Quantum Mechanical (gamow) Tunneling

Type of physical science: Nuclear physics

Field of study: Thermonuclear reactions

When treated quantum mechanically, particles are able to pass through regions that are classically forbidden by conservation of energy. This quantum mechanical, or Gamow, tunneling is not only historically important, since its application to the α (alpha) decay of heavy nuclei provided one of the first confirmations of the quantum theory, but also applicable to the emission of electrons from metal surfaces, the operation of many devices used in modern electronics, and the understanding of fundamental processes such as nuclear fission and fusion and the decay of black holes.

Overview

Tunneling is a quantum mechanical phenomenon whereby a particle penetrates a region that is classically inaccessible. The classical behavior of a particle that encounters a potential energy barrier is quite different from the corresponding quantum behavior. Classically, the particle either passes over the barrier or is stopped by the barrier and caused to rebound in the direction from which it came. The determining factor is the size of the particle's total energy in comparison with the height of the barrier. The quantum treatment of the encounter indicates that the particle has a chance of passing beyond the barrier and a chance of rebounding from the barrier. The size of the particle's total energy in relation to the barrier's height and width determines the relative probabilities of these two possibilities, but no definite statement can be made about which one will take place in a given encounter.

In terms of the motion of an individual particle, the principle of conservation of energy requires that the sum of the kinetic energy, or energy of motion, and the potential energy, or energy of position, of the particle must remain constant. As an example, consider a ball that starts from rest at point A, the top of a hill, where its energy is completely gravitational potential energy. As the ball begins to move down the track to the right, its potential energy is gradually converted into kinetic energy. As the ball travels over a small hill at point B, some of this kinetic energy is converted back into potential energy, and the ball slows; however, it has sufficient total energy to get over the top, since it started its motion from a point higher than B. On the other side of point B is another hill, taller than the one at point A. The top of this hill is point D. Ignoring the small frictional losses, the ball will arrive at point C, which between B and D and at the same height as point A. At C, having converted all of its kinetic energy back into potential energy, the ball has insufficient total energy to climb over the hill and arrive at point E on the other side. The region between C and E is said to be classically forbidden. In this case, the ball will simply roll back and forth between points A and C. If the ball were initially started from a point higher than D, then it would pass over the large hill and arrive at point E.

For a given starting point, either all balls pass over the barrier or all rebound from the barrier. When treated quantum mechanically, however, the encounter between a particle and a potential barrier can yield quite different results. In this case, the barrier is most often an electromagnetic barrier rather than a gravitational barrier. According to a 1923 proposal by Louis de Broglie, every particle has an associated matter wave whose behavior governs the behavior of the particle.

The probability of finding the particle at any point is directly related to the amplitude of the associated matter wave at that point. The behavior of the matter wave is in turn determined by the wave equation derived by Erwin Schrödinger in 1926, which plays the same role in quantum mechanical descriptions of motion that Sir Isaac Newton's laws play in classical descriptions of motion. At any point at which the Schrödinger equation predicts a nonzero value for this matter wave, there is a finite probability of finding the associated particle. One of the many counterintuitive results of quantum mechanics is that the Schrödinger equation predicts a nonzero value for the matter wave inside classically forbidden regions. The amplitude of the matter wave decreases rapidly with increasing penetration into a classically forbidden region.

Now let the ball in the above example represent a particle such as an electron moving on a track, which represents the electromagnetic interaction that the electron experiences. When the electron impinges on the large barrier CDE, a detailed examination of the Schrödinger equation shows that a nonzero matter wave exists in the classically forbidden region between C and E as well as in the regions to the left of C and to the right of E. The electron has a chance of appearing on the far side of the barrier and a chance of being repelled by the barrier and moving back in the direction from which it came. For a given encounter of the electron with the barrier, no definite prediction can be made regarding which of these two possibilities will occur. Only their relative probabilities are calculable.

The Heisenberg uncertainty principle provides another way of understanding how this quantum mechanical barrier penetration comes about without relying on the highly technical details of the solution of the Schrödinger equation. The uncertainty principle states that the product of the uncertainty in the energy transferred to or from a particle (Δ E) and the uncertainty in the time at which the transfer took place (Δ t) must exceed a minimum constant value: (Δ E) (Δ t) ≥ constant. This principle implies that it is possible to violate strict conservation of energy for brief intervals of time. That is, an amount of energy Δ E may be borrowed against the principle of conservation of energy for a time interval that may not exceed Δ t. As the amount of energy borrowed increases, the time interval allowed for its repayment decreases. The electron encountering the barrier may be able to borrow sufficient energy to allow it to pass over the barrier and appear on the other side, giving the appearance of having tunneled through the barrier. With increasing barrier height, the amount of energy that must be borrowed to enable the particle to pass over the barrier increases, and thus the time for which energy may be borrowed becomes prohibitively short, not allowing the particle sufficient time to surmount the barrier before being required to return the borrowed energy. Similarly, as the width of the barrier increases, the time for which the energy must be borrowed increases, with the result that only a very small amount of energy can be borrowed. Thus tunneling is most efficacious for low or narrow barriers and practically inoperative for high, wide barriers.

Applications

The phenomenon of tunneling, or barrier penetration, finds a wide range of applications, from the tunneling of electrons exploited in devices such as the scanning tunneling microscope and the Josephson junction to the tunneling of nuclear particles involved in the fusion processes that power stars and the tunneling of particles out of black holes.

The first application on the principles of barrier penetration came soon after the development of the principles of quantum theory, on which the prediction of barrier penetration was based. One of the unresolved mysteries of the early period of nuclear physics involved the radioactive decay of nuclei such as uranium by the emission of α (alpha) particles. In the case of uranium, the energy of the alpha particles emitted by the radioactive nuclei was measured to be about four million electronvolts. (In the context of this discussion, an energy of one electronvolt is the energy acquired by an electron rolling down a potential hill one volt high.) A number of experiments using alpha particles as projectiles had established that the potential barrier surrounding the interior of the nucleus was at least nine million electronvolts high. The puzzle, then, was how to account for alpha particles that had an energy of only four million electronvolts getting over the potential barrier with a height of more than twice that value. In addition, the length of time required on average for alpha-particle emission by radioactive nuclei varies greatly even for nuclei that are otherwise very similar (three ten-millionths of a second for polonium and ten thousand million years for thorium). In 1928, Soviet-born physicist George Gamow and, independently, American Edward Uhler Condon and British-born Ronald Wilfred Gurney developed an explanation in terms of quantum mechanical tunneling. Inside the nucleus, protons and neutrons exert attractive forces on one another as a result of the strong nuclear force. This attractive force tends to keep the particles that make up the nucleus bound together, although they move relatively freely while inside the nucleus. Just outside the nuclear surface, a strong repulsive electrical force acts between the positively charged alpha particle and the positively charged nucleus. The attractive strong nuclear force does not act outside the nucleus because of its extremely short range. Although differing in detail, the resulting potential within and around the nucleus is qualitatively similar to that in the example with the ball. When an alpha particle (a tightly bound configuration of two protons and two neutrons) moving inside the nucleus approaches the nuclear surface, it may tunnel through to the other side of the potential barrier, appearing outside the nucleus as a radioactively emitted alpha particle. As described, the ability of the alpha particle to tunnel through the nuclear surface barrier is strongly dependent on even modest variations in the height and width of the barrier, accounting for the wide range of alpha decay times.

Fusion, another fundamental nuclear process, in which two low-mass nuclei come together and fuse into a single, more massive nucleus, can also be treated as a barrier penetration. The energy that results from fusion processes, which are responsible for energy generation within stars, is being developed as a commercial energy source. To initiate the fusion process, two colliding nuclei must be brought sufficiently close that their mutual attraction by means of the strong nuclear force will cause the nuclei to merge. This is extremely difficult to accomplish because of the very strong mutual repulsion of the two positively charged nuclei. Even at the high temperatures that exist in the cores of stars, the energies of the colliding nuclei are much lower than these repulsive potential barriers surrounding the nuclei. Thus fusion is achieved within stars only by means of quantum mechanical tunneling through this barrier. Attempts to achieve controlled fusion in the laboratory also rely on maintaining a sufficiently high temperature and density in a plasma so that tunneling to produce fusion has an appreciable probability.

A number of applications are based on the tunneling of electrons. One representative example is the scanning tunneling microscope (STM), which is able to achieve magnifications of up to one hundred million. The STM was invented in 1981 by Gerd Binnig and Heinrich Rohrer, who won the 1986 Nobel Prize in Physics for their achievement. The electrons that move about freely within a metal have a small probability of being found just outside the surface of the metal, having tunneled through the confining barrier at the surface. The STM works by placing a needlelike probe near the surface, where it collects these electrons in the form of a tiny electrical current. The tip of the probe must be only a single atom in width. Since the size of the electrical current is extremely sensitive to the probe's distance from the surface, it can be used to construct an accurate three-dimensional contour map of the metal surface. While the STM works best with conducting materials such as metal, it can be used to study organic materials as well. Due to its precision, it can also be used to manipulate the atoms in the sample being studied.

Another application of quantum tunneling is the tunnel diode, also called the Esaki diode, which is a type of semiconductor that makes use of the tunneling effect in order to operate at very high speed. The diode was invented in 1958 by Leo Esaki, who shared the 1973 Nobel Prize in Physics for its development. The tunnel diode's high speed makes it ideal for microwave radio frequency applications, and it has been shown to maintain stable performance for much longer than other types of semiconductors. However, they also have lower power capability and are difficult to reliably reproduce. While tunnel diodes are still ideal for some applications, such as in oscillators and frequency converters and detectors, improvements in other forms of semiconductor technology mean that tunnel diodes are typically only used in certain specific circumstances.

The behavior of the ammonia molecule is easily understood in terms of the ideas of barrier penetration. The ammonia molecule consists of a single nitrogen atom that is located either above or below the plane of the triangle formed by three hydrogen atoms. These two positions of the nitrogen atom are completely equivalent and, as in the example of the ball on the track, are separated by a potential barrier. The nitrogen atom that starts out in one of these positions will not remain there indefinitely but will tunnel through to the other position. This flip-flop process continues indefinitely, with a very precise frequency that depends on the tunneling rate, as in the alpha decay example.

A final example that illustrates the wide range of applicability of the tunneling concept is provided by black holes. A black hole is often described as an object that prevents anything, even light, from escaping from its powerful gravitational pull. This was the prevailing view until the 1974 prediction by Stephen W. Hawking that black holes should emit particles by means of a process involving tunneling. When a particle-antiparticle pair is formed only at the surface, or event horizon, of the black hole, one may be pulled into the black hole while the other escapes by tunneling through the gravitational potential barrier that surrounds the hole. Since the thickness of the barrier is proportional to the mass of the black hole, as the hole emits particles, the barrier becomes progressively smaller and the particles tunnel out more rapidly. Eventually, the black hole radiates away all of its material in a large explosion. Estimates indicate that small black holes evaporate within ten billion years of their formation, while more massive black holes require times far in excess of the present age of the universe to evaporate.

Context

The development of quantum mechanics and the investigation of the wave properties of particles occurred partly in response to the so-called crisis of classical physics. Near the beginning of the twentieth century, a number of phenomena were observed that could not be understood by means of the application of classical physics. The new quantum theory had a number of early successes, but these were mainly related to phenomena that the theory had been developed specifically to explain. The quantum theory predicted a number of additional physical ideas that were initially unobserved. Gamow, Condon, and Gurney's application of the concept of barrier penetration by a matter wave to the alpha decay of radioactive nuclei provided very important support to the fledgling theory. It was even more significant, however, that their work showed that the quantum theory, originally developed to explain atomic phenomena, could be successfully applied in other environments, such as the interior of the nucleus, and that the concept of a matter wave could be applied to particles other than the electron. In short, the prediction and observation of quantum mechanical barrier penetration constituted an important step in the building of a complete quantum picture of the physical universe. The deterministic world of classical physics was sharply contrasted with the probabilistic world of quantum physics.

The concept of quantum mechanical tunneling continues to be applied to gain an understanding of a number of diverse phenomena. The detailed description of many of the astrophysical processes that occur in stars and in the cosmos at large, the development of ever more sensitive devices for studying the surfaces of materials, and the microelectronic applications such as Josephson junctions and superconducting quantum interference devices (SQUIDs) all find their basis in the principle of quantum mechanical tunneling. It can only be assumed that new applications of this concept to these fields, as well as to others still unimagined, will be forthcoming.

Principal terms

BLACK HOLE: a region of space in which such a large quantity of mass is concentrated that the escape velocity at the surface exceeds that of light

CONSERVATION OF ENERGY: the principle, applicable in classical and quantum physics, that states that the total energy of a system must remain constant

KINETIC ENERGY: the energy that an object has as a result of its motion; kinetic energy is proportional to the mass of the object and its speed squared

MATTER WAVE: a quantum mechanical wave associated with any particle that governs the behavior of the particle and is itself determined by solving a quantum mechanical wave equation or a Schrödinger equation

POTENTIAL BARRIER: any region of space in which a force acts on an impinging particle (or wave), causing it to reduce its kinetic energy and correspondingly increase its potential energy

POTENTIAL ENERGY: the energy that an object has as a result of its position in a field of some kind of force, such as gravitational or electromagnetic

QUANTUM MECHANICS: the set of rules that must be applied in order to understand the behavior of particles on the atomic and subatomic scales

RADIOACTIVE NUCLEI: atomic nuclei that are rendered unstable by an excess or a deficiency of neutrons or by excitation and stabilize themselves by emitting one of three types of decay particles: α (alpha), β (beta), or γ (gamma)

Bibliography

Boorse, Henry A., L. Motz, and J. H. Weaver. The Atomic Scientists. New York: Wiley, 1989. Print. This readable book presents the history of the development of atomic theory in the form of biographical sketches of the people who developed it. In particular, the large chapter on wave mechanics presents a detailed qualitative description of matter waves and their behavior.

Feynman, Richard P. The Character of Physical Law. Cambridge: MIT P, 1965. Print. This book, consisting of seven lectures presented at Cornell University, still presents a unique explanation of quantum theory.

Gamow, George. Mr. Tompkins in Paperback. New York: Cambridge UP, 1965. Print. This small book, which has been popular since its initial publication, presents an account of the strange consequences of relativity and quantum theory in a world in which the constants of nature have been altered. Gamow was one of the first people to apply quantum theory to tunneling effects.

Hey, Anthony, and Patrick Walters. The Quantum Universe. New York: Cambridge UP, 1987. Print. This volume presents all the important topics that compose the foundations of quantum physics. It does so in a qualitative way, with only an occasional elementary use of mathematics, and presents modern applications that illustrate each of the topics treated. An outstanding feature of this book is the wealth of photographs and diagrams that facilitate understanding of the textual material.

Nakamura, Hiroki, and Gennady Mil'nikov. Quantum Mechanical Tunneling in Chemical Physics. Boca Raton: CRC, 2013. Print.

Phipps, Kristian. Quantum Tunnelling and Its Applications. Delhi: Learning, 2012. E-book.

Polkinghorne, John C. The Quantum World. New York: Longman, 1984. Print. This small book is similar in its selection of topics to the volume by Hey and Walters, but is somewhat more sophisticated in its treatment of those topics. Nevertheless, it takes a qualitative approach that should be accessible to the interested layperson.

Prutchi, David, and Shanni R. Prutchi. Exploring Quantum Physics through Hands-On Projects. Hoboken: Wiley, 2012. Print.

United States. Natl. Research Council. Panel on Condensed-Matter Physics. Condensed-Matter Physics. Washington: NAP, 1986. Print. This is one volume of a multivolume set of reports from a committee of distinguished physicists. This volume has descriptions of a number of examples of quantum mechanical tunneling, including other microscopes based on electron tunneling, Josephson junctions, SQUIDs, and other tunneling applications.

Weaver, Jefferson Hane, Lloyd Motz, and Dale McAdoo.The World of Physics. Vol. 3. New York: Simon, 1987. Print. This is the final volume of a set composed of samples of the literature of physics from antiquity to the late 1980s. Volume 3 has a number of short articles by physicists who have done work on tunneling phenomena. The phenomena described are the types of tunneling involved in black holes and alpha decay.

Path of ball

Forces on Charges and Currents

The Interpretation of Quantum Mechanics

Scanning Tunneling Microscopy and Atomic Force Microscopy

Thermonuclear Reactions in Stars