A Coupled Volterra Integral Equation Approach to Solving the Time-Dependent Schr\"odinger Equation*

Most approaches to solving the time-dependent Schr\"odinger (TDSE) involving time-dependent interactions invoke the short-time approximation.  In essence, one propagates the TDSE over a sufficiently short time interval that one can ignore the time dependence of the interaction.  It is possible to improve these approaches, for example, as is done in the Magnus expansion, but they often require the evaluation of commutators which are not always easy to compute.  A new and simple approach to overcome both of the aforementioned problems, is based on rewriting the TDSE as;

\begin{align}

\ket{\Psi(t)} = \exp(-iH_0(t-t_0))\ket{\Psi(t_0)}  - i \exp(-iH_0t) \int_{t_0}^{t}  \exp(iH_0t^\prime) V(t^\prime) \ket{\Psi(t^\prime)}  \hspace{.2cm}  t_0 \le t \le t_f

\end{align}

where $H = H_0 + V(t)$ and the spatial variables have been surpressed for notational convemience.  Since we have chosen $H_0$ to be time independent, the problem reduces to an exact propagation of the TDSE when $V(t)=0$.  If the time step is sufficiently small so that the integral may be ignored, the equation reduces to the calculation of the action of a matrix exponential on a known vector which may be evaluated a using a variety of well known methods.

Using Lagrange interpolation of the integrand, we compute a set of integration weights $w_{p,i}$ and perform the integration to get,

\begin{align}

\ket{\Psi(t_p)} = \exp(-iH_0(t_p-t_0))\ket{\Psi(t_0)} - i \exp(-iH_0t_p) \sum_q w_{p,q}  \exp(iH_0t_q) V(t_q) \ket{\Psi(t_q)} \\

\end{align}

We have successfully solved this set of algebraic equations using both the Gauss-Seidel and the Jacobi iterative methods.  The former converges more rapidly but requires the solution of a set of coupled equations at each iteration.  Details and examples will be given in the poster.