The reflection of narrow slow quantum packets from mirrors

We study the reflection of narrow (in space) quantum wavepackets from nonabsorptive mirrors (totally or partially reflecting). If the initial mean value of the momentum component perpendicular to the mirror surface is less than the momentum uncertainty, then the mean value of this component gradually increases with time at the expense of shrinking the packet in the momentum space. As a consequence, very slow particles moving initially in the direction parallel to the mirror surface will be deflected to appreciable angles, even when they pass at macroscopical distances from the mirror. We give analytical expressions describing the asymptotical behaviour of wavefunctions and density matrices in the coordinate and momentum representations for arbitrary narrow initial packets. We show that the asymptotical mean values and variances do not depend on the phases of the complex reflection and transmission coefficients. Moreover, they are insensitive to the concrete form of the reflective potential in the case of totally reflecting mirrors. For partially reflecting mirrors we introduce the concept of conditional wavefunctions and mean values. The dependences of the asymptotical values of different quantities characterizing the packet (e.g. the 'momentum transformation coefficient' and the 'invariant uncertainty product') on the parameters of the reflecting potential (the height and width or characteristic length of the transition region) are analysed in the examples of the potentials of Epstein's type and their limit cases (the ideal reflecting wall and the delta potential). A possibility for the verification of the effect of quantum deflection in experiments with ultracold atoms is briefly discussed.