Quantization in polar coordinates and the phase operator

Betweenness in Polar Coordinates

betweenness, geometry, lines, polar coordinates, trilinear coordinates We consider straight line drawings. From a graph theoretic point of view these are plane graphs with a set of preferred paths. This graph is expressed in geometric terms through the consideration of the betweenness of the points of intersection along each of the lines. If we fix a reference system for polar coordinates, we c...

Non-hermitian radial momentum operator and path integrals in polar coordinates

A salient feature of the Schrödinger equation is that the classical radial momentum term pr in polar coordinates is replaced by the operator P̂ † r P̂r, where the operator P̂r is not hermitian in general. This fact has important implications for the path integral and semi-classical approximations. When one defines a formal hermitian radial momentum operator p̂r = (1/2) ( ( ~̂xr )~̂ p + ~̂p( ~̂x r ) ) ,...

The Laplacian in Spherical Polar Coordinates

The Strain Compatibility Equations in Polar Coordinates

The same cannot be said for 3D (spherical) polars. However, they can be derived from the Riemann tensor. Curiously, this method does not require the explicit expressions for the strains in terms of the polar displacements to be known. In 3D there are six independent compatibility equations. It turns out that one of them is identical to Equ.1 (where is now the polar angle to the z-axis, and the ...

Constrained Contouring in the Polar Coordinates

A constrained contour is an outline of a region of interest, obtained by linking edges under the constraints of connectivity, smoothness, image context and subjective or user speciied constraints. We discuss a constrained contouring algorithm in polar coordinates to trace closed contours. The algorithm nds optimal contour locations in all radial direction according to the constraints of context...