Fermi's Trick and Symplectic Capacities: A Geometric Picture of Quantum States

TL;DR

This paper explores the geometric representation of quantum states using symplectic capacities, extending the concept of quantum blobs to excited states of the generalized harmonic oscillator, revealing new links between symplectic topology and quantum mechanics.

Contribution

It introduces a novel application of Ekeland–Hofer symplectic capacities to characterize Fermi functions of excited quantum states, bridging symplectic topology and quantum physics.

Findings

Fermi's function for Gaussian states can be characterized by symplectic capacities.

Symplectic topology provides new insights into quantum state geometry.

Extension of quantum blobs to excited states of harmonic oscillators.

Abstract

We extend the notion of quantum blob studied in previous work to excited states of the generalized harmonic oscillator in n dimensions. This extension is made possible by Fermi's observation in 1930 that the state of a quantum system may be defined in two different (but equivalent) ways, namely by its wavefunction {\Psi} or by a certain function g_{F} on phase space canonically associated with {\Psi}. We study Fermi's function when {\Psi} is a Gaussian (generalized coherent state). A striking result is that we can use the Ekeland--Hofer symplectic capacities to characterize the Fermi functions of the excited states of the generalized harmonic oscillator, leading to new insight on the relationship between symplectic topology and quantum mechanics.