Ch. 5. Quantum Mechanics Ch. 5. Quantum Mechanics
Erwin Schrödinger
(1887-1961)
2
2
2
2
( , , , ) 2
ˆ ˆ
ˆ ( , , , ) ( , , , ) ( , , , ) ˆ ( , , , ) 2
( , , , ) ( , , , ) ( , , , ) ( , , , ) 2
p U x y z t H m
i H i
t
x y z t U x y z t x y z t H x y z t m
x y z t U x y z t x y z t i x y z t
m t
+ =
= − ∇ = ∂
∂
Ψ + Ψ = Ψ
− ∇ Ψ + Ψ = ∂ Ψ
∂ p
p
= G =
= =
Schrödinger’s Equation
Schrödinger’s Equation
We consider the case of time-independent potential energy only.
2 2
2 ( , ) ( , ) ( , ) ( , )
2 x t U x t x t i x t
m x t
∂ ∂
− Ψ + Ψ = Ψ
∂ ∂
= =
( , ) ( ) U x t = U x
2 2
2 ( , ) ( ) ( , ) ( , )
2 x t U x x t i x t
m x t
∂ ∂
− Ψ + Ψ = Ψ
∂ ∂
= =
One-Dimensional Schrödinger’s Equation
One-Dimensional Schrödinger’s Equation
Try ( , ) ( ) ( ).
Then, ( ) .
i E t
x t x t
t e
ψ φ
φ −
Ψ =
= =
2 2
2
( ) ( ) ( ) ( ) 2
d x
U x x E x m dx
ψ ψ ψ
− = + =
Time-Independent Schrödinger’s Equation
Time-Independent Schrödinger’s Equation
Finite and single-valued
Normalized (except for the case of considering plane waves)
Continuous
First-order spatial derivative should be continuous (except for the case of infinite potential energy)
Wave Functions
Wave Functions
Waves of probability
function density
y probabilit )
, , ,
( x y z t
2Ψ
Max Born (1882-1970)
Waves of What?
Waves of What?
One-Dimensional Quantum Well
One-Dimensional Quantum Well
그림
5.4
무한히 딱딱한 벽을 가진 상자에 해당하는 양끝에서 무한히 높은 장벽을 가 진 네모 퍼텐션 우물.
One-Dimensional Quantum Well
One-Dimensional Quantum Well
2 2 2 2 2 2
, 1, 2, 3,
( ) 2 sin for 0
2 2
n
n
k n n
L
x n x x L
L L
k n
E m mL
π ψ π
π
= =
⎛ ⎞
= ⎜ ⎝ ⎟ ⎠ < <
= =
"
= =
One-Dimensional Quantum Well
One-Dimensional Quantum Well
One-Dimensional Quantum Well
One-Dimensional Quantum Well
2 2 2 2 2 2
1 1
2 1
( , ) ( ) ( ) 2 sin , 1, 2, 3,
2 2
( , ) ( , ) 2 sin
where 1.
n
n
iE t
n n n
n
iE t
n n n
n n
n n
x t x t n x e n
L L
k n
E m mL
x t C x t C n x e
L L
C
ψ φ π
π
π
−
∞ ∞ −
= =
∞
=
⎛ ⎞
Ψ = = ⎜ ⎟ =
⎝ ⎠
= =
⎛ ⎞
Ψ = Ψ = ⎜ ⎟
⎝ ⎠
=
∑ ∑
∑
=
=
"
= =
One-Dimensional Quantum Well – General Solutions One-Dimensional Quantum Well –
General Solutions
표
5.1
몇가지 관측가능량에 대한 연산자Operators
Operators
Operators and Observables Operators and Observables
* ˆ
Q ∞ Q dx
= ∫ −∞ Ψ Ψ
모든 물리량에는 해당하는
Hermitian Operator
가 있고 그 고유치(eigenvalue)
는 실수이며,
그 물리량을 측정할 때 측정될 수 있는 값들은 그 고유치들이다.
기대치
그림
5.7
유한한 장벽을 가진 네모 퍼텐셜 우물.
갇혀 있는 입자의 에너지E
가 장벽 높이U
보다 작다.
Finite Potential Well
Finite Potential Well
그림
5.8
유한 퍼텐셜 우물에서의 파동함수와 확률밀도.
Finite Potential Well
Finite Potential Well
Bandgap Engineering
Bandgap Engineering
MBE (Molecular Beam Epitaxy)
MBE (Molecular Beam Epitaxy)
Quantum Well Devices Quantum Well Devices
Multiple quantum well lasers
Quantum Well Devices Quantum Well Devices
Quantum well solar cells
Quantum well modulator
More Quantum – Quantum Wires and Quantum Dots More 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 transistor
http://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
