Quantum Wave Packet (1D Schrödinger)

Follow a packet

Select a potential and Run. The example packet starts at x₀ = 48 with k₀ = 3 and σ = 5. The default barrier has height 5 and width 1.25, so reflection and a smaller transmitted packet are visible. Pause and Step inspect one Δt; Reset clears the result and keeps inputs. Changing a model parameter requires a new Run. Playback rate and phase colour may change during playback.

Model and initial state

The equation is iψₜ = −½ψₓₓ + Vψ with ħ = m = 1 on a box of length 160. Initially ψ is proportional to exp[−(x−x₀)²/(2σ²)]exp[ik₀(x−x₀)]. Density has standard deviation σ/√2. The free mean energy is k₀²/2 + 1/(4σ²), not just the central-momentum energy. The displayed initial energy also includes the mean potential. These are model units, not a specified material or particle in SI units.

Barriers, wells and steps begin at x = 100. V₀ is positive; a well uses −V₀. Width affects only a barrier or well. The actual width is rounded to complete grid cells and reported in the details. The harmonic potential is ½ω²(x−80)². Its Ground state example sets x₀ = 80, k₀ = 0 and σ = 1/√ω, with energy ω/2.

Probability and boundaries

All probability shares use the original unit norm. Remaining plus absorbed probability should total one, up to rounding. Absorbed left and right tails are accumulated separately. Regional left/inside/right probabilities are not automatically final reflection and transmission coefficients: initially the incoming packet is on the left. After outgoing packets separate, left/right totals including absorbed tails estimate scattering fractions. The step has no finite inside interval. Mean position and spread describe only the remaining density and are omitted below norm 10⁻¹⁰; the mean is not an unwrapped coordinate across a periodic boundary.

FFT propagation is periodic. Absorbing mode adds a smooth imaginary potential in the outer 16 units at both ends, reducing wrap-around but not guaranteeing reflection-free boundaries. A message marks edge contact. Absorption is applied symmetrically with the potential and scales with Δt; removed mass is never renormalized away. Periodic mode preserves norm and deliberately permits return across the boundary.

Accuracy and display

Strang splitting uses potential half steps around a full Fourier kinetic step. Automatic Δt is 0.04, 0.01 or 0.0025 for 512, 1024 or 2048 points. Manual steps may be one eighth of that value up to that value. This supported accuracy/work range is not a CFL stability theorem. Compare finer grids and half steps, especially at discontinuous potentials; a stable norm alone does not prove an accurate scattering fraction.

The density scale never shrinks; potential and energy use a separate labelled scale. Phase colour is cyclic and hidden where density is negligible. Playback targets the selected model time units per real second, may slow under load, and pauses when the tool is hidden. These are finite wave packets, not exact monochromatic scattering states.

References: Split-operator derivation; MIT: Gaussian wave packets.