Quantum tunnelling

Zjawisko tunelowe
Z Wikipedii
http://pl.wikipedia.org/wiki/Zjawisko_tunelowe
Zjawisko tunelowe zwane teŜ efektem tunelowym - zjawisko kwantowe przejścia cząstki
przez barierę potencjału o wysokości (energii potencjalnej) większej niŜ energia cząstki. To
zjawisko, charakterystyczne dla mechaniki kwantowej, jest z punktu widzenia fizyki
klasycznej paradoksem łamiącym klasycznie rozumianą zasadę zachowania energii, gdyŜ
cząstka przez pewien czas przebywa w obszarze zabronionym przez zasadę zachowania
energii.
Zjawisko to zostało przewidziane teoretycznie w 1928 roku przez R.H. Fowlera i L.
Nordheima. Wkrótce potem wytłumaczono nim zjawisko emisji cząstek α w procesie rozpadu
promieniotwórczego jąder atomowych.
Zjawisko jest odpowiedzialne za wiele procesów szczególnie zachodzących z niewielką
szybkością, zanim dany proces zajdzie ze znacznie większą szybkością, gdy energia cząstek
przekroczy barierę potencjału.
Warto wspomnieć, Ŝe zjawisku tunelowemu zawdzięczamy Ŝycie na ziemi, gdyŜ fuzja
jądrowa będąca źródłem energii Słońca zachodzi w warunkach zjawiska tunelowego. Energie
zjonizowanej plazmy słonecznej są bowiem zbyt niskie aby pokonać barierę odpychania
kulombowskiego jąder atomów wodoru. Bez zjawiska tunelowego jądra nie mogłyby się
zbliŜyć wystarczająco aby połączyć się w jedno jądro. Na szczęście dzięki efektowi
tunelowemu nie jest to wcale konieczne.
Eksperymentalnie zjawisko to zostało potwierdzone na początku lat 60. We współczesnej
technice dzięki zjawisku tunelowemu funkcjonują urządzenia takie jak dioda tunelowa czy
skaningowy mikroskop tunelowy.
Wyjaśnienie [edytuj]
Cząsteczka o energii E znajduje się w obszarze otoczonym obszarem o energii potencjalnej
odpowiadającej wykresowi. Z punktu widzenia fizyki klasycznej energia cząstki jest sumą
energii kinetycznej i potencjalnej
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PoniewaŜ energia kinetyczna jest nieujemna, prawo zachowania energii dopuszcza tylko ruch
w obszarach gdzie
Ruch przez barierę (obszar, w którym E − U(x) < 0) jest zabroniony. W mechanice kwantowej
jest moŜliwe przeniknięcie przez barierę z pewnym określonym prawdopodobieństwem.
Prawdopodobieństwo to określa równanie falowe Schrödingera mechaniki kwantowej
W obszarze między x1 a x2 ruch jest ograniczony, dobrym przybliŜeniem jest uwaŜanie go za
ruch zdeformowanego oscylatora harmonicznego w potencjale
Barierę moŜna przybliŜyć potencjałem
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Daleko od bariery (gdy x > x3) ruch jest swobodny – U(x)=0. Rozwiązaniem równania
jest fala płaska
o energii
JeŜeli cząstka napotyka na stałą barierę o wysokości U>0, to równanie Schrödingera ma w
tym obszarze postać
Rozwiązaniem jest zanikająca amplituda prawdopodobieństwa
Wnikanie cząstki w barierę potencjału opisana jest przez parametr penetracji
gdzie
jest zasięgiem penetracji.
Źródło: "http://pl.wikipedia.org/wiki/Zjawisko_tunelowe"
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Kategoria: Mechanika kwantowa
Quantum tunnelling
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Quantum mechanics
Uncertainty principle
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[show]Background
[hide]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.[1] A barrier, in
terms of quantum tunnelling, may be a form of energy state analogous to a "hill" or incline in
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classical mechanics, which classically suggests that passage through or over such a barrier
would be impossible without sufficient energy.
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". 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
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1 History
2 Semi-classical calculation
3 See also
4 In Popular Culture
5 References
o 5.1 Notes
o 5.2 Books
[edit] History
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.
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After attending a seminar by Gamow, Max Born recognized the generality of quantummechanical 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.[2]
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.[3]
[edit] 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).
Now let us recast the wave function Ψ(x) as the exponential of a function.
Ψ(x) = eΦ(x)
Now let us separate Φ'(x) into real and imaginary parts using real valued functions A and B.
Φ'(x) = A(x) + iB(x)
,
because the pure imaginary part needs to vanish due to the real-valued right-hand side:
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Next we want to take the semiclassical approximation to solve this. That means we expand
each function as a power series in . From the equations we can already see that the power
series must start with at least an order of
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.
A0(x)B0(x) = 0
If the amplitude varies slowly as compared to the phase, we set A0(x) = 0 and get
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
On the other hand, if the phase varies slowly as compared to the amplitude, we set B0(x) = 0
and get
Which is obviously only valid when you have more potential than energy - tunnelling motion.
Grinding out the next order of the expansion yields
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.
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In a specific tunneling problem, we might already suspect that the transition amplitude be
proportional to
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 x1. Now because we are near E = V(x1), we can easily
expand
in a power series.
Let us only approximate to linear order
This differential equation looks deceptively simple. Its solutions are Airy functions.
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.
Now we can easily construct global solutions and solve tunneling problems.
The transmission coefficient,
potential barrier is found to be
, for a particle tunneling through a single
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Where x1,x2 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
, 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.
[edit] See also
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Josephson effect
SQUID
Tunnel diode
WKB approximation
Scanning tunnelling microscope
Finite potential barrier (QM)
Flash memory
Delta potential barrier (QM)
Ferroelectric tunnel junction
Quantum Tunneling Composite
[edit] In Popular Culture
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In The Simpsons episode "Future-Drama", Homer and Bart drive through a mountain,
and the mountain is labeled "Quantum tunnel." 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".
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.
Kitty Pryde, a character in Marvel Comics, uses the tunneling phenomenon to pass
through walls.
[edit] References
[edit] Notes
1. ^ Razavy, Mohsen. (2003)., p1
2. ^ A. Vilenkin (2003)
3. ^ Seed: The Quantum Shortcut
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[edit] Books
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Razavy, Mohsen (2003). Quantum Theory of Tunneling. World Scientific. ISBN 981238-019-1.
Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice
Hall. ISBN 0-13-805326-X.
Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN
0-8053-8714-5.
Vilenkin, Alexander (2003). "Particle creation in a tunneling universe". Phys.Rev. D
68: 023520.
Retrieved from "http://en.wikipedia.org/wiki/Quantum_tunnelling"
Categories: Physics | Particle physics | Quantum mechanics
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