Abstract
From the known coordinate representation of these operators, a unified treatment of the abstract operators for curvilinear coordinates and their canonically conjugate momenta is given for systems in three dimensions. A configuration representation, corresponding to classical configuration space, exists in which description is simplified; the three-dimensional ket space factors into a direct product of one-dimensional spaces. Four cases are examined, according to the range of the continuous curvilinear coordinate. In addition to normalization of momentum eigenstates to the Kronecker delta for finite range of coordinate and normalization to the Dirac delta for range on the entire real axis, normalization occurs to δ (or δ+) for range on the positive (or negative) part of the real axis. The domain of Hermiticity of curvilinear coordinate and conjugate momentum operators in each case is the entire three-dimension ket space. The fundamental commutation relations hold in all representations. Statements to the contrary for the radial and azimuth coordinates in spherical polar coordinates are seen to be erroneous.
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Leaf, B. Curvilinear coordinate and momentum operators in configuration representation. Found Phys 10, 581–599 (1980). https://doi.org/10.1007/BF00715041
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DOI: https://doi.org/10.1007/BF00715041