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} $\mathbf{x}\mathbf{p}$\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., $z$-axis component of the angular momentum. The posmom has two parity symmetries, specifically, invariant under two operations $m_{x}$ and $m_{y}$ representing mirror symmetry about $x$ and $y$ 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 $S^{1}$ in two-dimensional flat space $R^{2}$ 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 $N-1$ ($N\geq2$) 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 $N-1$ ($N\geq 2$) curved hypersurface embedded in $N$ 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 $N-1$ ($N\geq 2$) dimensional hypersurface embedded in an $N$ dimensional Euclidean space, two different components $p_{i}$ and $p_{j}$ ($i,j=1,2,3,...N$) of the Cartesian momentum of the particle are not mutually commutative, and explicitly commutation relations $[p_{i},p_{j}]\left( \neq 0\right)$ 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