Estimate barrier transmission using practical quantum models. Enter energy, height, width, and particle mass. Export a neat summary for assignments today.
For a one-dimensional rectangular barrier with height V0 and width a, the tunneling regime is E < V0. The decay constant is κ = sqrt(2m(V0−E))/ħ.
A practical estimate is the WKB form: T ≈ exp(−2κa). The calculator also shows a common finite-barrier transmission form using sinh(κa) for E < V0, and sin(k2 a) for E > V0.
These forms assume a constant barrier and nonrelativistic motion. Use them for learning, comparisons, and quick checks.
| Case | E (eV) | V0 (eV) | a (nm) | T (approx) |
|---|---|---|---|---|
| Moderate barrier | 0.5 | 1.0 | 0.5 | ~1e-3 |
| Thicker barrier | 0.5 | 1.0 | 1.0 | ~1e-6 |
| Near threshold | 0.95 | 1.0 | 0.5 | ~1e-1 |
Quantum tunneling is the passage of a particle through a classically forbidden barrier. It governs leakage in thin oxides, charge transport in scanning tunneling microscopy, and electron transfer in nanojunctions. Even when the particle energy is below the barrier height, a finite transmission probability can exist and can dominate performance.
The key inputs are particle energy E, barrier height V0, width a, and mass m. The calculator supports common energy units (eV family and joules), several length scales (pm to µm), and mass given in electron masses or kilograms.
When E < V0, the result reflects true tunneling and includes the decay constant κ. When E > V0, transmission oscillates with width due to wave interference, and the tool reports a finite-barrier transmission that can dip below one even above the barrier.
In the tunneling regime, the WKB estimate scales as T ≈ exp(−2κa). This exponential sensitivity means doubling the width can reduce transmission by orders of magnitude. For an electron with V0−E = 0.5 eV, increasing width from 0.5 nm to 1.0 nm can drop T from around 10⁻³ toward 10⁻⁶, depending on exact parameters.
The calculator shows a widely used finite-barrier closed form for E < V0 using sinh(κa), plus the WKB estimate for quick intuition. WKB is excellent for thick barriers or large κa, while the finite-barrier form is often closer for moderate widths.
If you enter a target transmission 0<T<1, the calculator solves for width using the WKB relation, producing a = −ln(T)/(2κ) when E < V0. This is useful for barrier design tasks where you know a desired leakage level.
Keep E and V0 in comparable units, and use realistic widths. Atomic-scale tunneling typically occurs around 0.1–2 nm. For heavier particles, κ increases, so transmission becomes much smaller at the same width and energy gap.
Use the CSV export for spreadsheets and batch comparisons, and the PDF export for lab notes. Each export includes inputs, regime, transmission, and key derived constants. This supports transparent reporting and quick parameter sweeps when exploring sensitivity.
It is the fraction of incident particles expected to pass through the barrier in a one-dimensional model. A value of 1 means full transmission, while very small values indicate strong suppression.
Finite barriers cause wave reflections at both interfaces. Interference between reflected waves can reduce transmission for certain widths, producing oscillations and dips even when the particle energy exceeds the barrier height.
Use an effective mass when modeling electrons in a solid, because band structure changes the inertial response. If unknown, start with 1 electron mass and then test sensitivity by adjusting the value.
WKB is strongest when the barrier is thick or high so that κa is large. In that case, the exponential dominates and the estimate closely matches more exact barrier models.
In the tunneling regime the dependence is exponential, T ≈ exp(−2κa). Small increases in width add directly to the exponent, so transmission can drop by orders of magnitude.
Yes. Enter a target transmission between 0 and 1 and leave the width empty. The calculator uses the WKB relation to estimate the required width when E is below the barrier height.
No. The formulas assume a rectangular barrier and stationary, nonrelativistic motion. For varying potentials, you would use a numerical solution or a generalized WKB integral across the barrier.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.