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Schrödinger conceived his wave equation having in mind de Broglie’s famous relation from which we learnt to attribute complementary behavior to quantum objects depending on the experimental situation in question. He also thought of a wave in the sense of classical waves, like electromagnetic waves and others. However, the space-time asymmetry of the equation with governs quantum phenomena lead the scientific community to investigate the new physics this specific wave was about to unveil. It turns out that in certain experimental condition classical light has its behavior dictated by a bidimensional Schrödinger equation for a free particle. This fact is well known for several years (Yariv, 1991; Snyder & Love, 1991; Berman, 1997; Marte & Stenholm, 1997). For this special kind of waves it is possible to define the analog of a Hilbert space and operators which do not commute (as reviewed in section 2) in such a way that the mathematical analogy becomes perfect. A natural question emerging in this context, and the case of the present investigation is the following: how far, in the sense of leaning new physics, can we take this analogy ? We have been able to show that the generalized uncertainty relation by Robertson and Schrödinger, naturally valid for paraxial waves, can shed new light on the physical context of a beautiful phenomenon, long discovered by Gouy (Gouy, 1890; 1891) which is an anomalous phase that light waves suffer in their passage by spatial confinement. This famous phase is directly related to the covariance between momentum and position and since for the “free particles" we are considering xx pp 2 xp constant we see that Gouy phase can be indirectly measured from the coordinate and momentum variances, quantities a lot easier to measure than covariance between x and p. On the other hand, as far as free atomic particles are concerned, experiments elaborated to test the uncertainty relation (Nairz et al., 2002) will reveal to us the matter wave equivalence of Gouy phase. Unfortunately the above quoted experiment was not designed to determine the phase and that is the reason why, so far, we have only an indirect evidence of the compatibility of theory and experiment. The last aim of our research is to try to encourage laboratories with facilities involving microwave cavities and atomic beams to perform an experiment to obtain the Gouy phase for matter waves. We believe that Gouy phase for matter waves could have important applications in the field of quantum information. The transversal wavefunction of an atom in a beam state can be treated not only as a continuous variable system, but also as an infinite-dimensional discrete system. 4

Gouy Phase and Matter Waves