About the Quantum Tunneling Simulation
This simulation solves 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 an FFT to apply the kinetic term exactly, then transforms back. You watch the probability density |ψ|² (blue), the real part of the wave function (green), and the barrier V(x) (yellow), while the panel reports the measured transmission and reflection coefficients.
Quantum tunneling is the purely quantum phenomenon in which a particle penetrates a barrier higher than its energy — something forbidden in classical mechanics. The transmission probability falls roughly exponentially with barrier height and width and with the particle's mass. This effect powers scanning tunnelling microscopes, tunnel and Esaki diodes, flash memory, and stellar nuclear fusion, and it explains alpha radioactive decay as an alpha particle leaking through the nuclear Coulomb barrier.
Frequently Asked Questions
What is quantum tunneling?
Quantum tunneling 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, which is both accurate and efficient for wave-packet propagation.
What do the colored curves mean?
The blue curve is the probability density |ψ|², showing where the particle is likely to be found. The green curve is the real part of the wave function, and the yellow bar is the potential energy barrier V(x).
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 tunneling, dropping roughly exponentially as height or width increases.
What do transmission T and reflection R mean?
T is the fraction of the wave packet's probability that ends up beyond the barrier, and R is the fraction that bounces back. They must sum to one, since the particle is certain to be either transmitted or reflected.
Why is the classical transmission shown separately?
Classically a particle is transmitted only if its energy exceeds the barrier height, giving 100% or 0%. Comparing this to the quantum T highlights the uniquely quantum behaviour: nonzero transmission even when the particle's energy is below the barrier.
What is resonance tunneling?
When the particle's energy is above the barrier, the waves reflected from the two edges can cancel each other, and for a plane wave whose wavelength inside the barrier fits a whole number of half-wavelengths across it, transmission spikes to 100%. A real wave packet carries a spread of energies rather than a single one, so the resonance preset shows a strong but partial enhancement — around 70% transmission, against roughly 25% at a nearby energy — instead of a perfect 100%.
How does an electron differ from a proton or alpha particle here?
Heavier particles tunnel far less readily, because the decay constant κ = √(2m(V₀−E))/ℏ grows with the square root of the mass. The heavy-particle preset sends a 5× heavier particle through the same barrier as the electron, and transmission collapses from roughly 25% to a couple of percent. A real proton (1836×) or alpha particle (7294×) facing an electron-volt barrier a fraction of a nanometre thick would transmit with a probability of order e⁻⁸⁰⁰ — effectively zero — which is exactly why alpha decay happens at mega-electron-volt energies over femtometre distances rather than on the scale shown here.
What real devices rely on quantum tunneling?
Scanning tunnelling microscopes, tunnel and Esaki diodes, and flash memory cells all exploit tunneling. It also enables nuclear fusion in stars and underlies alpha radioactive decay.
Why does the wave packet spread out over time?
A Gaussian wave packet contains a range of momenta that travel at slightly different speeds, so it disperses as it propagates — a normal feature of free quantum evolution that you can see even before it reaches the barrier.