Tunneling in Quantum Mechanics

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.

Barrier Penetration

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.

Formula for Rectangular Barriers

For a simple rectangular potential barrier, the tunneling probability can be approximated using the expression $P\approx e^{-2\kappa a}$. Here $a$ is the barrier width, and $\kappa$ is given by $\kappa =\sqrt{\frac{2m(V-E)}{{\hslash }^2}}$. In this expression, $m$ is the particle mass, $V$ is the barrier height, $E$ is the particle energy, and $\hslash$ is the reduced Planck constant. The probability decays exponentially with both barrier width and the square root of the energy difference.

Using the Calculator

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 $\kappa$, 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.

Energy and Thickness

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.

Quantum Devices

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.

Tunneling in Nuclear Physics

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.

Wave Function Attenuation

Inside the barrier, the wave function amplitude decreases exponentially. The parameter $\kappa$ sets the decay length, so a larger $\kappa$ 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.

Conceptual Interpretation

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.

Looking Ahead

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.

Worked Example

To see the numbers in action, consider an electron approaching a 5 eV barrier that is 0.5 nm thick. Suppose the electron’s energy is 1 eV. Plugging these values into the calculator gives a tunneling probability on the order of ${10}^{-}$, meaning roughly one in one hundred million electrons would penetrate the barrier. Doubling the width to 1 nm drops the probability to around ${10}^{-}$, illustrating how sensitive tunneling is to distance. Such small probabilities still matter in semiconductor junctions, where trillions of electrons attempt to cross every second.

The example also highlights the effect of mass. If the particle were a proton instead of an electron, the probability would decrease exponentially because the higher mass increases the value of $\kappa$. This mass dependence explains why light particles such as electrons and neutrons exhibit tunneling more readily than heavier nuclei.

Limits of the WKB Approximation

The formula used here stems from the Wentzel–Kramers–Brillouin (WKB) approximation, which assumes the barrier varies slowly compared to the particle’s wavelength. For very thin barriers or cases where the energy is close to the barrier height, the approximation becomes less accurate. In such situations, solving the full Schrödinger equation or using more advanced techniques yields better results. Nevertheless, the WKB expression provides valuable insight and reasonable estimates for many practical problems.

Additionally, the calculator treats the barrier as perfectly rectangular and one-dimensional. Real-world barriers may have sloped edges, variable heights, or multiple layers. These complexities can cause resonant tunneling or interference effects that the simple formula does not capture. When designing actual devices, engineers often rely on numerical simulations or experimental measurements to account for these subtleties.

Historical Perspective

Quantum tunneling was first recognized in the 1920s when physicists such as George Gamow and Ronald Gurney applied wave mechanics to radioactive decay. Their calculations explained how alpha particles could escape atomic nuclei despite lacking the energy to overcome the nuclear potential barrier. Later, the development of the tunnel diode in the 1950s offered direct technological evidence of tunneling. These milestones cemented the concept as a fundamental aspect of quantum theory and a cornerstone of modern electronics.

Tunneling and Temperature

Temperature influences tunneling indirectly by altering the energy distribution of particles. In solids, higher temperatures broaden the range of electron energies, increasing the likelihood that some electrons approach the barrier with energies closer to its height. Thermal activation thus combines with tunneling to enable conduction in materials that would otherwise act as insulators at low temperatures. Superconductors, by contrast, exhibit tunneling through Josephson junctions even near absolute zero, where thermal effects are minimal.

Practical Use of the Calculator

When exploring scenarios with the calculator, try adjusting one variable at a time to build intuition. Increase the barrier width slowly and watch the probability fall off, or experiment with heavier particles to see how mass suppresses tunneling. Because the computation runs entirely in your browser, you can test extreme values without risk. Educators may project the tool during lectures to visualize concepts, while students can use it to check homework problems and explore beyond textbook examples.