Lecture physics a2 schrodinger equation and applications

SCHRÖDINGER EQUATION AND APPLICATIONS Tran Thi Ngoc Dung – Huynh Quang Linh – Physics A2 HCMUT 2016 CONTENTS I Schrödinger equation II Applications of Schrödinger equation 1 Particle in a 1 D infinite[.]

SCHRÖDINGER EQUATION

AND APPLICATIONS

Tran Thi Ngoc Dung – Huynh Quang Linh – Physics A2 HCMUT 2016

CONTENTS

I Schrödinger equation

II Applications of Schrödinger equation

1. Particle in a 1-D infinite potential well

2. Tunnel effect

) r p Et ( i

oe )

t , r (













De Brogile wave function of a free

particle of energy E, momentum p:

Wave function of a particle moving in a

field that having potential energy U(r) is:

) Et (

i

e ) r ( )

t , r







0 ) r ( )) r ( U E

( m 2 ) r

(









satisfies the time-independent

Schrödinger equation

)

r



If 1, 2 are the solutions of Schrödinger equation, =C 11 +C 2 2 is also

I Schrödinger Equation

Schrodinger equation

in Quantum Mechanics

Newton 2 nd law

in Classical mechanics

Solving Schrodinger equation

Wave function that describes the state of the particle, and the

possible energy levels of the particle

0 ) r (

mE

2 ) r (

2  









( ) ( )

2

2

FOR A FREE PARTICLE

( , )

( , )

( , ); ( , )

( , )

x y z

x y z

x y z

Et p r Et p x p y p z

i

Et p x p y p z o

x

i

Et p x p y p z

x o

x

i

x

p i

x

r t

   

   





 

2 2 2

2

2

( ) ( )

2

2

( , ) ( , )

2 2

2

2 ( ) ( ) 0

p

m

mE

mE

     

Derive Schrödinger Equation

0 )

r (

mE

2 )

r







+ For a free particle

E: is the Kinetic energy of the free particle

0 )

r ( )) r ( U E

(

m

2 )

r







+ For a particle in a region of potential energy U(r),

E is the energy of the particle, and KE is E-U

Derive Schrödinger Equation (cont.)

 ( x ) E

) x ( )) r ( U )

x

( dx

d m 2

) r ( E )

r ( )) r (

U m

2

) r ( m

2

Energy

Total PE

KE

2

2 2

2



















































Schrödinger Equation (cont.)

REVIEW about wave fuction

The statistic meaning of de Broglie Wave of a particle

dV

| ) t , r (

|





probability of finding the

particle per unit volume=

probabilty density

)

(

| ) t , r (

2

probability of finding the

particle in a volume dV

probability of finding the

particle over all space =1 (the

particle is certainly found)

1 dV

| ) t , r (

|







probability of finding the particle

V

2

dV

| ) t , r (

|

Normalized Condition of the wave function / Điều kiện chuẩn hóa của hàm sóng

Constraints on Wavefunction

In order to represent a physically observable system, the wavefunction must satisfy certain constraints:

- Must be a single-valued function

- M ust be normalizable This implies that the wavefunction approaches zero as x approaches infinity.

- Must be a continuous function of x.

- the first derivative of (x,t) must be continuous

O a x

U

Particle in a 1-D infinite potential energy well











a x , 0 x

a x 0

0 U

Particle can move freely inside the well, but it can not overcome the

potential barrier to get outside

For example: Electron in the metal can move freely, but it needs energy for escaping the metal

II Application of Schrodinger equation

1 Particle in a 1-D infinite potential energy well

(www.micro.uiuc.edu)

This is a basic problem in “Nano-science” It‟s a simplified (1D) model for an electron confined in a quantum structure (e.g., “quantum dot”), which scientists/engineers make, e.g., at the UIUC Microelectronics Laboratory !

KE term

PE term

Total E term

U = 0 for 0 < x < L

U =  everywhere else

) ( )

( ) (

) (

2 2

2 2

x E

x x

U dx

x d

(www.kfa-juelich.de/isi/) (newt.phys.unsw.edu.au)

„Quantum dots‟

U(x)