# 7.6 The quantum tunneling of particles through potential barriers

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By the end of this section, you will be able to:
• Describe how a quantum particle may tunnel across a potential barrier
• Identify important physical parameters that affect the tunneling probability
• Identify the physical phenomena where quantum tunneling is observed
• Explain how quantum tunneling is utilized in modern technologies

Quantum tunneling is a phenomenon in which particles penetrate a potential energy barrier with a height greater than the total energy of the particles. The phenomenon is interesting and important because it violates the principles of classical mechanics. Quantum tunneling is important in models of the Sun and has a wide range of applications, such as the scanning tunneling microscope and the tunnel diode.

## Tunneling and potential energy

To illustrate quantum tunneling    , consider a ball rolling along a surface with a kinetic energy of 100 J. As the ball rolls, it encounters a hill. The potential energy of the ball placed atop the hill is 10 J. Therefore, the ball (with 100 J of kinetic energy) easily rolls over the hill and continues on. In classical mechanics, the probability that the ball passes over the hill is exactly 1—it makes it over every time. If, however, the height of the hill is increased—a ball placed atop the hill has a potential energy of 200 J—the ball proceeds only part of the way up the hill, stops, and returns in the direction it came. The total energy of the ball is converted entirely into potential energy before it can reach the top of the hill. We do not expect, even after repeated attempts, for the 100-J ball to ever be found beyond the hill. Therefore, the probability that the ball passes over the hill is exactly 0, and probability it is turned back or “reflected” by the hill is exactly 1. The ball never makes it over the hill. The existence of the ball beyond the hill is an impossibility or “energetically forbidden.”

However, according to quantum mechanics, the ball has a wave function and this function is defined over all space. The wave function may be highly localized, but there is always a chance that as the ball encounters the hill, the ball will suddenly be found beyond it. Indeed, this probability is appreciable if the “wave packet” of the ball is wider than the barrier.

View this interactive simulation for a simulation of tunneling.

In the language of quantum mechanics, the hill is characterized by a potential barrier    . A finite-height square barrier is described by the following potential-energy function:

$U\left(x\right)=\left\{\begin{array}{cc}\hfill 0,& \text{when}\phantom{\rule{0.2em}{0ex}}x<0\hfill \\ \hfill {U}_{0},& \text{when}\phantom{\rule{0.2em}{0ex}}0\le x\le L\hfill \\ \hfill 0,& \text{when}\phantom{\rule{0.2em}{0ex}}x>L.\hfill \end{array}$

The potential barrier is illustrated in [link] . When the height ${U}_{0}$ of the barrier is infinite, the wave packet representing an incident quantum particle is unable to penetrate it, and the quantum particle bounces back from the barrier boundary, just like a classical particle. When the width L of the barrier is infinite and its height is finite, a part of the wave packet representing an incident quantum particle can filter through the barrier boundary and eventually perish after traveling some distance inside the barrier.

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