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Physical Review Research

Alternative momentum concept for a quantum mechanical particle in a box

M.H. Al-Hashimi, U.‐J. Wiese

Published Oct 26, 2021• 11 citations
Publisher

Abstract

For a particle in a box, the operator p = -i∂ x is not self-adjoint.We provide an alternative construction of a momentum operator pR + i pI , which has two self-adjoint components pR and pI .This leads to a description of momentum measurements performed on a particle that is strictly limited to the interior of a box of size L, which yields quantized momentum values π n/L with n ∈ Z.

DOI
10.1103/physrevresearch.3.l042008

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For a particle in a box, the operator p = -i∂ x is not self-adjoint.We provide an alternative construction of a momentum operator pR + i pI , which has two self-adjoint components pR and pI .This leads to a description of momentum measurements performed on a particle that is strictly limited to the interior of a box of size L, which yields quantized momentum values π n/L with n ∈ Z.
For a particle in a box, the operator −i∂x- i \partial_x−i∂x​ is not Hermitean. We provide an alternative construction of a momentum operator p=pR+ipIp = p_R + i p_Ip=pR​+ipI​, which has a Hermitean component pRp_RpR​ that can be extended to a self-adjoint operator, as well as an anti-Hermitean component ipIi p_IipI​. This leads to a description of momentum measurements performed on a particle that is strictly limited to the interior of a box.
We construct a discrete non-Hermitian momentum operator, which implements faithfully the non-self-adjoint nature of momentum for a particle in a box. Its eigenfunctions are strictly limited to the interior of the box in the continuum limit, with the quarter wave as first nontrivial eigenstate. We show how to construct the corresponding Hermitian Hamiltonian for the infinite well as a concrete example to realize unitary dynamics. The resulting Hilbert space can be decomposed into a physical and unphysical subspace, which are mutually orthogonal. The physical subspace in the continuum limit reproduces that of the continuum theory and we give numerical evidence that the correct probability distributions for momentum and energy are recovered. Published by the American Physical Society 2024
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 the recent construction of a self-adjoint momentum operator for a particle confined in a one-dimensional interval, we extend the construction to arbitrarily shaped regions in any number of dimensions. Different components of the momentum vector do not commute with each other unless very special conditions are met. As such, momentum measurements should be considered one direction at a time. We also extend other results, such as the Ehrenfest theorem and the interpretation of the Heisenberg uncertainty relation to higher dimensions.
Based on the recent construction of a self-adjoint momentum operator for a particle confined in a one-dimensional interval, we extend the construction to arbitrarily shaped regions in any number of dimensions. Different components of the momentum vector do not commute with each other unless very special conditions are met. As such, momentum measurements should be considered one direction at a time. We also extend other results, such as the Ehrenfest theorem and the interpretation of the Heisenberg uncertainty relation to higher dimensions.