Quantum mechanical pure states with gaussian wave functions

Andre G. Campos, Renán Cabrera, Denys I. Bondar +1 more

In nonrelativistic quantum mechanics, Hudson's theorem states that a Gaussian wave function is the only pure state corresponding to a positive Wigner function (WF). We explicitly construct non-Gaussian Dirac spinors with positive relativistic WFs. These pure relativistic states are coherent superpositions of particles and antiparticles, while the existence of positive WFs exclusive composed of particles is conjectured. These observations may open new directions in relativistic quantum information theory.

G. Ottonello

Nobuharu Nakajima

A method of reconstructing a wave function from part of the Q function in a pure state of light is proposed. This method involves the use of a noniterative phase-retrieval method based on Gaussian filtering, which allows one to determine a convolution of the wave function with a known Gaussian function for the vacuum state from measured data of the Q function along only two parallel lines in phase space. By using a deconvolution process, the wave function is reconstructed from the convolution, provided that the resolution of the reconstructed wave function is limited by the extent of the Fourier transform of the Gaussian function. Numerical simulations demonstrate the applicability of this method to the reconstruction of the wave function from the Q function obtained from the photoelectric counting statistics in unbalanced homodyne detection.

X WANG, T HIROSHIMA, A TOMITA +1 more

Giuliano Benenti, Giuliano Strini

A pure quantum state can be equivalently represented by means of its wave function ψ(q) or by the Fermi function gF(q,p), with q and p coordinates and conjugate momenta of the system of interest. We show that a Gaussian wave packet can be conveniently visualized in phase space by the curve gF(q,p)=0. The change in time of the gF=0 curve is calculated for a Gaussian packet evolving freely or under a constant or a harmonic force, and the spreading or shrinking of the packet is easily interpreted in phase space. We also discuss a gedanken prism microscope experiment for measuring the position-momentum correlation. This gedanken experiment, together with the well-known Heisenberg microscope and von Neumann velocimeter, is sufficient to fully determine the state of a Gaussian packet.

Lipo Wang, Robert F. O’Connell