quantum tunnelling - History and consequences, Semiclassical calculation

A feature specific to quantum systems, in which particles have a certain probability of existing beyond barriers, ie they are able to penetrate barriers that classically would be expected to constrain them. The feature arises from the wave description of quantum systems. Alpha decay, the tunnelling electron microscope, and the Josephson junction all display quantum tunnelling.

Quantum tunnelling (or tunneling) is the quantum-mechanical effect of transitioning through a classically-forbidden energy state.

Consider rolling a ball up a hill. For a quantum particle moving against a potential energy "hill", the wave function describing the particle can extend to the other side of the hill. This wave represents the probability of finding the particle in a certain location, meaning that the particle has the possibility of being detected on the other side of the hill.

As this is a quantum and non-classical effect, it can generally only be seen in nanoscopic phenomena — where the wave behavior of particles is more pronounced.

Availability of states is necessary for tunneling to occur. Analogously, a particle can tunnel through the barrier, but unless there are states available within the barrier, the particle can only tunnel to the other side of the barrier.

History and consequences

In the early 1900s, radioactive materials were known to have characteristic exponential decay rates or half lives.

Alpha decay via tunnelling was also solved concurrently by Ronald Gurney and Edward Condon.

After attending a seminar by Gamow, Max Born recognized the generality of quantum-mechanical tunnelling.

Quantum tunnelling was later applied to other situations, such as the cold emission of electrons, and perhaps most importantly semiconductor and superconductor physics.

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.

Very recently 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.

Semiclassical calculation

Let us consider the time-independent Schrödinger equation for one particle, in one dimension, under the influence of a hill potential V(x).

Now let us recast the wave function Ψ(x) as the exponential of a function.

Ψ(x) = e

Now let us separate Φ'(x) into real and imaginary parts.

B'(x) − 2A(x)B(x) = 0

Next we want to take the semiclassical approximation to solve this.

The constraints on the lowest order terms are as follows.

A0(x)B0(x) = 0

If the amplitude varies slowly as compared to the phase, we set A0(x) = 0 and get

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

On the other hand, if the phase varies slowly as compared to the amplitude, we set B0(x) = 0 and get

Which is obviously only valid when you have more potential than energy - tunnelling motion. Grinding out 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 E = V(x).

In a specific tunnelling problem, we might already suspect that the transition amplitude be proportional to and thus the tunnelling be exponentially dampened by large deviations from classically permitable motion.

But to be complete we must find the approximate solutions everywhere and match coefficients to make a global approximate solution.

Let us label a classical turning point x1.

Let us only approximate to linear order

This differential equation looks deceptively simple.

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.

Fortunately the Bessel function solutions will asymptote into sine, cosine and exponential functions in the proper limits.

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

The transmission coefficient, , for a particle tunnelling through a single potential barrier is found to be

Where x1,x2 are the 2 classical turning points for the potential barrier.