QUANTUM

TUNNELING EFFECT

Contents:

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
and feel a repulsive potential V?
that starts at x=0 and stays flat(constant) 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 are given by
(4.15) and
(4.17), respectively but for 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.