123 Tunneling

One of the most remarkable predictions of quantum mechanics is that particles can sometimes pass through energy barriers that they do not have enough energy to overcome according to classical physics. This phenomenon, known as quantum tunneling, has no classical counterpart and plays an essential role in nuclear physics, semiconductor electronics, and many modern technologies.

To understand the basic idea, imagine placing a marble inside a smooth bowl, as shown in Figure 123.1. The marble can roll freely within the bowl, but unless it gains enough energy to climb over the rim, it remains trapped indefinitely. According to classical physics, the marble can never escape unless additional energy is supplied.

Atomic nuclei behave similarly. Protons and neutrons are bound inside the nucleus by the attractive strong nuclear force. Removing one of these particles requires energy because it is trapped within a region of low potential energy. On average, nucleons are bound by approximately 8 MeV per nucleon.

Figure 123.2 shows a simplified potential energy diagram for an alpha particle inside a nucleus. The attractive nuclear force creates a deep potential well that confines the particle. Outside the nucleus, the positively charged alpha particle experiences the repulsive Coulomb force from the remaining protons, producing a potential energy barrier. Classically, the alpha particle does not possess enough kinetic energy to overcome this barrier, so it should remain trapped inside the nucleus forever.

Yet alpha decay clearly occurs. Radioactive nuclei spontaneously emit alpha particles even though those particles appear to lack sufficient energy to escape. The explanation comes from quantum mechanics.

In quantum theory, particles are described by wave functions rather than by precise trajectories. Instead of being confined to a single location, a particle has a probability of being found at different positions in space. When the wave function reaches a potential energy barrier, it does not stop abruptly. Instead, it extends into the barrier while decreasing exponentially.

Because the wave function remains nonzero inside the barrier, there is a small—but finite—probability that the particle can be found on the opposite side. When this occurs, the particle is said to have tunneled through the barrier. This process is called quantum tunneling or barrier penetration.

The idea that quantum tunneling explains alpha decay was first developed in the late 1920s by George Gamow and independently by Ronald Gurney and Edward Condon. Their work successfully explained both why alpha decay occurs and why different radioactive nuclei have dramatically different half-lives.

The theory predicts that the probability of tunneling depends very strongly on the thickness and height of the potential barrier. If an alpha particle has slightly more energy, it begins closer to the top of the barrier, making the barrier effectively thinner. The wave function then decreases less rapidly inside the barrier, increasing the probability that the particle will tunnel through. As a result, nuclei that emit higher-energy alpha particles generally have much shorter half-lives than nuclei that emit lower-energy alpha particles. Remarkably, this simple quantum model accurately explains alpha-decay half-lives spanning more than 17 orders of magnitude.

Quantum Tunneling Beyond Nuclear Physics

Quantum tunneling is not limited to radioactive decay. The same phenomenon occurs whenever quantum particles encounter sufficiently thin potential barriers. Electrons, for example, can tunnel between two conducting objects separated by an extremely small gap, even though they lack the classical energy required to cross it.

One important application is the scanning tunneling microscope (STM). In an STM, a sharp conducting probe is brought within a few atomic diameters of a sample surface. Electrons tunnel between the probe and the sample, producing a tiny electric current. Because the tunneling probability changes exponentially with the width of the gap, even an atomic-scale change in surface height produces a measurable change in current. By scanning the probe across the surface while monitoring this current, the microscope can generate images that resolve individual atoms.

Interactive Exploration: Quantum Tunneling and Wave Packets

According to classical physics, a particle cannot pass through a potential energy barrier unless it has enough energy to climb over it. Quantum mechanics predicts a very different outcome. Because particles are described by wave functions, there is a finite probability that a particle can be found on the opposite side of a barrier even when its energy is lower than the barrier height. This phenomenon is known as quantum tunneling.

In this interactive simulation, you will investigate how wave packets behave when they encounter potential energy barriers. By changing the barrier's height and width, you can observe how these properties influence the probability that a particle tunnels through instead of being reflected.