Time Propagation

Time propagation asks how a represented quantum state changes over a finite interval. The choice of algorithm depends on whether the Hamiltonian is fixed, sparse, separable, or strongly time-dependent. This chapter develops the numerical methods and their verification, with particular attention to the fact that stable or norm-preserving evolution can still have large phase errors.

Enter this chapter

Start with the numerical time-dependent Schrödinger equation to identify the inner product, physical invariants, and independent error sources. Then use matrix exponentials as the finite-dimensional reference problem.

The time-evolution operator explains the propagator and time ordering. Numerical propagation begins after the Hamiltonian, basis, and boundary conditions have been specified; it cannot repair a wrong representation of those ingredients.

Match the propagator to the operator

Hamiltonian structure	Starting method	Error that needs its own check
Moderate fixed Hermitian matrix	Matrix exponential	Matrix-function accuracy and basis error
Large sparse or matrix-free fixed operator	Krylov evolution	Subspace defect and restart accumulation
Fixed Hermitian operator with safe spectral bounds	Chebyshev propagation	Polynomial tail and spectral interval
Separately simple kinetic and local potential terms	Split operator	Commutators, Fourier resolution, and box effects
Sparse linear system with reusable factorization	Crank–Nicolson	Phase dispersion and linear-solve residual
General smooth time dependence	Runge–Kutta or Magnus	Time resolution, stability, and time ordering

These are starting points, not universal rankings. An expensive operator action can favor a different method from a cheap sparse multiply. Benchmark total work at a fixed accuracy in the requested observable and at a fixed represented Hamiltonian.

From algorithms to physical trajectories

Wave-packet propagation combines spatial resolution, free dispersion, probability current, and finite-box diagnostics. Driven two-level systems tests carrier conventions and rotating frames without a spatial grid. Pulse simulation adds waveform interpolation, discontinuities, ordering, and filtering.

Adiabatic-dynamics numerics compares propagated states with instantaneous eigenspaces and separates physical leakage from solver error. Floquet numerics computes period propagators, quasienergy branches, harmonic cutoffs, and micromotion. Both require eigenvector or eigenspace tracking in addition to a time integrator.

A periodic drive, an adiabatic sweep, and a short pulse can share the same propagator implementation while demanding different physical diagnostics. Compare the quantities relevant to the question: quasienergies, leakage from an instantaneous eigenspace, or the final state after a pulse.

Benchmarks from simple phases to driven motion

First test a constant scalar Hamiltonian, which produces only phase. Next test a two-level rotation, which has a known transition probability and relative phase. Then use a noncommuting driven Hamiltonian to exercise time ordering. For grid methods, add a free Gaussian packet before studying a barrier or nonlinear potential.

At each level, compare several step sizes with a fixed final time. If a method has global order $p$ in its smooth asymptotic regime, successive differences under step halving should decrease by about $2^p$. Norm drift and energy drift are additional diagnostics, not substitutes for this comparison.

For example, Crank–Nicolson gives an eigenmode phase $-2\arctan(hE/2\hbar)$ per step, whereas the exact phase is $-hE/\hbar$. Both have unit modulus. A test that inspects only normalization misses the entire dispersive error.

Keep the error sources distinct

A useful result separates spatial or basis refinement, time-step refinement, inner exponential or linear-solve tolerance, waveform resolution, and boundary effects. Report the final time: a small per-step phase error can accumulate over many oscillations. For explicitly driven systems, test the correct work–energy balance rather than demanding constant energy.

For non-Hermitian absorbers or open-system generators, norm or purity can change physically. Use the corresponding loss or master-equation balance, and do not transfer closed-system unitarity claims unchanged.

Exercises

A split-operator simulation has constant norm but its transmission changes when the box is enlarged. Which convergence study is missing?

Solution

The box or boundary treatment is not converged. Periodic wraparound, premature absorption, or insufficient spatial separation can change transmission while every split step remains unitary. Repeat box and absorber tests independently of the time-step study.

References

• Blanes, S., F. Casas, J. A. Oteo, and J. Ros, “The Magnus expansion and some of its applications,” Physics Reports 470, 151–238 (2009), doi:10.1016/j.physrep.2008.11.001 — time-dependent exponential methods.
• Leforestier, C., et al., “A comparison of different propagation schemes for the time dependent Schrödinger equation,” Journal of Computational Physics 94, 59–80 (1991), doi:10.1016/0021-9991(91)90137-A — comparative propagation benchmarks.