Quantum-Classical Correspondence through the Momentum Function

A. O. Bolivar

Ricardo Antonio Mosna, Ian P. Hamilton, Luigi Delle Site

We propose an approach to quantum–classical correspondence based on a deformation of the momentum and kinetic operators of quantum mechanics. Making use of the factorization method, we construct classical versions of the momentum and kinetic operators which, in addition to the standard quantum expressions, contain terms that are functionals of the N -particle density. We show that this implementation of quantum–classical correspondence is related to Witten's deformation of the exterior derivative and Laplacian, introduced in the context of supersymmetric quantum mechanics. The corresponding deformed action is also shown to be related to the Fisher information. Finally, we briefly consider the possible relevance of our approach to the construction of kinetic-energy density functionals.

Jan Mostowski, Joanna Pietraszewicz

The Wigner function is a quantum analogue of the classical joined distribution of position and momentum. As such is should be a good tool to study quantum-classical correspondence. In this paper, the classical limit of the Wigner function is shown using the quantum harmonic oscillator as an example. The Wigner function is found exactly for all states. The semi-classical wavefunctions for highly excited states are used as the approach to the classical limit. Therefore, one can found the classical limit of the Wigner function for highly excited states and shown that it gives the classical microcanonical ensemble.

Sumita Singh, Smriti P Suman, Vijay A. Singh

Abstract The correspondence principle provides a prescription to connect quantum physics to classical. It asserts that the physical quantities evaluated quantum mechanically approach their respective classical values for large quantum numbers. This has been shown for the pedagogically important cases of the particle in a box and a harmonic oscillator. However, a particle in a constant field has a wave function related to the Airy function and has at best been treated numerically. Employing energy eigenstates we obtain the expectation values of the position, the momentum and their moments upto fourth order, rigorously and without resorting to numerical or graphical techniques. We compare them with the corresponding classical values. We also examine the uncertainty product for the system.

Yu.I. Zaparovannyi