Exponent rules everything
Every extra nanometer of width can slash T by orders of magnitude because T ~ exp(-2κa).
Double-edged
Quantum Tunneling Probability Calculator
Enter mass, barrier height, width, and particle energy to estimate transmission through a 1D rectangular barrier. Uses the standard T ≈ exp(-2κa) approximation for E < V₀.
Tunneling probability
Tunneling probability T
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Decimal form
Tunneling probability
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Percentage form
Suppression factor
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Exponent 2κa (unitless)
Barrier region κ
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κ = √(2m(V₀−E))/ħ (1/m)
Probability bar (capped at 1%)
How this estimate works
This quantum tunneling calculator gives a quick, intuitive estimate of how likely a particle is to pass through a barrier it would not clear in classical physics. It uses the textbook 1D rectangular barrier model to translate your inputs into a tunneling probability, letting you see how mass, energy, barrier height, and barrier width work together. The result is a compact picture of exponential suppression, ideal for learning, sanity checks, and homework-level intuition.
The core idea is simple: when a particle’s energy E is below a barrier height V₀, the wavefunction does not stop at the barrier. Instead, it decays inside the barrier, which means there is still a small but finite chance of appearing on the other side. That decay rate is captured by κ = √(2m(V₀−E))/ħ, and the transmission probability is approximated by T ≈ exp(-2κa), where a is the barrier width. If E is at or above the barrier, classical transmission is allowed and the calculator reports T ≈ 1 for a simplified comparison.
To use the calculator, enter the particle mass in kilograms, the particle energy and barrier height in electron volts, and the barrier width in nanometers. Then click Calculate. The tool automatically converts units and reports the tunneling probability, the suppression exponent 2κa, and the decay constant κ. The probability bar provides a visual sense of scale so you can compare scenarios at a glance.
This estimate is especially helpful when exploring how sensitive tunneling is to distance. Even a small increase in width or barrier height can drive the probability down by orders of magnitude. That behavior shows up in real-world systems such as alpha decay in nuclear physics, electron tunneling in scanning tunneling microscopes, and thin insulating layers in semiconductor devices. While the model is simplified—no resonant tunneling, no multiple reflections, and no 3D effects—it captures the essential physics and provides a solid starting point for deeper Schrödinger equation treatments.
Quick tunneling facts
Mass matters
Heavier particles tunnel far less. Electrons zip through thin barriers; alphas face huge suppression.
m in κ
Alpha decay is tunneling
An alpha trapped by the nuclear Coulomb barrier sneaks out via tunneling, setting decay lifetimes.
Nuclear
STM relies on it
Scanning tunneling microscopes measure current from electrons tunneling across a tiny vacuum gap.
Instrumentation
Not faster-than-light
Tunneling doesn’t move particles superluminally; group delays can be short without causal signaling.
Causality
Why tunneling is mind-bending (and useful)
Classically, a particle with energy below a barrier stops. Quantum mechanically, its wavefunction decays but does not vanish inside the barrier, leaving a finite chance to emerge on the other side. That probability is the tunneling transmission T. When 2κa is small, T can be sizable; when it is large, T plummets exponentially.
This behavior underpins many phenomena: alpha decay from nuclei, electron transport in tunnel diodes, Josephson junctions in superconducting qubits, and the contrast mechanism of scanning tunneling microscopes. In coursework, it’s the go-to example of how quantum amplitudes differ from classical probabilities.
Use the presets to see both extremes. The electron example shows a modest barrier giving a non-negligible T. The alpha example shows how quickly T collapses when mass and barrier height grow, explaining the long half-lives associated with nuclear tunneling.