VIBRATE / SPLIT / ESCAPE

When a Molecular Bond Breaks

Kick a bound vibrational wavepacket into motion. Watch part of it return while another part escapes along a Morse potential, then compare the same preparation with an ideal spring.

One dimensionless nuclear coordinate on a fixed electronic potential. The initial state is a momentum-kicked Morse ground state. This is not a laser-pulse calculation or a calibrated molecular species. Probability beyond a finite stretch is not an asymptotic dissociation yield.

THE BOND / THE WAVE / THE ESCAPE

PROBABILITY BEYOND STRETCH SIX

0\%

EVOLUTION TIME

\tau=0

READ THE PROBABILITY

What stays. What leaves.

Every instant on one map

Time runs downward. Drag the white cursor. Gold marks stretch six; it is an observation boundary, not a detector or a wall.

01.5

A probability profile

Cyan: now. Gray: initial preparation. Gold: pinned prediction. The plot has an adaptive vertical scale.

ENERGY DRIFT

0

NORM DRIFT

0

EDGE WEIGHT

0

HIGH-FREQUENCY WEIGHT

0

INDEPENDENT SIMULATED MEASUREMENTS

Stop time. Prepare again. Measure.

Sample one stretch coordinate from each of 768 independently prepared molecules. The histogram comes only from those records; its axis includes every sample.

Ideal joint position readout, without blur or loss. Moving the time cursor preserves old records; changing preparation clears them. Samples are not a wavefunction reconstruction or a continuous movie of one molecule.

Equations, coordinates and boundaries

x=q/\ell,\quad\tau=\frac{\hbar t}{m\ell^2},\quad E_* =\frac{\hbar^2}{m\ell^2}

Length, time and energy are dimensionless. No specific molecular mass or experimental time scale is assigned. Positive stretch points outward; in the two-coordinate model the second coordinate is positive upward in the flat map.

\begin{gathered}i\partial_\tau\psi=H\psi,\quad H=-\tfrac12\partial_x^2+V(x)\\V(x)=D(1-e^{-ax})^2\\V_{\mathrm{spring}}(x)=Da^2x^2\end{gathered}

The exact Morse ground state is prepared and multiplied by a momentum-kick phase. The stretch coordinate extends over the real line as an effective vibration model. Rotation, electronic transitions, pulse preparation and environmental loss are omitted. The far-region fraction is a finite-time position probability, not a continuum spectral projection.

\begin{gathered}\psi(x,0)\propto z^{\lambda-1/2}e^{-z/2}e^{ip_0x}\\z=2\lambda e^{-ax},\quad\lambda=\sqrt{2D}/a\end{gathered}

A periodic Fourier grid and symmetric split-step propagation solve the declared Hamiltonian. No absorption or propagation renormalization is used. Edge occupation, high-wave-number weight and relative energy drift are checked; the view warns if the chosen run loses numerical resolution.

Sources establish the model context; this illustrative parameter set is not a fit or a reproduction of those experimental data.

Start with the essentials

Focus question: An ideal spring always returns. A molecular bond can let go.
One-sentence intuition: Anharmonicity makes the outer wall flatten. A kicked bound state contains many energies, so part of the wavepacket can escape while another part returns.

Core mathematical model

Hamiltonian

H=-\frac12\partial_x^2+D(1-e^{-ax})^2

Dimensionless units set the effective mass and reduced Planck constant to one. All propagation is unitary; no time-step renormalization is used.

Bound energy levels

E_n=a\sqrt{2D}(n+\tfrac12)-\frac{a^2}{2}(n+\tfrac12)^2

The number of bound levels is finite. These are analytic reference energies, not peaks inferred from the position samples.

Common difficulties

A landscape is not a trajectory

Typical misconceptionThe bright spot is one atom moving on a literal surface.

Better mental modelIt is a nuclear configuration probability. Energy is plotted vertically only as a visualization.

Scope

Typical misconceptionThe simulation predicts a particular molecule.

Better mental modelOne dimensionless nuclear coordinate on a fixed electronic potential. The initial state is a momentum-kicked Morse ground state. This is not a laser-pulse calculation or a calibrated molecular species. Probability beyond a finite stretch is not an asymptotic dissociation yield.

Run the experiment

01
Compare
Start with the first two presets and select the same time in both. Pin a frame before changing the preparation.
What to observe: The outgoing probability grows as the kick increases; an ideal spring returns it.
02
Read the axes
Switch between landscape and density views, then inspect the data row.
What to observe: Potential height, density brightness and probability relief are distinct quantities with explicit display scales.
03
Acquire
Sample independent preparations at one chosen instant. Move time and confirm the old records stay fixed. Export the raw CSV.
What to observe: A finite sample fluctuates. It does not reconstruct the complex wavefunction or reveal a continuous path.