In classical physics, a particle with less energy than a barrier cannot cross it. Quantum mechanics, however, allows for the counterintuitive phenomenon of tunneling. Because particles have wave-like properties, there is a finite probability that they will appear on the other side of a barrier even if they lack sufficient energy to climb over it. This effect underlies processes as diverse as nuclear fusion in stars and electron flow in semiconductors.
When a wave encounters a barrier, its amplitude decays exponentially inside the barrier region. The rate of decay depends on the particle’s mass, the difference between the barrier height and the particle’s energy, and the barrier’s width. The probability of the particle emerging on the far side decreases rapidly as the barrier becomes taller or thicker. Nonetheless, if the barrier is thin enough, tunneling can occur with measurable likelihood.
For a simple rectangular potential barrier, the tunneling probability can be approximated using the expression . Here is the barrier width, and is given by . In this expression, is the particle mass, is the barrier height, is the particle energy, and is the reduced Planck constant. The probability decays exponentially with both barrier width and the square root of the energy difference.
Enter the particle mass in units of the electron mass, along with its energy, the barrier height, and the barrier width in nanometers. The script converts these values to SI units, computes , and evaluates the exponential formula above. The result represents the approximate probability of the particle appearing on the far side of the barrier. Note that this approximation assumes the barrier is much higher than the particle energy and that the barrier is one-dimensional.
As you adjust the particle energy, you’ll notice that the probability rises sharply once the energy approaches the barrier height. Similarly, even small increases in barrier width cause the probability to plummet. This sensitivity explains why tunneling is significant only on the atomic or subatomic scale. A barrier just a few nanometers thick can prevent electrons from flowing, while a thinner layer may allow appreciable current to pass.
Modern electronics harness tunneling in various ways. Tunnel diodes rely on quantum penetration through very thin depletion regions to achieve rapid switching. Flash memory cells trap electrons behind potential barriers that they can only escape via tunneling when enough voltage is applied. In scanning tunneling microscopes, a sharp tip is brought extremely close to a surface, and the tunneling current reveals atomic-scale details. Understanding how the probability responds to different parameters is essential for designing these technologies.
The fusion of hydrogen nuclei inside stars occurs at temperatures far lower than classical physics would suggest. Quantum tunneling allows protons to overcome their mutual electrostatic repulsion and merge, releasing energy that powers the stars. Likewise, radioactive alpha decay involves a helium nucleus tunneling out of a larger nucleus. These processes highlight how tunneling shapes the very existence of the universe.
Inside the barrier, the wave function amplitude decreases exponentially. The parameter sets the decay length, so a larger means the wave dies out faster. When you compute the probability in this calculator, you can see how quickly the exponential factor suppresses tunneling for wider or higher barriers. Although the formula is approximate, it captures the essence of how quantum waves behave in classically forbidden regions.
Quantum tunneling challenges our everyday intuition. Rather than picturing a particle as an object bouncing off a wall, it is more accurate to envision a spread-out wave that leaks through the barrier. The probability of finding the particle beyond the barrier reflects how much of that wave penetrates. Thinking in terms of waves rather than bullets can help demystify this remarkable phenomenon.
As technology pushes to smaller scales, tunneling will continue to play a central role. Future quantum computers may rely on precisely controlled tunneling events to manipulate qubits. By experimenting with different masses, energies, and widths in this calculator, you can appreciate the delicate balance between wave mechanics and potential barriers that governs quantum behavior.
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