As the system evolves in time, this probability may change, so we take its derivative with respect to time \(t\), and when necessary substitute in the other side of the Schrödinger equation to get:
\[\begin{aligned} - \pd{P}{t} - &= \int_{V} \psi \pd{\psi^*}{t} + \psi^* \pd{\psi}{t} \dd[3]{\vec{r}} + \pdv{P}{t} + &= \int_{V} \psi \pdv{\psi^*}{t} + \psi^* \pdv{\psi}{t} \dd[3]{\vec{r}} = \frac{i}{\hbar} \int_{V} \psi (\hat{H} \psi^*) - \psi^* (\hat{H} \psi) \dd[3]{\vec{r}} \\ &= \frac{i}{\hbar} \int_{V} \psi \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi^* + V(\vec{r}) \psi^* \Big) @@ -83,15 +83,15 @@ \end{aligned}\]
Returning to the derivation of \(\vec{J}\), we now have the following equation:
\[\begin{aligned} - \pd{P}{t} - = \int_{V} \pd{|\psi|^2}{t} \dd[3]{\vec{r}} + \pdv{P}{t} + = \int_{V} \pdv{|\psi|^2}{t} \dd[3]{\vec{r}} = - \int_{V} \nabla \cdot \vec{J} \dd[3]{\vec{r}} \end{aligned}\]
By removing the integrals, we thus arrive at the continuity equation for \(\vec{J}\):
\[\begin{aligned} \boxed{ \nabla \cdot \vec{J} - = - \pd{|\psi|^2}{t} + = - \pdv{|\psi|^2}{t} } \end{aligned}\]
This states that probability is conserved, and is reminiscent of charge conservation in electromagnetism. In other words, the probability at a point can only change by letting it “flow” towards or away from it. Thus \(\vec{J}\) represents the flow of probability, which is analogous to the motion of a particle.
-- cgit v1.2.3