--- title: "Through a Conical Intersection Interactive Lab | Sandbox Physics" description: "Two energy landscapes touch. The wavepacket takes more than one way. Send a nuclear wavepacket toward a two-state conical intersection. Compare a central arrival, an off-centre passage and a zero-mixing control using surface populations and independently sampled configurations." canonical_url: "https://sandboxphysics.com/labs/molecular-conical-intersection/" language: "en" page_type: "lab" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/labs/molecular-conical-intersection/" last_modified: "2026-10-21" --- TWO SURFACES / ONE WAVEPACKET # Through a Conical Intersection Send a nuclear wavepacket toward a two-state conical intersection. Compare a central arrival, an off-centre passage and a zero-mixing control using surface populations and independently sampled configurations. > Interactive simulation: open this page in a web browser to change the parameters and run the model. Dimensionless two-coordinate linear vibronic-coupling model, evolved as a two-component wavefunction. The axes are nuclear configuration coordinates; height is electronic potential energy. This is not an atom trajectory, a particular molecular reaction or a direct measurement of geometric phase. NUCLEAR CONFIGURATION / ELECTRONIC ENERGY LOWER ENERGY SURFACE POPULATION $0\%$ EVOLUTION TIME $\tau=0$ READ THE PROBABILITY ## The surface is a model. The dots are samples. ### The nuclear configuration plane Cyan indicates the lower energy surface, coral the upper. The total brightness is square-root density with a fixed scale. These are nuclear coordinates, not electron tracks. Lower 0 0.7 Upper 0 0.7 ### Two bases, two different readings Solid cyan: local lower-surface population. Dashed violet: fixed channel A population. These are different projections. Gold is the pinned lower-surface population. ENERGY DRIFT $0$ NORM DRIFT $0$ EDGE WEIGHT $0$ HIGH-FREQUENCY WEIGHT $0$ INDEPENDENT SIMULATED MEASUREMENTS ### Stop time. Prepare again. Measure. Sample nuclear configuration and fixed electronic channel from each of 768 independent preparations. Violet dots are channel A; gold dots are B. These are fixed electronic channels, not energy surfaces. 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. $$ \begin{gathered}H=-\tfrac12\nabla^2 I+V_0 I+x\sigma_z+\kappa y\sigma_x\\V_0=\tfrac12\omega^2(x^2+y^2),\quad\omega=0.65\\V_\pm=V_0\pm\sqrt{x^2+\kappa^2y^2}\end{gathered} $$ The full two-component state is propagated in a fixed diabatic basis. Lower-surface density uses the local spectral projector; at an exactly degenerate grid point its weight is assigned equally. Setting mixing to zero removes electronic-channel transfer and changes the point intersection to a line crossing. No claim of measuring a Berry phase is made. $$ \begin{gathered}\Psi(x,y,0)=\begin{pmatrix}0\\g(x,y)\end{pmatrix}\\g\propto e^{-[(x+4)^2+(y-y_0)^2]/(2s^2)}e^{ip_0x},\quad s=0.7\end{gathered} $$ 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. - [Li, Joubert-Doriol & Izmaylov — Linear vibronic coupling](https://arxiv.org/abs/1707.08266) - [Valahu et al. — Trapped-ion conical-intersection experiment](https://arxiv.org/html/2211.07320) Sources establish the model context; this illustrative parameter set is not a fit or a reproduction of those experimental data. Physics tutorial ## Through a Conical Intersection **Background** Send a nuclear wavepacket toward a two-state conical intersection. Compare a central arrival, an off-centre passage and a zero-mixing control using surface populations and independently sampled configurations. **Why it matters** Connect the energy landscape to the changing probability, then ask what a measurement actually records. Start with the essentials - **Focus question**: Two energy landscapes touch. The wavepacket takes more than one way. - **One-sentence intuition**: Electronic mixing and nuclear motion evolve together. A fixed electronic channel is different from the locally lower or upper energy state. ### Core mathematical model #### Hamiltonian $$ H=-\frac12\nabla^2 I+\frac{\omega^2(x^2+y^2)}2I+x\sigma_z+\kappa y\sigma_x $$ Dimensionless units set the effective mass and reduced Planck constant to one. All propagation is unitary; no time-step renormalization is used. #### Energy surfaces $$ V_\pm=\frac{\omega^2(x^2+y^2)}2\pm\sqrt{x^2+\kappa^2y^2} $$ With nonzero mixing the surfaces meet at one point. Setting mixing to zero produces a crossing line, so it changes the model geometry as well as the dynamics. ### Common difficulties #### A landscape is not a trajectory **Typical misconception** The bright spot is one atom moving on a literal surface. **Better mental model** It is a nuclear configuration probability. Energy is plotted vertically only as a visualization. #### Scope **Typical misconception** The simulation predicts a particular molecule. **Better mental model** Dimensionless two-coordinate linear vibronic-coupling model, evolved as a two-component wavefunction. The axes are nuclear configuration coordinates; height is electronic potential energy. This is not an atom trajectory, a particular molecular reaction or a direct measurement of geometric phase. ### Run the experiment 1. 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:** An off-centre preparation changes both the spatial pattern and the populations. 2. 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. 3. 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. Physical conclusion Dimensionless two-coordinate linear vibronic-coupling model, evolved as a two-component wavefunction. The axes are nuclear configuration coordinates; height is electronic potential energy. This is not an atom trajectory, a particular molecular reaction or a direct measurement of geometric phase. Quantum & fields · Starter path ## Start from a single measurement Predict outcomes and collect counts, building from one qubit to wave packets, bound states and two-photon interference. [Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare it with the frequency of acquired shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/) 1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/) 2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/) 3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/) 4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/) 5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/) 6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/) [View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)