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

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.