Three-dimensional Quantum Slit Diffraction and Diffraction in Time

We derive a self-consistent formula for the quantum propagator of the quantum diffraction by a reflecting (or non-absorbing) screen with an arbitrary aperture, in the transmission region. To achieve this, we use the Green function approach (Brukner and Zeilinger, Phys. Rev. A. 56(5), 3804–3824 1997) that we extend to mixed boundary conditions (including Dirichlet and Neumann) on the shutter’s screen and to any initial wave functions, such as Gaussian wave packets. To illustrate our results, we apply this method to the famous rectangular slit diffraction problem. It allows us to take into account the effect of the quantum nature of the motion perpendicular to the screen where the diffraction-in-time phenomenon appears in this direction. Then we derive corrections to the standard semi-classical formula and the diffraction pattern. We also point out situations in which this might be observable. In particular, we discuss the diffraction in space and time in the presence of gravity.