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\(\begin{eqnarray}\lefteqn{i\hbar{\partial\over\partial t}G(\mathbf x,t;\mathbf x{'},t{'})}&&\\
&=& \langle0|i\hbar{\partial\over\partial t} \psi(\mathbf x,t)\psi^{\dagger}(\mathbf x{'}, t{'})|0\rangle\theta(t-t{'}) +i\hbar\langle0|\psi(\mathbf x,t)\psi^\dagger(\mathbf x{'},t{'})|0\rangle \delta(t-t{'})\\
&&+i\hbar \langle0|\psi^{\dagger}(\mathbf x{'}, t{'}){\partial\over\partial t} \psi(\mathbf x,t) |0\rangle\theta(t{'}-t)-i\hbar\langle0|\psi^\dagger(\mathbf x{'}, t{'})\psi(\mathbf x,t)|0\rangle\delta(t{'}-t)
\\&& \qquad\qquad \mbox{\HighLight{we have used \(\quad\dd{t}\theta(t-t{'})=\delta(t-t{'})= -\dd{t} \theta(t{'}-t)\) }} \\
&=&-{\hbar^2\over2m}\nabla^2\langle0|\psi(\mathbf x,t)\psi^\dagger(\mathbf x{'} , t{'})|0\rangle\theta(t-t{'})-{\hbar^2\over2m}\nabla^2\langle0| \psi^\dagger(\mathbf x{'}, t{'}) \psi(\mathbf x,t)|0\rangle\theta(t{'}-t)\\
&&+i\hbar\langle0|\big[\psi(\mathbf x,t),\psi^\dagger(\mathbf x{'}, t{'})\big]|0\rangle\delta(t-t{'}) \end{eqnarray}\)
\begin{eqnarray}x+y&=&1 \\a+b&=3&\\ u+v&=&10\end{eqnarray}
\[\begin{eqnarray*}x+y&=&1 \\a+b&=3&\\ u+v&=&10\end{eqnarray*}\]
\(\begin{eqnarray}x+y&=&1 \\a+b&=3&\\ u+v&=&10\end{eqnarray}\)
\[u_i(\mathbf x)=\frac{e^{\mathbf k_i\dot{\mathbf x}}}{(2\pi)^{3/2}}, \quad u_f(\mathbf x)=\frac{e^{\mathbf k_f\dot{\mathbf x}}}{(2\pi)^{3/2}}.\]
