Quantum tunnelling

Quantum mechanics

$\Delta x \, \Delta p \ge \frac{\hbar}{2}$

Fundamental concepts
Quantum state · Wave function Superposition · Entanglement Measurement · Uncertainty Exclusion · Duality Decoherence · Ehrenfest theorem · Tunneling

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In quantum mechanics, quantum tunneling is a micro nanoscopic phenomenon in which a particle violates the principles of classical mechanics by penetrating or passing through a potential barrier or impedance higher than the kinetic energy of the particle. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain Quantum mechanics (QM or quantum theory) is a physical science dealing with the behavior of Matter and Energy on the scale of Atoms The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of Quantum mechanics. In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system Quantum superposition is the fundamental law of Quantum mechanics. Quantum entanglement is a quantum mechanical Phenomenon in which the Quantum states of two or more objects are linked together so that one object The framework of Quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925 In Physics and Chemistry, wave–particle duality is the concept that all Matter and Energy exhibits both Wave -like and In Quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior The Ehrenfest theorem, named after Paul Ehrenfest, relates the time Derivative of the expectation value for a quantum mechanical operator Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Nanotechnology, a particle is defined as a small object that behaves as a whole unit in terms of its transport and properties Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects The kinetic energy of an object is the extra Energy which it possesses due to its motion ^[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 classically suggests that passage through or over such a barrier would be impossible without sufficient energy. A quantum mechanical system or particle that is bound, confined spacially can only take on certain discrete values of energy as opposed to classical particles which The kinetic energy of an object is the extra Energy which it possesses due to its motion

Calculated using Mathematica, by the Crank-Nicolson method of finite differences.

On the quantum scale, objects exhibit wave-like behaviour; in quantum theory, quanta moving against a potential energy "hill" can be described by their wave-function, which represents the probability amplitude of finding that particle in a certain location at either side of the "hill". In Physics and Chemistry, wave–particle duality is the concept that all Matter and Energy exhibits both Wave -like and A potential well is the region surrounding a Local minimum of Potential energy. A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system 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".

1 History
2 Semi-classical calculation
3 See also
4 In Popular Culture
5 References
- 5.1 Notes
- 5.2 Books

History

By 1928, George Gamow had solved the theory of the alpha decay of a nucleus via tunneling. Year 1928 ( MCMXXVIII) was a Leap year starting on Sunday (link will display full calendar of the Gregorian calendar. George Gamow (pronounced as ˈgamof ( March 4, 1904 &ndash August 19, 1968), born Georgiy Antonovich Gamov (Георгий Антонович Alpha decay is a type of radioactive decay in which an Atomic nucleus emits an Alpha particle (two protons and two neutrons bound together into a particle The nucleus of an Atom is the very dense region consisting of Nucleons ( Protons and Neutrons, at the center of an atom Classically, the particle is confined to the nucleus because of the high energy requirement to escape the very strong potential. The Mathematical study of potentials is known as Potential theory; it is the study of Harmonic functions on Manifolds This mathematical 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. Edward Uhler Condon ( March 2 1902 &ndash March 26 1974) was a distinguished American Nuclear physicist, a pioneer 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. Seminar is generally a form of Academic instruction either at a University or offered by a commercial or professional organization Max Born (11 December 1882 &ndash 5 January 1970 was a German Physicist and Mathematician who was instrumental in the development of Quantum 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. Nuclear physics is the field of Physics that studies the building blocks and interactions of Atomic nuclei. Today the theory of tunneling is even applied to the early cosmology of the universe. Physical cosmology, as a branch of Astronomy, is the study of the large-scale structure of the Universe and is concerned with fundamental questions about its The Universe is defined as everything that Physically Exists: the entirety of Space and Time, all forms of Matter, Energy ^[2]

Quantum tunneling was later applied to other situations, such as the cold emission of electrons, and perhaps most importantly semiconductor and superconductor physics. Field emission (FE is the emission of electrons from the surface of a condensed phase into another phase due to the presence of high electric fields The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J A semiconductor' is a Solid material that has Electrical conductivity in between a conductor and an insulator; it can vary over that Superconductivity is a phenomenon occurring in certain Materials generally at very low Temperatures characterized by exactly zero electrical resistance Phenomena such as field emission, important to flash memory, are explained by quantum tunneling. Field emission (FE is the emission of electrons from the surface of a condensed phase into another phase due to the presence of high electric fields Flash memory is non-volatile computer memory that can be electrically erased and reprogrammed 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. Scanning tunneling microscope (STM is a powerful technique for viewing surfaces at the atomic level A microscope ( Greek: ( micron) = small + ( skopein) = to look or see is an instrument for viewing objects that are Electron tunneling microscopes overcome the limiting effects of conventional microscopes (optical aberrations, wavelength limitations) by scanning the surface of an object with tunneling electrons. Aberrations are departures of the performance of an optical system from the predictions of Paraxial optics. In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J

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. Enzymes are Biomolecules that catalyze ( ie increase the rates of Chemical reactions Almost all enzymes are Proteins ^[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 $V (x)$ . In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it

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

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

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 using real valued functions A and B.

Φ'(x) = A (x) + i B (x)

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

because the pure imaginary part needs to vanish due to the real-valued right-hand side:

$i\left(B'(x) - 2 A(x) B(x)\right) = 0.$

Next we want to take the semiclassical approximation to solve this. In Physics, the adjective semiclassical has different precise meanings depending on the context That means we expand each function as a power series in $\hbar$ . In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + 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 Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta.

$A(x) = \frac{1}{\hbar} \sum_{k=0}^\infty \hbar^k A_k(x)$

$B(x) = \frac{1}{\hbar} \sum_{k=0}^\infty \hbar^k B_k(x).$

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

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

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

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

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

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

Which is obviously only valid when you have more potential than energy - tunnelling 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 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 $\frac{2m}{\hbar^2}\left(V(x)-E\right)$ in a power series.

$\frac{2m}{\hbar^2}\left(V(x)-E\right) = v_1 (x - x_1) + v_2 (x - x_1)^2 + \cdots$

Let us only approximate to linear order $\frac{2m}{\hbar^2}\left(V(x)-E\right) = v_1 (x - x_1)$

$\frac{d^2}{dx^2} \Psi(x) = v_1 (x - x_1) \Psi(x)$

This differential equation looks deceptively simple. Its solutions are Airy functions. In Mathematics, the Airy function Ai( x) is a Special function named after the British astronomer George Biddell Airy.

$\Psi(x) = C_A Ai\left( \sqrt[3]{v_1} (x - x_1) \right) + C_B Bi\left( \sqrt[3]{v_1} (x - x_1) \right)$

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 Airy 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

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

Where $x 1, x 2$ are the 2 classical turning points for the potential barrier. See also Reflection coefficient The transmission coefficient is used in physics and electrical engineering when Wave propagation in a medium containing If we take the classical limit of all other physical parameters much larger than Planck'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. In Quantum mechanics, the finite potential barrier is a standard one-dimensional problem that demonstrates the phenomenon of Quantum tunnelling.

In Popular Culture

In The Simpsons episode "Future-Drama", Homer and Bart drive through a mountain, and the mountain is labeled "Quantum tunnel. The Josephson effect is the phenomenon of current flow across two weakly coupled Superconductors, separated by a very thin insulating barrier Squid are marine Cephalopods of the order Teuthida, which comprises around 300 species A tunnel Diode or Esaki diode is a type of Semiconductor diode which is capable of very fast operation well into the Microwave frequency In Physics, the WKB (Wentzel–Kramers–Brillouin approximation also known as WKBJ (Wentzel–Kramers–Brillouin–Jeffreys approximation is the most familiar Scanning tunneling microscope (STM is a powerful technique for viewing surfaces at the atomic level In Quantum mechanics, the finite potential barrier is a standard one-dimensional problem that demonstrates the phenomenon of Quantum tunnelling. Flash memory is non-volatile computer memory that can be electrically erased and reprogrammed The Delta potential well is a common theoretical problem of Quantum mechanics. Ferroelectricity is a physical property of a material whereby it exhibits a spontaneous electric polarization, the direction of which can be switched between equivalent " Future-Drama " is the fifteenth episode of the sixteenth season of The Simpsons. " It was likely a joke referring to this phenomenon.
In the science fiction show Sliders, the main characters travel to parallel universes using "quantum tunneling through an Einstein-Rosen-Podolsky bridge". Sliders is an American Science fiction television series that ran for five seasons from 1995 to 2000.
In the science fiction serial Zeta Disconnect, the gateway that the main character uses to travel through time is referred to several times as a "quantum tunnel".
In the video game Supreme Commander, humans use quantum tunneling as a means of teleportation, and thus as a way to colonize distant areas.
In the Michael Crichton novel Timeline, the characters use quantum tunneling as a means for experimental time travel. John Michael Crichton, ˈkraɪtən, (born October 23 1942 is an American author Film producer, Film director, Medical doctor, and Television producer A timeline is a graphical representation of a Chronological sequence of events also referred to as a Chronology.
Kitty Pryde, a character in Marvel Comics, uses the tunneling phenomenon to pass through walls. Katherine "Kitty" Pryde is a Fictional character that appears in Comic books published by Marvel Comics. Marvel Comics is an American comic book company owned by Marvel Publishing Inc

References

Notes

^ Razavy, Mohsen. (2003). , p1
^ A. Vilenkin (2003)
^ Seed: The Quantum Shortcut

Books

Razavy, Mohsen (2003). Quantum Theory of Tunneling. World Scientific. ISBN 981-238-019-1.
Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed. ). Prentice Hall. ISBN 0-13-805326-X.
Liboff, Richard L. (2002). Richard L Liboff is a US physicist who has authored five books and nearly 150 other publications in variety of fields including Plasma physics, Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5.
Vilenkin, Alexander (2003). "Particle creation in a tunneling universe". Phys. Rev. D 68: 023520.

Dictionary

-noun

(physics) Any of several quantum mechanical effects in which a particle disobeys the laws of classical mechanics

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