111 research outputs found
Uniform Semiclassical Approximation for the Wigner Symbol in Terms of Rotation Matrices
A new uniform asymptotic approximation for the Wigner symbol is given in
terms of Wigner rotation matrices (-matrices). The approximation is uniform
in the sense that it applies for all values of the quantum numbers, even those
near caustics. The derivation of the new approximation is not given, but the
geometrical ideas supporting it are discussed and numerical tests are
presented, including comparisons with the exact -symbol and with the
Ponzano-Regge approximation.Comment: 44 pages plus 20 figure
Adiabatic motion of a neutral spinning particle in an inhomogeneous magnetic field
The motion of a neutral particle with a magnetic moment in an inhomogeneous magnetic field is considered. This situation, occurring, for example, in a Stern-Gerlach experiment, is investigated from classical and semiclassical points of view. It is assumed that the magnetic field is strong or slowly varying in space, i.e., that adiabatic conditions hold. To the classical model, a systematic Lie-transform perturbation technique is applied up to second order in the adiabatic-expansion parameter. The averaged classical Hamiltonian contains not only terms representing fictitious electric and magnetic fields but also an additional velocity-dependent potential. The Hamiltonian of the quantum-mechanical system is diagonalized by means of a systematic WKB analysis for coupled wave equations up to second order in the adiabaticity parameter, which is coupled to Planck’s constant. An exact term-by-term correspondence with the averaged classical Hamiltonian is established, thus confirming the relevance of the additional velocity-dependent second-order contribution
Symplectic and Semiclassical Aspects of the Schl\"afli Identity
The Schl\"afli identity, which is important in Regge calculus and loop
quantum gravity, is examined from a symplectic and semiclassical standpoint in
the special case of flat, 3-dimensional space. In this case a proof is given,
based on symplectic geometry. A series of symplectic and Lagrangian manifolds
related to the Schl\"afli identity, including several versions of a Lagrangian
manifold of tetrahedra, are discussed. Semiclassical interpretations of the
various steps are provided. Possible generalizations to 3-dimensional spaces of
constant (nonzero) curvature, involving Poisson-Lie groups and q-deformed spin
networks, are discussed.Comment: 40 pages, 8 figure
Diagonalization of multicomponent wave equations with a Born-Oppenheimer example
A general method to decouple multicomponent linear wave equations is presented. First, the Weyl calculus is used to transform operator relations into relations between c-number valued matrices. Then it is shown that the symbol representing the wave operator can be diagonalized systematically up to arbitrary order in an appropriate expansion parameter. After transforming the symbols back to operators, the original problem is reduced to solving a set of linear uncoupled scalar wave equations. The procedure is exemplified for a particle with a Born-Oppenheimer-type Hamiltonian valid through second order in h. The resulting effective scalar Hamiltonians are seen to contain an additional velocity-dependent potential. This contribution has not been reported in recent studies investigating the adiabatic motion of a neutral particle moving in an inhomogeneous magnetic field. Finally, the relation of the general method to standard quantum-mechanical perturbation theory is discussed
Semiclassical Time Evolution and Trace Formula for Relativistic Spin-1/2 Particles
We investigate the Dirac equation in the semiclassical limit \hbar --> 0. A
semiclassical propagator and a trace formula are derived and are shown to be
determined by the classical orbits of a relativistic point particle. In
addition, two phase factors enter, one of which can be calculated from the
Thomas precession of a classical spin transported along the particle orbits.
For the second factor we provide an interpretation in terms of dynamical and
geometric phases.Comment: 8 pages, no figure
Kinematic Orbits and the Structure of the Internal Space for Systems of Five or More Bodies
The internal space for a molecule, atom, or other n-body system can be
conveniently parameterised by 3n-9 kinematic angles and three kinematic
invariants. For a fixed set of kinematic invariants, the kinematic angles
parameterise a subspace, called a kinematic orbit, of the n-body internal
space. Building on an earlier analysis of the three- and four-body problems, we
derive the form of these kinematic orbits (that is, their topology) for the
general n-body problem. The case n=5 is studied in detail, along with the
previously studied cases n=3,4.Comment: 38 pages, submitted to J. Phys.
Boundary Conditions on Internal Three-Body Wave Functions
For a three-body system, a quantum wave function with definite
and quantum numbers may be expressed in terms of an internal wave
function which is a function of three internal coordinates. This
article provides necessary and sufficient constraints on to
ensure that the external wave function is analytic. These
constraints effectively amount to boundary conditions on and its
derivatives at the boundary of the internal space. Such conditions find
similarities in the (planar) two-body problem where the wave function (to
lowest order) has the form at the origin. We expect the boundary
conditions to prove useful for constructing singularity free three-body basis
sets for the case of nonvanishing angular momentum.Comment: 41 pages, submitted to Phys. Rev.
Total Angular Momentum Conservation in Ab Initio Born-Oppenheimer Molecular Dynamics
We prove both analytically and numerically that the total angular momentum of
a molecular system undergoing adiabatic Born-Oppenheimer dynamics is conserved
only when pseudo-magnetic Berry forces are taken into account. This finding
sheds light on the nature of Berry forces for molecular systems with spin-orbit
coupling and highlights how ab initio Born-Oppenheimer molecular dynamics
simulations can successfully capture the entanglement of spin and nuclear
degrees of freedom as modulated by electronic interactions
Semiclassical Analysis of the Wigner -Symbol with Small and Large Angular Momenta
We derive a new asymptotic formula for the Wigner -symbol, in the limit
of one small and eight large angular momenta, using a novel gauge-invariant
factorization for the asymptotic solution of a set of coupled wave equations.
Our factorization eliminates the geometric phases completely, using
gauge-invariant non-canonical coordinates, parallel transports of spinors, and
quantum rotation matrices. Our derivation generalizes to higher -symbols.
We display without proof some new asymptotic formulas for the -symbol and
the -symbol in the appendices. This work contributes a new asymptotic
formula of the Wigner -symbol to the quantum theory of angular momentum,
and serves as an example of a new general method for deriving asymptotic
formulas for -symbols.Comment: 18 pages, 16 figures. To appear in Phys. Rev.
Quantum dynamics and breakdown of classical realism in nonlinear oscillators
The dynamics of a quantum nonlinear oscillator is studied in terms of its
quasi-flow, a dynamical mapping of the classical phase plane that represents
the time-evolution of the quantum observables. Explicit expressions are derived
for the deformation of the classical flow by the quantum nonlinearity in the
semiclassical limit. The breakdown of the classical trajectories under the
quantum nonlinear dynamics is quantified by the mismatch of the quasi-flow
carried by different observables. It is shown that the failure of classical
realism can give rise to a dynamical violation of Bell's inequalities.Comment: RevTeX 4 pages, no figure
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