Correlation Between Position and Momentum. A Phase-Space View

Thomas Schürmann, Ingo Hoffmann

Marco A. Tapia-Valerdi, I. Ramos-Prieto, Francisco Soto +2 more

It is generally thought that, in the same way that position and momentum are canonical variables in quantum mechanics, the number operator should have a canonical operator, i.e., there should be a possibility of defining a well-defined phase operator associated with it. From the point of view of the phase space, it becomes clearer as position and momentum uncertainties are well reflected. Indeed, the same occurs in the phase space for phase and intensity. In this contribution, we do a small survey on the possibilities of defining a phase operator via discrete Fourier operators and introduce some possible applications.

Koustav D. Chakladar, Santanu Mondal, Jayanta K. Saha

Abstract Despite extensive studies on the position-space electronic distributions for the ground state of the He atom trapped in an impenetrable spherical cavity, the corresponding momentum-space analysis has received little attention. The present article tries to bridge this gap by adopting an even-powered expansion of correlated coordinates in a Hylleraas basis consistent with the Dirichlet boundary condition. The corresponding momentum-space wavefunction is determined by the 3D Fourier transform of the explicitly correlated position-space wavefunctions. The analytic expression of the momentum-space wavefunction, and thereby the electronic density, allows us to examine the asymptotic limits of the same. With increased compression, the two-particle density distribution changes, and in the high compression regime, a crystalline-like structure in the contour map starts to appear. The corresponding topological complexity varies w.r.t. the cavity radius by maintaining symmetry along the equal momenta line. As a consequence, the one-particle momentum-space density and the derived Compton profile show oscillatory structure in the high-momentum regime. The evaluation of Shannon entropy provides further insights into the problem from the quantum information theoretic point of view. The estimation of position and momentum entanglement entropy measures is carried out here for the first time to exhibit the complete picture of the momentum correlation. Our investigation yields that despite showing all the contrasting differences in various density distributions and statistical information measures of the conjugate position and momentum spaces, the quantum correlation measure in terms of entanglement entropy remains invariant. Additionally, the Compton profiles are theoretically evaluated and subsequently compared with available theoretical and experimental results, demonstrating overall good agreement and consistency.

Andrei Yu. Khrennikov

Spiros Konstantogiannis

Kyu-Won Park, Soojoon Lee, Kabgyun Jeong

Avoided level crossings, commonly associated with quantum chaos, are typically interpreted as signatures of eigenstate hybridization and spatial delocalization, often viewed as ergodic spreading. We show that, contrary to this expectation, increasing chaos in quantum billiards enhances mutual information between conjugate phase space variables, revealing nontrivial correlations. Using an information-theoretic decomposition of eigenstate entropy, we demonstrate that spatial delocalization may coincide with increased mutual information between position and momentum. These correlations track classical invariant structures in phase space and persist beyond the semiclassical regime, suggesting a robust information-theoretic manifestation of quantum-classical correspondence.