Transformations In Phase Space Research Articles

The Condon reflection principle in collision dynamics

The final internal state distributions obtained by collision dynamics and electronic spectroscopy are shown, by semi-classical arguments, to have a common origin in the properties of the relevant classical transformation in phase space. Both cases allow for the development of a Condon reflection spectrum, which mirrors the shape of the initial internal wavefunction, ultimately because the initial classical orbit encloses an area of (2n 1 + 1)πħ, which is conserved during the transition. The appearance of such structure is therefore independent of any specific Franck-Condon feature in the scattering mechanism; but it is shown to require a direct rather than a compound mechanism, and the absence of significant competition between different channels (reactive, inelastic and quenching). Such structure may in principle be observed in any type of channel, but the absence of sufficient interaction may preclude its observation in purely inelastic scattering except at very high collision energies. Cases suggested ...

Covariant formulation of non-equilibrium statistical thermodynamics

The Fokker Planck equation is considered as the master equation of macroscopic fluctuation theories. The transformation properties of this equation and quantities related to it under general coordinate transformations in phase space are studied. It is argued that all relations expressing physical properties should be manifestly covariant, i.e. independent of the special system of coordinates used. The covariance of the Langevin-equations and the Fokker Planck equation is demonstrated. The diffusion matrix of the Fokker Planck equation is used as a contravariant metric tensor in phase space. Covariant drift vectors associated to the Langevin- and the Fokker Planck equation are found. It is shown that special coordinates exist in which the covariant drift vector of the Fokker Planck equation and the usual non-covariant drift vector are equal. The physical property of detailed balance and the equivalent potential conditions are given in covariant form. Finally, the covariant formulation is used to study how macroscopic forces couple to a system in a non-equilibrium steady state. A general fluctuation-dissipation theorem for the linear response to such forces is obtained.

Symmetries and constraints in generalized Hamiltonian dynamics

A general analysis of symmetries and constraints for singular Lagrangian systems is given. It is shown that symmetry transformations can be expressed as canonical transformations in phase space, even for such systems. The relation of symmetries to generators, constraints, commutators, and Dirac brackets is clarified.

Fluxes of =50-keV protons and =30-keV electrons at ∼35 RE , 1. Velocity anisotropies and plasma flow in the magnetotail

The NOAA/APL energetic particle experiment (EPE) on Imp 7 (Explorer 47) is capable of measuring hot plasma flow with a temporal resolution of 20 s if the densities exceed approx.0.1 cm/sup -3/. The lowest-energy electron channel (30--90 keV) can detect plasma with kTapprox.1 keV, while the lowest-energy proton channel (50--200 keV) can detect plasmas at similar temperatures if they have a bulk velocity of approximately-greater-than50 km s/sup -1/. The bulk velocities can be deduced from the measured proton angular distributions (16 sectors in the ecliptic) by using straightforward expressions derived from phase space transformations of a Maxwellian distribution. Representative examples are given for quiet time flow in the magnetosheath (Vapprox.500 km s/sup -1/, kTapprox. =5 keV) and the magnetotail (Vapprox.70 km s/sup -1/, kT/subp/approx.10 keV), with T/subp//T/sube/>3 in both cases. A detailed example of a class of high-intensity events that occur in the plasma sheet in the dusk sector of the magnetosphere is also presented. The 1-hour event (0520--0620 UT, October 3, 1973) was associated with a 1500-..gamma.. depression in the H component of the geomagnetic field at Barrow, Alaska. Tailward velocities of 1260 km s/sup -1/ were deduced during the expansive phase of the substorm, and sunward velocities ofmore » approx.950 km s/sup -1/ during the recovery. Comparison with higher-energy proton measurements on the same spacecraft over the first year of data (September 1972 to October 1973) reveals that the larger events often have a nonthermal tail, although bulk velocities, densities, and temperatures deduced from the smaller events are consistent with previous measurements of plasma flow by other workers. (AIP)« less

Realizations of the central extension of the inhomogeneous symplectic algebra as time dependent invariance algebras of nonrelativistic quantum systems

We interpret the one-element central extension of the Lie algebra of the group of real inhomogeneous linear canonical transformations of phase space as a time dependent invariance algebra in the Schrödinger picture. By limiting to a two-dimensional space quantum system, we exhibit all the nonconjugate Hamiltonians contained in the symplectic algebra which, in the Heisenberg picture, plays the role of dynamical algebra and contains several known degeneracy and spectrum generating algebras.

Symmetrized double-monochromator systems

A system of three triplet lenses is investigated with regard to the trajectory symmetrization in magnetic double-monochromator systems. Double-dispersive and non-dispersive telescopic modes can be realized for mirror-symmetric as well as antisymmetric arrangements without changing the hardware geometry. Beside the symmetry fit certain asymmetry constraints can be adjusted in order to match special phase-space transformation properties. The resulting ion-optical structures are discussed with the aid of geometrical light optics as well as first-order matrix theory.

Homogeneous canonical transformations in phase space: On necessary conditions for the existence of symplectic actions of simple lie groups on differentiable manifolds

The problem of the existence of homogeneous canonical transformations in phase space is dealt with in the framework of differential geometry. Necessary theorems for the existence of symplectic actions of simple Lie groups on differentiable manifolds are proved.

Canonical transformations applied to the free Landau electron

It is shown that the problem of a free eletron in a uniform magnetic field provides an interesting example for the use of recently developed canonical transformation methods in quantum mechanics. The Schrödinger equation is immediately solved by a unitary transformation corresponding to a finite linear symplectic transformation in phase space. The natural invariance and dynamical groups involve affine and not only linear symplectic transformations, and the Weyl group turns out to be an appropriate invariance group to account completely for the infinite degeneracies.

Canonical transformations and quadratic hamiltonians

It is shown that there exist symmetry transformations in phase space that preserve Hamilton’s canonical equations of motion for one Hamiltonian, but not for all. Examples of these « canonoid » transformations are given and their relation to canonical transformations is developed. It is demonstrated that a sufficient condition for a canonoid transformation to be canonical is that it preserve Hamilton’s equations for all Hamiltonians quadratic in theq’s andp’s.

Linear Canonical Transformations and Their Unitary Representations

We show that the group of linear canonical transformations in a 2N-dimensional phase space is the real symplectic group Sp(2N), and discuss its unitary representation in quantum mechanics when the N coordinates are diagonal. We show that this Sp(2N) group is the well-known dynamical group of the N-dimensional harmonic oscillator. Finally, we study the case of n particles in a q-dimensional oscillator potential, for which N = nq, and discuss the chain of groups Sp(2nq)⊃Sp(2n)×O(q). An application to the calculation of matrix elements is given in a following paper.

Collisional Transfer Contributions in the Quantum Theory of Transport Coefficients

The quantum-mechanical BBGKY equations are solved in terms of the phase-space transformation functions. Introducing the molecular-chaos assumption, the pair distribution function is obtained. In the binary-collision approximation, this leads to a generalized Boltzmann equation, in which the collisional-transfer contributions are included. This equation is solved by the method of Chapman and Enskog, and expressions for the transport coefficients are obtained, which are given entirely in terms of the solutions of the two-particle Schrödinger equation. The classical limit (ℏ→0) of the expressions are also derived. The heat flux is found to contain contributions which are proportional to the gradient in density. The significance of these terms, which are of the order ℏ4, is discussed.

Lorentz transformations in phase space and in physical space

The relation between “tensorial” Lorentz transformations in physical space and “canonical” Lorentz transformations in phase space is examined in detail. It is shown in particular that the Lorentz-Liouville equation transforms the phase-space distribution function in such a way that the relevant average values computed with it, transform precisely as vectors or tensors. This has been proved explicitly in the case of the current-density vector and the energy-momentum tensor. An important fact which comes out of the theory is that the tensorial variance is not attached intrinsically to a given function but is determined by the dynamical nature of the system For instance, the average identified with the energy-momentum tensor for free particles, transforms as a tensor in a free particle system, but not in a system of interacting particles.

Thermal Velocity Limit to Current Density in a Beam

The limiting current density that can be achieved with thermal velocities in the source emission is calculated by phase-space ellipse transformations. The results check those of Pierce, and are applicable to a wider range of problems. Simple lens systems can realize the ellipse transformations which give the limiting density. Under certain conditions the maximum current density is obtained between the object plane and the image plane, a situation analogous to the ``waist'' formation in a beam transport system used with particle accelerators.

On the kinetic theory of dense fluids. VIII. some comments on the formal computation of the non-equilibrium distribution function of a fluid

In this paper we consider two problems: 1. (a) The formal cluster expansion of the integro-differential equation determining the non-equilibrium n body distribution function, and 2. (b) Methods of computing the phase space transformation function. Two formal solutions for the phase space transformation function are presented, one exact for the case of weak interactions, and one approximate and connected closely with the concepts of local equilibrium and Brownian motion. The cluster expansion leads to collision kernels corresponding to multiple body encounters. The operators are so defined that at least one interaction must occur between a group of m molecules and a separate group of n molecules for there to be a non-vanishing contribution to the transport equation for the m th order distribution function. Interactions wholly within the group m or the group n make no contribution. To mitigate the difficulties in computing multiple body trajectories it is suggested that the approximate solutions for the transformation functions be used in the theory of dense fluids.

Statistical Mechanical Theory of Transport Processes. XII. Dense Rigid Sphere Fluids

The theory of transport in a dense fluid of rigid spheres is developed from classical statistical mechanics by the use of phase space transformation functions. A modified Maxwell-Boltzmann integro-differential equation for the distribution function in μ space is derived, and the difference between this equation and the Enskog equation is discussed. To obtain a formulation of the stress tensor and heat flux solely in terms of binary collisions, it is necessary to allow the time of coarse graining to be very short. The implications of this are discussed with relation to the general principles of the statistical mechanics of transport. The viscosity and thermal conductivity of the dense rigid sphere fluid are calculated. The viscosity is the same as that first computed by Enskog, but the thermal conductivity differs from his calculations.

General Relativity and Particle Dynamics

In order to adapt the Hamiltonian dynamics of a system of classical (nonquantum) particles to general relativity it seems necessary to generalize the transformations used from contact transformations of phase space to extended point transformations of a joint observer and phase space. When this is done the curvature of observer space is seen to arise from the physical interaction of particles without the introduction of new gravitational field variables.In order to extend this theory to quantized particle and field theory it seems necessary to extend the unitary transformations of quantum mechanics in a similar manner, to linear transformations of matrices maintaining the trace of the product of two matrices, the Hermitian nature of a matrix, and the unit matrix. Such transformations do not preserve the phases of wave functions or the product of two matrices, but do preserve the characteristic values of matrices and the traces of the products of any number of matrices. That observers should be designated by $q$ numbers, not $c$ numbers, makes it difficult to interpret the theory except in the classical limit.

Statistical Mechanical Theory of Transport Processes. IX. Contribution to the Theory of Brownian Motion

The equations of Brownian motion are derived from Liouville's equation in a treatment which parallels Kirkwood's statistical mechanical analysis but which is based upon the theory of phase space transformation functions. The transformation function obeys Liouville's equation and thus expresses the equations of motion; it is obtained as a solution of that equation by a consistent method of successive approximations which treats the forces as perturbations. Retention of first-order terms only in this solution leads to the Chandrasekhar equation and yields a tractable relation between the friction constant and the intermolecular forces. The restrictions imposed by the present derivation on the molecular interpretation of the equations of Brownian motion are discussed.

Dirac Bracket Transformations in Phase Space

The purpose of this paper is twofold. One is to analyze the group-theoretical significance of the Dirac bracket and to examine, in particular, the apparent ambiguities in the presence of both first-class and second-class constraints. The other is to prepare the ground for the utilization of the Dirac bracket for the quantization of generally covariant theories. It is shown that the Dirac bracket represents the commutator of infinitesimal transformations in phase space which are not canonical but form a group, in that they are the only transformations that preserve the form of all the constraints of a theory as well as the canonical form of the equations of motion. This group of transformations possesses an invariant subgroup: those transformations that correspond to coordinate transformations, gauge transformations and the like.From this subgroup, we can construct the factor group. All members of the original group have generators, but the generators of the invariant subgroup are zero. There is, then, a one-to-one correspondence between the nonvanishing generators and the members of the factor group. The Dirac bracket is uniquely defined for all dynamical variables that can serve as generators; excluded are variables that have no invariant significance (such as the divergence of the electromagnetic vector potential). If it is possible to identify all these permissible generators, then the theory can be reformulated in terms of these and will be free of constraints. It is proposed to adopt the set of generators and the Dirac brackets between them as the point of departure for the formulation of covariant quantum theories.

The Statistical-Mechanical Theory of Transport Processes. VIII. Quantum Theory of Transport in Gases

The quantum-mechanical analog of the Maxwell-Boltzmann equation of transport in gases of low density is derived from the quantum-mechanical equation for the motion in phase space of the Wigner function. The derivation is based upon the theory of phase-space transformation functions which is developed in this article. Although the present treatment resembles the derivation of Mori and Ono, it is simpler and more general. Particularly, the assumption of random a priori phases, an integral part of Mori and Ono's work, need not be applied explicitly. The Uhling and Uhlenbeck equation is verified in the Born collision approximation.

Quantum mechanics as a statistical theory

An attempt is made to interpret quantum mechanics as a statistical theory, or more exactly as a form of non-deterministic statistical dynamics. The paper falls into three parts. In the first, the distribution functions of the complete set of dynamical variables specifying a mechanical system (phase-space distributions), which are fundamental in any form of statistical dynamics, are expressed in terms of the wave vectors of quantum theory. This is shown to be equivalent to specifying a theory of functions of non-commuting operators, and may hence be considered as an interpretation of quantum kinematics. In the second part, the laws governing the transformation with time of these phase-space distributions are derived from the equations of motion of quantum dynamics and found to be of the required form for a dynamical stochastic process. It is shown that these phase-space transformation equations can be used as an alternative to the Schrödinger equation in the solution of quantum mechanical problems, such as the evolution with time of wave packets, collision problems and the calculation of transition probabilities in perturbed systems; an approximation method is derived for this purpose. The third part, quantum statistics, deals with the phase-space distribution of members of large assemblies, with a view to applications of quantum mechanics to kinetic theories of matter. Finally, the limitations of the theory, its uniqueness and the possibilities of experimental verification are discussed.