[QUE/QFT-01005] QFT-PROBLEM

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The equation of motion of free Schrodinger field obey the equation \begin{equation} i\hbar \dd[\psi]{t} = -\frac{\hbar^2}{2m}\nabla^2 \psi . \end{equation} The Green function of this equation obeys the partial differential equation \begin{equation} i\hbar \dd{t}G(x,x{'}; t,t{'}) + \frac{\hbar^2}{2m}\nabla^2 G(x,x{'}; t,t{'})= \delta(x-x{'})\delta(t-t{'}) . \end{equation} Taking Fourier transform of the above equation, and using suitable contour in the complex plane show that the retarded Green function is given by \begin{equation} G(x,x{'}; t,t{'}) = \Big(\frac{m}{2\pi i \hbar (t-t{'})}\Big)^{1/2} \exp\left(\frac{im(x-x{'})^2}{2\hbar{(t-t{'})}}\right)\theta(t-t{'}). \end{equation}