Introduction to Quantum Tunnelling

Credit - JohnsonMartin: https://pixabay.com/illustrations/wormhole-space-time-light-tunnel-739872/

Wave-particle duality is a key aspect of quantum mechanics. The wave characteristics of a particle are mathematically described by a quantity called the wave function. The square of the modulus (absolute value) of the wave function at a given position represents the probability of finding the particle at that point.¹ Quantum mechanics is inherently probabilistic. It is only when an observation is made that the wave function collapses. This feature is most apparent in the double-slit experiment which was talked about in the previous post (https://phys-talk.blogspot.com/2020/09/the-problem-with-quantum-mechanics.html). The wave-like nature of particles paves the way for an interesting phenomenon known as quantum tunnelling.

Going through walls

Quantum tunnelling is the phenomenon of particles passing through ‘seemingly impassable force barriers.’ ² Consider a ball rolling up a hill: if the ball does not have sufficient kinetic energy, it will not be able to overcome the potential energy barrier and roll onto the other side. Let us now imagine an electron travelling towards a potential barrier. If the electron has less kinetic energy that the potential energy barrier, surely, like the ball and hill scenario, it will fail to reach the other side? As is often the case with physics on the quantum scale, things are not quite so simple. It turns out that sometimes, the electron will end up on the other side, “tunnelling” through the potential energy barrier.

As mentioned at the beginning of this post, particles can be depicted using a wavefunction which provides information on how likely it is to find the particle at that point:

When the wave function passes through the potential barrier it decays exponentially with most of the wave packet energy reflecting off the barrier. The key point is that the wave function does not decay to zero if the potential barrier is thin enough (this is explained in the next paragraph) and some of the wave packet is able to pass through. This results in a non-zero wave function as shown in the above diagram. Thus, there is a chance, albeit small, of finding the electron on the opposite side when observed (wave function collapses). Usually, when considering physical processes, a vast number of tiny particles are involved. Quantum tunnelling can be a relatively significant effect.

Evanescent waves

When electromagnetic waves reflect off a boundary, they do not stop abruptly at the boundary. Instead, they quickly decay as non-propagating waves. These waves are known as evanescent waves. Maxwell’s equations show that they decay exponentially. This explains why a non-zero wave function is possible when passing through a potential barrier – an evanescent wave forms at the boundary.

Sources:

1. Khan Academy: https://www.khanacademy.org/science/physics/quantum-physics/atoms-and-electrons/v/quantum-wavefunction

2. Britannica: https://www.britannica.com/science/tunneling

Up and Atom YouTube Channel: https://www.youtube.com/watch?v=WPZLRtyvEqo