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arXiv:1411.0203

A self-adjoint decomposition of radial momentum implies that Dirac's introduction of the operator is insightful

Q. H. Liu, Shifang Xiao

Published Nov 2, 2014• 0 citations
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Abstract

With acceptance of the Dirac's observation that the \textit{canonical quantization entails using Cartesian coordinates, }we examine the\textit{\ }% operator $\mathbf{e}_{r}P_{r}$ rather than $P_{r}$ itself and demonstrate that there is a decomposition of $\mathbf{e}_{r}P_{r}$ into two self-adjoint but non-commutative parts, in which one is the total momentum and another is the transverse one. This study renders the operator $P_{r}$ indirectly measurable and physically meaningful.

DOI
10.48550/arxiv.1411.0203

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With acceptance of the Dirac's observation that the \textit{canonical quantization entails using Cartesian coordinates, }we examine the\textit{\ }% operator erPr\mathbf{e}_{r}P_{r}er​Pr​ rather than PrP_{r}Pr​ itself and demonstrate that there is a decomposition of erPr\mathbf{e}_{r}P_{r}er​Pr​ into two self-adjoint but non-commutative parts, in which one is the total momentum and another is the transverse one. This study renders the operator PrP_{r}Pr​ indirectly measurable and physically meaningful.
With acceptance of the Dirac's observation that the canonical quantization entails using Cartesian coordinates, we examine the operator erPr rather than Pr itself and demonstrate that there is a decomposition of erPr into a difference of two self-adjoint but noncommutative operators, in which one is the total momentum and another is the transverse one. This study renders the operator Pr indirectly measurable and physically meaningful, offering an explanation of why the mean value of Pr over a quantum mechanical state makes sense and supporting Dirac's claim that Pr "is real and is the true momentum conjugate to r".
For a particle moving on a half-line or in an interval the operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}=\ensuremath{-}i{\ensuremath{\partial}}_{x}$ is not self-adjoint and thus does not qualify as the physical momentum. Consequently canonical quantization based on $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}$ fails. Based upon an alternative concept for a self-adjoint momentum operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{R}$, we show that canonical quantization can indeed be implemented on the half-line and on an interval. Both the Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$ and the momentum operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{R}$ are endowed with self-adjoint extension parameters that characterize the corresponding domains $D(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H})$ and $D({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{R})$ in the Hilbert space. When one replaces Poisson brackets by commutators, one obtains meaningful results only if the corresponding operator domains are properly taken into account. The alternative concept for the momentum is used to describe the results of momentum measurements of a quantum mechanical particle that is reflected at impenetrable boundaries, either at the end of the half-line or at the two ends of an interval.
For a particle moving on a half-line or in an interval the operator pˆ = −i∂x is not self-adjoint and thus does not qualify as the physical momentum. Consequently canonical quantization based on pˆ fails. Based upon an alternative concept for a self-adjoint momentum operator pˆR, we show that canonical quantization can indeed be implemented on the half-line and on an interval. Both the Hamiltonian Hˆ and the momentum operator pˆR are endowed with self-adjoint extension parameters that characterize the corresponding domains D(Hˆ ) and D(pˆR ) in the Hilbert space. When one replaces Poisson brackets by commutators, one obtains meaningful results only if the corresponding operator domains are properly taken into account. The alternative concept for the momentum is used to describe the results of momentum measurements of a quantum mechanical particle that is reflected at impenetrable boundaries, either at the end of the half-line or at the two ends of an interval.
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.
The dynamical system of a point particle constrained on a torus is quantized à la Dirac with two kinds of coordinate systems respectively; the Cartesian and toric coordinate systems. In the Cartesian coordinate system, it is difficult to express momentum operators in coordinate representation owing to the complication in structure of the commutation relations between canonical variables. In the toric coordinate system, the commutation relations have a simple form and their solutions in coordinate representation are easily obtained with, furthermore, two quantum Hamiltonians turning up. A problem comes out when the coordinate system is transformed, after quantization, from the Cartesian to the toric coordinate system.