an introduction to quantum tunneling and its application
TRANSCRIPT
7/22/2019 An Introduction to Quantum Tunneling and Its Application
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INTRODUCTIONTO
QUANTUM TUNNELING
ANDITS APPLICATIONS
SCIF1121
Student: Hoang Bao
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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
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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
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1. A brief history of Quantum Tunneling
G. Gamow E. Condon M. Born
(1904-1968) (1902-1974) (1882-1970)
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1. A brief history of Quantum Tunneling
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
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1. A brief history of Quantum Tunneling
L. Esaki I. Giaever B. Josephson
(1925- ) (1929- ) (1940- )
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2. Some primarily concepts of Quantum
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
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2. Some primarily concepts of Quantum
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
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2. Some primarily concepts of Quantum
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
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2. Some primarily concepts of Quantum
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
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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)
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3.Mathematical theory of Quantum Tunneling
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
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3.Mathematical theory of Quantum Tunneling
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
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3.Mathematical theory of Quantum Tunneling
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
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3.Mathematical theory of Quantum Tunneling
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
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4.Quantum Tunneling applications:
Theoretical, Experimental and Technical.
The theoretical explanation of alpha decay
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4.Quantum Tunneling applications:
Theoretical, Experimental and Technical.
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
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4.Quantum Tunneling applications:
Theoretical, Experimental and Technical.
Potential curved proposed by G.Gamow
The potential well of nuclear binding energy
Or1 r2 r
V(r)
E
The Coulomb repulsion tail
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4.Quantum Tunneling applications:
Theoretical, Experimental and Technical.
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
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4.Quantum Tunneling applications:
Theoretical, Experimental and Technical.
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
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4.Quantum Tunneling applications:
Theoretical, Experimental and Technical.
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
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4.Quantum Tunneling applications:
Theoretical, Experimental and Technical.
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4.Quantum Tunneling applications:
Theoretical, Experimental and Technical.
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,…
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4.Quantum Tunneling applications:
Theoretical, Experimental and Technical.
http://upload.wikimedia.org/wikipedia/commons/f/f9/ScanningTunnelingMicroscope_schematic.png
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4.Quantum Tunneling applications:
Theoretical, Experimental and Technical.
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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!
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THANK YOU FOR YOUR ATTENTION!