QUANTUM A PhD student Leo Elski observe the operation

QUANTUM
TUNNELING EFFECT

Contents:

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01: History

02: Theory

03: Mathematical
formulism

04: Applications

History:
As
quantum mechanics was evolved in 20th century, among all success of
quantum mechanics, “QUANTUM TUNNELLING” is much more impressive start of
quantum mechanics. There were five scientists in the history of quantum
mechanics. They received a Nobel prizes for their work in semiconductors,
superconductors and for the invention of scanning tunneling microscopy which
were successful. History of quantum tunneling begin in 1923 with the De-Broglie
postulating that the particles also exhibit wave like properties with a
wavelength which is given as;  .

This wavelength led to
the possibility of matter wave being able to pass through classically forbidden
region. It was not able to quantitatively described until formulation of the
“ERWIN SCHRODINGER” formulated his equations and the probabilistic
interpretation of the Schrodinger equations by “MAX BORN” for which he won the
Nobel prize for physics in 1954.

Tunneling – History,
Applications – Particle, Wave, Quantum, and Particles – JRank Articles http://science.jrank.org/pages/7028/Tunneling.html#ixzz51oZw2qH9

In 1928 EDARD CONDON,
GEORGE GAMOW, and RONALD QURNEY noted that alpha particles emissions were due
to quantum tunneling. A PhD student Leo Elski observe the operation of tunnel
diode in heavily doped germanium junctions for the use in transistor when he
was researching semiconductor. He received the Nobel prize due to his work for
physics in 1973. http://profmattstrassler.com/articles-and-posts/particle-physicsbasics/tunneling-a-quantum-process/

 In the early days of quantum mechanics, many
researches were set in research centers for the research into theoretical
physics; these centers are GOETTINGEN, LEIPZIG, and BERLIN in GERMANY,
GOPENHAGEN, DENMARK; COMBRIDGE in both England and Massachusetts; Princeton, New
jersey and Pasadena, California.

Friedrich Hund (1896- 1997) was the first scientist who make use
of the quantum Mechanics phenomenon of tunneling and he submitted two papers
related to the theory of molecular spectra, his first paper published in 1927(F. HUND, Z.PHYS. 40, 742(1927)). His papers in which he
explained that the electron moving in and out of the 2 dimensional potential
barrier well were submitted. Merzbacher,
E. (2002). The
early history of quantum tunneling. Physics
Today, 55(8),
44-50.

One of the Biggest
breakthrough in relation to quantum tunneling applications came in 1981 by GERD
BINNING and HEINRICH ROTTRER with the advent of the scanning Tunneling
Microscopy (STM) which allowed scientists to observe particles.

THEORY:

“The penetration of
matter waves and transmission of particles through a potential barrier”

Tunneling,
also known as the effect, is a quantum mechanics phenomenon by which a tiny
particle can penetrate a barrier through which macromolecules cannot pass.
Quantum tunneling effect possess some unique characteristics. For example, much
more particles pass through the thin barrier than thick barrier in the same way
low barrier allow particles to be penetrate than that of high ones. Quantum
tunneling effect don’t occur in the macroscopic world. This quantum tunneling
was only possible in microscopic word. Just as electron atoms can also tunnel
but those things which cannot be seen by naked eye will not have this effect.
For microscopic particles, the barrier height is described in terms of energy.
https://www.britannica.com/science/tunneling https://www.sciencenews.org/article/quantum-tunneling-takes-time-new-study-shows

Examples:
To
explain the tunneling effect of quantum mechanics, let us consider that if we
roll a ball very slowly towards a cement speed bumps, we confidently say that
it does not cross the barrier and bounce back. By considering the concept of
wave nature and probability make it possible that the ball tunnels through this
sand bar. Thus, treating a microscopic particle with the mathematical model of
a ball clearly tells us tunneling effect is impossible (in classical
mechanics), while using the mathematical model of a wave clearly states that
particles will always have a chance to tunnel.

Trixler, F. (2013).
Quantum tunneling to the origin and evolution of life. Current organic
chemistry, 17(16), 1758-1770.

Process: consider
a particle with an energy E present in the inner region of one dimensional
potential well V. In the classical point of view, if particle energy E is
always less than potential V then particle will remain in the inner region and
is unable to cross the barrier. If E is greater than V, then it can easily
cross the barrier. In quantum mechanics, if particle have an energy less than V
it can cross the barrier but the probability is less depending on the
difference of E and V. There is a possibility that the particle will tunnel
through barrier and have energy as before.

This process relies on
Heisenberg uncertainty principle. Because this process of tunneling relies on
probability and several variable that effect the probability are;

The mass of object, the
thickness of barrier, the ability to penetrate the barrier.

https://www.researchgate.net/profile/Carl_Bender/publication/47637745_Quantum_tunneling_as_a_classical_anomaly/links/02e7e518a833689db1000000.pdf

 Mathematical Formulism of Quantum Tunneling
Effect:

As the concept of tunneling is that “penetration of
matter waves and a transmission of particles through the potential barrier”, so
we can explain it by using the mathematical formulism of potential step.

A particle that is free everywhere, but beyond a
particular point, say x=0, the potential increase sharply (i.e., it becomes
repulsive or attractive). A potential of this type is called a potential step.

                                          

                                                                                  

 

In this problem we have to analyze the dynamics of a
flux of particles (all having the same mass m and moving with the same
velocity) moving from left to the right. We are going to consider two cases,
depending on whether the energy of the particles is larger or smaller than V?. 

(a)  
Case
E > V?

The particles are free for x 0. Let us analyze the
dynamics of this flux of particles classically and then quantum mechanically.

                   Classically, the particles
approach the potential step or barrier from the left with a constant momentum .As the particles enter
this region , where the potential
now is ,they slow down to a
momentum ; they will then
conserve this momentum as they travel to the right. Since the particles have
sufficient energy to penetrate into the region , there will be total  transmission: all the particles will emerge
to the right with a smaller kinetic energy .This is then a simple
scattering problem in one dimension.

     
Quantum mechanically, the dynamics of the particle is regulated by the
Schrodinger equation, which is given in these two regions by

           

 

                                                      
                          (4.15)

                                                      
                                              (4.16)

Where and .The most general
solutions to these two equations are plane waves:

                                          
                                      (4.17)                                                           

                                                                                
(4.18)

Where  and  represent waves moving in the positive -direction, but   and  correspond to waves moving in the negative -direction. We are
interested in the case where the particles are initially incident on the
potential step from the left, they can be reflected or transmitted at  = 0. Since no wave is reflected from the
region  > 0 to the left, the constant D must
vanish. Since we are dealing with stationary states, the complete wave function
is thus given by

 

                                   (4.19)

Where   ,  and ,represent the incident ,the reflected and the transmitted waves , respectively; they travel to
the right, the left, and the right. Note that the probability densityshown in the lower left
plot of right.

      Let us now evaluate
the reflection and transmission
coefficients, R and T, as defined by

 

                                     (4.20)

R represent the ratio of the reflected to the incident beams and
T the ratio of the transmitted to the incident beams. To calculate R and T , we
need to find       ,  since the incident wave is ,the incident current density is given by

 

                                                         (4.21)

Similarly, since the reflected and
transmitted waves are  and ,

We can verify that the reflected and
transmitted fluxes are

  

                                                   =                         (4.22)

A combination of (4.20) to (4.22)
yields

                                                          

   
                                                                                                                      
(4.23)

Thus, the calculation of R and T is
reduced to determining the constants B and C. For this we need to use the
boundary conditions of the wave function at x = 0. Since both the wave function
and its first derivative are continuous at x = 0,

      

                                                                        
(4.24)

Equations (4.17) and (4.18) yield 

                                                      (4.25)

Hence

                             ,                            .                                                
(4.26)

As for the constant A, it can be
determined from the normalization condition of the wave function, but we don’t
need it here, since R and T are expressed in terms of ratios. A combination 0f
(4.23) with (4.26) leads to

                        

                 ,                                                        (4.27)

Where  .The sum of R and T is equal to 1, as it should be.

       
In contrast to classical mechanics, which states that none of the
particle get reflected, equation (4.27) shows that the quantum mechanical
reflection coefficient R is not zero: there are particle that get reflected in
spite of their energies being higher than the step .This effect must be attributed to the wavelength behavior of the particles.

       
From (4.27) we see that as  gets smaller and smaller,  also gets smaller and smaller so
that when  the transmission coefficient  becomes zero and . On the other hand, when , we have ; hence  and . This is expected since, when the incident particles have very high
energies, the potential step is so weak that it produces no noticeable effect
on their motion.

Remark: physical meaning of the boundary
conditions

Throughout this chapter, we will
encounter at numerous times the use of the boundary condition of the wave function
and its first derivatives as in Eq (4.24).

 We can make two observations:

·        
Since the probability
density  of finding the particle in any small region
varies continuously from one point to another, the wave function  must, therefore, be a continuous function of ; thus we must have

·    
Since the
linear momentum of the particles, , must be a continuous function of x as the particles,
, must also be a continuous function of x, notably at
x = 0. Hence, we must have  

 

(b)  Case

Classically the
particles arriving at the potential step from the left(with the momentum  ) will come to a stop at x=0 and
then all will bounce back to the left with the magnitude of their momenta
unchanged. None of the particles will make it into the right side of the
barrier x=0, there is total reflection of the particles. So the motion of the
particles is reversed by the potential barrier.

           Quantum mechanically, the picture
will be somewhere different. In this case, the Schrodinger equation and the
wave function in the region x 0 the Schrodinger equation is given by

 

 

                                                                   (4.28)

 

Where .This equation’s solution is

 

 

                                        (4.29)

Since the wave function must be finite
everywhere, and since the term  diverges when    the constant  has to be zero. Thus the complete
wave function is

 

 

                                       (4.30)

    
Let us now evaluate, as we did in the previous case, the reflected and
the transmitted coefficients. First we should note that the transmitted
coefficient, which corresponds to the transmitted wave function , is zero since  is purely real function  and therefore

 

                                                      (4.31)

Hence, the reflected coefficient  must be equal to 1. We can obtain
this result by applying the continuity conditions at x = 0 for (4.17) and (4.29):

 

                                      ,                                         
                           (4.32)

Thus, the reflected coefficient is
given by

                       

                                                                                                                  
(4.33)

We therefore have total reflection, as
in the classical case.

  
 There is, however, a difference
with the classical case: while none of the particles can be found classically
in the region x > 0, quantum mechanically there is a nonzero probability that the wave function   penetrates this classically forbidden region. To see this, that the relative
probability density

 

                                                                   (4.34)

Is appreciable near x = 0 and falls
exponentially to small values as x becomes large; the behavior of the
probability density.

(Quantum mechanics ,)

Applications of quantum tunneling:

Radioactive decay-particle tunneling
out of a nucleus causing radioactive decay.

Cold emission-occurs in semi- and
superconductors. The electron jumps from surface of a metal to follow a
voltage. If the electric field is large enough and thin, it allows the electron
to tunnel through, important for devices.

Touch screens/ artificial skin-because
it provides a smaller insulating gap.

Importance of quantum tunneling:

·        
Tunneling plays
an essential role in several physics chemical, and biological phenomena, such
as radioactive decay or the manifestation of large kinetic isotopes effects in
chemicals of enzymatic reactions.

·        
In a case,
scientist measured electrons escaping from atoms without having the necessary
energy to do so. The outcome is that tunneling occurs in less than a few
hundred attoseconds (10^-18sec). this phenomenon initiates many fast processes,
which are very in nature. Quantum tunneling is important because it is a
fundamental process of nature which is responsible for many things on which
life itself is dependent.

·        
It has been
hypothesized that the very beginning of the universe was caused by a tunneling
event, allowing the universe to pass from a “state of no geometry” (no space or
time) to a state in which space, time, matter, and life could exist.

Future of quantum
tunneling:

·        
In the present
day, the world is moving to a world of nanotechnology. In this technology, we
will definite need a thorough understanding at what occurs at the atomic scale.
This includes the quantum tunneling. As has been already said, transistor
scanning tunneling microscope. This technology already exists, soon enough it
will be in every part of our everyday.

 

 

 

 

 

 

 

 

 

 

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