Lagrangian form of Schr\"odinger equation

Lagrangian formulation of quantum mechanical Schr\"odinger equation is developed in general and illustrated in the eigenbasis of the Hamiltonian and in the coordinate representation. The Lagrangian formulation of physically plausible quantum system results in a well defined second order equation on a real vector space. The Klein-Gordon equation for a real field is shown to be the Lagrangian form of the corresponding Schr\"odinger equation.Comment: To appear in Foundation of Physic

Evolution of the wave function of an atom hit by a photon in a three-grating interferometer

In 1995, Chapman et al. (1995 Phys. Rev. Lett. 75 2783) showed experimentally that the interference contrast in a three-grating atom interferometer does not vanish under the presence of scattering events with photons, as required by the complementarity principle. In this work we provide an analytical study of this experiment, determining the evolution of the atom wave function along the three-grating Mach-Zehnder interferometer under the assumption that the atom is hit by a photon after passing through the first grating. The consideration of a transverse wave function in momentum representation is essential in this study. As is shown, the number of atoms transmitted through the third grating is given by a simple periodic function of the lateral shift along this grating, both in the absence and in the presence of photon scattering. Moreover, the relative contrast (laser on/laser off) is shown to be a simple analytical function of the ratio d_p/\lambda_i, where d_p is the distance between atomic paths at the scattering locus and \lambda_i the scattered photon wavelength. We argue that this dependence, being in agreement with experimental results, can be regarded to show compatibility of the wave and corpuscle properties of atoms.Comment: 8 pages, 4 figure

Trajectory-based interpretation of Young's experiment, the Arago-Fresnel laws and the Poisson-Arago spot for photons and massive particles

We present a trajectory based interpretation for Young's experiment, the Arago-Fresnel laws and the Poisson-Arago spot. This approach is based on the equation of the trajectory associated with the quantum probability current density in the case of massive particles, and the Poynting vector for the electromagnetic field in the case of photons. Both the form and properties of the evaluated photon trajectories are in good agreement with the averaged trajectories of single photons observed recently in Young's experiment by Steinberg's group at the University of Toronto. In the case of the Arago-Fresnel laws for polarized light, the trajectory interpretation presented here differs from those interpretations based on the concept of "which-way" (or "which-slit") information and quantum erasure. More specifically, the observer's information about the slit that photons went through is not relevant to the existence of interference; what is relevant is the form of the electromagnetic energy density and its evolution, which will model consequently the distribution of trajectories and their topology. Finally, we also show that the distributions of end points of a large number of evaluated photon trajectories are in agreement with the distributions measured at the screen behind a circular disc, clearly giving rise to the Poisson-Arago spot.Comment: 8 pages, 5 figure

Should particle trajectories comply with the transverse momentum distribution?

The momentum distributions associated with both the wave function of a particle behind a grating and the corresponding Bohmian trajectories are investigated and compared. Near the grating, it is observed that the former does not depend on the distance from the grating, while the latter changes with this distance. However, as one moves further apart from the grating, in the far field, both distributions become identical.Comment: 10 pages, 7 figure

Electron Transport in Metallic Grains

We discuss electron transport in individual nanometer-scale metallic grains at dilution refrigerator temperatures. In the weak coupling regime, the grains exhibit Coulomb blockade and discrete energy levels. Electron-electron interactions lead to clustering and broadening of quasiparticle states. Magnetic field dependences of tunneling resonances directly reveal Kramers degeneracy and Lande g-factors. In grains of Au, which have strong spin-orbit interaction, g-factors are strongly suppressed from the free electron value. We have recently studied grains in the strong coupling regime. Coulomb blockade persists in this regime. It leads to a suppression in sample conductance at zero bias voltage at low temperatures. The conductance fluctuates with the applied magnetic field near zero bias voltage. We present evidence that the fluctuations are induced by electron spin. This paper reviews the evolving progress in interpreting these observations.Comment: Since the original submission, we have gathered data in weak coupling regime, showing that our initial speculation - that the conductance dip is due to EE interactions in the Altshuler-Aronov sense - is not correct. Instead, it is caused by the Coulomb Blockade. Conference Proceedings, "Quantum Transport and Quantum Coherence" Localisation 2002, Sophia University, Tokyo, August 16-19, to be published in the supplement of the Journal of the Physical Society of Japa