# Bulletin of the American Physical Society

# 53rd Annual Meeting of the APS Division of Atomic, Molecular and Optical Physics

## Volume 67, Number 7

## Monday–Friday, May 30–June 3 2022; Orlando, Florida

### Session V01: Poster Session III (4:00-6:00pm, EDT)

4:00 PM,
Thursday, June 2, 2022

Room: Grand Ballroom C

### Abstract: V01.00146 : A Coupled Volterra Integral Equation Approach to Solving the Time-Dependent Schr\"odinger Equation*

#### Presenter:

Barry I Schneider

(National Institute of Standards and Tech)

#### Authors:

Barry I Schneider

(National Institute of Standards and Tech)

Ryan Schneider

(University of California, San Diego)

Heman Gharibnejad

(National Institute of Standards and Tech)

\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.

*This work was supported by the National Institute of Standards and Technolgy and also from funds from the Mathematical Sciences Graduate Intership program of the National Science Foundation.

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