# Quantum Tunneling Probability Calculator

Rectangular-barrier WKB T ≈ e^(−2κa) vs exact transmission, reflection R = 1−T, and over-barrier partial reflection for electron, proton, or custom mass.

> Canonical page: https://elysiatools.com/en/tools/tunneling-probability

- **Category:** Science & Education

- **Keywords:** quantum tunneling, wkb approximation, transmission coefficient, rectangular barrier, alpha decay, quantum mechanics

## 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.

## Inputs

- **Particle** (select)
- **Custom mass (kg)** (number): e.g. 9.109e-31
- **Barrier height V₀ (eV)** (number): e.g. 1
- **Barrier width a (nm)** (number): e.g. 1
- **Particle energy E (eV)** (number): e.g. 0.5

## 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

- Select a particle preset such as electron or proton, or choose custom mass and enter a mass value in kilograms.
- Enter the barrier height V₀ in eV, barrier width a in nm, and incident particle energy E in eV.
- The 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.

## Frequently asked questions

### 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)).

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