ch. 5. quantum mechanicsocw.snu.ac.kr/sites/default/files/note/2802.pdf · 2018. 1. 30. · quantum...

Seoul National UniversitySeoul National UniversityCe nte r for Active Plasm onics

Application Syste m s

Ce nte r for Active Plasm onics


Ch. 5. Quantum MechanicsCh. 5. Quantum Mechanics

Erwin Schrödinger(1887-1961)





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





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





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





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





Waves of probability

functiondensityyprobabilit),,,( 2tzyxΨ

Max Born (1882-1970)

Waves of What?Waves of What?





One-Dimensional Quantum WellOne-Dimensional Quantum Well





그림

5.4 무한히

딱딱한

벽을

가진

상자에 해당하는

양끝에서

무한히

높은

장벽을

가

진

네모

퍼텐션

우물.






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

= =

⎛ ⎞= < <⎜ ⎟⎝ ⎠

= =






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





표

5.1 몇가지

관측가능량에

대한

연산자

OperatorsOperators





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





그림

5.7 유한한

장벽을

가진

네모

퍼텐셜

우물. 갇혀 있는

입자의

에너지

E가 장벽 높이 U보다

작다.

Finite Potential WellFinite Potential Well





그림

5.8 유한

퍼텐셜

우물에서의

파동함수와

확률밀도. 입자는

우물

바깥에서도

발견될

일정한

확률을

가진다.

Finite Potential WellFinite Potential Well





Bandgap EngineeringBandgap Engineering





MBE (Molecular Beam Epitaxy)MBE (Molecular Beam Epitaxy)

http://people.deas.harvard.edu/~jones/ap216/images/bandgap_engineering/bandgap_engineering.html





Quantum Well DevicesQuantum Well Devices

Multiple quantum well lasers





Quantum Well DevicesQuantum Well Devices

Quantum well solar cells

Quantum well modulator





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

ch. 5. quantum mechanicsocw.snu.ac.kr/sites/default/files/note/2802.pdf · 2018. 1. 30. · quantum...

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