Quantum tunnelling

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Quantum mechanics

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

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

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In quantum mechanics, quantum tunnelling (or the tunnel effect) is a nanoscopic phenomenon in which a particle violates the principles of classical mechanics by penetrating 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 classically suggests that passage through or over such a barrier would be impossible without sufficient energy.

Reflection and tunneling of an electron wavepacket directed at a potential barrier. The bright spot moving to the left is the reflected part of the wavepacket. A very dim spot can be seen moving to the right of the barrier. This is the small fraction of the wavepacket that tunnels through the classically forbidden barrier. Also notice the interference fringes between the incoming and reflected waves.

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".

1 History
2 Semi-classical calculation
3 In popular culture
4 See also
5 Notes
6 Further reading

[edit] History

By 1928, George Gamow had solved the theory of the alpha decay of a nucleus via tunnelling. 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 tunnelling 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 tunnelling. He realized that the tunnelling 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 tunnelling is even applied to the early cosmology of the universe.^[2]

Quantum tunnelling 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 tunnelling. Tunnelling 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-tunnelling microscopes (see scanning tunnelling microscope) which can resolve objects that are too small to see using conventional microscopes. Electron tunnelling microscopes overcome the limiting effects of conventional microscopes (optical aberrations, wavelength limitations) by scanning the surface of an object with tunnelling electrons.

Quantum tunnelling has been shown to be a mechanism used by enzymes to enhance reaction rates. It has been demonstrated that enzymes use tunnelling to transfer both electrons and nuclei such as hydrogen and deuterium. It has even been shown, in the enzyme glucose oxidase, that oxygen nuclei can tunnel under physiological conditions. ^[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 $V (x)$ .

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

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

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

$\Psi(x) = e^{\Phi(x)} \,$

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

Now we separate $Φ'(x)$ into real and imaginary parts using real valued functions A and B.

$\Phi'(x) = A(x) + i B(x) \,$

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

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. That means we expand each function as a power series in $\hbar$ . From the equations we can 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.

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

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

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 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 only valid when you have more potential than energy - tunnelling motion. Resolving 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 tunnelling problem, we might suspect that the transition amplitude is proportional to $e^{-\int dx \sqrt{\frac{2m}{\hbar^2} \left( V(x) - E \right)}}$ and thus the tunnelling is 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 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.

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

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 are 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 construct global solutions and solve tunnelling problems.

The transmission coefficient, $\left| \frac{C_{\mbox{outgoing}}}{C_{\mbox{incoming}}} \right|^2$ , for a particle tunnelling 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. 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.

[edit] In popular culture

In The Simpsons episode "Future-Drama", Homer and Bart drive through a mountain, and the mountain is labeled "Quantum tunnel."
In the science fiction show Sliders, the main characters travel to parallel universes using "quantum tunnelling 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 tunnelling as a means of teleportation, and thus as a way to colonize distant areas.
In the Michael Crichton novel Timeline, the characters use quantum tunnelling as a means for experimental time travel.
Kitty Pryde, a character in Marvel Comics, uses the tunnelling phenomenon to pass through walls.

[edit] See also

[edit] Notes

^ Razavy, Mohsen. (2003)., p1
^ A. Vilenkin (2003)
^ Quantum catalysis in enzymes: beyond the transition state theory paradigm

[edit] Further reading

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). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5.
Vilenkin, Alexander (2003). "Particle creation in a tunneling universe". Phys.Rev. D 68: 023520. doi:10.1103/PhysRevD.68.023520, http://arxiv.org/abs/gr-qc/0210034.

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