# Lecture 16

### Tunneling

(c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed and made available online by work supported jointly by University of Illinois, the National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the sponsoring agencies.

### Tunneling

 We will consider a problem in which a particle of mass

m

and energy

E

hits a potential barrier of height

V

and width

L. V

is greater than

E

. Classically, the particle cannot overcome the barrier, but quantum mechanically it can “tunnel” through it.

### Tunneling

  The Schr ödinger equation to solve: 2

d

2 Y 2

m dx

2 +

V

Y =

E

Y Energy

E

is

given

(continuous) and assumed

smaller

than

V.

### Tunneling

 The situation we describe:    A particle flies in from the left.

Sometimes

it is bounced back by the left barrier;

sometimes

it passes through it. The particle can be bounced back by the right barrier.

What is the ratio of transmission versus reflection |

A

| 2 /|

A

’| 2 ?

### Tunneling

 The shape of the potential naturally divides the space into three regions. In each region, the potential is flat.

 For a flat potential,

e ikx

and cos) and

e kx

(or equivalently sin are the most promising functional forms.

e ikx

: -

d

2

dx

2

e ikx e kx

: -

d

2

dx

2 = ( ) 2

e ikx e kx

= -

k

2

e kx

=

k

2

e ikx

### Tunneling

 Our intuitive picture of the solution:

e ikx

represents an incoming particle, with momentum +

ħk e –ikx

is a reflected particle, with momentum –

ħk e ikx

is a transmitted particle, with momentum +

ħk e –κx , e +κx

is a decaying function which connects the left and right wave functions.

### Tunneling

 The wave function

within

V

>

E

and

e –κx

### and

e +κx

the barrier: Here,

### are more suitable.

2 2

m d

2 Y Middle

dx

2  2

d e

x dx

2  2

d e ikx dx

2    2

e

x

 2

k e ikx

= (

E

-

V

) Y Middle Negative Positive

 Left 

Ae ikx

Be

ikx

2 2

m d

2 Y Left

dx

2 =

k

2 2

m

2 Y Left

E

 Middle 

Ce

x

De

 

x

2 2

m d

2 Y Middle

dx

2 +

V

Y Middle = æ

V

ææ k 2 2

m

2 æ ææ Y Middle

= E < V

 Right  

ikx

2 2

m d

2 Y Right

dx

2 =

k

2 2

m

2 Y Right

E

### Boundary conditions

 The wave functions must be

continuous

and

smooth

(no infinity in potentials).

Continuous

C D

Smooth  

C

 

D Ce

L

Continuous 

De

 

L

 

ikL

Ce

L

Smooth  

De

 

L

 

ikL k

2 2

m

2 =

E

We know

k

and

κ

.

V

k 2 2

m

2 =

E

### Tunneling

  Energy, thus,

k

and

κ

are known.

We have five unknowns:

A, B, C, D, A

΄.

 We have four boundary conditions.

 The fifth condition comes from the normalization, determining all five unknowns.  We want to determine the transmission probability

|A

΄

|

2 / |

A

| 2 . In fact, with the four boundary conditions alone (without normalization), we can find four

ratios

:

B/A, C/A, D/A, A

΄

/A

.

### Tunneling

A

¢

A

2 2 µ

e

2 k

L V

k 2 2

m

2 =

E

® k = 2

m

(

V

-

E

)  For a high, wide barrier, the transmission probability

decreases

exponentially with 

Thickness of barrier

,  

Square root of particle mass

, Square root of energy deficit V

–E

.

### Tunneling

 Tunneling occurs when the particle does

not

have enough energy to overcome the barrier.  Classically, this is impossible. Within the barrier, the kinetic energy would be negative.

 Quantum mechanically, this is possible according to the Schrödinger equation.  (Some argue that the particle can acquire a higher energy necessary to cross the barrier momentarily thanks to the time-energy uncertainty principle).

### Permeation

 We have already seen (in harmonic oscillator problem) that a wave function can permeate in the classically forbidden region. This is by the same mechanism that tunneling occurs.

### Scanning tunneling microscope

Gold (100) surface Public image from Wikipedia

### Summary

 A particle of energy

E

can barrier of potential height

V

tunnel

>

E

.

through a  A particle of energy

E

can also

permeate

the potential wall of

V

>

E

.

into  The thinner the barrier or the lower the barrier height or the lighter the particle, the more likely the particle can

tunnel

.