Interpretations
Interpretation of the Wavefunction
Wavefunction \(\Psi(\vec{x},t)\), w/ Probability Density \(\rho(\vec{x},t)=|\Psi(\vec{x},t)|^2\), \(e\rho(\vec{x},t)\) gives the charge density.
\(\int_{V_0} d\vec{x}\rho(\vec{x},t) =\) probability to find the particle in the volume.
Probability flux: \(\vec{j}(\vec{x},t)=-\frac{i\hbar}{2m}(\Psi^*\vec{\nabla}\Psi-(\vec{\nabla}\Psi^*)\Psi) = -\frac{i\hbar}{2m}2i\Im\left[\Psi^*\vec{\nabla}\Psi\right] = \frac{\hbar}{m}\Im\left[\Psi^*\vec{\nabla}\Psi\right]\).
\(\int_\mathbb{R}d\vec{x} j(\vec{x},t) = \frac{\hbar}{m}\Im\int_\mathbb{R}d\vec{x}\Psi(\vec{x},t)^*\vec{\nabla}\Psi(\vec{x},t) = \frac{1}{2m}\int_\mathbb{R}d\vec{x}\Psi^*(\hat{P}\Psi) + (\hat{P}^*\Psi^*)\Psi = \frac{1}{m}\Re\langle \hat{P}\rangle_t = \frac{1}{m}\langle\hat{P}\rangle_t\).
How is the probability density and flux related?
\(\frac{\partial\rho}{\partial t}=\frac{\partial}{\partial t}\Psi^*\Psi = \frac{i}{\hbar}\hat{H}\Psi^*\Psi - i\hbar\Psi^*\hat{H}\Psi = \frac{i}{\hbar}\left[\frac{-\hbar^2}{2m}(\vec{\nabla^2}\Psi^*)\Psi + (\hat{V}\Psi^*)\Psi - \Psi^*\frac{\hbar^2}{2m}(\vec{\nabla}^2\Psi) - \Psi^*\hat{V}\Psi\right] = \frac{i\hbar}{2m}\left(\Psi^*\vec{\nabla}^2\Psi - \Psi\vec{\nabla}^2\Psi\right) = -\frac{\hbar}{m}\Im(\Psi^*\vec{\nabla}^2\Psi)\).
The divergence of the probability flux is, \(\vec{\nabla}\cdot\vec{j}(\vec{x},t) = -\frac{i\hbar}{2m}2i\Im(\Psi^*\vec{\nabla}^2\Psi) = -\frac{\partial}{\partial t}\rho\).
Continuity Equation - Probability Conservation Law
\(\frac{\partial}{\partial t}\rho + \vec{\nabla}\cdot\vec{j} = 0\).