|
|
|
|
Quantum tunnelling
|
| |
|
| |
In quantum mechanics, wave-mechanical tunneling (also called quantum-mechanical tunneling, quantum tunneling, and the tunnel effect) is an evanescent-wave phenomenon that occurs because the behaviour of particles is governed by Schroedinger's wave-equation. All wave-equations exhibit evanescent-wave phenomena if the conditions are right. Wave-phenomena exactly analogous to those to which quantum mechanics gives the name "tunneling" can occur with Maxwell's wave-equation (both with light and with microwaves), and with the common non-dispersive wave-equation that applies (for example) to waves on strings.
Discussion
Ask a question about 'Quantum tunnelling' Start a new discussion about 'Quantum tunnelling' Answer questions from other users |
Encyclopedia
In quantum mechanics, wave-mechanical tunneling (also called quantum-mechanical tunneling, quantum tunneling, and the tunnel effect) is an evanescent-wave phenomenon that occurs because the behaviour of particles is governed by Schroedinger's wave-equation. All wave-equations exhibit evanescent-wave phenomena if the conditions are right. Wave-phenomena exactly analogous to those to which quantum mechanics gives the name "tunneling" can occur with Maxwell's wave-equation (both with light and with microwaves), and with the common non-dispersive wave-equation that applies (for example) to waves on strings. For these phenomena to occur there has to be situation where a thin slab of "medium 2" is sandwiched between two slabs of "medium 1", and the properties of these media have to be such that the wave-equation has "traveling-wave" solutions in medium 1, but "real exponential solutions" (rising and falling) in medium 2. If conditions are right, amplitude from a traveling wave, incident onto medium 2 from medium 1, can "leak through" medium 2 and emerge as a traveling wave in the second slab of medium 1 on the far side. If the second slab of medium 1 is not present, then the traveling wave incident on medium 2 is totally reflected, although it does penetrate into medium 2 to some extent. Depending on the wave-equation being used, the leaked amplitude is interpreted physically as traveling energy or 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 the probability that the particle tunnels through the barrier.
Some basics
The scale on which these "tunneling-like phenomena" occur depends on the wavelength of the traveling wave. For electrons the thickness of "medium 2" (called in this context "the tunneling barrier") is typically a few nanometres; for alpha-particles tunneling out of a nucleus the thickness is very much less; for the analogous phenomenon involving light the thickness is very much greater.
With Schroedinger's wave-equation, the characteristic that defines the two media discussed above is the kinetic energy of the particle if it could be considered as an object that could be located at a point. In medium 1 the kinetic energy would be positive, in medium 2 the kinetic energy would be negative. There is no inconsistency in this, because physics knows that the nature of the physical world is such that particles are not objects that can physically be located at a point: they are always spread out ("delocalised") to some extent, and the kinetic energy of the delocalised object is always positive.
What is true is that it is sometimes mathematically convenient to treat particles as behaving like points, particular in the context of Newton's Second Law and classical mechanics generally. In the past, people thought that the success of classical mechanics meant that particles could always and in all circumstances be treated as if they were located at points. But there never was any convincing experimental evidence that this was true when very small objects and very small distances are involved, and we now know that this viewpoint was mistaken. However, because it is still traditional to teach students early on in their careers that particles behave like points, it sometimes comes as a big surprise for people to discover that it is very well established that traveling physical particles always physically obey a wave-equation (even when it is convenient to use the mathematics of moving points). Clearly, a hypothetical classical point particle analysed according to Newton's Laws could not enter a region where its kinetic energy would be negative. But, a delocalised object that always has positive kinetic energy, and obeys a wave-equation, can leak through such a region if conditions are right.
An electron approaching a barrier has to be represented as a wave-train. This can sometimes be quite long – which makes animations difficult. If it were legitimate to represent the electron by a short wave-train, then you could represent tunneling as in the animation alongside.
It is sometimes said that tunneling occurs only in quantum mechanics. Unfortunately, this statement is bit of linguistic conjuring trick. As indicated above, "tunneling-type" evanescent-wave phenomena occur in other contexts too. But, until recently, it has only been in quantum mechanics that such evanescent-wave-phenomena are called "tunneling".
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.
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.
Semi-classical calculation
Let us consider the time-independent Schrödinger equation for one particle, in one dimension, under the influence of a hill potential .
Now let us recast the wave function as the exponential of a function.
Now we separate into real and imaginary parts using real valued functions A and B.
,
because the pure imaginary part needs to vanish due to the real-valued right-hand side:
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 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.
If the amplitude varies slowly as compared to the phase, we set and get
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
On the other hand, if the phase varies slowly as compared to the amplitude, we set and get
which is only valid when you have more potential than energy - tunnelling motion. Resolving 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 . 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 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 .
Let us label a classical turning point . Now because we are near , we can 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 are able to find a relationship between and .
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 construct global solutions and solve tunnelling problems.
The transmission coefficient, , for a particle tunnelling through a single potential barrier is found to be
Where 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.
See also
Further reading
|
| |
|
|
