Wave-mechanical tunneling

Wave-mechanical tunneling (also called quantum-mechanical tunneling, quantum tunneling, and the tunnel effect) is an evanescent wave coupling effect that occurs in the context of quantum mechanics because the behaviour of particles is governed by Schrödinger's wave-equation. All wave equations exhibit evanescent wave coupling effects if the conditions are right. Wave coupling effects mathematically equivalent to those called "tunneling" in quantum mechanics can occur with Maxwell's wave-equation (both with light and with microwaves), and with the common non-dispersive wave-equation often applied (for example) to waves on strings and to acoustics.For these effects to occur there has to be a situation where a thin region of "medium type 2" is sandwiched between two regions of "medium type 1", and the properties of these media have to be such that the wave equation has "traveling-wave" solutions in medium type 1, but "real exponential solutions" (rising and falling) in medium type 2. In optics, medium type 1 might be glass, medium type 2 might be vacuum. In quantum mechanics, in connection with motion of a particle, medium type 1 is a region of space where the particle total energy is greater than its potential energy, medium type 2 is a region of space (known as the "barrier") where the particle total energy is less than its potential energy - for further explanation see the section on "Schrödinger equation - tunneling basics" below. Schematic representation of quantum tunneling through a barrier. The energy of the tunneled particle is the same, only the quantum amplitude (and hence the probability of the process) is decreased. Plotting done with Wolfram's Mathematica 6.0If conditions are right, amplitude from a traveling wave, incident onto medium type 2 from medium type 1, can "leak through" medium type 2 and emerge as a traveling wave in the second region of medium type 1 on the far side. If the second region of medium type 1 is not present, then the traveling wave incident on medium type 2 is totally reflected, although it does penetrate into medium type 2 to some extent. Depending on the wave equation being used, the leaked amplitude is interpreted physically as traveling energy or as a traveling particle, and, numerically, the ratio of the square of the leaked amplitude to the square of the incident amplitude gives the proportion of incident energy transmitted out the far side, or (in the case of the Schrödinger equation) the probability that the particle "tunnels" through the barrier.

Further information:

1. N. Fröman and P.-O. Fröman (1965). JWKB Approximation: Contributions to the Theory. North-Holland, Amsterdam
2. Razavy, Mohsen (2003). Quantum Theory of Tunneling. World Scientific
3. Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall
4. Vilenkin, Alexander (2003). "Particle creation in a tunneling universe". Phys.Rev. D 68: 023520
5. Блохинцев Д. И., Основы квантовой механики, 4 изд., М., 1963; Ландау Л. Д., Лифшиц Е. М., Квантовая механика. Нерелятивистская теория, 3 изд., М., 1974 (Теоретическая физика, т. 3)
6. Article Wave-mechanical tunneling from Wikipedia, the Free Enciclopedia. Available under the license Creative Commons Attribution-Share Alike.

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