Non-Hermitian Radial Momentum Operator and Path Integrals in Polar Coordinates(Particles and Fields)
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A salient feature of the Schrodinger equation is that the classical radial momentum term P^2_r in polar coordinates is replaced by the operator P^^^^†_rP^^^_r, where the operator P^^^_r is not hermitian in general. This fact has important implications for the path integral and semiclassical approximations. When one defines a formal hermitian radial momentum operator P^^^_r=(1/2)((<x^^→>/r)^^^<P^^^→>^^^+<P^^^→>^^^(<x^^→>/r)^^^), the relation P^^^^†_rP^^^_r=P^^^^2_r+ℏ^2(d-1)(d-3)/(4r^2) holds in d-dimensional space and this extra potential appears in the path integral formulated in polar coordinates. The extra potential, which influences the classical solutions in the semi-classical treatment such as in the analysis of solitons and collective modes, vanishes for d=3 and attractive for d=2 and repulsive for all other cases d≥4. This extra term induced by the non-hermitian operator is a purely quantum effect, and it is somewhat analogous to the quantum anomaly in chiral gauge theory.
- 理論物理学刊行会の論文
- 2008-08-25
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