Quantum Tunneling Probability Calculator | Online Free Tool | Elysia Tools

Tool usage guide

Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Category: Science & Education
Input types: select, number
Output type: text
Sample coverage: 4
API ready: Yes

Overview

Calculate quantum tunneling transmission and reflection coefficients through a one-dimensional rectangular potential barrier. The tool computes the decay constant, compares the exact analytical solution with the WKB approximation for electrons, protons, or custom particle masses, and handles both sub-barrier tunneling and over-barrier quantum reflection.

When to use

Determining tunneling transmission probabilities through thin potential barriers in nanoscale semiconductor devices or quantum physics coursework.
Comparing the accuracy of the exponential WKB approximation against the exact rectangular barrier formula.
Analyzing quantum reflection and transmission coefficients for particles traveling above a potential barrier where E exceeds V₀.

How it works

1Select a particle preset such as electron or proton, or choose custom mass and enter a mass value in kilograms.
2Enter the barrier height V₀ in eV, barrier width a in nm, and incident particle energy E in eV.
3The calculator determines the decay constant κ (or wave number k′), computing the exact transmission coefficient T, the WKB estimate T ≈ e^(−2κa), and the reflection coefficient R = 1 − T.

Use cases

Semiconductor device modeling to estimate gate dielectric tunneling leakage currents.
Undergraduate quantum mechanics assignments evaluating 1D finite rectangular barriers.
Nuclear and physical chemistry studies analyzing proton transfer and alpha decay tunneling rates.

Examples

1. Sub-Barrier Electron Tunneling Analysis

Physics Student

Background

Analyzing a 1 nm insulating barrier with a 1.0 eV height for a quantum mechanics problem.

Problem

Calculate transmission for an incident electron at E = 0.5 eV and check how close the WKB approximation is to the exact solution.

How to use

Select electron, set Barrier height V₀ to 1 eV, Barrier width a to 1 nm, and Particle energy E to 0.5 eV.

Outcome

Yields κa = 3.6226, an exact transmission T = 2.85×10⁻³ (about 1 in 350 electrons), and a WKB approximation T ≈ 7.14×10⁻⁴.

2. Over-Barrier Quantum Reflection Evaluation

Device Engineer

Background

Investigating ballistic carrier transport over a 1.0 eV potential barrier in a 1 nm heterostructure.

Problem

Determine the non-classical reflection rate for electrons with energy E = 1.5 eV crossing the barrier.

How to use

Select electron, set Barrier height V₀ to 1 eV, Barrier width a to 1 nm, and Particle energy E to 1.5 eV.

Outcome

Calculates an exact transmission T = 0.9334 and an over-barrier quantum reflection R = 6.66%.

FAQ

What is the difference between the WKB approximation and the exact solution?

The WKB formula T ≈ e^(−2κa) approximates transmission for thick barriers where κa ≫ 1, whereas the exact rectangular formula accounts for wave boundary matching at both interfaces.

Why does reflection occur when particle energy E is greater than barrier height V₀?

Wave impedance changes at the potential boundaries cause quantum reflection even when classical mechanics predicts complete transmission.

How does particle mass affect tunneling probability?

Heavier particles have a larger decay constant κ, causing transmission probability to drop exponentially faster with increasing barrier width.

What units are used for input parameters?

Barrier height and particle energy use electronvolts (eV), barrier width uses nanometers (nm), and custom mass uses kilograms (kg).

What happens when energy E exactly equals barrier height V₀?

The calculator evaluates the finite limiting transmission formula T = 1 / (1 + m V₀ a² / (2 ħ²)).

Quantum Tunneling Probability Calculator

Run this tool

Examples that match this tool

Continue with connected tools and hubs

Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Overview

When to use

How it works

Use cases

Examples

1. Sub-Barrier Electron Tunneling Analysis

2. Over-Barrier Quantum Reflection Evaluation

FAQ

Android Image Processing Java Samples

Android Image Processing Kotlin Samples

Web Image Processing Python Samples

Web Image Processing Rust Samples

Ampère Force Calculator

Coulomb Force Calculator

Electric Potential Calculator

Faraday Induction EMF Calculator