This is the 2D companion to the 3D split-step Schrödinger simulator: instead of animating the full wavepacket, it computes the WKB tunneling transmission probability for the same double-well hydrogen-bond potential directly and numerically, using real physical constants (ħ, proton mass) rather than natural units.
V(x) = V₀·[(x/a)² − 1]² (double well, barrier peak V₀ at x=0)
E₀ = ½ħω, ω = √(8V₀ / m a²) (zero-point energy, harmonic well approx.)
x_t = a·√(1 − √(E₀/V₀)) (inner turning point, V(x_t) = E₀)
T ≈ exp[ −2 ∫_{−x_t}^{x_t} √(2m(V(x)−E₀)) / ħ dx ] (Simpson's rule, 400 steps)
The attempt frequency ν = ω/2π is the proton's real oscillation frequency in one well (~10¹³–10¹⁴ Hz for a light X–H stretch), and the tunneling rate is ν·T — the same golden-rule-style combination used throughout tunneling spectroscopy. Because ν is far too fast to watch directly, the animation compresses time: it makes one visible "attempt" roughly every half second, using the exact same computed T as that attempt's success probability, and separately reports how much real physical time (attempts ÷ ν) those attempts represent — so the displayed T, ν, rate and half-life are the true WKB numbers even though the on-screen pacing is artistic.
- Barrier height V₀ — how strongly the bond resists the proton crossing; T falls off roughly like exp(−const·√V₀) as V₀ grows.
- Barrier half-width a — distance from the barrier peak to a well minimum; T falls off exponentially as a grows, the classic WKB width sensitivity.
- Isotope — swapping ¹H for ²H (deuterium) doubles the tunneling mass, which appears under the square root in both the WKB exponent and ω — exponentially suppressing T. This is the real kinetic isotope effect used to test the tunneling-mutation hypothesis experimentally.
- Base pair — A–T (2 H-bonds) vs G–C (3 H-bonds) set representative barrier/width presets; G–C's extra bond makes for a shorter, stiffer, higher barrier.
The plot shows V(x) (line), the harmonic-oscillator zero-point level E₀ (dashed), and a qualitative |ψ(x)|² built from the same WKB decay factor exp[−∫κ(x')dx'] bridging the two wells — so the height of the connecting "tunneling hump" in the classically forbidden zone is not decorative, it is drawn directly from the computed κ(x) integral and shrinks or grows exactly as T does.