Quantum Dynamics

In the following we develop numerical methods for solving the time-dependent Schrödinger equation:

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

with an initial condition:

$$|\psi(t=0)\rangle=|\psi(0)\rangle$$

Assuming that the Hamiltonian \(\hat{H}\) is time-independent allows us to write the solution in terms of the exponential propagator:

$$|\psi(t)\rangle= \exp\left(-\frac{\mathrm{i}}{\hbar}\hat{H}t\right)|\psi(0)\rangle$$

Here, the propagator is given in the form of the operator exponential which is defined by the following infinite operator series:

$$\exp \left(-\frac{\mathrm{i}}{\hbar}\hat{H}t\right) = \sum_{n=0}^{\infty}\frac{1}{n!}\left(-\frac{\mathrm{i}t}{\hbar}\right)^{n}\hat{H}^{n}$$

In order to develop a numerical scheme for the propagation of the TDSE we first divide the time-interval \([0,t]\) into \(N\) sub-intervalse such that:

$$t = N \cdot \Delta t$$

If \(N\) is very large, \(\Delta t\) will be very small and thus the propagation over the time interval \(\Delta t\) might be performed in multiple steps by using some short time approximation for the propagator. Since the full propagation time can now be achieved by repeating \(N\) propagation steps, we have:

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