Quantum Mechanical Current Density

Let’s we define wave function of a particle as below:

\[\Psi = Aexp(j\varphi(t))\]

We can write Schrodinger equation of the particle as below:

\[i\hbar\frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m}\triangledown^2\Psi + \hat{V}\Psi\]

\[\hat{H} = -\frac{\hbar^2}{2m}\triangledown^2 + \hat{V}\]

\[i\hbar\frac{\partial\Psi}{\partial t} = \hat{H}\Psi\]

\[i\hbar\Psi^*\frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m}\Psi^*\triangledown^2\Psi + \hat{V}\Psi^*\Psi\]

\[-i\hbar\Psi\frac{\partial\Psi^*}{\partial t} = -\frac{\hbar^2}{2m}\Psi\triangledown^2\Psi^* + \hat{V}\Psi\Psi^*\]

\[\hat{H}\Psi = -\hat{H}\Psi^*\]