Exact time evolution of the pair distribution function for an entangled two-electron initial state

Abstract

Based on the correlated ground-state wave function of an exactly solvable interacting one-dimensional two-electron model Hamiltonian we address the switch-off of confining and interparticle interactions to calculate the exact time-evolving wave function from a prescribed correlated initial state. Using this evolving wave function, the time-dependent pair probability function $\mathcal{R}({x}_{1},{x}_{2},t)\ensuremath{\equiv}{n}_{2}({x}_{1},{x}_{2},t)/[n({x}_{1},t)\phantom{\rule{0.16em}{0ex}}n({x}_{2},t)]$ is determined via the pair density ${n}_{2}({x}_{1},{x}_{2},t)$ and single-particle density $n(x,t)$. It is found that $\mathcal{R}(0,0,t=\ensuremath{\infty})=\mathcal{R}(0,0,t=0)>1$, and $\mathcal{R}({x}_{1},{x}_{2},{t}^{*})=1$ at a finite ${t}^{*}$ for $\ensuremath{\Lambda}\ensuremath{\ne}0$ interparticle interaction strength in the initial two-electron model. By expanding $n(x,t)$ in an infinite sum of closed-shell products of time-dependent normalized single-particle states and time-dependent occupation numbers ${P}_{k}(\ensuremath{\Lambda},t)$, the von Neumann entropy $S(\ensuremath{\Lambda},t)=\ensuremath{-}{\ensuremath{\sum}}_{k=0}^{\ensuremath{\infty}}{P}_{k}(t)\mathrm{ln}{P}_{k}(t)$ is calculated as well. The such-defined information entropy is zero at ${t}^{*}(\ensuremath{\Lambda})$ and its maximum in time is $S(\ensuremath{\Lambda},t=\ensuremath{\infty})=S(\ensuremath{\Lambda},t=0)$.