What It Demonstrates
This is a plain 2D-canvas rendering of a real numerical solve of the
1-D time-dependent Schrödinger equation, using the
split-step Fourier method: at every step the wave
function is multiplied by the potential phase in position space,
Fourier-transformed, multiplied by the exact kinetic-energy phase in
momentum space, then transformed back.
- The blue curve is the probability density |ψ(x)|².
- The green curve is the real part of the wave function Re(ψ).
- The yellow bar is the potential barrier V(x).
How to Use
Pick a preset or drag the sliders. Higher/wider barriers and heavier
particles suppress transmission; the panel reports the transmission
T and reflection R measured by integrating |ψ|² on each side of the
barrier once the packet has cleared it.
Did You Know?
The same physics underlies the scanning tunnelling microscope, the
tunnel diode, and alpha radioactive decay, where a nucleus's alpha
particle leaks through the nuclear Coulomb barrier.
About the 2D Quantum Tunnelling Simulation
This is a plain 2D-canvas version of a real numerical solve of the one-dimensional time-dependent Schrödinger equation for a Gaussian wave packet approaching a rectangular potential barrier. It uses the split-step Fourier method: each time step applies the potential in position space, transforms to momentum space with a fast Fourier transform to apply the kinetic term exactly, then transforms back. The panel reports the measured transmission and reflection coefficients, integrated from the actual |ψ|² distribution once the packet has cleared the barrier.
Quantum tunnelling is the purely quantum phenomenon in which a particle penetrates a barrier higher than its energy, something forbidden in classical mechanics. Transmission probability falls roughly exponentially with barrier height and width and with particle mass — try the presets to see the effect of each.
Frequently Asked Questions
What is quantum tunnelling?
Quantum tunnelling is when a particle passes through a potential barrier that, according to classical physics, it has too little energy to surmount. Because the wave function extends into and beyond the barrier, there is a finite probability the particle appears on the far side.
What algorithm does this simulation use?
It uses the split-step Fourier method to integrate the time-dependent Schrödinger equation: the potential phase is applied in real space and the kinetic phase in momentum space via a fast Fourier transform.
How is this different from the 3D version of this simulation?
Both solve the same 1-D Schrödinger equation with the split-step Fourier method. This page renders the wave function on a plain 2D canvas; the companion page renders it with WebGL. The physics and the transmission/reflection numbers are the same numerical method.
What determines whether a particle tunnels through?
The transmission probability depends on barrier height relative to particle energy, barrier width, and particle mass. Lower, thinner barriers and lighter particles give a much higher chance of tunnelling, dropping roughly exponentially as height or width increases.
What is resonance tunnelling?
When the particle's energy is above the barrier, waves reflected from its two edges can interfere, and transmission can rise sharply near certain barrier widths — the resonance preset shows this enhancement.