Landscape of uncertainty in Hilbert space for one-particle states

Article English OPEN
Weigert, S. (1996)

The functional of uncertainty J[¿] 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, 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 J[¿] depends nonlinearly on the state ¿¿>.
  • References (13)
    13 references, page 1 of 2

    @1# M. M. Nieto and D. R. Truax, Phys. Rev. Lett. 71, 2843 ~1993!.

    @2# R. Jackiw, J. Math. Phys. 9, 339 ~1968!.

    @3# From now on the product of the variances of position and momentum operator will be referred to as ''uncertainty.''

    @4# C. Aragone, E. Chalbaud, and S. Salamo, J. Math. Phys. 17, 1963 ~1976!.

    @5# T. Opatrny´, J. Phys. A ~to be published!.

    @6# Ph. M. Morse and H. Feshbach, Methods of Theoretical Physics ~McGraw Hill, New York, 1953!, Vol. I.

    @7# For vanishing Dq or Dp, Eq. ~9! is easily seen to define position and momentum eigenstates, respectively.

    @8# D. Stoler, Phys. Rev. D 1, 3217 ~1970!.

    @9# R. F. Bishop and A. Vourdas, Phys. Rev. A 50, 4488 ~1994!.

    @10# A. Perelomov, Generalized Coherent States and Their Applications ~Springer, Berlin, 1986!.

  • Metrics
    No metrics available
Share - Bookmark