Visualise quantum wavefunctions evolving in real time. A wave packet propagates, tunnels through barriers and reflects — the Schrödinger equation brought to life in interactive simulation.
The time-dependent Schrödinger equation iℏ∂ψ/∂t = Ĥψ is solved numerically. The probability density |ψ|² shows where the particle is likely to be found.
Set up potential barriers and wells. Launch a wave packet and watch it propagate, reflect and tunnel. The probability density evolves according to quantum mechanics.
Quantum tunnelling is exploited in tunnel diodes, scanning tunnelling microscopes and flash memory. The Sun shines because protons tunnel through the Coulomb barrier to fuse — without tunnelling, the Sun would be cold.
This simulation solves the time-dependent Schrödinger equation, iℏ∂ψ/∂t = -ℏ²/2m ∂²ψ/∂x² + V(x)ψ, on a 1200-point line using natural units (ℏ = m = 1). A Gaussian wave packet is launched into one of six selectable potentials and advanced in small explicit time steps, so you watch the complex wavefunction propagate, spread, reflect and tunnel. The shaded curve is the probability density |ψ|², the only thing experiments can actually measure.
A normalised Gaussian wave packet evolving under the 1D TDSE. The Laplacian is approximated by a three-point finite difference and integrated forward in time (an explicit FTCS scheme with small dt), letting you see interference, reflection and barrier tunnelling for wells, barriers, a harmonic trap, a double well and a potential step.
Pick a potential from the drop-down, then press "↺ Wave packet" to relaunch. The k₀ slider sets the packet's initial momentum (speed and direction), σ sets its spatial width, and Potential sets the barrier or well height. Toggle Re(ψ), Im(ψ) and |ψ|² to view the real part, imaginary part or probability density.
A particle can pass through a barrier taller than its own energy — quantum tunnelling. The probability falls off exponentially with barrier width and height, which is precisely why scanning tunnelling microscopes can resolve single atoms from tiny changes in tip-to-surface gap.
It is the fundamental equation of quantum mechanics, describing how a particle's wavefunction ψ changes over time. The time-dependent form shown here, iℏ∂ψ/∂t = -ℏ²/2m ∂²ψ/∂x² + V(x)ψ, plays the same role for quantum systems that Newton's laws play for classical ones. Its solutions are complex-valued, and the measurable quantity is the probability density |ψ|².
It works on a fixed grid of 1200 points in units where ℏ = m = 1. The spatial second derivative is replaced by a three-point finite-difference Laplacian, and the wavefunction is stepped forward in time with a small explicit (FTCS) update of the real and imaginary parts. Several substeps run between each frame to keep the animation smooth.
k₀ sets the central wavenumber of the initial Gaussian packet, which determines its momentum and therefore how fast and in which direction it travels. σ sets the packet's width: a narrow packet (small σ) has well-defined position but a broad spread of momenta, so it disperses quickly, illustrating the Heisenberg uncertainty principle directly.
Qualitatively, yes. When the packet meets a barrier taller than its energy, part of it reflects and a smaller part emerges on the far side, exactly as real tunnelling predicts. The model is a simplified 1D, fixed-grid solver, so absolute energies and exact transmission ratios are illustrative rather than laboratory-precise, but the behaviour and trends are correct.
The wavefunction itself is complex and cannot be observed directly. Born's rule states that |ψ|² gives the probability of finding the particle at each position, and the wavefunction is normalised so this integrates to one. The real and imaginary parts encode the phase, which drives interference, but every measurable outcome ultimately comes from |ψ|².