Physics – Quantum Physics
Scientific paper
2011-09-24
Physics
Quantum Physics
17 pages. Phys. Rev. A. 2011 (to appear)
Scientific paper
In Dirac's canonical quantization theory on systems with second-class constraints, the commutators between the position, momentum and Hamiltonian form a set of algebraic relations that are fundamental in construction of both the quantum momentum and the Hamiltonian. For a free particle on a two-dimensional sphere or a spherical top, results show that the well-known canonical momentum p_{{\theta}} breaks one of the relations, while three components of the momentum expressed in the three-dimensional Cartesian system of axes as p_{i} (i=1,2,3) are satisfactory all around. This momentum is not only geometrically invariant but also self-adjoint, and we call it geometric momentum. The nontrivial commutators between p_{i} generate three components of the orbital angular momentum; thus the geometric momentum is fundamental to the angular one. We note that there are five different forms of the geometric momentum proposed in the current literature, but only one of them turns out to be meaningful.
Liu Qing-Hua
Tang Lei-Han
Xun D. M.
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Anonymous visitor
a very interesting subject, in fact one which has been unresolved for several decades, namely the issue of properly defining quantum mechanics on a surface in real three dimensional Euclidean space. The paper at hand provides an interesting way to resolve this issue.
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