Quantum Lab 02 · forbidden region and transmission

Quantum Tunneling Chamber

Why can a particle sometimes be detected beyond a barrier it classically cannot cross? Inside the barrier the wave's amplitude decays but never quite reaches zero. A barrier of finite width leaves a little amplitude on the far side. Over many repeated trials, the detector beyond it clicks with a small, steady probability.

Send a wave packet at a rectangular barrier that is higher than the particle's energy. Change the barrier height, the width and the particle energy to see the amplitude enter the classically forbidden region and decay inside it. Repeat the measurement many times and a stable transmission probability appears.

Transmission

T

$14.75\%$

Detected beyond wall0

Measurement trials0

Penetration depth

\ell_\kappa

$0.41\,L_0$

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: The amplitude inside the barrier is not zero. If the barrier has finite width, some amplitude is left on the far side, so a few of many repeated trials are detected beyond it.

Core mathematical model

Time-dependent Schrödinger equation

i\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

\psi(x)\propto e^{-\kappa x},\qquad \kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar}

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

Transmission through a rectangular barrier

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

At fixed energy and barrier height, increasing the width $a$ makes the transmission drop quickly.

Common difficulties

The particle does not borrow energy

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

Better mental modelIn steady-state scattering from a fixed barrier, the incoming and transmitted parts of the wave have the same energy. Tunneling happens because the quantum state is spread out in space, and energy is conserved throughout.

The glowing wave is not the particle’s path

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

Better mental modelThe waveform shows the probability amplitude. Each detector flash is one measurement at one place. Nothing in the experiment measures a continuous path inside the barrier.

Run the experiment

01
Build the statistics first
Choose “Thin barrier,” launch a stream, and watch both detectors.
What to observe: You cannot predict any single result, but the fraction detected beyond the wall gets closer and closer to the predicted transmission.
02
Make the wall thicker
Hold energy and barrier height fixed while increasing $a$ .
What to observe: The amplitude decays more inside the thicker wall, and transmission events become exponentially rarer.
03
Approach the barrier top
Raise $E$ while keeping $E<V_0$ .
What to observe: The decay constant shrinks, the wave reaches deeper into the barrier, and detections beyond the wall become much more likely.

Quantum Tunneling Chamber

Quantum tunneling: how impossible becomes improbable

Core mathematical model

Time-dependent Schrödinger equation

Decay in the forbidden region

Transmission through a rectangular barrier

Common difficulties

The particle does not borrow energy

The glowing wave is not the particle’s path

Run the experiment

Build the statistics first

Make the wall thicker

Approach the barrier top

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