The same 1D time-dependent Schrödinger equation as the 3D version, integrated on this page with a plain explicit finite-difference (FTCS) scheme in natural units ℏ = m = 1:
iℏ ∂ψ/∂t = -ℏ²/2m ∂²ψ/∂x² + V(x)ψ
The top pane draws the potential (amber), the current wavefunction curve and phase-coloured probability bars — pick |ψ|² for probability density, or Re(ψ)/Im(ψ) to see the complex wave itself. Below it, a scrolling space-time strip stacks every frame's |ψ|² as a new column, so the packet's trajectory — reflection, spreading, or the faint transmitted tail that leaks through a barrier — becomes a visible streak instead of something you have to remember from a moving dot.
The readout panel tracks conserved and derived quantities live: the norm ∫|ψ|²dx (should stay near 1, drifting only from the explicit scheme's discretisation error), the mean position ⟨x⟩ and mean momentum ⟨p⟩ = ℏ·Im∫ψ*∂ψ/∂x dx, the fraction of probability that has crossed to the right half of the grid, and a numerical estimate of the total energy.