202011110657 Sidenote of Separation of Variables

Two-particle system: $\displaystyle{i\hbar\frac{\partial \Psi}{\partial t}=H\Psi}$

in which $\displaystyle{H=-\frac{\hbar^2}{2m_1}\nabla_1^2-\frac{\hbar^2}{2m_2}\nabla_2^2+V(\mathbf{r_1},\mathbf{r_2},t)}$ .

Assuming that the potential is time-independent.

Let $\Psi (\mathbf{r}_1,\mathbf{r}_2,t)=\psi (\mathbf{r}_1,\mathbf{r}_2)\varphi (t)$ ,

$\begin{aligned} & i\hbar \frac{\partial \psi (\mathbf{r}_1,\mathbf{r}_2)\varphi (t)}{\partial t} \\ = & -\frac{\hbar^2}{2m_1}\nabla_1^2\bigg( \psi (\mathbf{r}_1,\mathbf{r}_2)\varphi (t)\bigg) -\frac{\hbar^2}{2m_2}\nabla_2^2\bigg( \psi (\mathbf{r}_1,\mathbf{r}_2)\varphi (t) \bigg) \\ &\qquad\enspace +V(\mathbf{r_1},\mathbf{r_2})\psi (\mathbf{r}_1,\mathbf{r}_2)\varphi (t) \\ & \\ & \textrm{\scriptsize OR} \\ & \\ & i\hbar\Big[ \frac{\partial \psi (\mathbf{r}_1,\mathbf{r}_2)}{\partial t}\varphi (t)+\psi (\mathbf{r}_1,\mathbf{r}_2)\frac{\partial \varphi (t)}{\partial t}\Big]\\ = & -\frac{\hbar^2}{2m_1}\varphi (t)\nabla_1^2\psi (\mathbf{r}_1,\mathbf{r}_2)-\frac{\hbar^2}{2m_2}\varphi (t)\nabla_2^2\psi (\mathbf{r}_1,\mathbf{r}_2) +V\psi \varphi \\ \end{aligned}$

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