NationMaster - Encyclopedia: Quantum tunneling

Quantum tunneling is the quantum-mechanical effect of transitioning through a classically-forbidden energy state. It can be generalized to other types of classically-forbidden transitions as well. Fig. ...

Consider rolling a basketball up a hill. If the ball is not given enough push, then the ball will not make it to the other side of the hill. In this case the ball does not have enough energy to roll over the hill. But in quantum mechanics, objects do not behave like classical objects, such as balls, do. In quantum mechanics objects frequently exhibit wavelike behavior and can be localized into a "particle" via measurement or interaction with the environment. The implication is that in the analogous quantum situation of a quantum particle moving against a potential hill, some of the wave can extend all the way through to the other side of the potential hill. Having some of the wave on the other side of the hill means that the quantum particle can be localized to the other side of the hill. This type of transition is not analagous to classical motion; it is called tunneling as if the particle were digging through the potential hill.

As this is a quantum and non-classical effect, it can generally only be seen in microscopic phenomena where the wave nature of particles is more pronounced.

1 History
2 Semiclassical calculation
3 Tunneling trivia
4 References
5 External link

History

In the early 1900's radioactive materials were known to have characteristic exponential decay rates or half lives and the radiation emissions were known to have certain characteristic energies. By 1928 George Gamow solved the theory of the alpha-decay of a nucleus via tunneling. Classically, the particle is confined to the nucleus because of a very strong potential. Classically, 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. Radioactive decay is the set of various processes by which unstable atomic nuclei (nuclides) emit subatomic particles. ... The term exponential may refer to any of several topics in mathematics: Exponential distribution Exponential function Exponential growth, exponential decay Exponential time Matrix exponential Exponential map (in differential geometry) All relate in some fashion to exponents. ... George Gamow (pronounced GAM-off) (March 4, 1904 â€“ August 19, 1968) , born Georgiy Antonovich Gamow (Ð“ÐµÐ¾Ñ€Ð³Ð¸Ð¹ ÐÐ½Ñ‚Ð¾Ð½Ð¾Ð²Ð¸Ñ‡ Ð“Ð°Ð¼Ð¾Ð²) was a Ukrainian born physicist and cosmologist. ... Alpha decay is a form of radioactive decay in which an atomic nucleus ejects an alpha particle and transforms into a nucleus with mass number 4 less and atomic number 2 less. ... Nucleus can mean: The Nuclear Envelope The nucleus is enveloped by a pair of membranes enclosing a lumen that is continuous with that of the endoplasmic reticulum. ... It has been suggested that this article or section be merged with Scalar potential. ... Half-Life is a science fiction first-person shooter computer game developed by Valve Software and published by Sierra Entertainment in 1998, based on a heavily-modified Quake game engine. ...

Alpha-decay via tunneling was also solved concurrently by Ronald Gurney and Edward Condon. Shortly thereafter both groups had also considered that particles could also tunnel into the nucleus. Edward Uhler Condon (March 2, 1902 â€“ March 26, 1974) was a distinguished nuclear physicist, a pioneer in quantum mechanics, a participant in the development of radar and nuclear weapons in World War II, research director of Corning Glass, director of the National Bureau of Standards, and president of the American...

After attending a seminar by Gamow, Max Born recognized the generality of quantum mechanical tunneling. He realised that tunneling phenomena 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. George Gamow (pronounced GAM-off) (March 4, 1904 â€“ August 19, 1968) , born Georgiy Antonovich Gamow (Ð“ÐµÐ¾Ñ€Ð³Ð¸Ð¹ ÐÐ½Ñ‚Ð¾Ð½Ð¾Ð²Ð¸Ñ‡ Ð“Ð°Ð¼Ð¾Ð²) was a Ukrainian born physicist and cosmologist. ... Max Born Max Born (born December 11, 1882 in Breslau, died January 5, 1970 in GÃ¶ttingen) was a German mathematician and physicist of Jewish heritage. ... Fig. ... Nuclear physics is the branch of physics concerned with the nucleus of the atom. ... Cosmology, from the Greek: κοσμολογία (cosmologia, κόσμος (cosmos) world + λογια (logia) discourse) is the study of the universe in its totality and by extension mans place in it. ... The deepest visible-light image of the cosmos. ...

Quantum tunneling was later applied to other situations such as the cold emission of electrons, and perhaps most importantly semiconductor and superconductor physics. Properties The electron (also called negatron, commonly represented as e−) is a subatomic particle. ... A semiconductor is a material with an electrical conductivity that is intermediate between that of an insulator and a conductor. ... Superconductivity is a phenomenon occurring in certain materials at low temperatures, characterised by the complete absence of electrical resistance and the damping of the interior magnetic field (the Meissner effect. ...

In 1970s, in the works of R.R. Dogonadze, M.V. Volkenshtein, Z.D. Urushadze and others were formulated the first Quantum-mechanical (physical) Model of Enzyme Catalysis. These works supported a theory that enzyme catalysis use quantum-mechanical effect such as tunneling. 1970 (MCMLXX) was a common year starting on Thursday. ... Revaz Dogonadze Revaz Dogonadze (November 21, 1931 - May 13, 1985) was a notable Georgian scientist, one of the founders of quantum electrochemistry, main author of the Quantum-Mechanical Theory of Kinetics of the Elementary Act of Chemical, Electrochemical and Biochemical Processes in Polar Liquids, Corresponding Member of the Georgian Academy... Mikhail Vladimirovich Volkenshtein (October 23, 1912 - February 18, 1992) was a notable Russian biophysicist, Corresponding Member of the Russian Academy of Sciences, Dr.Sci. ...

Semiclassical calculation

Let us consider the time-independent Schrödinger equation for one particle, in one dimension, under the influence of a hill potential $V (x)$ .

$-frac{hbar^2}{2m} frac{d^2}{dx^2} Psi(x) + V(x) Psi(x) = E Psi(x)$

$frac{d^2}{dx^2} Psi(x) = frac{2m}{hbar^2} left( V(x) - E right) Psi(x)$

Now let us recast the wave function $Ψ(x)$ as the exponential of a function.

Ψ(x) = e Φ(x)

$Phi''(x) + Phi'(x)^2 = frac{2m}{hbar^2} left( V(x) - E right)$

Now let us separate $Φ'(x)$ into real and imaginary parts.

$Phi'(x) = A(x) + imath B(x)$

$A'(x) + A(x)^2 - B(x)^2 = frac{2m}{hbar^2} left( V(x) - E right)$

B'(x) - 2 A (x) B (x) = 0

Next we want to take the semiclassical 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 plank's constant as possible.

$A(x) = frac{1}{hbar} sum_{i=0}^infty hbar^i A_i(x)$

$B(x) = frac{1}{hbar} sum_{i=0}^infty hbar^i B_i(x)$

The constraints on the lowest order terms are as follows.

$A_0(x)^2 - B_0(x)^2 = 2m left( V(x) - E right)$

A 0 (x) B 0 (x) = 0

If the amplitude varries slowly as compared to the phase, we set $A 0 (x) = 0$ and get

$B_0(x) = pm sqrt{ 2m left( E - V(x) right) }$

Which is obviously only valid when you have more energy than potential - classical motion. After the same proceedure on the next order of the expansion we get

$Psi(x) approx C frac{ e^{imath int dx sqrt{frac{2m}{hbar^2} left( E - V(x) right)} + theta} }{sqrt[4]{frac{2m}{hbar^2} left( E - V(x) right)}}$

On the other hand, if the phase varries slowly as compared to the amplitude, we set $B 0 (x) = 0$ and get

$A_0(x) = pm sqrt{ 2m left( V(x) - E right) }$

Which is obviously only valid when you have more potential than energy - tunneling motion. Grinding out the next order of the expansion yields

$Psi(x) approx frac{ C_{+} e^{+int dx sqrt{frac{2m}{hbar^2} left( V(x) - E right)}} + C_{-} e^{-int dx sqrt{frac{2m}{hbar^2} left( V(x) - E right)}}}{sqrt[4]{frac{2m}{hbar^2} left( V(x) - E right)}}$

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 $e^{-int dx sqrt{frac{2m}{hbar^2} left( V(x) - E right)}}$ and thus the tunneling be exponentially dampened by large deviations from classically permitable 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 $frac{2m}{hbar^2}left(V(x)-Eright)$ in a power series.

$frac{2m}{hbar^2}left(V(x)-Eright) = U_1 (x - x_1) + U_2 (x - x_1)^2 + cdots$

Let us only approximate to linear order $frac{2m}{hbar^2}left(V(x)-Eright) = U_1 (x - x_1)$

$frac{d^2}{dx^2} Psi(x) = U_1 (x - x_1) Psi(x)$

This differential equation looks deceptively simple. It takes some trickery to transform this into a Bessel equation. The solution is as follows. In mathematics, Bessel functions, first defined by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real number Î± (the order). ...

$Psi(x) = sqrt{x - x_1} left( C_{+frac{1}{3}} J_{+frac{1}{3}}left(frac{2}{3}sqrt{U_1}(x - x_1)^{frac{1}{3}}right) + C_{-frac{1}{3}} J_{-frac{1}{3}}left(frac{2}{3}sqrt{U_1}(x - x_1)^{frac{1}{3}}right) right)$

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 $C,θ$ 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.

$C_{+} = frac{1}{2} C cos{left(theta - frac{pi}{4}right)}$

$C_{-} = - C sin{left(theta - frac{pi}{4}right)}$

Now we can easily construct global solutions and solve tunneling problems.

The transmission coefficient, $left| frac{C_{mbox{outgoing}}}{C_{mbox{incoming}}} right|^2$ , for a particle tunneling through a single potential barrier is found to be In telecommunication, the term transmission coefficient has the following meanings: 1. ...

$T = frac{e^{-2int_{x_1}^{x_2} dx sqrt{frac{2m}{hbar^2} left( V(x) - E right)}}}{ left( 1 + frac{1}{4} e^{-2int_{x_1}^{x_2} dx sqrt{frac{2m}{hbar^2} left( V(x) - E right)}} right)^2}$

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 plank's constant, abbreviated as $hbar rightarrow 0$ , 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.

Tunneling trivia

One of the major applications is in electron-tunneling microscopes (see scanning tunneling microscope) used to see 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. Image of substitutional Cr impurities (small bumps) in the Fe(001) surface. ... Aberration in optical systems (lenses, prisms, mirrors or series of them intended to produce a sharp image) generally leads to blurring of the image. ... The wavelength is the distance between repeating units of a wave pattern. ... Properties The electron is a fundamental subatomic particle which carries a negative electric charge. ...

Tunneling is a source of major current leakage in Very-large-scale integration (VLSI) electronics. This results in the substantial power drain and heating effects that plague high-speed and mobile technology. Very-large-scale integration (VLSI) of systems of transistor-based circuits into integrated circuits on a single chip first occurred in the 1980s as part of the semiconductor and communication technologies that were being developed. ...

See also: Tunnel diode It has been suggested that Esaki diode be merged into this article or section. ...

References

Razavy, Moshen (2003). Quantum Theory of Tunneling, World Scientific. ISBN 9812380191.
Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.), Prentice Hall. ISBN 013805326X.
Liboff, Richard L. (2002). Introductory Quantum Mechanics, Addison-Wesley. ISBN 0805387145.
R.R. Dogonadze and Z.D. Urushadze, Semi-classical Method of Calculation of Rates of Chemical Reactions Proceeding in Polar Liquids.- J.Electroanal.Chem., 32, 1971, pp. 235-245
M.V. Volkenshtein, R.R. Dogonadze, A.K. Madumarov, Z.D. Urushadze and Yu.I. Kharkats, Theory of Enzyme Catalysis.-J. Molekuliarnaya Biologia, Moscow, 6, 1972, pp. 431-439 (in Russian, English summary)

External link

http://www.physicspost.com/imageview.php?imageId=184
Quantum Tunneling Simulation at Visual Quantum Mechanics
"About the History of the Quantum-Mechanical Theory of Enzyme Catalysis" by Prof. Dr. Zurab D. Urushadze.- Issued by the Georgian National Section of EUROSCIENCE, 2005

Category: Quantum mechanics

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Contents

History var ACE_AR = {Site: '756743', Size: '300250'};

Semiclassical calculation

Tunneling trivia

References

External link

History