Quantum tunnelling

In quantum mechanics, quantum tunneling is a micro and 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 classical mechanics, which classically suggests that passage through or over such a barrier would be impossible without sufficient energy.

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".

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.

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.^[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]

Semi-classical calculation

Let us consider the time-independent Schrödinger equation for one particle, in one dimension, under the influence of a hill potential Failed to parse (Missing texvc executable; please see math/README to configure.): V(x) .

Failed to parse (Missing texvc executable; please see math/README to configure.): -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} \Psi(x) + V(x) \Psi(x) = E \Psi(x)

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): \Psi(x)

as the exponential of a function.

Failed to parse (Missing texvc executable; please see math/README to configure.): \Psi(x) = e^{\Phi(x)}

Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi''(x) + \Phi'(x)^2 = \frac{2m}{\hbar^2} \left( V(x) - E \right)

Now let us separate Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi'(x)

into real and imaginary parts using real valued functions A and B.

Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi'(x) = A(x) + i B(x)

Failed to parse (Missing texvc executable; please see math/README to configure.): 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \hbar . From the equations we can already see that the power series must start with at least an order of Failed to parse (Missing texvc executable; please see math/README to configure.): \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.

Failed to parse (Missing texvc executable; please see math/README to configure.): A(x) = \frac{1}{\hbar} \sum_{k=0}^\infty \hbar^k A_k(x)

Failed to parse (Missing texvc executable; please see math/README to configure.): B(x) = \frac{1}{\hbar} \sum_{k=0}^\infty \hbar^k B_k(x)

The constraints on the lowest order terms are as follows.

Failed to parse (Missing texvc executable; please see math/README to configure.): A_0(x)^2 - B_0(x)^2 = 2m \left( V(x) - E \right)

Failed to parse (Missing texvc executable; please see math/README to configure.): A_0(x) B_0(x) = 0

If the amplitude varies slowly as compared to the phase, we set Failed to parse (Missing texvc executable; please see math/README to configure.): A_0(x) = 0

and get

Failed to parse (Missing texvc executable; please see math/README to configure.): 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): B_0(x) = 0

and get

Failed to parse (Missing texvc executable; please see math/README to configure.): 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): E=V(x) .

Let us label a classical turning point Failed to parse (Missing texvc executable; please see math/README to configure.): x_1 . Now because we are near Failed to parse (Missing texvc executable; please see math/README to configure.): E=V(x_1) , we can easily expand Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{2m}{\hbar^2}\left(V(x)-E\right)

in a power series.

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{2m}{\hbar^2}\left(V(x)-E\right) = v_1 (x - x_1)

Failed to parse (Missing texvc executable; please see math/README to configure.): \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.

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): C,\theta

and Failed to parse (Missing texvc executable; please see math/README to configure.): 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.

Failed to parse (Missing texvc executable; please see math/README to configure.): C_{+} = \frac{1}{2} C \cos{\left(\theta - \frac{\pi}{4}\right)}

Failed to parse (Missing texvc executable; please see math/README to configure.): C_{-} = - C \sin{\left(\theta - \frac{\pi}{4}\right)}

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

The transmission coefficient, Failed to parse (Missing texvc executable; please see math/README to configure.): \left| \frac{C_{\mbox{outgoing}}}{C_{\mbox{incoming}}} \right|^2 , for a particle tunneling through a single potential barrier is found to be

Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \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.

See also

• 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

References

Notes

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). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5. 
• Vilenkin, Alexander (2003). "Particle creation in a tunneling universe". Phys.Rev. D 68: 023520. bg:Тунелен преход

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