Second Order Difference

The instability of the forward Euler scheme can be traced back to the fact that this scheme is not invariant with respect to the time reversal. In the following, we develop the second order difference scheme by combining the forward and backward Euler propagation steps. For the forward step we have as before:

$$|\psi(t+\Delta t)\rangle=|\psi(t)\rangle-\frac{\mathrm{i}\Delta t}{\hbar}\hat{H}|\psi(t)\rangle$$

The backward step can be obtained by inserting \(-\Delta t\) into the forward expression:

$$|\psi(t-\Delta t)\rangle=|\psi(t)\rangle+\frac{\mathrm{i}\Delta t}{\hbar}\hat{H}|\psi(t)\rangle$$

By subtracting these two equations we get:

$$|\psi(t+\Delta t)\rangle=|\psi(t-\Delta t)\rangle-2\frac{\mathrm{i}\Delta t}{\hbar}\hat{H}|\psi(t)\rangle$$

which represents the second order difference (SOD) propagator. This is a two step propagator since at each step be need the wavefunction at the time \(t\) and \(t - \Delta t\).

Written in terms of the wavefunction grid components, the SOD scheme can be formulated as:

$$\psi_{i}^{n+1}=\psi_{i}^{n-1}-2\frac{\mathrm{i}}{\hbar}\Delta t\sum_{j}H_{ij}\psi_{j}^{n}$$