Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
Ch. 5. Quantum MechanicsCh. 5. Quantum Mechanics
Erwin Schrödinger(1887-1961)
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
2
2
22
( , , , )2
ˆˆ
ˆ ˆ( , , , ) ( , , , ) ( , , , ) ( , , , )2
( , , , ) ( , , , ) ( , , , ) ( , , , )2
p U x y z t Hm
i H it
x y z t U x y z t x y z t H x y z tm
x y z t U x y z t x y z t i x y z tm t
+ =
∂= − ∇ =
∂
Ψ + Ψ = Ψ
∂− ∇ Ψ + Ψ = Ψ
∂
p
p
Schrödinger’s EquationSchrödinger’s Equation
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
We consider the case of time-independent potential energy only.
2 2
2 ( , ) ( , ) ( , ) ( , )2
x t U x t x t i x tm x t
∂ ∂− Ψ + Ψ = Ψ
∂ ∂
( , ) ( )U x t U x=
2 2
2 ( , ) ( ) ( , ) ( , )2
x t U x x t i x tm x t
∂ ∂− Ψ + Ψ = Ψ
∂ ∂
One-Dimensional Schrödinger’s EquationOne-Dimensional Schrödinger’s Equation
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
Try ( , ) ( ) ( ).
Then, ( ) .Ei t
x t x t
t e
ψ φ
φ−
Ψ =
=
2 2
2
( ) ( ) ( ) ( )2
d x U x x E xm dx
ψ ψ ψ− + =
Time-Independent Schrödinger’s EquationTime-Independent Schrödinger’s Equation
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
Finite and single-valuedNormalized (except for the case of considering plane waves)ContinuousFirst-order spatial derivative should be continuous
(except for the case of infinite potential energy)
Wave FunctionsWave Functions
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
Waves of probability
functiondensityyprobabilit),,,( 2tzyxΨ
Max Born (1882-1970)
Waves of What?Waves of What?
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
One-Dimensional Quantum WellOne-Dimensional Quantum Well
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
그림
5.4 무한히
딱딱한
벽을
가진
상자에 해당하는
양끝에서
무한히
높은
장벽을
가
진
네모
퍼텐션
우물.
One-Dimensional Quantum WellOne-Dimensional Quantum Well
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
2 2 2 2 2
2
, 1, 2,3,
2( ) sin for 0
2 2
n
n
nk nL
n xx x LL L
k nEm mL
π
πψ
π
= =
⎛ ⎞= < <⎜ ⎟⎝ ⎠
= =
One-Dimensional Quantum WellOne-Dimensional Quantum Well
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
One-Dimensional Quantum WellOne-Dimensional Quantum Well
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
2 2 2 2 2
2
1 1
2
1
2( , ) ( ) ( ) sin , 1,2,3,
2 22( , ) ( , ) sin
where 1.
n
n
Ei t
n n n
n
Ei t
n n nn n
nn
n xx t x t e nL L
k nEm mL
n xx t C x t C eL L
C
πψ φ
π
π
−
∞ ∞ −
= =
∞
=
⎛ ⎞Ψ = = =⎜ ⎟⎝ ⎠
= =
⎛ ⎞Ψ = Ψ = ⎜ ⎟⎝ ⎠
=
∑ ∑
∑
One-Dimensional Quantum Well – General Solutions
One-Dimensional Quantum Well – General Solutions
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
표
5.1 몇가지
관측가능량에
대한
연산자
OperatorsOperators
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
Operators and ObservablesOperators and Observables
* ˆQ Q dx∞
−∞= Ψ Ψ∫
모든 물리량에는 해당하는 Hermitian Operator가 있고 그 고유치(eigenvalue)는 실수이며, 그 물리량을 측정할
때 측정될 수 있는 값들은 그 고유치들이다.
기대치
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
그림
5.7 유한한
장벽을
가진
네모
퍼텐셜
우물. 갇혀 있는
입자의
에너지
E가 장벽 높이 U보다
작다.
Finite Potential WellFinite Potential Well
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
그림
5.8 유한
퍼텐셜
우물에서의
파동함수와
확률밀도. 입자는
우물
바깥에서도
발견될
일정한
확률을
가진다.
Finite Potential WellFinite Potential Well
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
Bandgap EngineeringBandgap Engineering
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
MBE (Molecular Beam Epitaxy)MBE (Molecular Beam Epitaxy)
http://people.deas.harvard.edu/~jones/ap216/images/bandgap_engineering/bandgap_engineering.html
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
Quantum Well DevicesQuantum Well Devices
Multiple quantum well lasers
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
Quantum Well DevicesQuantum Well Devices
Quantum well solar cells
Quantum well modulator
Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics
Application Syste m s
Ce nte r for Active Plasm onics
Application Syste m s
More Quantum – Quantum Wires and Quantum DotsMore Quantum – Quantum Wires and Quantum Dots
Quantum wire laser operating through the eye of a needle
http://wwwrsphysse.anu.edu.au/ admin/pgbrochure/nano.html
V-groove quantum wire field effect transistorhttp://www.shef.ac.uk/eee/research/smd/research/quantum_fet.html
Carbon nanotube
Average pore diameter is 52 nm.
http://www.people.vcu.edu/~sbandy/project1.html