Adjust the barrier height, width, and particle energy to compute the quantum tunneling transmission probability T in real time. Includes wavefunction visualization plus barrier-width and energy sweeps.
Particle Type
Barrier height V₀
eV
Particle energy E
eV
Barrier width d
nm
Particle mass m1 mₑ
T = 1.2×10⁻⁵
Results
Transmission T
1.2e-5
—
Reflection R
~1
—
κ (decay constant)
7.24
nm⁻¹
ΔE = V₀ − E
2.0
eV
Wavefunction
Transmission vs Barrier Width
Transmission vs Particle Energy
Theory & Key Formulas
$\kappa = \frac{\sqrt{2m(V_0-E)}}{\hbar}$ — decay constant inside the barrier (units of m⁻¹).
$T \approx e^{-2\kappa d}$ — WKB transmission probability through a rectangular barrier of width $d$.
If a particle's energy is below the barrier height, classically it can't cross. So how does quantum mechanics let it through?
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In quantum mechanics the particle is described by a wavefunction. Inside the barrier the wave doesn't vanish — it decays exponentially as $e^{-\kappa x}$. If the barrier is thin enough, a non-zero amplitude leaks out the other side, so there's a finite probability of transmission. The thinner and lower the barrier, the more leaks through.
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How does scanning tunneling microscopy (STM) actually image individual atoms with this?
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A sharp metallic tip is held about 1 nm above the sample so that electrons tunnel across the vacuum gap. Because $T$ depends exponentially on the gap, a 0.1 nm change in tip height changes the current by roughly an order of magnitude. The "T vs barrier width" tab here shows that steepness directly.
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Does my phone's flash memory really rely on tunneling too?
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Yes — NAND flash uses Fowler–Nordheim tunneling to push electrons through an oxide barrier into a floating gate during write/erase. Without quantum tunneling there would be no reliable way to inject and remove charge through that insulator.
FAQ
Q. How is alpha decay related to tunneling?
A. Alpha particles escape heavy nuclei by tunneling through the Coulomb barrier. Gamow's 1928 theory predicted the exponential dependence of half-life on alpha energy (Geiger–Nuttall law) using exactly this WKB formula.
Q. Is tunneling required for fusion in the Sun?
A. Yes. At the Sun's core temperature (~15 million K, ~1.3 keV) classical kinetic energy is far below the proton–proton Coulomb barrier. Quantum tunneling enables fusion at lower energies; the "Gamow peak" combines the Maxwell–Boltzmann tail with the tunneling factor.
Q. How large is a typical tunneling current?
A. STM tunneling currents are usually 0.1–10 nA. NAND flash program currents reach tens of µA when bias is high, but in retention (no bias) the leakage is essentially zero. The current is roughly proportional to T × (electron density × velocity × charge).
Q. What does WKB mean?
A. The Wentzel–Kramers–Brillouin approximation evaluates the wavefunction for slowly varying potentials. The expression $T \approx e^{-2\kappa d}$ used here is the WKB result for a rectangular barrier; for smooth barriers the exponent becomes the integral $\int \sqrt{2m(V-E)}/\hbar\,dx$.
What is the Quantum Tunneling Simulator?
This tool computes the quantum-mechanical probability that a particle penetrates a one-dimensional rectangular potential barrier. By adjusting the height $V_0$, width $d$, energy $E$, and mass $m$, you can see how the transmission $T \approx e^{-2\kappa d}$ depends exponentially on every parameter. The wavefunction view shows the incident wave on the left, the evanescent decay inside the barrier, and the transmitted wave on the right.
For barriers thinner than a few nanometers and light particles such as electrons, the transmission probability is large enough to dominate device-level physics — from STM imaging to NAND flash retention.
Physical Model & Key Equations
For a rectangular barrier with $E < V_0$, the exact transmission is $T = \left[1 + \frac{V_0^2}{4E(V_0-E)}\sinh^2(\kappa a)\right]^{-1}$ with $\kappa = \sqrt{2m(V_0-E)}/\hbar$. In the thick-barrier limit ($\kappa a \gg 1$), this simplifies to the WKB form $T \approx e^{-2\kappa a}$ used here.
Each slider corresponds to a physical parameter. Mass enters through $\kappa$, so heavier particles (proton: 1836 mₑ, α: 7294 mₑ) tunnel exponentially less than electrons for the same barrier.
Real-World Applications
Semiconductor industry: NAND flash memory, tunnel diodes, and resonant-tunneling structures rely on engineered barrier widths between 2 and 10 nm. Designers use simple WKB estimates to set program/erase voltages and project endurance.
Surface science: STM uses the exponential gap dependence to map surface topography with sub-Ångström vertical resolution. Scanning tunneling spectroscopy adds bias sweeps to probe local density of states.
Nuclear & astrophysics: α decay half-lives across many orders of magnitude are reproduced by combining the Coulomb barrier with this WKB formula. The same machinery underlies stellar fusion rates.
Common Misconceptions and Cautions
"Tunneling is set only by barrier height and width": in fact $T$ also depends exponentially on $\sqrt{m(V_0-E)}$, so small changes in mass or energy dramatically change the result.
"STM works at any tip distance": only at gaps below ~1 nm is the tunneling current measurable. The exponential decay is what enables atomic-scale imaging — but it also makes mechanical stability critical.
"Flash memory traps charge perfectly": real oxides leak via tunneling. The non-zero retention probability sets the upper limit on data-retention time, especially at scaled nodes and elevated temperature.