About the wave and gates
Choose N = 2, 3, 4 or 5. The box and mass stay fixed: T = Trev/(2N), and the inverse is a positive-time wait of (2N−1)T. There is one scalar wave and no spin channel.
Each box labels an orthogonal logical mode built from N sine eigenmodes along each axis, with N selectable from 2 to 5. Packets overlap, so logical-mode probability differs from probability inside its geometrical box.
Labels use binary position registers for 2×2 and 4×4, and |x,y⟩ for 3×3 and 5×5. Columns increase left to right; rows increase bottom to top. Select the initial packet and goal directly on the wave using the guide beside the title. The default input is the lower-left packet |0,0⟩ (|0000⟩ at 4×4). The reference pulse acts on the selected input.
ℋ = −ℏ²∇²/(2m)
A = U(T), A† = U((2N−1)T), U(2NT) = I
Preparation and forward mixing wait for T = πℏ/(NE₁). The inverse waits forward for (2N−1)T under the same positive kinetic Hamiltonian. The fixed revival time is 19.2 gate-time seconds. At the default 4×4 size, T is 2.4 and 7T is 16.8. Both axes evolve simultaneously.
Preparation gives equal probabilities with definite relative phases. Oracle and reference pulses remain ideal mode-projector phase operations. The first optimal peak takes 1, 2, 3 or 3 iterations for 2×2, 3×3, 4×4 or 5×5 respectively.
Free mixing: j = (ℏ/m) Im(ψ*∇ψ)
Phase pulses: j = ∇χ, ∇²χ = −∂ρ/∂t
Tracer transport: v = j / ρ
During free evolution the arrows show the Schrödinger probability current. During the ideal nonlocal phase pulses they show the chosen conserved curl-free current. Yellow tracers move with v = j / ρ to visualize the evolving density. They are visualization samples, and adding more tracers changes only the sampling detail. Every stage has zero normal flux at the outer walls.
Compare tracer fractions with “Inside goal box.” Arrow lengths show current strength on a compressed scale. Pause and manual gate checkpoints hold the wave, current, tracers, and trails still for inspection.