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

Tunneling

 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.

Ammonia inversion

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

.