Exploring Quantum Tunneling Through a Potential Barrier

In classical physics a particle approaching a barrier with insufficient energy simply reflects; it cannot crest the barrier and emerge on the other side. Quantum mechanics, however, allows for a remarkable possibility: even when the particle's energy is less than the barrier height, there remains a finite probability that it will appear beyond the barrier. This phenomenon, called quantum tunneling, stems from the wave-like nature of matter and the probabilistic interpretation of its behavior. Rather than being confined strictly to the region permitted by classical energy considerations, the particle's wavefunction penetrates the forbidden region with an exponentially decaying amplitude. If the barrier is thin enough, a portion of the wave will leak through and continue propagating, leading to a measurable transmission probability.

To quantify this effect for a simple one-dimensional system, physicists often consider a rectangular potential barrier of height and width . A particle with mass and kinetic energy approaches from the left. The time-independent Schrödinger equation can be solved in each region—before, inside, and after the barrier—with boundary conditions enforcing continuity of the wavefunction and its derivative. For energies below the barrier (), the wavefunction inside the barrier decays exponentially. The solution yields a transmission coefficient

,

where . The hyperbolic sine reflects the exponential behavior inside the barrier. As the width increases, grows and the term blows up, driving the denominator larger and the transmission toward zero. Similarly, higher barriers (larger ) reduce the probability. When the particle's energy approaches the barrier height, the expression approaches unity, connecting smoothly to the classical limit where the particle surmounts the barrier with little reflection.

The calculator provided here implements this standard formula. Users supply the particle mass in units of the electron mass , the particle energy in electron-volts, the barrier height in electron-volts, and the barrier width in nanometers. Internally, the script converts these values to SI units, computes , evaluates the hyperbolic sine, and returns the transmission probability. If the input energy exceeds the barrier height, the code still applies the formula but notes that the situation now corresponds to above-barrier scattering where oscillatory solutions occur; in that case, approaches one as expected.

Quantum tunneling has profound implications across physics. In nuclear fusion, tunneling enables positively charged nuclei to overcome their mutual electrostatic repulsion, allowing stars—including our Sun—to shine. Alpha decay, a form of radioactive decay, occurs when an alpha particle tunnels out of the nucleus through a barrier created by the strong nuclear force and Coulomb repulsion. In electronics, tunneling underlies devices such as the tunnel diode and the scanning tunneling microscope (STM). The STM, for instance, measures tunneling current between a sharp tip and a surface, allowing imaging of individual atoms. In chemistry and biology, tunneling can facilitate reactions at low temperatures by allowing particles to bypass energy barriers that classical mechanisms would not allow. Even phenomena like proton tunneling in DNA have been proposed as contributors to mutation.

While the rectangular barrier is an idealization, it captures the essential physics and provides insight into how parameters influence the tunneling probability. The exponential dependence on barrier width and height means that small changes can drastically alter transmission. For example, a barrier only a nanometer thick can significantly impede electrons in a semiconductor, enabling devices like flash memory where thin insulating layers retain charge for years. Conversely, designing barriers of precise dimensions allows control over tunneling rates, a principle exploited in resonant tunneling diodes and quantum cascade lasers.

The table below illustrates how transmission probability varies with different energies and barrier widths for an electron encountering a 5 eV barrier. The stark contrast between the probabilities for thin and thick barriers underscores the sensitivity of tunneling to physical dimensions.

Energy (eV)	Width (nm)	Transmission Probability
1	0.5	0.065
1	1.0	0.004
2	0.5	0.23
2	1.0	0.031
4	0.5	0.71

Students can use this calculator to develop intuition about tunneling. By experimenting with different masses, energies, and barrier characteristics, one quickly sees how lighter particles or higher energies enhance transmission. Doubling the width typically reduces the probability by orders of magnitude, while doubling the particle energy can compensate by raising the exponent's argument. These relations are vital in fields from semiconductor engineering to quantum computing, where controlling tunneling is essential for device functionality.

Tunneling also introduces the concept of quantum indeterminacy. Even if a particle has a fixed energy well below the barrier, there is always some chance—however tiny—that it will appear on the other side. In macroscopic systems this probability is negligible, which is why we do not witness everyday objects tunneling through walls. Yet on the atomic scale the effect becomes significant and even dominant. The delicate interplay of wavefunctions and potential landscapes gives rise to behavior that defies classical intuition. Recognizing this departure from classical determinism is a cornerstone of modern physics education.

As research progresses into nanoscale materials and quantum information technologies, precise control over tunneling will become ever more crucial. Devices like Josephson junctions in superconducting circuits rely on tunneling of Cooper pairs, enabling quantum bits (qubits) used in superconducting quantum computers. Understanding and modeling tunneling probabilities is thus not merely an academic exercise but a practical requirement for cutting-edge technology. The calculator provided here aims to demystify the fundamental relationships governing tunneling, allowing learners to explore the rich and counterintuitive consequences of quantum theory.