327 research outputs found
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
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
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
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
- …