Wavefunction evolution

Wavefunction evolution advances a complex state through time. A display of probability density is useful, but it is a derived view: two states can have the same |ψ|² while differing in phase, and that phase can change later interference. A simulator must preserve enough data to reproduce both views.

Think like a programmer

Store complex samples as { real, imaginary } or parallel typed arrays. Treat probability density, phase, norm, and energy diagnostics as derived data. A render loop may read a density chart every frame, but the model owns the initial state, potential, grid spacing, time step, boundary policy, and integrator used to create it.

Model checklist

Inputs: Initial complex state ψ, potential V(x), spatial grid Δx, time step Δt, boundary policy, and integrator.
State: Real and imaginary amplitude samples plus elapsed simulation time.
Rule: Apply an approximation to unitary time evolution, then derive density, phase, and diagnostics.
Output: Later ψ(x,t), |ψ|², phase view, norm, and convergence comparisons.
Check: Norm stays near one; changing only a global phase leaves probability density unchanged; refined time steps agree over a fixed duration.

Ideal closed-system evolution is written

\[|\psi(t)\rangle=e^{-i\hat Ht/\hbar}|\psi(0)\rangle\]

For a position sample, the displayable probability density is

\[\rho(x,t)=|\psi(x,t)|^2=\operatorname{Re}(\psi)^2+\operatorname{Im}(\psi)^2\]

Multiplying the entire state by a constant phase exp(iφ) changes neither ρ nor the norm. That gives a sharp unit test: construct a state, rotate every complex sample by the same angle, and compare the two density arrays. In contrast, a position-dependent phase can change momentum information and later interference even when the density initially looks identical.

const probability = state.map((sample) => sample.real ** 2 + sample.imaginary ** 2);
const norm = probability.reduce((sum, value) => sum + value * dx, 0);

Log norm - 1 rather than silently rescaling every animation frame. A rescale may be appropriate at a clearly stated algorithm boundary, but blind per-frame normalization can hide instability. Compare a fixed physical duration at Δt and Δt / 2; a simulation that only looks smooth has not passed a convergence check.

Try this experiment

Prediction: A global phase rotation changes the real and imaginary traces but leaves every probability-density sample unchanged.

Imagine a complex state with one nonzero sample 1 + 0i. Multiply it by i. Predict its real component, imaginary component, and density before calculating. Then explain what extra experiment would reveal a relative phase difference between two paths.

Where this model breaks

This lesson assumes a closed, one-particle nonrelativistic model. It omits measurement back-action, dissipation, decoherence, many-body state growth, spin, quantum fields, and detector response. Finite grids can cause reflections, aliasing, and phase error; a density-only chart cannot expose all of them.

Summary

Wavefunction evolution is complex-state evolution, not a succession of probability pictures. Retain phase-carrying state, derive density deliberately, monitor norm and refinement, and use global-phase invariance as a focused regression test.

Glossary

Unitary: ideal norm-preserving evolution of an isolated quantum state.
Phase: angular component of a complex amplitude.
Potential: spatial energy input to a Hamiltonian model.
Global phase: common complex rotation of every state component that leaves probabilities unchanged.
Probability density: squared complex magnitude used to predict position outcomes.

Self-check

Why cannot a density plot fully represent a wavefunction?
What happens to density under a global phase rotation?
Why should norm drift be logged rather than hidden?

Sources

D. J. Griffiths and D. J. Schroeter, Introduction to Quantum Mechanics, time evolution and phase.
OpenStax, University Physics Volume 3, Wave Functions.

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]

Inputs: Quantities you set or measure, with units and useful bounds.
State: Values the program must retain to reproduce the next result.
Rule: The relationship or update that turns inputs and state into a result.
Check: A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Wavefunction Evolution, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Quantum Beginnings, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Wavefunction Evolution as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

Input: a measured value or chosen parameter supplied to a model.
State: the smallest set of values needed to continue or reproduce a model.
Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

Which values are inputs, and which values must remain state?
What observable result would tell you the model is behaving as expected?
Which assumption would you test first before applying the model to a real system?

Model review: turn Wavefunction Evolution into a test

Evolve complex state with replayable settings and expose both phase and probability diagnostics.

Name the inputs and units that the quantum beginnings model needs.
Separate the state you must keep from values you can calculate when needed.
Write one rule that maps the current state and inputs to an observable result.
Choose a limiting case, unit check, invariant, or known result before trusting an output.
State one assumption you would change before using this simplified model for a real decision.

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