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arXiv:quant-ph/0009046 · European Journal of Physics

On the connection between the radial momentum operator and the Hamiltonian in n dimensions

Gil Paz

Published Sep 11, 2000• 24 citations
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Abstract

The radial momentum operator in quantum mechanics is usually obtained through canonical quantization of the (symmetrical form of the) classical radial momentum. We show that the well known connection between the Hamiltonian of a free particle and the radial momentum operator $\hat{H}=\hat{P}_{r}^2/2m+ $\hat{L}^2$}/2mr^{2}$ is true only in one or three dimensions. In general, an extra term of the form $\hbar^{2}(n-1)(n-3)/ 2m \cdot 4r^{2}$ has to be added to the Hamiltonian.

DOI
10.48550/arxiv.quant-ph/0009046

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The radial momentum operator in quantum mechanics is usually obtained through canonical quantization of the (symmetrical form of the) classical radial momentum. We show that the well known connection between the Hamiltonian of a free particle and the radial momentum operator H^=P^r2/2m+\hat{H}=\hat{P}_{r}^2/2m+ H^=P^r2​/2m+\hat{L}^2}/2mr^{2} is true only in one or three dimensions. In general, an extra term of the form ℏ2(n−1)(n−3)/2m⋅4r2\hbar^{2}(n-1)(n-3)/ 2m \cdot 4r^{2}ℏ2(n−1)(n−3)/2m⋅4r2 has to be added to the Hamiltonian.
A salient feature of the Schrödinger equation is that the classical radial momentum term p2r in polar coordinates is replaced by the operator , where the operator is not hermitian in general. This fact has important implications for the path integral and semi-classical approximations. When one defines a formal hermitian radial momentum operator , the relation 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.
When a non-relativistic quantum particle moving under a radial potential V(r) has either an extra conserved vectorial operator or an extra conserved tensorial operator then, in the first case, the corresponding operator must be proportional to the quantum Laplace-Runge-Lenz vector and the compatible potential has to be the Coulomb potential. While, in the second case, the compatible potential is the harmonic oscillator one and the conserved quantity has to be a tensor formed by a symmetric combination of the position and the momentum operators, which is the quantum version of the classically conserved tensor. Thus, the superintegrable condition on classical 3D radial problems producing only closed, periodic and plane orbits still manages, even in the quantum case, to impose strong restrictions on the interaction potentials.
A straightforward derivation is given for the Hermitian operator pr ≡ 1r ħi δδrr corresponding to the radial component of linear momentum in the central force problem. The purpose is purely pedagogical; we believe that the radial term of the Schrödinger equation in spherical coordinates is more comprehensible when it appears in the form (pr2/2m)ψ = [(1r ħi δδrr)2/2m]ψ than when it appears, as it does in most textbooks, in the form −ħ22mr2 δδr[r2δδr]ψ. The latter expression is usually arrived at by a tedious transformation of variable done “in the appendix”; instead one may derive pr with a minimum of algebra and with the focus where it belongs, on the concepts of quantum mechanics.
Synopsis: Direct product spaces are encountered in the generalization to higher dimensions, and the connection to wave function separability is discussed. The spherical coordinate basis is introduced in preparation for discussing spherically symmetric potentials. Angular momentum operators and properties are examined, and a complete set of compatible observables is determined. The angular momentum eigenvalues and eigenstates are determined using ladder operators, and the connection to spherical harmonics is made. The stage is set for understanding the Schrödinger radial equation.
Synopsis: Direct product spaces are encountered in the generalization to higher dimensions, and the connection to wave function separability is discussed. ƒe spherical coordinate basis is introduced in preparation for discussing spherically symmetric potentials. Angular momentum operators and properties are examined, and a complete set of compatible observables is determined. ƒe angular momentum eigenvalues and eigenstates are determined using ladder operators and the connection to spherical harmonics is made. ƒe stage is set for understanding the Schrödinger radial equation.
In quantum mechanics, physical observables are represented by operators on a certain Hilbert space. The question of how such operators commute, has been a matter of discussion. In the standard perspective, the operators corresponding to the position and momentum of a system are assumed to satisfy the canonical commutation relations. It is known that these relations imply that the Hamilton and Heisenberg equations of motion are compatible as operator equations. However, Wigner showed that the inverse statement is not true. Therefore, it is a much weaker constraint to impose the compatibility of the equations of motion. For any physical system, this results in a set of compatibility conditions, which form the core of Wigner quantization. The key to finding operators that are subject to the compatibility conditions is provided by Lie superalgebras and their representations. Lie superalgebras can be defined as algebras generated by odd elements satisfying particular superbracket relations. Using these defining relations, Lie superalgebra generators can be found that obey the compatibility conditions. The Lie superalgebra elements act as operators on a vector space if we consider Lie superalgebra representations. Various physical systems are investigated in this thesis in the context of Wigner quantization. For each of these systems solutions are found in terms of Lie superalgebra generators, after which specific representations are examined. In such representations, the focus lies on determining particular physical properties of the system. Most of the studied systems are harmonic oscillator models. First, we examine a set-up of coupled harmonic oscillators, for which the interaction is represented by an interaction matrix. Then we focus on the angular momentum content of a 3N3N3N-dimensional harmonic oscillator. Finally, our attention goes to two one-dimensional systems, namely the free particle and the Berry-Keating-Connes Hamiltonian. The latter of these Hamiltonians is notorious for its possible connection with the Riemann hypothesis. All of the aforementioned Hamiltonians have been extensively investigated in the context of canonical quantization, so that our results can be compared to the well-known canonical case.