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arXiv:2103.01715 · Physical Review Research

Canonical quantization on the half-line and in an interval based upon an alternative concept for the momentum in a space with boundaries

M. H. Al-Hashimi, U.‐J. Wiese

Published Jul 21, 2021• 14 citations
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Abstract

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.

DOI
10.1103/physrevresearch.3.033079

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For a particle moving on a half-line or in an interval the operator p\ifmmode^\elseˆ\fi=\ensuremath−i\ensuremath∂x\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}=\ensuremath{-}i{\ensuremath{\partial}}_{x}p\ifmmode^\elseˆ\fi​=\ensuremath−i\ensuremath∂x​ is not self-adjoint and thus does not qualify as the physical momentum. Consequently canonical quantization based on p\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}p\ifmmode^\elseˆ\fi​ fails. Based upon an alternative concept for a self-adjoint momentum operator p\ifmmode^\elseˆ\fiR{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{R}p\ifmmode^\elseˆ\fi​R​, we show that canonical quantization can indeed be implemented on the half-line and on an interval. Both the Hamiltonian H\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}H\ifmmode^\elseˆ\fi​ and the momentum operator p\ifmmode^\elseˆ\fiR{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{R}p\ifmmode^\elseˆ\fi​R​ are endowed with self-adjoint extension parameters that characterize the corresponding domains D(H\ifmmode^\elseˆ\fi)D(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H})D(H\ifmmode^\elseˆ\fi​) and D(p\ifmmode^\elseˆ\fiR)D({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{R})D(p\ifmmode^\elseˆ\fi​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.
It is known that the momentum operator canonically conjugated to the position operator for a particle moving in some bounded interval of the line (with Dirichlet boundary conditions) is not essentially self-adjoint: it has a continuous set of self-adjoint extensions. We prove that essential self-adjointness can be recovered by symmetrically weighting the momentum operator with a positive bounded function approximating the indicator function of the considered interval. This weighted momentum operator is consistently obtained from a similarly weighted classical momentum through the so-called Weyl-Heisenberg covariant integral quantization of functions or distributions.
We cannot perform the projective measurement of a momentum on a half line since it is not an observable.Nevertheless, we would like to obtain some physical information of the momentum on a half line.We define an optimality for mea- surement as minimizing the variance between an inferred outcome of the measured system before a measuring process and a measurement outcome of the probe system after the measuring process, restricting our attention to the covariant measurement studied by Holevo.Extending the domain of the momentum operator on a half line by introducing a two dimensional Hilbert space to be tensored, we make it self- adjoint and explicitly construct a model Hamiltonian for the measured and probe systems.By taking the partial trace over the newly introduced Hilbert space, the optimal covariant positive operator valued measure (POVM) of a momentum on a half line is reproduced.We physically describe the measuring process to optimally evaluate the momentum of a particle on a half line.
The similarity between classical and quantum physics is large enough to make an investigation of quantization methods a worthwhile endeavour. As history has shown, Dirac's canonical quantization method works reasonably well in the case of conventional quantum mechanics over Rn but it may fail in non-trivial phase spaces and also suffer from ordering problems. Affine quantization is an alternative method, similar to the canonical quantization, that may offer a positive result in situations for which canonical quantization fails. In this paper we revisit the affine quantization method on the half-line. We formulate and solve some simple models, the free particle and the harmonic oscillator.