You are in the Physics lab.
Thin the barrier or raise the energy and watch the transmission jump
Transmission T2.4%
Reflection R97.6%
RegimeTunneling (E < V₀)
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.
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 that arrives at a region where the potential rises to V₀. If E is below V₀ there is no travelling wave inside the barrier — instead the wave function decays exponentially, e−κx, with κ set by how far the energy falls short of the barrier. It does not reach zero instantly, so it still has some value at the far wall.
2
A fraction gets through
Whatever amplitude survives to the far side continues as a travelling wave again, smaller than the one that came in. The transmission probability T is that survival, and it depends steeply on the width: because the decay is exponential, a slightly thicker barrier can cut the chance of tunneling by orders of magnitude. This tool computes T exactly from the rectangular-barrier formula.
3
Even over the top, not guaranteed
Raise the energy above the barrier and classically the particle always passes — but the wave still meets a sudden change in potential and part of it reflects. So the transmission climbs toward, but does not reach, 100 percent, and it ripples with energy and width. Over-barrier reflection has no classical analogue; it is a reminder that 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. Classically, a particle with energy above the barrier always passes; 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.
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