1,708 research outputs found
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
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
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 (-fermions) are introduced as particles
with creation and annihilation operators obeying the simple nonlinear
anticommutation relation . The
()-order nilpotency of these operators follows from the existence of
unique -vacuum. Supposing appropreate ()-order nilpotent para-Grassmann
variables and integration rules the sets of -fermion number states, 'right'
and 'left' ladder operator coherent states (CS) and displacement-operator-like
CS are constructed. The matrix realization of the related
para-Grassmann algebra is provided. General -order nilpotent ladder
operators of finite dimensional systems are expressed as polynomials in terms
of -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
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