Every other bench in this drift froze a state in place. Here we let one move. Launch a Gaussian wave packet and integrate the time-dependent Schrödinger equation — a unitary split-step that does the kinetic kick in momentum space (the butterfly's FFT) and the potential kick in position space. The bump crawls at the group velocity v_g=k₀ while the colour stripes — the phase — race ahead at v_φ=k₀/2. Spread it free, split it on a wall, slosh it in a bowl.
Hit Launch. In Free the bump crawls right and slowly broadens — watch the hollow ⟨x⟩±Δx ring widen, exactly σ(t)=σ₀√(1+(t/2σ₀²)²). The faint dotted track is the Newtonian particle picture x₀+k₀t; the gold dot sits on it (Ehrenfest). Switch to Barrier and fire at a wall taller than ⟨E⟩: classically nothing gets through, yet a transmitted lobe peels off and the readout shows ∫|ψ|² past the wall — the same wall the Tunnelling bench draws as a static T(E) curve, here finally seen happening. Bowl sloshes the packet side to side; launch it at the coherent width and it swings without breathing.
Scope & honesty. This is a numerical solve of the TDSE on a grid by a unitary split-step integrator (ħ=m=1); conservation laws hold to a stated tolerance (norm < 1e-9, energy/Ehrenfest ~1e-3 over the run), not the last bit — grid spacing & time step set the floor, and the self-test shows the drift. The free packet is checked against its exact closed form; the others are trusted only through the conservation laws. The kinetic step's FFT is imported char-for-char from the estate's own butterfly bench.
↘ next drift over · The same wall, finally seen happening Quantum Tunnelling → draws transmission as a static T(E) curve — a number with no picture. Here that number is the lobe peeling off the wall. The self-test ties them: the dynamical transmitted ∫|ψ|² lands inside the energy-spread band of that bench's exact T(⟨E⟩). ↗ one bench over · the coherent-state rhyme The Harmonic Oscillator → hangs a standing ladder in the bowl. Launch a packet at the coherent width σ=1/√(2ω) and it swings like a classical ball without spreading — the breathing-free state that bridges the standing ladder and this moving picture.