Sakurai says that the propagator is simply the Green's function for the time-dependent wave equation satisfying
[−ℏ22m▽″2+V(x″)−ih∂∂t]K(x″,t;x′,t0)=−iℏδ3(x″−x′)δ(t−t0)
with the boundary condition
K(x″,t;x′,t0)=0
for $t
I don't have any idea about where the −iℏδ3(x″−x′)δ(t−t0) term comes from, and the propagator must be equal to zero when $t
Answer
Hint : Check if this "modified" Schrodinger equation is satisfied by the "modified" propagator \begin{equation} \widetilde{K}(\mathbf{x''},t \; \boldsymbol{;} \;\mathbf{x'},t_{0})=\theta(t-t_{0})\;K(\mathbf{x''},t;\mathbf{x'},t_0) \tag{01} \end{equation} where \;\theta(t-t_{0})\; the unit step function with property \begin{equation} \dfrac{\partial \theta (t-t_{0}) }{\partial t}=\dfrac{d \theta (t-t_{0}) }{d t}=\delta (t-t_{0}) \tag{02} \end{equation}
Note that
\begin{equation} \dfrac{\partial \widetilde{K}}{\partial t}=\dfrac{\partial (\theta K) }{\partial t}=\theta\;\dfrac{\partial K}{\partial t}+K\;\dfrac{\partial \theta }{\partial t} \tag{03} \end{equation} and
\begin{equation} K\;\dfrac{\partial \theta }{\partial t}=K(\mathbf{x''},t \; \boldsymbol{;} \;\mathbf{x'},t_{0}) \delta (t-t_{0})=K(\mathbf{x''},t_{0} \; \boldsymbol{;} \;\mathbf{x'},t_{0}) \delta (t-t_{0})=\delta^{3}(\mathbf{x''}-\mathbf{x'})\delta (t-t_{0}) \tag{04} \end{equation}
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