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American Journal of Physics

Momentum operators for curvilinear coordinate systems

Boris Leaf

Published Sep 1, 1979• 13 citations
Publisher

Abstract

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.

DOI
10.1119/1.11701

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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.
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.
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.
Since the conformation of physical systems is often advantageously described with the help of generalized (i.e. curvilinear) coordinates, the following questions are raised and general answers given to them: (i) What are the components of the momentum operators, those of the adjoint momentum operators and those of the hermitian momentum operators? (ii) What is the general form of the hamiltonian operator? (iii) What do the general forms of the momentum and hamiltonian operators become when expressed in terms of quasi-momentum operators (e.g. angular momentum component operators)? (iv) How are the answers to the above questions affected by an arbitrary choice of normalization convention for the total wave-function? A complete set of formulae (whatever the independent choices of coordinates, quasi-momentum operators and normalization) is given and various ‘historical’ formulae are shown to be particular instances of the general formulae proposed.
Based on (i) the spectral resolution definition of the momentum operator, (ii) the linearity of correspondence between physical observables and quantum Hermitian operators, (iii) the definition of conjugate coordinate-momentum variables in classical mechanics, and (iv) the invariance of the classical Hamiltonian to canonical transformations (transformations that change from one set of conjugate variables to another), we show that the $[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x},\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}]$ coordinate-momentum commutator must have the value ihihih where hhh is a real nonzero number (which we can identify with the experimentally determined $\ensuremath{\hbar}$). The results are then extended to include all generalized coordinates and their conjugate momenta as well as the Cartesian special-relativistic case.
Geometrical properties of space are described by relations between quantum observables, functions of generalized coordinate and momentum operators of a physical system. Space comprises the set of eigenvalues of the coordinate observables, which may be a discrete set of points instead of the continuum of classical geometry. Differentiation on coordinates and momenta is defined by commutation operations. Well-known formulas from classical geometry for gradient and divergence operators, connections, covariant differentiation, tangent vectors, geodesics, and the Riemann curvature tensor are derived as relations between quantum operators. The quantum formulation retains the symmetry between coordinate and momentum representations required by the principle of complementarity.