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Yinshan Academic Journal

Different Coordinate Momentum Operator

Qiao Wen-hua

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

Correspondence of operator-ih▽ and momentum operator in the quantum mechanics is discussed,and pointed out the simple correspondence in cartesian coordinate does not apply to general curvilinear coordinate,due to Ermi nature of mechanical quantity operator,we can't give corresponding momentum operator directly through simple correspondence under general curvilinear coordinate.

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Journal of Yibin University•Jan 1, 2011•0 citations
Correspondence of operator-ih▽ and momentum operator in the quantum mechanics is discussed,and pointed out the simple correspondence in cartesian coordinate does not apply to general curvilinear coordinate,due to Ermi nature of mechanical quantity operator,we can't give corresponding momentum operator directly through simple correspondence under general curvilinear coordinate.
In quantum mechanics textbooks the momentum operator is defined in the Cartesian coordinates and rarely the form of the momentum operator in spherical polar coordinates is discussed. Consequently one always generalizes the Cartesian prescription to other coordinates and falls in a trap. In this work we introduce the difficulties one faces when the question of the momentum operator in general curvilinear coordinates arises. We have tried to elucidate the points related to the definition of the momentum operator taking spherical polar coordinates as our specimen coordinate system and proposed an elementary method in which we can ascertain the form of the momentum operator in general coordinate systems.
Using the transformation matrix between the unit basis vectors in deferent orthogonal curvilinear coordinate systems,we deduce the general form of the transformation between the angular momentum operators in these coordinate systems.We give the representations of Hamilton operator  and angular momentum operators in the orthogonal curvilinear coordinate system.We discuss the transformation matrix of unit basis vectors among the spherical coordinate system,cylindrical coordinates system and rectangular coordinates system,and the transformation between the angular momentum operators.Furthermore,we discuss the treatment of 2operator in different coordinate systems.
Expressions for the covariant and contravariant components of the quantum momentum operator (h//i) ∇ are formulated for curvilinear coordinates in three-dimensional space. The classical formalism is modified to take account of the noncommutivity of coordinates and momenta in the quantum case. These components are shown to be simply related to canonical momentum operators conjugate to the curvilinear coordinates. Expressions for components of angular momenta are also obtained. These results are specialized for orthogonal curvilinear coordinates and illustrated for spherical polar coordinates.
We construct the Hermitian vector and canonical components of the momentum operator in 3D Euclidean space spanned by general curvilinear coordinates (GCC's) using a simple, natural and unified approach based on identifying the momentum operator in any coordinate system as mass times the velocity operator. When this latter is calculated by applying the Heisenberg equation of motion, it returns (−iℏ-i\hbar−iℏ times) the gradient operator plus an additional zero-valued sum, which when distributed among the components of the gradient, it makes each the Hermitian vector component of the momentum operator in GCC's. The canonical components follow immediately upon symmetrizing each of these vector components in the corresponding base vector. For accessability by wider audiences, we first develop the formalism for the simple polar coordinates and then we develop the case for GCC's.