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".
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.
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.