5 research outputs found
Semiclassical Dynamics of Electrons in Magnetic Bloch Bands: a Hamiltonian Approach
y formally diagonalizing with accuracy the Hamiltonian of electrons
in a crystal subject to electromagnetic perturbations, we resolve the debate on
the Hamiltonian nature of semiclassical equations of motion with Berry-phase
corrections, and therefore confirm the validity of the Liouville theorem. We
show that both the position and momentum operators acquire a Berry-phase
dependence, leading to a non-canonical Hamiltonian dynamics. The equations of
motion turn out to be identical to the ones previously derived in the context
of electron wave-packets dynamics.Comment: 4 page
Inverse problem and Bertrand's theorem
The Bertrand's theorem can be formulated as the solution of an inverse
problem for a classical unidimensional motion. We show that the solutions of
these problems, if restricted to a given class, can be obtained by solving a
numerical equation. This permit a particulary compact and elegant proof of
Bertrand's theorem.Comment: 11 pages, 3 figure
From Feynman Proof of Maxwell Equations to Noncommutative Quantum Mechanics
In 1990, Dyson published a proof due to Feynman of the Maxwell equations
assuming only the commutation relations between position and velocity. With
this minimal assumption, Feynman never supposed the existence of Hamiltonian or
Lagrangian formalism. In the present communication, we review the study of a
relativistic particle using ``Feynman brackets.'' We show that Poincar\'e's
magnetic angular momentum and Dirac magnetic monopole are the consequences of
the structure of the Lorentz Lie algebra defined by the Feynman's brackets.
Then, we extend these ideas to the dual momentum space by considering
noncommutative quantum mechanics. In this context, we show that the
noncommutativity of the coordinates is responsible for a new effect called the
spin Hall effect. We also show its relation with the Berry phase notion. As a
practical application, we found an unusual spin-orbit contribution of a
nonrelativistic particle that could be experimentally tested. Another practical
application is the Berry phase effect on the propagation of light in
inhomogeneous media.Comment: Presented at the 3rd Feynman Festival (Collage Park, Maryland,
U.S.A., August 2006