Quantum correlation of two non-interaction spinless particles: exact solution of the time-dependent <PERSON> equation
I would like to get some comment about the following obtained result.
I consider the one-dimensional time-dependent <PERSON> equation for two non-interaction particles with masses $m_1$ and $m_2$: $$ i \hbar \frac{\partial \Psi}{\partial t} = - \frac{\hbar^2}{2m_1} \frac{\partial^2 \Psi}{\partial x_1^2} - \frac{\hbar^2}{2m_2} \frac{\partial^2 \Psi}{\partial x_2^2}. $$ Let us assume that the wave-function $\Psi(x_1, x_2, t)$ has the gaussian form at the initial instant $t=0$: $$ \Psi(x_1,x_2,0) = \frac{1}{\sqrt{2\pi \sigma_1 \sigma_2}\,(1-r^2)^{1/4}}\, \exp \left[ - \frac{1}{1-r^2} \left( \frac{x_1^2}{4\sigma_1^2} + \frac{(x_2-x_0)^2}{4\sigma_2^2} - \frac{r x_1 (x_2 - x_0)}{2\sigma_1\sigma_2}\right) + \frac{i}{\hbar}\, p_0 x_2 \right]. $$ In my understanding, it describes the situation when the first particle is placed at the origin, while the second one is moving along $x$-axis with the velocity $v_0 = p_0/m_2$. Here $\sigma_1$ and $\sigma_2$ are the initial standard deviations of the particle coordinates, and $r$ is the initial correlation coefficient, $|r|<1$. 955
Thus, if $r \neq 0$, then we have a two-particle entangled state at $t=0$. Without loss of generality, we will assume that $r>0$.
I have exactly solved the time-dependent <PERSON> equation with the given initial condition and obtained an explicit form of the wave-function $\Psi(x_1,x_2,t)$ (the corresponding expression is rather cumbersome and I do not write it here). Then I have calculated the time-dependent correlation coefficient: $$ r_t := \frac{\langle x_1 x_2 \rangle - \langle x_1 \rangle \langle x_2 \rangle}{\delta x_1 \delta x_2} = \frac{r \sigma_1 \sigma_2 - \frac{r \hbar^2 t^2}{4 m_1 m_2 (1-r^2) \sigma_1 \sigma_2}}{\sqrt{\sigma_1^2 + \frac{\hbar^2 t^2}{4 m_1^2 (1-r^2)\sigma_1^2}} \, \sqrt{\sigma_2^2 + \frac{\hbar^2 t^2}{4 m_2^2 (1-r^2) \sigma_2^2}}}, $$ where $\langle \hat{A} \rangle := \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \overline{\Psi(x_1, x_2, t)} \, \hat{A} \Psi(x_1, x_2, t) dx_1 dx_2$, $\delta x_i = \sqrt{\langle x_i^2 \rangle - \langle x_i \rangle^2}$, $i=1,2$. An analysis of this expression shows that: 1) the function $r_t$ has a maximum at $t=0$, which equals $r$; 2) for $t>0$ the function $r_t$ monotonically decreases and asymptotically approaches the value $r_{\infty} = -r$; 3) the function $r_t$ vanishes at the instant $t_0 = 2 \sigma_1 \sigma_2 \sqrt{m_1 m_2(1-r^2)}/\hbar$. 66