Forward Euler Approximation

For short time steps \(\Delta t\), the infinite series representing the propagator can be truncated at low order leading to a polynomial representation of the quantum propagator. If we restrict ourselves to the terms linear in \(\Delta t\), we obtain the linear version of the short-time propagator given by:

$$\exp\left(-\frac{\mathrm{i}}{\hbar}\hat{H}\Delta t\right) = 1 - \frac{\mathrm{i} \Delta t}{\hbar}\hat{H}$$

This allows us in principle to propagate the initial wavefunction by repeated application of the short-time propagator:

$$|\psi(t)\rangle = \left[1 - \frac{\mathrm{i}\Delta t}{\hbar}\hat{H}\right]^{n} |\psi(0)\rangle$$

For a single propagation step we have the following expression, which allows us to advance the wavefunction in time:

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

By using the same discretization procedure that we used in the numerical calculation of eigensates in the frame of the Fourier grid approximation, we can transform this into a matrix equation involving the grid components of the wavefunction:

$$|\psi_{i}(t+\Delta t)\rangle = |\psi_{i}(t)\rangle - \frac{\mathrm{i}\Delta t}{\hbar}\sum_{j=0}^{N-1}H_{ij}|\psi_{j}(t)\rangle$$

Here, \(\psi_{i}\) and \(\psi_{j}\) represent the values of the discretely sampled wavefunction at the grid points \(i\) and \(j\), respectively and \(H_{ij}\) is the previously defined discrete representation of the Hamiltonian:

$$ H_{ij} = \frac{\left(-1\right)^{i-j}}{N} \sum_{k=0}^{N-1} \frac{p_{k}^{2}}{2m} \exp\left(\frac{2\pi \mathrm{i}ki}{N}\right) \exp\left(-\frac{2\pi \mathrm{i}kj}{N} \right) + V(x_{i})\delta_{ij} $$

The product involving the Hamiltonian matrix and the wavefunction vector can be in the grid representation expressed as:

$$ H_{ij} = \frac{\left(-1\right)^{i-j}}{N} \sum_{k=0}^{N-1} \frac{p_{k}^{2}}{2m} \exp\left(\mathrm{i}\frac{2\pi ki}{N}\right) \exp\left(-\mathrm{i}\frac{2\pi kj}{N}\right) + V(x_{i})\delta_{ij}$$