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