Split-operator Fourier time evolution

For N equally spaced positions on a ring of length N times a, use a unitary forward transform with a negative exponential and an inverse with a positive exponential. For even N, label modes from zero upward and then use negative indices, assigning the Nyquist entry consistently. The momentum for mode m is two pi hbar m divided by the ring length. Reversing transform signs without relabeling momentum changes propagation and phase interpretation.

The kinetic operator in this Fourier representation uses the continuum quadratic momentum evaluated at the retained modes. It is not the centered finite-difference kinetic matrix from the first lesson, whose dispersion follows a cosine. Both define finite Hermitian models, but substituting one dispersion into the other's benchmark compares different systems. At a fixed grid, sufficiently high-frequency signals also alias into the retained band; increasing time accuracy alone cannot recover missing spatial information.

For an autonomous Hamiltonian T plus V, a symmetric Strang step applies a potential half-step, a kinetic full step and another potential half-step. With finite bounded Hermitian matrices, its local error is third order in timestep and its accumulated error over a fixed smooth evolution interval is second order. Nested commutators determine the leading splitting error. If T and V commute, the split becomes exact apart from arithmetic and representation error.

A time-dependent potential requires an appropriate time-centered prescription to retain the desired order; merely evaluating every factor at the old time can lower it. A product of unitary factors still remains unitary even when it uses a poor approximation to time ordering. Adaptive comparisons should therefore examine the state or observable difference under step subdivision, not just norm drift. A small change in one occupation can also hide a relative-phase error in another observable.

A Fourier grid is periodic even when the physical problem was intended to be an open line. A packet reaching the edge can reappear at the opposite edge. Enlarging the domain and using carefully tested absorption can delay or suppress that artifact, but a generic FFT does not provide an outgoing boundary condition. Potential multiplication can generate unresolved Fourier content, especially for abrupt features, so both bandwidth and the real-space sampling of the potential need convergence checks.

Commutator bounds provide a small-matrix diagnostic of operator splitting. For a first-order Lie product of finite Hermitian pieces, the operator-norm error is bounded by one half timestep squared over hbar squared times the commutator norm. The symmetric method has different nested-commutator constants, so its leading coefficient cannot be inferred from that first-order bound alone. These distinctions are useful when comparing methods rather than labeling every norm-preserving FFT routine exact quantum dynamics.