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Sandbox Physics

Quantum Lab 02 · forbidden region and transmission

Quantum Tunneling Chamber

Launch a wave packet at a rectangular barrier higher than the particle energy. Change the barrier height, width, and particle energy to see amplitude enter and decay through the classically forbidden region, then let repeated measurements reveal a stable transmission probability.

Interactive modelQuantum Tunneling Chamber
Transmission TT14.75%14.75\%
Detected beyond wall0
Measurement trials0
Penetration depth ℓκ\ell_\kappa0.41 L00.41\,L_0

Physics tutorial

Quantum tunneling: how impossible becomes improbable

BackgroundA classical particle cannot cross a barrier higher than its energy. A quantum state must instead satisfy a continuous wave equation, so amplitude enters the barrier and decays exponentially.

Why it mattersThis measurable quantum effect helps enable stellar fusion, scanning tunneling microscopy, and many electronic devices.

Start with the essentials

Focus question
If the particle lacks enough energy to climb the barrier, why can a detector beyond it still click?
One-sentence intuition
Amplitude inside the barrier is not zero. A barrier of finite width leaves nonzero amplitude beyond it, so repeated measurements include a small number of transmission events.

Core mathematical model

Time-dependent Schrödinger equation

iℏ∂ψ∂t=(−ℏ22m∂2∂x2+V(x))ψi\hbar\frac{\partial\psi}{\partial t}=\left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V(x)\right)\psi

The potential changes the spatial evolution of the wavefunction, but a finite barrier does not abruptly cut it to zero.

Decay in the forbidden region

ψ(x)∝e−κx,κ=2m(V0−E)ℏ\psi(x)\propto e^{-\kappa x},\qquad \kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar}

When E<V0E<V_0, amplitude decays exponentially inside the barrier instead of propagating as an oscillation.

Transmission through a rectangular barrier

T=[1+V02sinh⁡2(κa)4E(V0−E)]−1T=\left[1+\frac{V_0^2\sinh^2(\kappa a)}{4E(V_0-E)}\right]^{-1}

At fixed energy and barrier height, increasing width aa rapidly suppresses transmission.

Common difficulties

The particle does not borrow energy

Typical misconceptionEnergy–time uncertainty lets the particle briefly borrow enough energy to cross.

Better mental modelIn stationary barrier scattering, incident and transmitted components have the same energy. Tunneling comes from the spatial extent of the quantum state without violating energy conservation.

The animation is not a hidden trajectory

Typical misconceptionThe luminous waveform traces the particle’s actual route through the wall.

Better mental modelThe waveform represents probability amplitude; detector flashes are localized measurement outcomes. No continuous path inside the barrier is being measured.

Run the experiment

  1. 01

    Build the statistics first

    Choose “Thin barrier,” launch a stream, and watch both detectors.

    What to observe: Individual results remain unpredictable while the fraction detected beyond the wall approaches the predicted transmission.
  2. 02

    Make the wall thicker

    Hold energy and barrier height fixed while increasing aa.

    What to observe: Amplitude decays farther inside the wall, and transmission events become exponentially rarer.
  3. 03

    Approach the barrier top

    Raise EE while keeping E<V0E<V_0.

    What to observe: The decay constant shrinks, penetration depth grows, and beyond-wall detections become much more likely.