Split-Operator Method

The split-operator method approximates evolution under $H=T+V$ by alternating propagators that are easy in different representations. For a Cartesian kinetic energy and local potential, Fourier transforms connect diagonal momentum-space and position-space steps. The method is unitary for real potentials on a consistent discrete Fourier grid, but it still has splitting, grid, and periodic-boundary errors.

Helpful background. Matrix exponentials explains exact finite-dimensional actions; wave-packet propagation develops box and flux diagnostics.

Symmetric kinetic–potential splitting

For a fixed Hamiltonian, define $A=-iV/\hbar$ and $B=-iT/\hbar$. A symmetric step is

$$S(h)=e^{hA/2}e^{hB}e^{hA/2}.$$

Expanding the exponentials shows agreement with $e^{h(A+B)}$ through terms of order $h^2$. Symmetry gives $S(-h)=S(h)^{-1}$, so the leading local error is $O(h^3)$ and the global error at fixed final time is generally $O(h^2)$ under sufficient regularity.

The leading error involves nested commutators of $T$ and $V$. A rapidly varying or nonsmooth potential can demand finer resolution and invalidate naive smooth-problem error estimates. If $[T,V]=0$, the splitting is exact for the represented operators.

On each step, multiply $\psi(x_j)$ by $e^{-ihV(x_j)/(2\hbar)}$, Fourier transform, multiply by $e^{-ih\hbar k_\ell^2/(2m)}$, inverse transform, and apply the second half potential step. Keep the FFT normalization and discrete norm convention consistent.

Fourier grids and physical boundaries

For $N$ evenly spaced points in a periodic interval of length $L$, the Fourier wave numbers are $k_\ell=2\pi\ell/L$ with positive and negative indices in the library’s ordering. The kinetic multiplier uses angular wave number, not ordinary spatial frequency. An incorrect factor of $2\pi$ changes dispersion while preserving norm.

The discrete Fourier representation is periodic. A packet leaving one side reenters at the other unless it is absorbed or the simulation ends earlier. An FFT is not an outgoing-boundary condition. Choose a large enough box, or use a tested absorber and account for its intentional norm loss.

The Nyquist scale is $\pi/\Delta x$. Keep substantial momentum weight away from it and inspect high-frequency tails; nonlinear operations or sharp potentials can transfer weight toward unresolved modes. Refining the time step does not remove this aliasing.

Driven potentials and higher order

For smooth $V(x,t)$, evaluate the potential at the time midpoint in both half steps to retain a second-order treatment of explicit time dependence. Using the beginning value throughout generally reduces the temporal order. More complicated time dependence or momentum-dependent couplings needs a derivation suited to the actual Hamiltonian.

Higher-order compositions can reduce splitting error but often use negative substeps. These can be problematic with absorbers or dissipative evolution, where reversing a damping step amplifies errors. Compare total cost at a fixed error tolerance rather than assuming a higher formal order is always better.

A free-particle benchmark

With $V=0$, the Fourier-grid propagation is exact for the represented periodic kinetic Hamiltonian at any step size: only diagonal phases remain. A plane wave with $k=2\pi\ell/L$ acquires phase $e^{-i\hbar k^2t/(2m)}$. A superposition of two such modes tests relative phase and interference.

For a smooth nonconstant potential, compare trajectories at $h,h/2,h/4$ with fixed grid and box. In the asymptotic second-order regime, differences between successive solutions should decrease by about four. Norm error can stay near roundoff in all three runs while their phase errors differ substantially.

Exercises

Why is a test with constant potential insufficient to establish second-order splitting convergence?

Solution

A constant potential commutes with $T$, so all commutator errors vanish and the splitting is exact for the represented problem. Use a nonconstant potential and a state exploring its gradients to measure the generic splitting error.

References

• 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 — quantum propagation comparisons.
• Strang, G., “On the construction and comparison of difference schemes,” SIAM Journal on Numerical Analysis 5, 506–517 (1968), doi:10.1137/0705041 — symmetric operator splitting.