Landscape of uncertainty in Hilbert space for one-particle states

Stefan Weigert
1996 Physical Review A. Atomic, Molecular, and Optical Physics  
The functional of uncertainty J͓͔ assigns to each state ͉͘ the product of the variances of the momentum and position operators. Its first and second variations are investigated. Each stationary point is located on one of a countable set of three-dimensional manifolds in Hilbert space. For a harmonic oscillator with given mass and frequency the extremals are identified as displaced squeezed energy eigenstates. The neighborhood of the stationary states is found to have the structure of a saddle,
more » ... cture of a saddle, thus completing the picture of the landscape of uncertainty in Hilbert space. This result follows from the diagonalization of the second variation of the uncertainty functional, which is not straightforward since J͓͔ depends nonlinearly on the state ͉͘.
doi:10.1103/physreva.53.2084 pmid:9913112 fatcat:c2eamkc7s5fz5oxgyheouuffra