝑖ℏ
ℏ
𝛹 𝑥, 𝑡
Ψ 𝑥, 𝑡
𝑉Ψ 𝑥, 𝑡 = 𝐸𝛹 𝑥, 𝑡
𝛿𝑉
𝐼𝑓 𝑡ℎ𝑒 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑜𝑓 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛 𝑖𝑠 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑜𝑓 𝑡𝑖𝑚𝑒
𝛿𝑡
0, 𝑠𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑟𝑦
𝛹 𝑥, 𝑡 𝑐𝑎𝑛 𝑏𝑒 𝑤𝑟𝑖𝑡𝑡𝑒𝑛 𝑎𝑠 𝜓 𝑥 𝜙 𝑡 ,변수분리방법
𝛹 𝑥, 𝑡
𝑖ℏ𝜓 𝑥
𝜓 𝑥 𝜙 𝑡
𝜙 𝑡
𝑖ℏ 𝜕
𝜙 𝑡
𝜙 𝑡 𝜕𝑡
𝜓 𝑥 exp
ℏ
𝜙 𝑡
𝑖𝜔𝑡
𝜓 𝑥
ℏ
1 𝜕
𝜓 𝑥
2𝑚 𝜓 𝑥 𝜕𝑥
𝑠𝑖𝑛𝑐𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
𝑉 𝑥 𝜓 𝑥 𝜙 𝑡 = 𝐸𝜓 𝑥 𝜙 𝑡
𝑉 𝑥
𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝐸 ·········· 16
𝑡𝑖𝑚𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
9
Time independent Schrodinger Equation
𝑖ℏ 𝜕
𝜙 𝑡
𝜙 𝑡 𝜕𝑡
𝑖ℏ
𝜙 𝑡
ℏ
1 𝜕
𝜓 𝑥
2𝑚 𝜓 𝑥 𝜕𝑥
𝐸𝜙 𝑡
𝜙 𝑡
𝑒
𝑉 𝑥
𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
/ℏ
𝛹 𝑥, 𝑡
𝐸 ·········· 16
𝜓 𝑥 exp
𝑖𝐸𝑡/ℏ
wavefunction
ℏ
1 𝜕
𝜓 𝑥
2𝑚 𝜓 𝑥 𝜕𝑥
𝑉 𝑥
𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝐸
𝜕 𝜓 2𝑚
𝐸 𝑉 𝜓 0
ℏ
𝜕𝑥
∶ 𝑜𝑛𝑒 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑡𝑖𝑚𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑆𝑐ℎ𝑟𝑜𝑑𝑖𝑛𝑔𝑒𝑟 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛
𝜕 𝜓 𝜕 𝜓 𝜕 𝜓 2𝑚
𝐸 𝑉 𝜓 0
ℏ
𝜕𝑥
𝜕𝑦
𝜕𝑧
∶ 𝑡ℎ𝑟𝑒𝑒 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑡𝑖𝑚𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑆𝑐ℎ𝑟𝑜𝑑𝑖𝑛𝑔𝑒𝑟 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛
10
Time independent Schrodinger Equation
Boundary
conditions
Crank
Potential
Energy
d  2me
 2 [ E  V ( x )]  0
2

dx
2
Energy
(x)
E = 전자의 에너지
(x) = 전자의 파동함수
11
Boundary conditions
The probability of finding the particle somewhere
is certain.
If the total energy and the potential are finite everywhere, the wave function
and its first derivative must have the following properties.
Uncertainty principle
finite
finite
12
Boundary conditions
13
Boundary conditions
is continuous.
is not continuous.
 Wave function with various potential function
 Behavior of electron under various potential can be explained.
 Semiconductor properties
14
Statistical Interpretation (1926, Born)
Since the total wave function is a complex function, it cannot by itself represent a real
physical quantity.
The probability of finding the particle between x and
x+dx at a given time. (1926, Max Born)
양자역학에 의하면 계에 대해 모든 정보를 알고 있어도 우리는 계에
대해 어떤 값을 측정하면 얻는 값에 대한 확률만 정확히 알 수 있다.
 Probability density function
Complex conjugate function
(
)
Time independent
 The position of a particle is found in terms of a probability. (in Q. M.)
The probability of finding the particle somewhere
is certain.
15
Statistical Interpretation (1926, Born)
(x,y,z,t)dxdydz = 시간 t일 때 x,y,z 위치에 있는
미소 체적 dx dy dz에서 전자를 발견할 확률
1차원에서,
(x,t) =단위 길이당 전자를 발견할 확률
(x,t) 는 의미가 없고,
|(x,t)|2 만이 의미를 가진다.
16