Showing papers on "Canonical coordinates published in 1986"

Coordinates for molecular dynamics: Orthogonal local systems

Abstract: Systems of orthogonal coordinates for the problem of the motion of three or more particles in classical or in quantum mechanics are considered from the viewpoint of applications to intramolecular dynamics and chemical kinetics. These systems, for which the kinetic energy of relative motion is diagonal, are generated by making extensive use of the concept of kinematic rotations, which act on coordinates of different particles and describe their rearrangements. An explicit representation of these rotations by mass dependent matrices allows to relate different particle couplings in the Jacobi scheme, and to build up alternative systems (such as those based on the Radau–Smith vectors or variants thereof): this makes it possible to obtain coordinates which, while being rigorously orthogonal, may approximate closely the local ones, which are based on actual interparticle distances and are in general nonorthogonal. It is also briefly shown that by defining as variables the parameters describing the kinematic rotations it is possible to obtain nonorthogonal systems of coordinates, which are useful in the treatment of collective modes.

Lagrangians for dissipative systems

Abstract: A Lagrangian and Hamiltonian formulation can be given for a damped harmonic oscillator with damping linear in the velocity. The canonical momentum is not equal to the kinetic momentum, and the Hamiltonian is not equal to the energy. On the other hand, a pendulum accreting mass has the same Lagrangian and equation of motion. However, in this case the canonical momentum is equal to the kinetic momentum, and the Hamiltonian is equal to the energy. No ambiguity arises if the physical situation is kept in mind.

Canonical transforms for paraxial wave optics

Abstract: Paraxial geometric optics in N dimensions is well known to be described by the inhomogeneous symplectic group I2N ∧ Sp(2N, ℜ). This applies to wave optics when we choose a particular (ray) representation of this group, corresponding to a true representation of its central extension and twofold cover \(\tilde \Gamma _N = W_N ^ \wedge Mp(2N,\Re )\). for wave optics, the representation distinguished by Nature is the oscillator one. There applies the theory of canonical integral transforms built in quantum mechanics. We translate the treatament of coherent states and other wave packets to lens and pupil systems. Some remarks are added on various topics, including a fundamental euclidean algebra and group for metaxial optics.

Baker–Campbell–Hausdorff relations for supergroups

Abstract: One‐parameter subgroups, the exponential mapping, and canonical coordinates have previously been defined for connected supergroups. In this paper, a technique for obtaining analytical expressions linking different coordinate schemes for supergroups is presented. These are the Baker–Campbell–Hausdorff relations. The method is illustrated in detail with the examples of supergroups based on the supersymmetric quantum‐mechanical superalgebra sqm (2) and on the Inonu–Wigner contraction isop(1/2) of the simple superalgebra osp(1/2).

Dirac’s theory of constraints in field theory and the canonical form of Hamiltonian differential operators

Abstract: A simple algorithm for constructing the canonical form of Hamiltonian systems of evolution equations with constant coefficient Hamiltonian differential operators is given. The result of the construction is equivalent to the canonical system derived using Dirac’s theory of constraints from the corresponding degenerate Lagrangian.

On the theory of one‐parameter subgroups of supergroups

Abstract: The one‐parameter subgroups of supergroups are defined and their primary properties are delineated. Parametrization schemes analogous to those used in Lie group theory are investigated. In particular, exponentiation is introduced and related to the one‐parameter subgroups. Canonical coordinates of three kinds are investigated, the third of which has no analog in conventional Lie theory. We illustrate our results with the examples of the supersymmetric quantum‐mechanical superalgebra sqm(2) and the simple superalgebra osp(1/2).

On stochastic Hamiltonian mechanics for diffusion processes

Abstract: In a close investigation into Nelson’s stochastic mechanics, a stochastic Hamiltonian mechanics for diffusion processes is formulated The key point is the observation that the equation of motion in Nelson’s mechanics can be put into a stochastic Hamilton-Jacobi equation and hence stochastic Hamilton’s equations result with a suitably defined Hamiltonian function The system of the diffusion processx(t) and the momentum processp(t) is called a Hamiltonian dynamical system, if for a given diffusion matrix these processes are determined by stochastic Hamilton’s equations and the continuity equation In the present Hamiltonian mechanics, canonical transformations are formulated with a method analogous to that in classical mechanics Particularly, inhomogeneous linear canonical trasformations are discussed in connection with the Fourier transform in quantum mechanics An example is given of the canonical transformations for a free fall

Action-angle formulation of angular momentum

Abstract: Angular-momentum theory is formulated in terms of quantum action-angle variables. The quantum canonical transformation between the spherical polar and action-angle coordinates and momenta is constructed, and the Hamiltonian and energies in terms of the action variables are determined. Various sets of action-angle variables are considered.

Active and passive views of gauge invariance in quantum mechanics

Abstract: For a quantum mechanical particle in a classical electromagnetic field, the active and passive views of gauge invariance are compared. In the active point of view, both the wavefunction and the potentials are transformed so that the Schrodinger equation is form invariant. In the passive point of view, a change in the representation of the canonical momentum is made which just cancels the change in the gauge of the potentials. The condition that a gauge‐invariant operator has the same expectation value in all gauges gives a functional equation, the solution of which shows that gauge‐invariant operators depend only on the kinetic momentum or coordinates.

Canonical formalism in equilibrium thermodynamics

Abstract: Gibbs’ principle is extended to systems subject to generalized constraints. It is shown that equilibrium thermodynamics can be presented under a canonical structure. Also, it is shown that the canonical phase space provides a natural representation for equilibrium states.

Bifurcation of solutions of a field‐reversed cylindrical model

Abstract: It is shown that the self‐consistent solutions of Vlasov steady states for the double delta distribution model of cylindrical field‐reversed configurations, generated by large ion orbits, have bifurcations within a definite interval of values for the total number of ions. The conjugate states have the same number of particles and the same radius of confinement but different azimuthal canonical momentum, orbit patterns, and magnetic fields. It is also shown that the states with a lower canonical momentum have a smaller energy content.

The rational canonical form of a matrix

Abstract: The purpose of this paper is to provide an efficient algorithmic means of determining the rational canonical form of a matrix using computational symbolic algebraic manipulation packages, and is in fact the practical implementation of a classical mathematical method.

Generalized gauge independence and the physical limitations on the von Neumann measurement postulate

Abstract: An analysis is presented of the significance and consequent limitations on the applicability of the von Neumann measurement postulate in quantum mechanics. Directly observable quantities, such as the expectation value of the velocity operator, are distinguished from mathematical constructs, such as the expectation value of the canonical momentum, which are not directly observable. A simple criterion to distinguish between the two types of operators is derived. The non-observability of the electromagnetic four-potentials is shown to imply the non-measurability of the canonical momentum. The concept of a mechanical gauge is introduced and discussed. Classically the Lagrangian is nonunique within a total time derivative. This may be interpreted as the freedom of choosing a “mechanical” (M) gauge function. In quantum mechanics it is often implicitly assumed that the M-gauge vanishes. However, the requirement that directly observable quantities be independent of the arbitrary mechanical gauge is shown to lead to results analogous to those derived from the requirement of electromagnetic gauge independence of observables. The significance of the above to the observability of transition amplitudes between field-free energy eigenstates in the presence (and absence) of electromagnetic fields is discussed. E- and M-gauge independent transition amplitudes between field-free energy eigenstates in the absence of electromagnetic fields are defined. It is shown that, in general, such measurable amplitudes cannot be defined in the presence of externally applied time-dependent fields. Transition amplitudes in the presence of time-independent fields are discussed. The path dependence of previous derivations of E-gauge independent Hamiltonians and/or transition amplitudes in the presence of electromagnetic fields are related to the inherent M-gauge dependence of these quantities in the presence of such fields.

Canonical quantization without conjugate momenta

Abstract: In the traditional form of canonical quantization, certain field components (not having “conjugate” momenta) must be regarded as noncanonical. This long-known distinction enters modern gauge theories, when they are canonically quantized as by Kugo and Ojima. We avoid that peculiarity by not using any conjugate “momenta” at all. In our formulation, canonical quantization can be related to Feynman's path integral.

Lagrangian formulation of a general Hamiltonian theory with constraints

Abstract: A Lagrangian formulation is constructed for a general Hamiltonian theory with constraints. A modification is proposed of the standard procedure of the “Hamiltonianization” of a Lagrangian theory in the case when the Lagrangian theory has primary constraints. The obtained results are used to establish the Lagrangian form of infinitesimally small canonical transformations in the Hamiltonian formalism.

Canonical transformations and the equivalence problem

Abstract: Several equivalence relations for Hamiltonian systems are studied. The relationship to the theory of canonical transformations is analyzed. In the hyperregular case, the results are transformed into the Lagrangian formulation. The gauge group of Lagrangian mechanics is obtained by looking at the generating functions for canonical fiber invariant transformations. An intrinsic proof of a theorem of Henneaux [M. Henneaux, Ann. Phys. (NY) 140, 45 (1982)] is given.