A ball rolled at a hill it cannot climb always rolls back — but a quantum particle sometimes appears on the other side. Because the particle is a wave, it does not stop at a barrier taller than its energy; it decays exponentially inside and, if the wall is thin enough, a sliver emerges beyond it. Set the particle's energy, the barrier's height and its width, and read the exact transmission probability — watch it collapse as the barrier thickens and leap as it thins. Even a particle with energy above the barrier can bounce back, a purely quantum twist. It all runs on your device.
Thin the barrier or raise the energy and watch the transmission jump
Transmission T2.4%
Reflection R97.6%
RegimeTunneling (E < V₀)
Decay constant5.12 nm⁻¹
Width in de-Broglie waves0.41 λ
de-Broglie wavelength1.23 nm
Transmission T against barrier width at the current energy and height, on a log scale (each gridline is ten times smaller). The amber dot is your setting; below the barrier the near-straight fall shows T dropping almost exponentially as the wall thickens, so thinning it a little sends the probability leaping back up.
A wall too tall, and yet a way through. Classically a particle with energy below the barrier is trapped, full stop. But its wave function does not vanish at the wall — it falls off as e−κx inside the barrier, so a thin enough wall leaves a little amplitude on the far side and a real chance of getting through. That is why transmission drops almost exponentially as the barrier thickens, and why even a particle sailing over the barrier can be reflected: it is a wave meeting an edge, not a ball rolling over a bump.
Reading the simulation
What the wave, the barrier and the log-scale curve are telling you.
1
Three regions of one wave
Left of the barrier the violet curve is the incident wave, oscillating with the de-Broglie wavelength set by the energy E (the dashed blue line). Inside the barrier — the shaded box of height V₀ and width a — it stops oscillating and decays as e−κx. Whatever amplitude reaches the far wall carries on as the transmitted wave, now shrunk to √T of the original. The width marker under the box reads a in nanometres.
2
The curve: exponential suppression
The lower panel plots the transmission T against barrier width on a log scale, so each gridline is ten times smaller than the one above. For E below V₀ the plot is a near-straight downward line — that straightness is the exponential fall: widen the barrier a little and T drops by a factor, widen it more and it drops by that factor again. The amber dot marks your current width, and the decay constant κ readout sets how steep the line is — a taller barrier makes κ larger and the fall steeper.
3
Over the top, still not certain
Raise E above V₀ (try the Over the top preset) and the regime readout flips to over-barrier. Classically the particle would always pass, yet T stays below 100% and the curve ripples — the wave reflects off the sudden step in potential even when it has energy to spare. Reflection and transmission always add to the whole: the T and R readouts sum to 100%.
How It Works
A wave that leaks through a wall, a probability that hangs on the width, and reflection where classical physics allows none.
1
The wave meets the barrier
A particle of energy E is a wave. Where the potential rises to V₀ above the particle's energy there is no travelling wave — instead the wave function decays exponentially, e−κx, with the decay constant κ = √(2m(V₀−E)) ⁄ ℏ set by how far the energy falls short. Because the decay is smooth, not sudden, the wave still has a small value at the far side of a thin enough wall.
2
A fraction gets through
Whatever survives to the far wall continues as a travelling wave of amplitude √T. This tool computes the transmission probability T exactly from the rectangular-barrier result, T = 1 ⁄ (1 + V₀²·sinh²(κa) ⁄ (4E(V₀−E))). The sinh² of κa is why T falls almost exponentially with the width a — a slightly thicker or taller barrier can cut the chance of tunneling by orders of magnitude.
3
Even over the top, not guaranteed
Raise the energy above the barrier and the sinh becomes an ordinary sin: the same formula with sin²(k₂a) in place of sinh². Classically the particle always passes, but the wave still meets a sudden change in potential and part of it reflects, so T climbs toward — yet ripples below — 100 percent. Over-barrier reflection has no classical analogue; at this scale everything is waves.
What is quantum tunneling?
Quantum tunneling is the ability of a particle to pass through a barrier that, by classical physics, it does not have enough energy to cross. Because a particle is described by a wave function, that wave does not stop dead at the barrier — it decays exponentially inside it, and if the barrier is thin enough a small but non-zero piece survives to the other side. The particle then has a real probability of being found beyond a wall it could never climb. This simulator shows the wave decaying through the barrier and a fraction emerging.
How does barrier width affect tunneling?
Very strongly — the transmission probability falls off roughly exponentially with the width of the barrier. Inside the barrier the wave function decays like e−κx, so doubling the width squares the (small) surviving amplitude and shrinks the probability enormously. A slightly thinner barrier can raise the chance of tunneling from a fraction of a percent to tens of percent. Try dragging the width slider in this tool and watch the transmission plummet or surge.
Can a particle with more energy than the barrier still be reflected?
Yes — a particle with more energy than the barrier can still be reflected: classically, such a particle always passes, but quantum mechanically it can still be reflected. Because the wave meets a sudden change in potential, part of it bounces back even when the energy exceeds the barrier height, so the transmission is high but not a full 100 percent, and it oscillates with energy and width. Set the energy above the barrier here and you will see the transmission stay below one — over-barrier reflection has no classical counterpart.
What is quantum tunneling used for?
Tunneling is behind a surprising amount of technology and nature. The scanning tunneling microscope images individual atoms by measuring the tiny tunneling current across a gap; flash memory stores data by tunneling electrons onto a floating gate; and alpha decay and the fusion that powers the Sun both happen only because particles tunnel through barriers they could not otherwise cross. The same rectangular-barrier physics shown here underlies all of them.
A ready-to-assign lab using the simulation above. Free to use — nothing to install or sign up for, and no student data leaves the browser.
High school, College10–20 minInteractive simulation
Learning objectives
Describe quantum tunneling through a barrier.
Relate tunneling probability to barrier height and width.
Contrast quantum and classical behavior.
Variables to change
Barrier height
Barrier width
Particle energy
Procedure
Send a particle at a barrier taller than its energy and watch part of the wavefunction pass through.
Increase the barrier width and observe the transmitted probability.
Increase the barrier height and observe again.
Observations
Record how the transmission probability changes with barrier width and height.
Questions
What is quantum tunneling?
How does barrier width affect tunneling?
How does barrier height affect tunneling?
Why can this happen at all?
Give a real application of tunneling.
Explanation
Quantum particles are waves, so their wavefunction leaks into a barrier and can emerge on the far side even without enough energy to go over it. This tunneling probability drops exponentially with barrier width and height, and underlies alpha decay and the scanning tunneling microscope.
Answer key (for instructors)
1. A particle passing through a barrier it classically lacks the energy to cross.
2. Wider barriers sharply reduce the tunneling probability (it falls exponentially with width).
3. Higher barriers reduce the tunneling probability.
4. The particle is described by a wavefunction that extends into and beyond the barrier, giving a nonzero probability of being found on the far side.
5. Scanning tunneling microscopes, alpha decay, and tunnel diodes all rely on it.
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