3D Quantum Tunneling

A wave packet hits a barrier and splits: part reflects, part gets through. The panel scores the measured transmission against the analytic answer.

Loading Schrodinger solver (WebGL 2.0)

Browser demo: educational Schrodinger solver, not a research result

A wave packet whose mean energy of motion along x is 3.4 meets a barrier of height 5 (the barrier only acts along x). The mean is below the barrier, but the packet spreads over momenta: about 17% of it has energy above the barrier and would pass even classically. The transmitted part therefore mixes sub-barrier tunnelling with above-barrier passage; it is not pure tunnelling. The solver is a unitary split-step Fourier method on a 64^3 grid (32^3 on phones), and the panel compares the measured transmitted probability with the analytic rectangular-barrier result averaged over the packet's momenta.

Schrodinger Equation

i hbar dpsi/dt = H psi, Strang split-step Fourier: every factor has modulus 1, so the norm is conserved for any dt

64^3 Grid

262,144 cells; periodic in y and z, absorbing layers at the x faces. Absorbed probability is counted as transmitted or reflected, so nothing is lost from the tally.

Phase Visualization

Colour = complex phase, brightness = |psi|^2 relative to its current peak

Quantum Tunneling Explained

In quantum mechanics, particles can "tunnel" through barriers that would be insurmountable classically. The wave function extends into the barrier, decaying exponentially, but can emerge on the other side.

What the numbers mean

For a plane wave of energy E below a barrier of height V0 and width a, the transmission is T = 1 / (1 + V0^2 sinh^2(kappa a) / (4 E (V0 - E))) with kappa = sqrt(2 m (V0 - E)) / hbar. A Gaussian packet carries a spread of momenta, so the theory line averages T over that spread. In a regression test the same solver on a fine 1D grid reproduces the averaged value to within 0.01; the 3D demo grid is coarser and typically lands within a few percent of it.

This page uses ordinary quantum mechanics only. There is no meta-calculus mode here.

Presets

  • Square Barrier: tunnelling with E below V0, measured against theory
  • Double Slit: interference behind two slits (the y direction is periodic)
  • Hydrogen-like: a 2p_z-shaped state in a softened Coulomb well (softening 0.5); it is not an eigenstate of that well, so it breathes
  • Harmonic: a displaced ground state oscillating in a harmonic well