--- title: "Three Ways Through an Avoided Crossing Interactive Lab | Sandbox Physics" description: "Do quantum waves, a mean force and hopping trajectories predict the same reaction branches?" canonical_url: "https://sandboxphysics.com/labs/nonadiabatic-avoided-crossing/" language: "en" page_type: "lab" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/labs/nonadiabatic-avoided-crossing/" --- A molecular wavepacket chooses its exits # Three Ways Through an Avoided Crossing **Do quantum waves, a mean force and hopping trajectories predict the same reaction branches?** A nuclear wavepacket can split between electronic states near an avoided crossing. Quantum evolution keeps both amplitudes; Ehrenfest follows their average force, and surface hopping samples separate paths. Their outgoing probabilities can differ. Send the same nuclear preparation through two coupled electronic states. Watch the quantum packet split, compare the mean-force swarm and count trajectories that switch surfaces. > Interactive simulation: open this page in a web browser to change the parameters and run the model. Model assumptions Tully’s one-coordinate, two-state model uses atomic units and an effective nuclear mass of 2000 electron masses. It is not calibrated to a molecule. The quantum packet and classical Wigner ensemble share the same initial Gaussian; their finite-time exit probabilities are compared at the same windows. Surface hopping retains electronic coherence and adds no decoherence correction. $t=0$ One nuclear coordinate. Height shows electronic energy; ribbon thickness shows density on a separate scale. Cyan is lower, coral is upper. At the common final time ## Follow the probability into the exits The exit windows begin four bohr from the crossing. Channel estimates include probability absorbed far beyond its exit; surviving and absorbed parts are saved separately. Probability between the windows is unresolved. Quantum upper exit $0$ Hopping minus quantum $0$ Hopping standard error $0$ Quantum unresolved $0$ ### Four exit probabilities Bars share the same probability scale. Thin whiskers show one sampling standard error for the trajectory estimates. ### Population reaching the right exit Quantum, mean-force and hopping predictions use the same two exit windows. An incoming packet on the left is not yet a reflected reaction product. Norm, energy, flux and measured cost Probability still on the grid $0$ Probability absorbed outside the exits $0$ Norm plus absorbed $0$ Quantum energy drift $0$ Largest trajectory energy drift $0$ Net outward window flux $0$ Quantum / mean force / hopping milliseconds $0$ Net outward flux subtracts incoming flux through the left window. Initially the packet is outside the windows: final exterior probability plus absorbed loss equals initial exterior probability plus net outward flux. Trigonometric interpolation evaluates current at the exact window planes; fractional-cell quadrature integrates their densities. Costs measure propagation on this device, excluding snapshots, figures and downloads. Quantum energy drift is meaningful before absorption; the absorbing layer removes probability beyond nineteen bohr. Model and numerical limits The quantum solver uses 512 periodic Fourier points on a forty-eight-bohr box and a Strang time step no larger than 0.8 atomic time units. It applies the exact local two-state exponential and an outer absorbing mask. No stepwise normalization is applied. The offline benchmark halves the step and doubles the grid. Ehrenfest trajectories follow the electronic mean force. Fewest-switches trajectories follow one adiabatic surface, propagate electronic amplitudes coherently, rescale momentum at accepted hops and reject energetically forbidden hops without velocity reversal. No decoherence correction is added. [Read the benchmark method](https://sandboxphysics.com/stories/nonadiabatic-dynamics/reproducible-benchmarks/) Physics tutorial ## Three Ways Through an Avoided Crossing **Background** Send the same nuclear preparation through two coupled electronic states. Watch the quantum packet split, compare the mean-force swarm and count trajectories that switch surfaces. **Why it matters** A change in electronic state changes the force on the nuclei. Approximations differ in how they keep those two changes together. Start with the essentials - **Focus question**: Do quantum waves, a mean force and hopping trajectories predict the same reaction branches? - **One-sentence intuition**: A mean-force trajectory feels both electronic branches; a hopping trajectory moves on one surface at a time. The quantum packet carries both amplitudes. ### Core mathematical model #### A nuclear coordinate coupled to two electronic states $$ H=-\frac{1}{2M}\partial_x^2 I+\begin{pmatrix}z(x)&w(x)\\w(x)&-z(x)\end{pmatrix},\quad M=2000 $$ Atomic units set the reduced Planck constant to one. Position is in bohr, energy in hartree and time in atomic time units. The matrix eigenvalues are the two adiabatic energies. #### The original simple avoided crossing $$ z(x)=0.01\,\operatorname{sgn}(x)(1-e^{-1.6|x|}),\qquad w(x)=0.005c\,e^{-x^2} $$ Electronic mixing multiplies the original off-diagonal coupling. The original Tully model has mixing equal to one. The initial packet is on the lower adiabatic surface. #### A shared Gaussian preparation $$ \chi(x)\propto e^{-(x+8)^2/(4\sigma_x^2)+ip_0x},\quad \sigma_p=\frac{1}{2\sigma_x} $$ The classical methods independently sample the positive Wigner distribution of this same minimum-uncertainty packet. Their initial electronic state is the local lower eigenvector. ### Common difficulties #### A population is not an exit probability **Typical misconception** Electronic population alone gives the reaction yield. **Better mental model** An exit also requires the nuclear packet to reach its chosen spatial window. During the animation the incoming packet is still on the left; the table uses the common final time and reports unresolved weight separately. ### Run the experiment 1. 01 #### Watch the split Choose Two outgoing branches and play the evolution. Compare the three rows at the same time. **What to observe:** The quantum density spreads onto both energy curves. Surface-hopping trajectories separate by their active surface; mean-force trajectories can travel between the two curves. 2. 02 #### Close an exit Choose An upper exit closes and inspect the late-time channel probabilities. **What to observe:** The upper asymptote requires enough total energy. A finite-width quantum packet contains an energy distribution; it can leave a small high-energy tail. 3. 03 #### Separate scatter and bias Open Ensemble and seed, increase Trajectories and save several independent seeds. Compare the final estimates with their standard errors. **What to observe:** Repeated estimates fluctuate less when more trajectories are used. The gap between each method and the quantum result need not shrink. Physical conclusion Tully’s one-coordinate, two-state model uses atomic units and an effective nuclear mass of 2000 electron masses. It is not calibrated to a molecule. The quantum packet and classical Wigner ensemble share the same initial Gaussian; their finite-time exit probabilities are compared at the same windows. Surface hopping retains electronic coherence and adds no decoherence correction. ## Related experiments and articles Nonadiabatic molecular dynamics · Reading path ### From energy surfaces to reaction branches Articles on this path that use this experiment, followed by the other experiments they suggest. 1. [Article **What changes when the electronic basis moves**](https://sandboxphysics.com/stories/nonadiabatic-dynamics/moving-electronic-basis/) 2. [Article **How much probability changes channel in one passage**](https://sandboxphysics.com/stories/nonadiabatic-dynamics/landau-zener-passage/) 3. [Article **Three dynamics methods for the same preparation**](https://sandboxphysics.com/stories/nonadiabatic-dynamics/three-dynamics/) Quantum & fields · Starter path ## Start from a single measurement Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference. [Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many 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)