104,098 research outputs found
Posmon spectrosopy of quantum state on a circle
Developing the analysis of the distribution of the particle's
position-momentum dot product, the so-called \textit{posmom} \textbf{,} to quantum states on a circular circle on two-dimensional
Cartesian coordinates, we give its posmometry (introduced recently by Y. A.
Bernard and P. M. W. Gill, Posmom: The Unobserved Observable, J. Phys. Chem.
Lett. 1\textbf{(}2010\textbf{)}1254) for eigenstates of the free motion on the
circle, i.e., -axis component of the angular momentum. The posmom has two
parity symmetries, specifically, invariant under two operations and
representing mirror symmetry about and axis respectively. The
complete eigenfunction set of the posmom is then four-valued and consists of
four basic parts each of them is defined within a distinct quadrant of the
circle. The results are not only potentially experimentally testable, but also
reflect a fact that the embedding of the circle in two-dimensional flat
space is physically reasonable.Comment: 9 pages, 6 figure
Geometric momentum for a particle constrained on a curved hypersurface
A strengthened canonical quantization scheme for the constrained motion on a
curved hypersurface is proposed with introduction of the second category of
fundamental commutation relations between Hamiltonian and positions/momenta,
whereas those between positions and moments are categorized into the first. As
an () dimensional hypersurface is embedded in an N dimensional
Euclidean space, we obtain the proper momentum that depends on the mean
curvature. For the surface is the spherical one, a long-standing problem on the
form of the geometric potential is resolved in a lucid and unambiguous manner,
which turns out to be identical to that given by the so-called confining
potential technique. In addition, a new dynamical group SO(N,1) symmetry for
the motion on the sphere is demonstrated.Comment: 5 pages, no figur
Generalized Centripetal Force Law and Quantization of Motion Constrained on 2D Surfaces
For a particle moves on a 2D surface f(x)=0 embedded in 3D Euclidean space,
the geometric momentum and potential are simultaneously admissible within the
Dirac canonical quantization scheme for constrained motion. In our approach,
not the full scheme but the symmetries indicated by classical brackets
[x,H]_{D} and [p,H]_{D} in addition to the fundamental ones [x,x]_{D},
[x,p]_{D} and [p,p]_{D} are utilized, where the subscript D stands for the
Dirac bracket. The generalized centripetal force law p=[p,H]_{D} for particle
on the 2D surface play the key role, and there is no simple relationship
between the force on a point of the surface and its curvatures of the point, in
sharp contrast to the motion on a curve.Comment: 11 pages, 1 figur
Distribution of xp in some molecular rotational states
Developing the analysis of the distribution of the so-called posmom xp to the
spherical harmonics that represents some molecular rotational states for such
as diatomic molecules and spherical cage molecules, we obtain posmometry
(introduced recently by Y. A. Bernard and P. M. W. Gill, Posmom: The Unobserved
Observable, J. Phys. Chem. Lett. 1(2010)1254) of the spherical harmonics and
demonstrate that it bears a striking resemblance to the momentum distributions
of the stationary states for a one-dimensional simple harmonic oscillator.Comment: 8 pages, 3 figure
Electron-acoustic_solitary_structures_in_two-electron-temperature_plasma_with_superthermal_electrons
The propagation of nonlinear electron- acoustic waves (EAWs) in an
unmagnetized collision- less plasma system consisting of a cold electron fluid,
superthermal hot electrons and stationary ions is investigated. A reductive
perturbation method is employed to obtain a modified Korteweg-de Vries (mKdV)
equa- tion for the first-order potential. The small amplitude electron-acoustic
solitary wave, e.g., soliton and dou- ble layer (DL) solutions are presented,
and the effects of superthermal electrons on the nature of the solitons are
also discussed. But the results shows that the weak stationary EA DLs cannot be
supported by the present model.Comment: Accepted for publication in Astrophysics & Space Scienc
Geometric momentum in the Monge parametrization of two dimensional sphere
A two dimensional surface can be considered as three dimensional shell whose
thickness is negligible in comparison with the dimension of the whole system.
The quantum mechanics on surface can be first formulated in the bulk and the
limit of vanishing thickness is then taken. The gradient operator and the
Laplace operator originally defined in bulk converges to the geometric ones on
the surface, and the so-called geometric momentum and geometric potential are
obtained. On the surface of two dimensional sphere the geometric momentum in
the Monge parametrization is explicitly explored. Dirac's theory on
second-class constrained motion is resorted to for accounting for the
commutator [x_{i},p_{j}]=i \hbar({\delta}_{ij}-x_{i}x_{j}/r^2) rather than
[x_{i},p_{j}]=i\hbar{\delta}_{ij} that does not hold true any more. This
geometric momentum is geometric invariant under parameters transformation, and
self-adjoint.Comment: 6 pages, no figur
Breakdown of Ehrenfest theorem for free particle constrained on a hypersurface
There is a belief that the Ehrenfest theorem holds true universally. We
demonstrate that for a classically nonrelativistic particle constrained on an
() curved hypersurface embedded in flat space, the theorem
breaks down.Comment: 3 pages, no figur
Curvature-induced noncommutativity of two different components of momentum for a particle on a hypersurface
As a nonrelativistic particle constrained to remain on an ()
dimensional hypersurface embedded in an dimensional Euclidean space, two
different components and () of the Cartesian
momentum of the particle are not mutually commutative, and explicitly
commutation relations depend on products
of positions and momenta in uncontrollable ways. The \textit{% generalized}
Dupin indicatrix of the hypersurface, a local analysis technique, is utilized
to explore the dependence of the noncommutativity on the curvatures on a
\textit{local point }of the hypersurface. The first finding is that the
noncommutativity can be grouped into two categories; one is the product of a
sectional curvature and the angular momentum, and another is the product of a
principal curvature and the momentum. The second finding is that, for a small
circle lying a \textit{tangential plane} covering the \textit{local point}, the
noncommutativity leads to a rotation operator and the amount of the rotation is
an angle anholonomy; and along each of the \textit{normal sectional curves}
centering the \textit{given point} the noncommutativity leads to a translation
plus an additional rotation and the amount of the rotation is one half of the
tangential angle change of the arc.Comment: 5 pages, no figure. two typos correcte
Discrete calculus of variations and Boltzmann distribution without Stirling's approximation
A \emph{double extrema form} of the calculus of variations is put forward in
which only the smallest one of the finite differences is physically meaningful
to represent the variational derivatives defined on the discrete points. The
most probable distribution for the Boltzmann system is then reproduced without
the Stirling's approximation, and free from other theoretical problems.Comment: 4 page
Transformation of perturbative series into complex phases and elimination of secular divergences in time-dependent perturbation theory in quantum mechanics
The difficulty that the probabilities infinitely increase with time as time
is long enough in time-dependent perturbation theory for some quantum systems
is resolved by means of simply transforming the perturbative series into
natural exponential functions of the re-summed perturbative series. Three
exactly solvable models are taken to check our new formulation, and excellent
agreements with the exact solution are achieved.Comment: 13 pages, revise
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