Wave-Packet Propagation

A wave-packet simulation combines an initial momentum distribution, a finite spatial representation, a propagator, and a measurement prescription. Each can change the observed scattering or spreading. The most useful first test is free Gaussian evolution, where center motion, width, and norm are known analytically.

Helpful background. Split-operator propagation supplies a common FFT implementation, and boundary conditions defines the spatial problem.

Gaussian initialization and resolution

In one dimension, use

$$\psi(x,0)=\frac{1}{(2\pi\sigma_x^2)^{1/4}} \exp\!\left[-\frac{(x-x_0)^2}{4\sigma_x^2} +\frac{ip_0(x-x_0)}{\hbar}\right].$$

Here $\sigma_x$ is the standard deviation of the probability density, not of the amplitude. The momentum standard deviation is $\sigma_p=\hbar/(2\sigma_x)$. On a uniform grid, normalize with $\Delta x\sum_j|\psi_j|^2=1$ if the stored array contains sampled wavefunction values. A weighted-amplitude convention instead absorbs $\sqrt{\Delta x}$ into the array; do not use both factors.

Choose a box in which initial tails are negligible and a mesh whose momentum window contains the packet. For an FFT grid, require substantial weight to stay well inside $|p|<\pi\hbar/\Delta x$. A narrow position packet has a broad momentum distribution, making this requirement harder.

Displacing the packet by a noninteger grid spacing and changing its carrier momentum are useful tests of interpolation and Fourier conventions.

Exact free spreading

For $H=p^2/(2m)$,

$$\langle x\rangle_t=x_0+\frac{p_0t}{m}, \qquad \sigma_x^2(t)=\sigma_x^2 \left[1+\left(\frac{\hbar t}{2m\sigma_x^2}\right)^2\right].$$

The momentum distribution and mean kinetic energy remain constant. These relations test different aspects of the calculation: center velocity detects momentum and unit conventions, spreading detects dispersion, and energy checks the represented kinetic operator.

On a periodic FFT grid, the discrete free propagator is exact for the represented Fourier modes. The continuum Gaussian formulas remain accurate only before periodic images and finite-window effects become relevant. A late-time disagreement can therefore be a box error even when reducing the time step changes nothing.

Probability current and scattering estimates

For a real local potential, the continuum current is

$$j(x,t)=\frac{\hbar}{m}\operatorname{Im} \left[\psi^*(x,t)\,\partial_x\psi(x,t)\right].$$

It satisfies $\partial_t|\psi|^2+\partial_xj=0$. A discrete current should match the chosen spatial discretization; combining a spectral propagator with an unrelated low-order derivative can introduce a separate diagnostic error.

After a localized barrier separates reflected and transmitted packets, integrate probability in left and right asymptotic regions. Account for probability still in the interaction region and any initially occupied bound states. A fixed Hermitian Hamiltonian cannot transfer a purely continuum state into its bound eigenspace; capture requires additional physics such as a drive or dissipation. At finite time, imposing $R+T=1$ by renormalization can hide an unfinished collision.

Time-integrated signed current through a surface measures net crossing probability under suitable geometry and separation assumptions. Recrossing, overlapping incident and reflected waves, and absorbers require careful interpretation. Count outgoing channels rather than integrating only positive current and assuming every crossing represents a different particle.

Packet bandwidth and absorbers

A packet has an energy distribution. Its transmitted probability approaches a weighted average of plane-wave transmission, not necessarily the transmission at its mean energy. To compare with a sharply energy-resolved prediction, narrow the momentum width and enlarge the box and propagation time accordingly.

For an absorber $H=H_0-iW$ with $W\geq0$,

$$\frac{d\|\psi\|^2}{dt}=-\frac{2}{\hbar}\langle W\rangle.$$

Here $\langle W\rangle=\langle\psi|W|\psi\rangle$ uses the unnormalized evolving state. Lost squared norm measures absorption, but it equals a desired asymptotic outgoing probability only after reflections and overlap with the physical interaction region are controlled. Vary absorber onset, width, and strength independently of the time step.

Exercises

With $\hbar=m=1$, $\sigma_x=1$, and $p_0=2$, find the center displacement and width at $t=2$.

Solution

The center moves by $p_0t/m=4$. The variance is $1+[2/(2)]^2=2$, so the standard deviation is $\sqrt2$. A code reporting width $2$ has confused variance with standard deviation.

References

• Kosloff, D., and R. Kosloff, “A Fourier method solution for the time dependent Schrödinger equation as a tool in molecular dynamics,” Journal of Computational Physics 52, 35–53 (1983), doi:10.1016/0021-9991(83)90015-3 — Fourier-grid wave-packet evolution.
• 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 — bound and scattering propagation tests.