A simple model of quantum particle is proposed in which the particle in a {it macroscopic} rest frame is represented by a {it microscopic d}-dimensional oscillator, {it s=(d-1)/2} being the spin of the particle. The state vectors are defined simply b
y complex combinations of coordinates and momenta. It is argued that the observables of the system are Hermitian forms (corresponding uniquely to Hermitian matrices). Quantum measurements transforms the equilibrium state obtained after preparation into a family of equilibrium states corresponding to the critical values of the measured observable appearing as values of a random quantity associated with the observable. Our main assumptions state that: i) in the process of measurement the measured observable tends to minimum, and ii) the mean value of every random quantity associated with an observable in some state is proportional to the value of the corresponding observable at the same state. This allows to obtain in a very simple manner the Born rule.
It is shown by examples that the position uncertainty on a circle, proposed recently by Kowalski and Rembielinski [J. Phys. A 35 (2002) 1405] is not consistent with the state localization. We argue that the relevant uncertainties and uncertainty rela
tions (URs) on a circle are that based on the Gram-Robertson matrix. Several of these generalized URs are displayed and related criterions for squeezed states are discussed.
The proposition 1 is incomplete. In some of the examples D(a,b) may not obey the triangle inequality. The paper is withdrawn for further elaboration.
A sufficient condition for a state |psi> to minimize the Robertson-Schr{o}dinger uncertainty relation for two observables A and B is obtained which for A with no discrete spectrum is also a necessary one. Such states, called generalized intelligent s
tates (GIS), exhibit arbitrarily strong squeezing (after Eberly) of A and B. Systems of GIS for the SU(1,1) and SU(2) groups are constructed and discussed. It is shown that SU(1,1) GIS contain all the Perelomov coherent states (CS) and the Barut and Girardello CS while the Bloch CS are subset of SU(2) GIS.
The three ways of generalization of canonical coherent states are briefly reviewed and compared with the emphasis laid on the (minimum) uncertainty way. The characteristic uncertainty relations, which include the Schroedinger and Robertson inequaliti
es, are extended to the case of several states. It is shown that the standard SU(1,1) and SU(2) coherent states are the unique states which minimize the second order characteristic inequality for the three generators. A set of states which minimize the Schroedinger inequality for the Hermitian components of the su_q(1,1) ladder operator is also constructed. It is noted that the characteristic uncertainty relations can be written in the alternative complementary form.
This paper aims to present the pure field part of the newly developed nonlinear {it Extended Electrodynamics} [1]-[3] in non-relativistic terms, i.e. in terms of the electric and magnetic vector fields (${mathbf E},{mathbf B}$), and to give explicitl
y those (3+1)-soliton solutions of the new equations which have the integral properties of photons. The set of solutions to the new equations contains all solutions to Maxwells equations as a subclass, as well as, new solutions, called nonlinear. The important characteristics {it scale factor}, {it amplitude function}, and {it phase function} of a nonlinear solution are defined in a coordinate free way and effectively used. The nonlinear solutions are identified through the non-zero values of two appropriately defined vector fields $vec{cal F}$ and $vec{cal M}$, as well as, through the finite values of the corresponding scale factors. The intrinsic angular momentum (spin) is also defined. A limited superposition principle (interference of nonlinear solutions), yielding the well known classical {it coherence} conditions, is found to exist.
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 Schrodinger 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 Schrodinger relations for the three invariants and for every pair of them simultaneously. The squeezing properties of the new states are briefly discussed.
New uncertainty relations for n observables are established. The relations take the invariant form of inequalities between the characteristic coefficients of order r, r = 1,2,...,n, of the uncertainty matrix and the matrix of mean commutators of the
observables. It is shown that the second and the third order characteristic inequalities for the three generators of SU(1,1) and SU(2) are minimized in the corresponding group-related coherent states with maximal symmetry.
It is shown that any two Hamiltonians H(t) and H(t) of N dimensional quantum systems can be related by means of time-dependent canonical transformations (CT). The dynamical symmetry group of system with Hamiltonian H(t) coincides with the invariance
group of H(t). Quadratic Hamiltonians can be diagonalized by means of linear time-dependent CT. The diagonalization can be explicitly carried out in the case of stationary and some nonstationary quadratic H. Linear CT can diagonalize the uncertainty matrix sigma(rho) for canonical variables p_k, q_j in any state rho, i.e., sigma(rho) is symplectically congruent to a diagonal uncertainty matrix. For multimode squeezed canonical coherent states (CCS) and squeezed Fock states with equal photon numbers in each mode sigma is symplectic itself. It is proved that the multimode Robertson uncertainty relation is minimized only in squeezed CCS.
Overcomplete families of states of the type of Barut-Girardello coherent states (BG CS) are constructed for noncompact algebras $u(p,q)$ and $sp(N,C)$ in quadratic bosonic representation. The $sp(N,C)$ BG CS are obtained in the form of multimode ordi
nary Schrodinger cat states. A set of such macroscopic superpositions is pointed out which is overcomplete in the whole $N$ mode Hilbert space (while the associated $sp(N,C)$ representation is reducible). The multimode squared amplitude Schrodinger cat states are introduced as macroscopic superpositions of the obtained $sp(N,C)$ BG CS.}
