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Planck Constant (planck + constant)

Distribution by Scientific Domains

Physics and Astronomy	50%

Selected Abstracts

Gauge-independent quantum dynamics on phase-space of charged scalar particles

FORTSCHRITTE DER PHYSIK/PROGRESS OF PHYSICS, Issue 2-3 2003
S. Varró
On the basis of the Hamiltonian form of the Klein-Gordon equation of a charged scalar particle field introduced by Feshbach and Villars, the gauge-invariant 2×2 Wigner matrix has been constructed whose diagonal elements describe positive and negative charge densities and the off-diagonal elements correspond to cross-densities in phase-space. The system of coupled transport equations has been derived in case of interaction with an arbitrary external electromagnetic field. A gauge-independent generalization of the free particle representation due to Feshbach and Villars is given, and on the basis of it both the nonrelativistic and the classical limits of the general relativistic quantum Boltzmann-Vlasov equation(RQBVE) is discussed. In the non-relativistic limit (p/mc,0) the set of equations of motion decouple to two independent quantum transport equations describing the dynamics of oppositely charged positon and negaton densities separately. In the classical limit(,,0) two relativistic Boltzmann-Vlasov equations result for the diagonal positon and negaton densities. It is obtained that, though in the latter equations the Planck constant , is absent, the real part of the cross-density does not vanish. [source]

Solutions to the nonlinear Schrödinger equation carrying momentum along a curve

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 9 2009
Fethi Mahmoudi
We prove existence of a special class of solutions to the (elliptic) nonlinear Schrödinger equation ,,2,, + V(x), = |,|p , 1, on a manifold or in Euclidean space. Here V represents the potential, p an exponent greater than 1, and , a small parameter corresponding to the Planck constant. As , tends to 0 (in the semiclassical limit) we exhibit complex-valued solutions that concentrate along closed curves and whose phases are highly oscillatory. Physically these solutions carry quantum-mechanical momentum along the limit curves. © 2008 Wiley Periodicals, Inc. [source]

Semiclassical limit for the Schrödinger-Poisson equation in a crystal

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 7 2001
Philippe Bechouche
We give a mathematically rigorous theory for the limit from a weakly nonlinear Schrödinger equation with both periodic and nonperiodic potential to the semiclassical version of the Vlasov equation. To this end we perform simultaneously a classical limit (vanishing Planck constant) and a homogenization limit of the periodic structure (vanishing lattice length taken proportional to the Planck constant). We introduce a new variant of Wigner transforms, namely the "Wigner Bloch series" as an adaption of the Wigner series for density matrices related to two different "energy bands." Another essential tool are estimates on the commutators of the projectors into the Floquet subspaces ("band subspaces") and the multiplicative potential operator that destroy the invariance of these band subspaces under the periodic Hamiltonian. We assume the initial data to be concentrated in isolated bands but allow for band crossing of the other bands which is the generic situation in more than one space dimension. The nonperiodic potential is obtained from a coupling to the Poisson equation, i.e., we take into account the self-consistent Coulomb interaction. Our results hold also for the easier linear case where this potential is given. We hence give the first rigorous derivation of the (nonlinear) "semiclassical equations" of solid state physics widely used to describe the dynamics of electrons in semiconductors. © 2001 John Wiley & Sons, Inc. [source]

Semiclassical expansion of quantum characteristics for many-body potential scattering problem

ANNALEN DER PHYSIK, Issue 9 2007
M.I. Krivoruchenko
Abstract In quantum mechanics, systems can be described in phase space in terms of the Wigner function and the star-product operation. Quantum characteristics, which appear in the Heisenberg picture as the Weyl's symbols of operators of canonical coordinates and momenta, can be used to solve the evolution equations for symbols of other operators acting in the Hilbert space. To any fixed order in the Planck's constant, many-body potential scattering problem simplifies to a statistical-mechanical problem of computing an ensemble of quantum characteristics and their derivatives with respect to the initial canonical coordinates and momenta. The reduction to a system of ordinary differential equations pertains rigorously at any fixed order in ,. We present semiclassical expansion of quantum characteristics for many-body scattering problem and provide tools for calculation of average values of time-dependent physical observables and cross sections. The method of quantum characteristics admits the consistent incorporation of specific quantum effects, such as non-locality and coherence in propagation of particles, into the semiclassical transport models. We formulate the principle of stationary action for quantum Hamilton's equations and give quantum-mechanical extensions of the Liouville theorem on conservation of the phase-space volume and the Poincaré theorem on conservation of 2p -forms. The lowest order quantum corrections to the Kepler periodic orbits are constructed. These corrections show the resonance behavior. [source]