In classical mechanics, it is known from Noether's theorem every continuous symmetry gives a conserved current \begin{equation} \partial_{\mu}J^{\mu}=0, \end{equation} where $J^{\mu}$ (generally) can be written in terms of fields and their conjugate momentum. The conserved charge $Q$ is \begin{equation} Q=\int d^3x J^0 \end{equation} In quantum case, our fields are promoted to operators. So $J^{\mu} \rightarrow \hat{J^{\mu}}$, $Q\rightarrow \hat{Q}$. Conservation of $\hat{Q}$ means $d\hat{Q}/dt=-i[\hat{Q},\hat{H}]=0$, where $\hat{H}$ is the Hamiltonian.

My question is: How to write $\partial_{\mu}J^{\mu}=0$ in quantum case? Do we have $\partial_{\mu}\hat{J^{\mu}}=0$ or $<\alpha|\partial_{\mu}\hat{J^{\mu}}|\alpha>=0$ for any ket $|\alpha>$?

In classical mechanics, it is known from Noether's theorem every continuous symmetry gives a conserved current \begin{equation} \partial_{\mu}J^{\mu}=0, \end{equation} where $J^{\mu}$ (generally) can be written in terms of fields and their conjugate momentum. The conserved charge $Q$ is \begin{equation} Q=\int d^3x J^0 \end{equation} In quantum case, our fields are promoted to operators. So $J^{\mu} \rightarrow \hat{J^{\mu}}$, $Q\rightarrow \hat{Q}$. Conservation of $\hat{Q}$ means $d\hat{Q}/dt=-i[\hat{Q},\hat{H}]=0$, where $\hat{H}$ is the Hamiltonian.

My question is: How to write $\partial_{\mu}J^{\mu}=0$ in quantum case? Do we have $\partial_{\mu}\hat{J^{\mu}}=0$ or $\langle\alpha|\partial_{\mu}\hat{J^{\mu}}|\alpha\rangle=0$ for any ket $|\alpha\rangle$?