An Introduction to Quantum Tunneling and Its Application

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INTRODUCTIONTO

QUANTUM TUNNELING

ANDITS APPLICATIONS

SCIF1121

Student: Hoang Bao



Content

1. A brief history of Quantum Tunneling.

2. Some primarily concepts of Quantum

Mechanics.

3. Mathematical theory of Quantum

Tunneling.

4. Quantum Tunneling applications:

Theoretical, Experimental and Technical.

5. Conclusion



1. A brief history of Quantum Tunneling

It was developed from the study ofradioactivity, with the works of H.

Becquerel, P. Curie, M. Curie and E.

Rutherford. Its first mathematical application for alpha

decay was owe to G. Gamow and

independently R. Gurney and E. Condonin 1928




G. Gamow E. Condon M. Born

(1904-1968) (1902-1974) (1882-1970)




M. Born, influenced by Gamow discovery,

developed a mathematical theory for

Quantum Mechanical Tunneling

In 1973, L. Esaki, I. Giaever and B.

Josephson received the Nobel Prize in

Physics for their prediction of the tunneling

of superconducting Cooper pairs




L. Esaki I. Giaever B. Josephson

(1925- ) (1929- ) (1940- )




Mechanics In classical mechanics, Newton‟s equation

allows us (in concept) to calculate preciselythe position of a particle at any instance.

Things are not so simple in Quantum mechanics. Youcannot know exactly where a particle is. Why?

It is well known (though not well understood) that aparticle also behaves like a wave. That wave isdescribed by the wave function

2

2

d m

dt

x F




Mechanics

But what does the wave function do? M. Born(remember him?) proposed his interpretation

The possibility that the

=particle is in the interval

(x,x+dx) at time t

So far so good. Now, how can we compute ?

Instead of Newton‟s equation, we use Schrödingerequation (if you are really enthusiastic, Landau and

Liftschitz provides the derivation of Schrödingerequation)

2

x,t dx

2 2

2i V

t 2m x




Mechanics

There is a special (and also very important)

class of solutions to the Schrödinger equation,

which is called the classed of stationary wavefunctions. Those have the form

is called the time – independent part of

the wave function

iEt

x e

x




Mechanics

Substituting this formula to the Schrödinger

equation, we get what is called the time –

independent Schrödinger equation

We will use this to study tunneling.

2 2

2 V x E *

2m x



3.Mathematical theory of Quantum Tunneling

For illustration, will study a simplified (but still good enough)case

The particle‟s energy is E<V0

0V x V 0,L

0 otherwise

V

O L

V0

x

E(I) (II) (III)




In region (I) and (III), (*) reads

It is known that a valid solution must have

In this case , we have

2 2

2

d E

2m dx

min E V

E 0

2

2

2d 2mE ,dx






If there is no particle coming toward the

“potential wall” in the region III, D must be

zero. If we now impose the boundary

conditions: and its first derivative arecontinuous at x=0 and x=L, we will get four

equations involving A,B,C,E,F.

Solving these (4) equations for B,C,E,F in

term of A (the incoming particle), we get theamplitudes of wave function in different

regions.

x




For our purpose, we only need to care about theamplitude of the tunneling wave

We can see that so there is a chance thatthe particle somehow leaks through the “potentialwall”.

2C 0

2

2

2020

0

AC

L 2m V E V 1 sinh

4E V E




Classically, a particle coming from the left thatdoes not have enough energy can never climbover the “potential wall”.

In quantum world, this can (and always) happen.

Isn‟t it crazy? Well, that‟s how our world work. Ifit‟s crazy, so are we.

This model is quite simple, for more complicatedpotential, we need a more sophisticatedtreatment: the Wentzel – Kramers – Brillouin

(WKB) approximation. Nevertheless, the Physicsis still the same



4.Quantum Tunneling applications:


The theoretical explanation of alpha decay





The alpha particle – two protons and two

neutrons is electrically repelled by the

leftover nucleus. Thus, there must be a

reason that they do not all decay. Gamow

proposed that the potential energy curve

created by the nucleus has the followingshape





Potential curved proposed by G.Gamow

The potential well of nuclear binding energy

Or1 r2 r

V(r)

E

The Coulomb repulsion tail





Gamow used the WKB approximation to

calculate the possibility that the alpha

particle leaks out of the binding well

He then calculated the nuclei‟ lifetimes,

which is inversely proportional to the

leaking possibility. He got

1

E e





Gamow‟s theory was supported experiments. Thefollowing graph show as a function of .

David Park, Introduction to the Quantum Theory, 3rd edition, McGraw Hill 1992

ln 1

E





Flash Memory : it is a special transistor.When the voltage difference between theSi Channel and the Floating Gate is

altered, electrons can leak in (data beingwritten) or out (data being erased) of theFloating Gate.

The Floating Gate can stored electrons(and hence data) for long time so thistransistor is used as an external memory









Scanning Tunneling Microscopy: Electrons(and other particles as well) that leak outfrom the surface of a metal contain the

information about electric carriers densityof that metal. Hence, by measuring theleaked electrons, we can study the innerstructure of metal.

Biophysicists use STM to studymicroscopic entities like cells, DNA,…





http://upload.wikimedia.org/wikipedia/commons/f/f9/ScanningTunnelingMicroscope_schematic.png







5. Conclusion

Quantum – scale particles can make „tunnels‟ tothe regions forbidden by Classical Mechanics.This is basically due to the fact that microscopicparticles can act as waves.

This phenomenon is well understood byphysicists and is now used widely in quantumelectronics and microscopic devices.

A small note: Instead of a barrier, we can have a

potential well and an incoming particle canbounce back from the well! Let’s imagine youdriving off a cliff and Quantum Mechanics savesyou by bouncing you back!



THANK YOU FOR YOUR ATTENTION!

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An Introduction to Quantum Tunneling and Its Application