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

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.