(a) Define the probability density $\rho(\mathbf{x}, t)$ and the probability current $\mathbf{J}(\mathbf{x}, t)$ for a quantum mechanical wave function $\psi(\mathbf{x}, t)$, where the three dimensional vector $\mathbf{x}$ defines spatial coordinates.

Given that the potential $V(\mathbf{x})$ is real, show that

$$\boldsymbol{\nabla} \cdot \mathbf{J}+\frac{\partial \rho}{\partial t}=0$$

(b) Write down the standard integral expressions for the expectation value $\langle A\rangle_{\psi}$ and the uncertainty $\Delta_{\psi} A$ of a quantum mechanical observable $A$ in a state with wavefunction $\psi(\mathbf{x})$. Give an expression for $\Delta_{\psi} A$ in terms of $\left\langle A^{2}\right\rangle_{\psi}$ and $\langle A\rangle_{\psi}$, and justify your answer.