We give a brief review of the theory of quantum and optical universal invariants, i.e., certain combinations of the second- and higher-order moments (variances) of quantum-mechanical operators or the transverse phase-space coordinates of optical paraxial beams that are preserved in time (or along the axis of the beam) independently of the concrete form of the coefficients of the Hamiltonian or the parameters of the optical system, provided that the effective Hamiltonian is either a generic quadratic form of the generalized coordinate-momenta operators or a linear combination of generators of certain finite-dimensional algebras. Using the phase-space representation of quantum mechanics (paraxial optics) in terms of the Wigner function, we elucidate the relation between the quantum invariants and the classical universal integral invariants of Poincaré and Cartan. The specific features of the Gaussian beams are discussed as examples.
We give a brief review of the theory of quantum and optical universal invariants, i.e., certain combinations of the second- and higher-order moments (variances) of quantum-mechanical operators or the transverse phase-space coordinates of optical paraxial beams that are preserved in time (or along the axis of the beam) independently of the concrete form of the coefficients of the Hamiltonian or the parameters of the optical system, provided that the effective Hamiltonian is either a generic quadratic form of the generalized coordinate-momenta operators or a linear combination of generators of certain finite-dimensional algebras. Using the phase-space representation of quantum mechanics (paraxial optics) in terms of the Wigner function, we elucidate the relation between the quantum invariants and the classical universal integral invariants of Poincaré and Cartan. The specific features of the Gaussian beams are discussed as examples.
Universal quantum integrals of motion are introduced, and their relation with the universal quantum invariants is established. The invariants concerned are certain combinations of the second- and higher-order moments (variances) of quantum-mechanical operators, which are preserved in time independently of the concrete form of the coefficients of the Schrödinger equation, provided the Hamiltonian is either a generic quadratic form of the coordinate and momenta operators, or a linear combination of generators of some finite-dimensional algebra (in particular, any semisimple Lie algebra). Using the phase space representation of quantum mechanics in terms of the Wigner function, the relations between the quantum invariants and the classical universal integral invariants by Poincaré and Cartan are elucidated. Examples of the `universal invariant solutions' of the Schrödinger equation, i.e. self-consistent eigenstates of the universal integrals of motion, are given. Applications to the physics of optical and particle beams are discussed.
Wigner's quasi probability and related functional and operator methods of quantum mechanics have recently played an important role in optics. We present an account of some of these developments. The symmetry structures underlying the ray and wave approaches to paraxial optics are explored in some detail, and the manner in which the Wigner phase-space representation captures the merits of both approaches is brought out. A fairly self-contained analysis of the second or intensity moments of general astigmatic partially coherent beams and of their behavior under transmission through astigmatic first-order optical systems is presented. Geometric representations of the intensity moments that render the quality parameters or polynomial invariants manifest are discussed, and the role of the optical uncertainty principle in assigning unbeatable physical bounds for these invariants is stressed. Measurement of the ten intensity moments of an astigmatic partially coherent beam is considered.
In the first part of the lectures dynamical invariants in classical mechanics and conventional quantum mechanics will be considered. In particular, we will begin with some remarks on classical mechanics and on quantization in order to establish the theory in the form that will be used later on. Starting from the time‐dependent Schrödinger equation, the dynamics of Gaussian wave packets and Ermakov invariants, the time‐dependent Green function/Feynman kernel, quantum‐classical connections, energetics and Lagrange—Hamilton formalism for quantum uncertainties, momentum space representation and the relation between the Wigner function and the Ermakov invariant will be discussed. The representation of canonical transformations in time‐independent and time‐dependent quantum mechanics, factorization of the Ermakov invariant and generalized creation/annihilation operators will be studied. Subsequently, the time‐independent Schrödinger equation, leading to nonlinear quantum mechanics related to Riccati/Ermakov systems as well as the occurrence of Riccati/Ermakov systems in the treatment of Bose—Einstein condensates via the so‐called moment method will be analyzed.In part two, irreversible dynamics of dissipative systems, classical and quantum mechanical descriptions and corresponding invariants will be treated. After some general remarks on classical and quantum mechanics with unitary time‐evolution and energy conservation, phenomenological Langevin and Fokker—Planck equations, master equations in classical and quantum mechanics and the system‐plus‐reservoir approach will be mentioned briefly. Then follows a more detailed discussion of modified Schrödinger equations and, particularly, of a nonlinear Schrödinger equation with complex logarithmic nonlinearity; its properties, solutions, invariants and energetics will be studied. Finally, a comparison with a classical description in expanding coordinates will lead to a non‐unitary connection between the logarithmic nonlinear Schrödinger equation, the Caldirola‐Kanai approach and the expanding coordinate approach.
In the first part of the lectures dynamical invariants in classical mechanics and conventional quantum mechanics will be considered. In particular, we will begin with some remarks on classical mechanics and on quantization in order to establish the theory in the form that will be used later on. Starting from the time‐dependent Schrödinger equation, the dynamics of Gaussian wave packets and Ermakov invariants, the time‐dependent Green function/Feynman kernel, quantum‐classical connections, energetics and Lagrange—Hamilton formalism for quantum uncertainties, momentum space representation and the relation between the Wigner function and the Ermakov invariant will be discussed. The representation of canonical transformations in time‐independent and time‐dependent quantum mechanics, factorization of the Ermakov invariant and generalized creation/annihilation operators will be studied. Subsequently, the time‐independent Schrödinger equation, leading to nonlinear quantum mechanics related to Riccati/Ermakov systems as well as the occurrence of Riccati/Ermakov systems in the treatment of Bose—Einstein condensates via the so‐called moment method will be analyzed.In part two, irreversible dynamics of dissipative systems, classical and quantum mechanical descriptions and corresponding invariants will be treated. After some general remarks on classical and quantum mechanics with unitary time‐evolution and energy conservation, phenomenological Langevin and Fokker—Planck equations, master equations in classical and quantum mechanics and the system‐plus‐reservoir approach will be mentioned briefly. Then follows a more detailed discussion of modified Schrödinger equations and, particularly, of a nonlinear Schrödinger equation with complex logarithmic nonlinearity; its properties, solutions, invariants and energetics will be studied. Finally, a comparison with a classical description in expanding coordinates will lead to a non‐unitary connection between the logarithmic nonlinear Schrödinger equation, the Caldirola‐Kanai approach and the expanding coordinate approach.
This paper deals with certain aspects of the phase-space formulation of quantum mechanics. Noting the ambiguities involved in quantizing the classical phase-space dynamics, a generalized quantum-mechanical phase space is introduced in terms of eigenvalues of two mutually incompatible complete sets of operators. General definitions are given for the phase-space distribution function corresponding to a given abstract state of the system and the phase-space function for an arbitrary operator. Our formulation goes over to the Wigner formulation when the complete sets correspond to position and momentum operators in Cartesian coordinates. Explicit examples are discussed with use of action-angle and polar coordinates for simple one-, two-, and three-dimensional systems. Symmetry changes in phase space are discussed by applying Gel'fand-Levitan transformations and it is shown that, in general, the degeneracies of motion in phase space are not reflected in the energy eigenvalues.
In geometric optics there is a natural distinction between the paraxial and aberration regimes, which contain respectively the linear and nonlinear canonical transformations of position and momentum in the phase space. In the Lie-theoretical presentation, linear inhomogeneous transformations are generated by linear and quadratic functions of the phase space, while aberrations of increasing order are generated by homogeneous functions with higher powers of these coordinates. In a way parallel but distinct from the Schrödinger quantization of continuous classical systems, we quantize the geometric optical model into discrete, finite-dimensional systems based on the Lie algebra , whose wavefunctions are N -point signals, phase space is a sphere and transformations are represented by the N × N unitary matrices that form the group . We factor this group into -linear and nonlinear unitary transformations of phase space and classify all its N 2 − 4 aberrations. This offers a new parametrization of based on a chosen subgroup.