The quantum potential: the breakdown of classical symplectic symmetry and the energy of localisation and dispersion

TL;DR

This paper explores how the quantum potential causes the breakdown of classical symplectic symmetry in quantum mechanics, analyzing its composition and effects in different representations and specific quantum systems.

Contribution

It derives general expressions for the quantum potential in both configuration and momentum spaces, revealing its role in symmetry breakdown and energy localization.

Findings

Quantum potential breaks classical symplectic symmetry.

Quantum potential can be expressed as sum of dispersion and localization energies.

Analysis of quantum systems illustrates the quantum potential's role in energy distribution.

Abstract

The composition of the quantum potential and its role in the breakdown of classical symplectic symmetry in quantum mechanics is investigated. General expressions are derived for the quantum potential in both configuration space and momentum space representations. By comparing the configuration space and momentum space representations of the causal interpretation of quantum mechanics, the quantum potential is shown to break the symplectic symmetry that exists between these two representations in classical mechanics. In addition, it is shown that the quantum potential in configuration space may be expressed as the sum of a momentum dispersion energy and a spatial localisation energy; a complementary expression for the quantum potential being found in the momentum representation. The composition and role of the quantum potential in both representations is analysed for a particle in a linear potential and for two eigenstates of the quantum harmonic oscillator.