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In quantum mechanics, 'quantum tunnelling' is a micro and nanoscopic phenomenon in which a particle violates principles of classical mechanics by penetrating or passing through a potential barrier or impedance higher than the kinetic energy of the particle.^[1] A barrier, in terms of quantum tunnelling may be a form of energy state; analogous to a "hill" or incline in classical mechanics, which would suggest that passage through or over such a barrier would be impossible without sufficient energy. However, because of differences in terms of scale and interaction between quantum and classical mechanics, practical applications of the latter would be inaccurate.
On the quantum scale, objects exhibit wave-like behavior; in quantum theory, quanta moving against a potential energy "hill" can be described by their wave-function, which represents the probability of finding that particle in a certain location at either side of the "hill". If this function describes the particle as being on the other side of the "hill", then there is the probability that it has moved ''through'', rather than ''over'' it, and has thus "''tunnelled''".

Contents

History

Beginnings of quantum theory

Relation to violations of special relativity

Semi-classical calculation

See also

References

Notes

Books

External links

History

Beginnings of quantum theory

By 1928, George Gamow had solved the theory of the alpha decay of a nucleus via tunneling. Classically, the particle is confined to the nucleus because of the high energy requirement to escape the very strong potential. Under this system, it takes an enormous amount of energy to pull apart the nucleus. In quantum mechanics, however, there is a probability the particle can tunnel through the potential and escape. Gamow solved a model potential for the nucleus and derived a relationship between the half-life of the particle and the energy of the emission.
Alpha decay via tunneling was also solved concurrently by Ronald Gurney and Edward Condon. Shortly thereafter, both groups considered whether particles could also tunnel into the nucleus.
After attending a seminar by Gamow, Max Born recognized the generality of quantum-mechanical tunneling. He realized that the tunneling phenomenon was not restricted to nuclear physics, but was a general result of quantum mechanics that applies to many different systems. Today the theory of tunneling is even applied to the early cosmology of the universe.
Quantum tunneling was later applied to other situations, such as the cold emission of electrons, and perhaps most importantly semiconductor and superconductor physics. Phenomena such as field emission, important to flash memory, are explained by quantum tunneling. Tunneling is a source of major current leakage in Very-large-scale integration (VLSI) electronics, and results in the substantial power drain and heating effects that plague high-speed and mobile technology.
Another major application is in electron-tunneling microscopes (see scanning tunneling microscope) which can resolve objects that are too small to see using conventional microscopes. Electron tunneling microscopes overcome the limiting effects of conventional microscopes (optical aberrations, wavelength limitations) by scanning the surface of an object with tunneling electrons.
It has been found that quantum tunneling may be the mechanism used by enzymes to speed up reactions in lifeforms to millions of times their normal speed^[2].

Relation to violations of special relativity

On 5 August, 2007, German physicists Dr Günter Nimtz and Dr Alfons Stahlhofen of The University of Koblenz and Landau published a paper entitled "Macroscopic violation of special relativity", in which they performed a closed and open-prism/Frustrated total internal reflection experiment; the latter being used to reproduce the case of quantum tunnelling through the presence of an air gap; both reflected and transmitted beams experienced a longitudinal shift in the plane of incidence, the Goos-Hänchen shift. ^[3]
The experiment conducted involved microwaves³ of 32.8mm wavelength³ measured by antennas of 350mm that moved parallel to the surface of the prism. They were passed through right-angled triangle perspex prisms of 0.4 x 0.4 m²³ to illustrate the FTIR concept, and all measurements were used in adherence to macroscopic measurements for quantum mechanical experiments.³ The beam was incident perpendicular to the first prism, and was reflected over the critical angle of the prism at 38.7^o, at under 45^o; the resulting measurements indicated that, the reflected and the transmitted rays both arrived at the detecting antennas at the same time, with the delay in reflection and transmission of the digital microwave pulse corresponding to the Goos-Hänchen shift along the perspex-air boundary, and the universal tunnelling time of 100ps.³ Because both beams had arrived at the same time, despite the tunnelling that had occurred with one beam, the speed was suggested to have exceeded boundaries of special relativity³ ; vis-a-vis the speed of light.
However, the incidence of tunnelling was also observed alongside possible photon evanescent modes, identified as virtual exchange particles by Nimtz in previous work^[4] are initially unobservable following from the uncertainty relation, but that can be observed within a localized region of photons at limited distance from its exponential tail.³

Semi-classical calculation

Let us consider the time-independent Schrödinger equation for one particle, in one dimension, under the influence of a hill potential $V(x)$.
:
:
ight) Psi(x)
Now let us recast the wave function $Psi(x)$ as the exponential of a function.
:$Psi(x)\; =\; e^\{Phi(x)\}$
:
ight)
Now let us separate $Phi\text{'}(x)$ into real and imaginary parts using real valued functions A and B.
:$Phi\text{'}(x)\; =\; A(x)\; +\; i\; B(x)$
:
ight),
because the pure imaginary part needs to vanish due to the real-valued right-hand side:
:$ileft(B\text{'}(x)\; -\; 2\; A(x)\; B(x)$
ight) = 0
Next we want to take the semi-classical approximation to solve this. That means we expand each function as a power series in $hbar$. From the equations we can already see that the power series must start with at least an order of $hbar^\{-1\}$ to satisfy the real part of the equation. But as we want a good classical limit, we also want to start with as high a power of Planck's constant as possible.
:
:
The constraints on the lowest order terms are as follows.
:$A\_0(x)^2\; -\; B\_0(x)^2\; =\; 2m\; left(\; V(x)\; -\; E$
ight)
:$A\_0(x)\; B\_0(x)\; =\; 0$
If the amplitude varies slowly as compared to the phase, we set $A\_0(x)\; =\; 0$ and get
:$B\_0(x)\; =\; pm\; sqrt\{\; 2m\; left(\; E\; -\; V(x)$
ight) }
Which is obviously only valid when you have more energy than potential - classical motion. After the same procedure on the next order of the expansion we get
:
ight)} + heta} }{sqrt[4]{rac{2m}{hbar^2} left( E - V(x)
ight)}}
On the other hand, if the phase varies slowly as compared to the amplitude, we set $B\_0(x)\; =\; 0$ and get
:$A\_0(x)\; =\; pm\; sqrt\{\; 2m\; left(\; V(x)\; -\; E$
ight) }
Which is obviously only valid when you have more potential than energy - tunnelling motion. Grinding out the next order of the expansion yields
:
ight)}} + C_{-} e^{-int dx sqrt{rac{2m}{hbar^2} left( V(x) - E
ight)}}}{sqrt[4]{rac{2m}{hbar^2} left( V(x) - E
ight)}}
It is apparent from the denominator, that both these approximate solutions are bad near the classical turning point $E\; =\; V(x)$. What we have are the approximate solutions away from the potential hill and beneath the potential hill. Away from the potential hill, the particle acts similarly to a free wave - the phase is oscillating. Beneath the potential hill, the particle undergoes exponential changes in amplitude.
In a specific tunneling problem, we might already suspect that the transition amplitude be proportional to
ight)}} and thus the tunneling be exponentially dampened by large deviations from classically allowable motion.
But to be complete we must find the approximate solutions everywhere and match coefficients to make a global approximate solution. We have yet to approximate the solution near the classical turning points $E=V(x)$.
Let us label a classical turning point $x\_1$. Now because we are near $E=V(x\_1)$, we can easily expand
ight) in a power series.
:
ight) = v_1 (x - x_1) + v_2 (x - x_1)^2 + cdots
Let us only approximate to linear order
ight) = v_1 (x - x_1)
:
This differential equation looks deceptively simple. It takes some trickery to transform this into a Bessel equation. The solution is as follows.
:
ight) + C_{-rac{1}{3}} J_{-rac{1}{3}}left(rac{2}{3}sqrt{v_1}(x - x_1)^{rac{1}{3}}
ight)
ight)
Hopefully this solution should connect the far away and beneath solutions. Given the 2 coefficients on one side of the classical turning point, we should be able to determine the 2 coefficients on the other side of the classical turning point by using this local solution to connect them. We should be able to find a relationship between $C,\; heta$ and $C\_\{+\},C\_\{-\}$.
Fortunately the Bessel function solutions will asymptote into sine, cosine and exponential functions in the proper limits. The relationship can be found as follows.
:
ight)}
:
ight)}
Now we can easily construct global solutions and solve tunneling problems.
The transmission coefficient,
ight|^2, for a particle tunneling through a single potential barrier is found to be
:
ight)}}}{ left( 1 + rac{1}{4} e^{-2int_{x_1}^{x_2} dx sqrt{rac{2m}{hbar^2} left( V(x) - E
ight)}}
ight)^2}
Where $x\_1,x\_2$ are the 2 classical turning points for the potential barrier. If we take the classical limit of all other physical parameters much larger than Planck's constant, abbreviated as $hbar$
ightarrow 0, we see that the transmission coefficient correctly goes to zero. This classical limit would have failed in the unphysical, but much simpler to solve, situation of a square potential.

See also

★ Josephson effect

★ SQUID

★ Tunnel diode

★ WKB approximation

★ Scanning tunnelling microscope

★ Finite potential barrier (QM)

★ Delta potential barrier (QM)

★ Ferroelectric tunnel junction

★ Quantum Tunneling Composite

References

Notes

1. Razavy, Mohsen. (2003)., p1
2. http://www.seedmagazine.com/news/2006/04/the_quantum_shortcut.php
3. Nimtz, Günter., Stahlhofen, Alfons. A., (2007) "Macroscopic violation of special relativity" Retrieved on 17 August, 2007 from http://www.arxiv.org/pdf/0708.0681 (preprint)
4. Stahlhofen, Alfons. A., Nimtz, Günter. (2006). "Evanescent modes are virtual photons" Europhysics Letters, Vol 76., p189-195 Retrieved on 17 August, 2007 from http://www.iop.org/EJ/abstract/-search=28068818.1/0295-5075/76/2/189 (Requires Full Membership)

Books

★ Quantum Theory of Tunneling, Razavy, Mohsen, , , World Scientific, 2003, ISBN 981-238-019-1

★ Introduction to Quantum Mechanics (2nd ed.), Griffiths, David J., , , Prentice Hall, 2004, ISBN 0-13-805326-X

★ Introductory Quantum Mechanics, Liboff, Richard L., , , Addison-Wesley, 2002, ISBN 0-8053-8714-5

External links

★ Quantum Tunnelling Made Easy - an easy to follow explanation for the lay person.