Remarks on the Extended Characteristic Uncertainty Relations

Three remarks concerning the form and the range of validity of the state-extended characteristic uncertainty relations (URs) are presented. A more general definition of the uncertainty matrix for pure and mixed states is suggested. Some new URs are provided.Comment: LaTex, 4 pages, no figure

Comment on "On the uncertainty relations and squeezed states for the quantum mechanics on a circle"

It is shown by examples that the position uncertainty on a circle, proposed recently by Kowalski and Rembieli\'nski [J. Phys. A 35 (2002) 1405] is not consistent with the state localization. We argue that the relevant uncertainties and uncertainty relations (UR's) on a circle are that based on the Gram-Robertson matrix. Several of these generalized UR's are displayed and related criterions for squeezed states are discussed.Comment: 5 pages, LaTex2e, 3 figures.ep

On the squeezed states for n observables

Three basic properties (eigenstate, orbit and intelligence) of the canonical squeezed states (SS) are extended to the case of arbitrary n observables. The SS for n observables X_i can be constructed as eigenstates of their linear complex combinations or as states which minimize the Robertson uncertainty relation. When X_i close a Lie algebra L the generalized SS could also be introduced as orbit of Aut(L^C). It is shown that for the nilpotent algebra h_N the three generalizations are equivalent. For the simple su(1,1) the family of eigenstates of uK_- + vK_+ (K_\pm being lowering and raising operators) is a family of ideal K_1-K_2 SS, but it cannot be represented as an Aut(su^C(1,1)) orbit although the SU(1,1) group related coherent states (CS) with symmetry are contained in it. Eigenstates |z,u,v,w;k> of general combination uK_- + vK_+ + wK_3 of the three generators K_j of SU(1,1) in the representations with Bargman index k = 1/2,1, ..., and k = 1/4,3/4 are constructed and discussed in greater detail. These are ideal SS for K_{1,2,3}. In the case of the one mode realization of su(1,1) the nonclassical properties (sub-Poissonian statistics, quadrature squeezing) of the generalized even CS |z,u,v;+> are demonstrated. The states |z,u,v,w;k=1/4,3/4> can exhibit strong both linear and quadratic squeezing.Comment: 25 pages, LaTex, 4 .pic and .ps figures. Improvements in text, discussion on generation scheme added. To appear in Phys. Script

Exact Solutions for the General Nonstationary Oscillator with a Singular Perturbation

Three linearly independent Hermitian invariants for the nonstationary generalized singular oscillator (SO) are constructed and their complex linear combination is diagonalized. The constructed family of eigenstates contains as subsets all previously obtained solutions for the SO and includes all Robertson and Schr\"odinger intelligent states for the three invariants. It is shown that the constructed analogues of the SU(1,1) group-related coherent states for the SO minimize the Robertson and Schr\"odinger relations for the three invariants and for every pair of them simultaneously. The squeezing properties of the new states are briefly discussed.Comment: 17 pages, Latex, no figures; final form to appear in J. Phys.

Nonlinear Fermions and Coherent States

Nonlinear fermions of degree $n$ ($n$-fermions) are introduced as particles with creation and annihilation operators obeying the simple nonlinear anticommutation relation $AA^\dagger + {A^\dagger}^n A^n = 1$. The ($n+1$)-order nilpotency of these operators follows from the existence of unique $A$-vacuum. Supposing appropreate ($n+1$)-order nilpotent para-Grassmann variables and integration rules the sets of $n$-fermion number states, 'right' and 'left' ladder operator coherent states (CS) and displacement-operator-like CS are constructed. The $(n+1)\times(n+1)$ matrix realization of the related para-Grassmann algebra is provided. General $(n+1)$-order nilpotent ladder operators of finite dimensional systems are expressed as polynomials in terms of $n$-fermion operators. Overcomplete sets of (normalized) 'right' and 'left' eigenstates of such general ladder operators are constructed and their properties briefly discussed.Comment: latex, 16 pages, no figure