Notebooks and Worked Examples

These worked computational guides show how to evolve a wave packet, compare quantum pictures, construct propagators, and calculate Wigner and Floquet quantities. Start from a model with an analytic solution, then change a numerical parameter and measure how the error responds.

Helpful background. How to use notebooks explains the format, and the time-evolution operator defines the propagator being computed. Floating-point arithmetic and convergence tests explain why plausible plots and preserved norms do not establish accuracy.

Choose a dynamics calculation

For a first exercise, try spin precession or free-packet evolution. The spin example uses small matrices and exposes picture and phase errors. The packet example adds spatial grids, boundaries, and dispersion. Follow the linked derivations when an implementation formula is unfamiliar.

Computational question	Notebook	Primary benchmark
How does a localized free packet translate and spread on a Fourier grid?	Wave-Packet Time Evolution	exact free Gaussian evolution
Do Schrödinger, Heisenberg, and interaction pictures give the same observable history?	Spin Precession in Three Pictures	analytic spin rotation and picture translation
How do full-line kernel convolution and periodic Fourier evolution represent the same free dynamics?	Free-Particle Propagator	exact Gaussian and free kernel
How do a spectral sum, closed kernel, and coherent-state trajectory agree for the oscillator?	Harmonic-Oscillator Propagator	Mehler kernel and coherent evolution
Why does exponentiating an integrated noncommuting Hamiltonian fail?	Time-Dependent Hamiltonian	exact rotating-frame propagator
How do first- and second-order product formulas reveal their convergence orders?	Trotter Evolution	exact finite-dimensional exponential
How does nonunitary imaginary-time evolution isolate low-energy states?	Imaginary-Time Projection	finite-difference diagonalization and oscillator spectrum
How is a continuous Wigner function represented without losing normalization, marginals, or negativity?	Wigner Function	analytic Gaussian and odd-cat parity value
How are eigenphases converted into tracked quasienergy branches and sampled dynamics?	Floquet Quasienergy	exact circular-drive Floquet operator

Measure convergence, not just conservation

A product of unitary steps preserves norm even if the factors are ordered incorrectly. An FFT evolution preserves norm even after a packet wraps around the periodic box. Compare an observable or operator with an analytic answer as well as testing conserved quantities.

For a fixed final time $T$, suppose a time-stepping method has error $e_N=C(T/N)^p$ after $N$ steps, once the leading error dominates. Doubling the number of steps gives

$$\frac{e_N}{e_{2N}}\approx2^p,\qquad p_{\rm observed}=\frac{\log(e_N/e_{2N})}{\log2}.$$

For example, errors $0.008$, $0.002$, and $0.0005$ at $N$, $2N$, and $4N$ suggest second-order convergence. This estimate is useful only if the box, spatial grid, final time, and error norm are held fixed, and spatial or roundoff errors have not taken over. Without an exact answer, differences between successive refinements can estimate the order, but agreement should also be checked against a different method when possible.

Use operator errors when comparing evolution operators. One initial state can miss an error acting on an orthogonal state.

Common Numerical Language

The pages use a consistent distinction among several errors:

• representation error: finite box, grid, basis, or spectral cutoff;
• discretization error: finite time step, quadrature spacing, or transform grid;
• algorithmic error: splitting, iteration, solver tolerance, or truncation of a formal expansion;
• roundoff error: finite-precision accumulation and conditioning;
• model error: replacing the intended Hamiltonian or domain by a simplified one.

Refining only one parameter can move error from one category to another. Increasing a Fourier grid size at fixed box length improves spatial resolution but does not delay periodic wraparound. Tightening an ODE tolerance does not help if the Hamiltonian is evaluated with the wrong absolute drive phase. A real-time oscillator kernel may require a distributional or regulated interpretation; increasing a spectral cutoff need not improve its pointwise values near a caustic.

Every convergence claim should therefore name what is held fixed and what is refined.

Save enough to repeat the calculation

Keep the model parameters, initial state, units, grid and boundary conditions, time steps or solver tolerances, numerical precision, and software versions with the results. Save the values behind error plots. If random sampling is used, save the generator and seed too. A short parameter block and a table of errors are more useful than an unexplained image.

For the physical interpretation of a numerical result, return to propagators, quantum pictures, or periodic driving. The software and benchmarks guide gives broader computational advice.

References

• E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration, 2nd ed. (Springer, 2006), doi:10.1007/3-540-30666-8.
• N. J. Higham, Functions of Matrices: Theory and Computation (SIAM, 2008), doi:10.1137/1.9780898717778.
• C. Leforestier 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.
• H. Tal-Ezer and R. Kosloff, “An Accurate and Efficient Scheme for Propagating the Time Dependent Schrödinger Equation,” Journal of Chemical Physics 81, 3967–3971 (1984), doi:10.1063/1.448136.