\mathbf{j} = \frac{-i\hbar}{2m}\left(\Psi^* \nabla \Psi - \Psi \nabla \Psi^*\right) {\displaystyle ={\frac {\hbar }{m}}\mathrm {Im} (\Psi ^{*}\nabla \Psi )=\mathrm {Re} (\Psi ^{*}{\frac {\hbar }{im}}\nabla \Psi )} = \frac\hbar m \mathrm{Im}(\Psi^*\nabla\Psi)=\mathrm{Re}(\Psi^* \frac{\hbar}{im} \nabla \Psi)

\hat{H} = \sum_{n=1}^{N}\frac{\hat{p}_n^2}{2m_n} + V(x_1,x_2,\cdots x_N,t)
= -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\frac{\partial^2}{\partial x_n^2} + V(x_1,x_2,\cdots x_N,t)