Quantum tunnelling
Position space · dimensionless units, ℏ = m = 1
Integrals of |ψ|² over each region. Interpret reflected and transmitted fractions after the packets have separated, with edge losses accounted for.
A barrier is not an absolute boundary.
The physics & the calculation Schrödinger equation · numerical limits ＋
A wavefunction, not a particle trajectory
We solve the nonrelativistic, time-dependent Schrödinger equation for a single particle in a fixed external potential. Units are scaled so that ℏ = m = 1. All lengths, times, momenta and energies shown are dimensionless.
The probability density is |ψ|². Real and imaginary parts can have either sign. Phase color encodes arg ψ; an overall phase has no observable effect. Bright areas indicate greater position probability, not a literal particle cloud.
In two dimensions the slice is instantaneous |ψ(xs, y)|², not accumulated detector counts. Click the plane to move it.
Initial states & exact checks
A Gaussian is prepared as ψ ∝ exp[−(x−x₀)²/(4σ²) + ip₀(x−x₀)], with Δx = σ and Δp = 1/(2σ). A localized packet has a spread of energies, so some transmission can come from above-barrier components.
For V = ½ω²x², a coherent state has σ = 1/√(2ω). A different Gaussian width gives a squeezed state. The optional exact-density guide uses the analytic free or harmonic Gaussian solution. In 2D a coherent packet follows the classical mean orbit while retaining its width.
Energy eigenstates have stationary density; only their complex phase evolves. The finite-well preset starts inside the well: bound components remain, while continuum components can escape.
Numerical method & boundaries
Second-order symmetric split-step Fourier evolution, with a fixed timestep and a uniform grid. Rectangular interfaces are cell averaged. The FFT grid is periodic. Optional smooth absorbing layers remove outgoing waves near the edges to approximate an open domain; they are numerical boundaries, not additional physical walls.
The solver never renormalizes away absorbed probability. Mean position, spread and energy are conditional on the part still in the grid. Density exposure changes only the display. Finite grids and timesteps introduce errors, particularly at sharp interfaces; compare the finer grid for sensitive results. Playback speed is a target and may be lower on slower devices. Repeat re-prepares the state once more than 99.5% has been absorbed.