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arxiv: 2606.24572 · v1 · pith:QTLFACPTnew · submitted 2026-06-23 · 🪐 quant-ph · math-ph· math.MP· physics.ed-ph

The Vector and Canonical Components of the Momentum Operator in 3D Euclidean Space Spanned by General Curvilinear Coordinates

M.S.Shikakhwa This is my paper

Pith reviewed 2026-06-26 00:03 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MPphysics.ed-ph
keywords momentum operatorcurvilinear coordinatesHeisenberg equationHermitian operatorsquantum mechanicsvelocity operatorgeneral coordinates
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The pith

The momentum operator in general curvilinear coordinates is mass times the velocity operator from the Heisenberg equation, with a zero sum distributed to make each component Hermitian.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows how to construct the Hermitian vector and canonical components of the momentum operator in three-dimensional Euclidean space using general curvilinear coordinates. It does so by equating the momentum operator to mass times the velocity operator and deriving the velocity from the Heisenberg equation of motion. This derivation yields the gradient operator plus a zero-valued sum that is then distributed among the gradient components to ensure each is Hermitian. The canonical components are then obtained by symmetrizing these vector components with the appropriate base vectors. The method is first illustrated with polar coordinates before being extended to the general case.

Core claim

Identifying the momentum operator as mass times the velocity operator, and calculating the velocity via the Heisenberg equation, returns -i ħ times the gradient operator plus an additional zero-valued sum. Distributing this sum among the components of the gradient makes each the Hermitian vector component of the momentum operator in GCC's. The canonical components follow immediately upon symmetrizing each of these vector components in the corresponding base vector.

What carries the argument

the zero-valued sum from the Heisenberg velocity calculation, distributed among the components of the gradient operator

If this is right

  • The resulting vector components of the momentum operator are Hermitian.
  • The canonical components are obtained by symmetrizing the vector components with the base vectors.
  • The construction applies to any general curvilinear coordinate system in 3D Euclidean space.
  • The same approach works uniformly for polar coordinates and more general systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This construction provides an alternative route to Hermitian momentum operators that does not explicitly invoke the volume element of the coordinate system.
  • The method could be tested by verifying Hermiticity in a specific curvilinear system such as spherical coordinates.

Load-bearing premise

The zero-valued sum can be arbitrarily distributed among the gradient components to enforce Hermiticity without reference to the coordinate metric or volume element.

What would settle it

An explicit calculation in spherical coordinates showing that the expectation value of the constructed momentum operator is not real for a physical state would falsify the claim.

read the original abstract

We construct the Hermitian vector and canonical components of the momentum operator in 3D Euclidean space spanned by general curvilinear coordinates (GCC's) using a simple, natural and unified approach based on identifying the momentum operator in any coordinate system as mass times the velocity operator. When this latter is calculated by applying the Heisenberg equation of motion, it returns ($-i\hbar$ times) the gradient operator plus an additional zero-valued sum, which when distributed among the components of the gradient, it makes each the Hermitian vector component of the momentum operator in GCC's. The canonical components follow immediately upon symmetrizing each of these vector components in the corresponding base vector. For accessability by wider audiences, we first develop the formalism for the simple polar coordinates and then we develop the case for GCC's.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript constructs the Hermitian vector and canonical components of the momentum operator in 3D Euclidean space spanned by general curvilinear coordinates (GCCs). It identifies the momentum operator as mass times the velocity operator obtained via the Heisenberg equation of motion, which yields -iħ times the gradient operator plus a zero-valued sum; this sum is distributed among the gradient components to produce Hermitian vector components, after which the canonical components are obtained by symmetrization with the corresponding base vectors. The formalism is first developed for polar coordinates before generalization to GCCs.

Significance. If the central construction is valid, the paper supplies a unified, physically motivated route to Hermitian momentum operators in curvilinear coordinates that avoids direct appeal to the volume element or Christoffel symbols. This could simplify operator constructions in quantum mechanics on non-Cartesian manifolds and make the results more accessible through the stepwise polar-to-GCC presentation.

major comments (1)
  1. [Abstract] Abstract (GCC construction paragraph): the step that distributes the zero-valued sum arising from the Heisenberg velocity calculation among the gradient components to enforce Hermiticity is load-bearing for the central claim, yet the manuscript provides no explicit verification that this redistribution reproduces the unique (up to boundary terms) self-adjoint form fixed by integration by parts against the inner product measure ∫ ψ* φ √g d³q. An arbitrary split supplies no information about √g or the metric and therefore does not guarantee the resulting operator is the correct Hermitian momentum on the coordinate manifold.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful comments on our manuscript. Below we provide a point-by-point response to the major comment.

read point-by-point responses

  1. Referee: [Abstract] Abstract (GCC construction paragraph): the step that distributes the zero-valued sum arising from the Heisenberg velocity calculation among the gradient components to enforce Hermiticity is load-bearing for the central claim, yet the manuscript provides no explicit verification that this redistribution reproduces the unique (up to boundary terms) self-adjoint form fixed by integration by parts against the inner product measure ∫ ψ* φ √g d³q. An arbitrary split supplies no information about √g or the metric and therefore does not guarantee the resulting operator is the correct Hermitian momentum on the coordinate manifold.

    Authors: The referee correctly notes that the manuscript does not include an explicit verification step comparing the redistributed gradient components to the self-adjoint form obtained via integration by parts. While the construction ensures Hermiticity by design through the distribution of the zero sum, and the polar coordinate case serves as a concrete example where the result matches known Hermitian operators, we acknowledge that a direct demonstration for the general curvilinear case would be beneficial. We will revise the manuscript to include such a verification, thereby confirming that the resulting operators are indeed the unique (up to boundary terms) self-adjoint ones with respect to the measure involving √g. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses standard Heisenberg identity and algebraic zero-sum adjustment

full rationale

The paper starts from the Heisenberg equation for velocity, sets p = m v by definition, obtains -iħ ∇ plus a null sum, and distributes the null sum to produce Hermitian components. This is an algebraic manipulation with no fitted parameters, no self-citation chains, and no reduction of the claimed result to its own inputs by construction. The approach remains independent of the final Hermitian form; external verification against the known curvilinear momentum operator is possible without circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on the standard quantum-mechanical identification of momentum with mass times velocity and the Heisenberg equation; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • standard math Heisenberg equation of motion: velocity operator equals (i/ħ) commutator of position with Hamiltonian
    Invoked to obtain velocity from which momentum follows; standard postulate of quantum mechanics.

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discussion (0)

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Reference graph

Works this paper leans on

22 extracted references · 1 canonical work pages

  1. [1]

    D. J. Griffiths and D. F. Schroeter,Introduction to Quantum Mechanics, 3rd ed. (Cambridge University Press, Cambridge, 2018)

    2018

  2. [2]

    Gasiorowicz,Quantum Physics, 3rd ed

    S. Gasiorowicz,Quantum Physics, 3rd ed. (Wiley, Hoboken, NJ, 2003)

    2003

  3. [3]

    J. J. Sakurai and J. Napolitano,Modern Quantum Mechanics, 2nd ed. (Addison–Wesley, San Francisco, 2010)

    2010

  4. [4]

    The quantum centripetal force on a free particle confined to the surface of a sphere and a cylinder,

    M. S. Shikakhwa, “The quantum centripetal force on a free particle confined to the surface of a sphere and a cylinder,” Physica E108, 249 (2019)

    2019

  5. [5]

    Symmetric surface momentum and centripetal force for a particle on a curved surface,

    M. S. Shikakhwa, “Symmetric surface momentum and centripetal force for a particle on a curved surface,” Commun. Theor. Phys.70, 263 (2018)

    2018

  6. [6]

    Constructing Hermitian Hamiltonians for spin-zero neutral and charged particles on a curved surface: physical approach,

    M. S. Shikakhwa and N. Chair, “Constructing Hermitian Hamiltonians for spin-zero neutral and charged particles on a curved surface: physical approach,” Eur. Phys. J. Plus137, 560 (2022)

    2022

  7. [7]

    Hamiltonian, geometricmomentumand force operators for a spin-zero particle on a curve: physical approach,

    M.S.ShikakhwaandN.Chair, “Hamiltonian, geometricmomentumand force operators for a spin-zero particle on a curve: physical approach,” Eur. Phys. J. Plus139, 559 (2024)

    2024

  8. [8]

    Geometric momentum: The proper momentum for a free particle on a two-dimensional sphere,

    Q. H. Liu, L. H. Tang, and D. M. Xun, “Geometric momentum: The proper momentum for a free particle on a two-dimensional sphere,” Physical Review A84, 042101 (2011)

    2011

  9. [9]

    Deformed algebras, position-dependent effective masses and curved spaces: An exactly solvable Coulomb prob- lem,

    C. Quesne and V. M. Tkachuk, “Deformed algebras, position-dependent effective masses and curved spaces: An exactly solvable Coulomb prob- lem,” J. Phys. A: Math. Gen.37, 4267 (2004); arXiv:math-ph/0403047

    Pith/arXiv arXiv 2004

  10. [10]

    Merzbacher,Quantum Mechanics, 3rd ed

    E. Merzbacher,Quantum Mechanics, 3rd ed. (Wiley, New York, 1998)

    1998

  11. [11]

    Park,Introduction to the Quantum Theory, 3rd ed

    D. Park,Introduction to the Quantum Theory, 3rd ed. (McGraw–Hill, New York, 1992)

    1992

  12. [12]

    Point transformations in quantum mechanics,

    B. S. DeWitt, “Point transformations in quantum mechanics,” Phys. Rev.85, 653 (1952). 21

    1952

  13. [13]

    Dynamical theory in curved spaces. I. A review of the classical and quantum action principles,

    B. S. DeWitt, “Dynamical theory in curved spaces. I. A review of the classical and quantum action principles,” Rev. Mod. Phys.29, 377 (1957)

    1957

  14. [14]

    Curvilinear momenta in quantum mechanics

    Cade R. Curvilinear momenta in quantum mechanics. Mathematical ProceedingsoftheCambridgePhilosophicalSociety.1951;47(2):451-453. doi:10.1017/S0305004100026803

    work page doi:10.1017/s0305004100026803 1951

  15. [15]

    Quantization in generalized coordinates,

    G. R. Gruber, “Quantization in generalized coordinates,” Found. Phys. 1, 227 (1971)

    1971

  16. [16]

    Momentum Operators in Curvilinear Coordinates,

    B. Leaf, “Momentum Operators in Curvilinear Coordinates,” Am. J. Phys.39, 1199 (1971)

    1971

  17. [17]

    Curvilinear coordinate and momentum operators in configu- ration representation,

    B. Leaf, “Curvilinear coordinate and momentum operators in configu- ration representation,” Found. Phys.9, 575 (1979)

    1979

  18. [18]

    Available online: http://homepages.engineering.auckland.ac.nz/pkel015/SolidMechanicsBooks/Part III/ (accessed July 2025)

    Piaras Kelly,Mechanics Lecture Notes Part III: Foundations of Continuum Mechanics, Department of Engineering Science, University of Auckland, New Zealand, 2015. Available online: http://homepages.engineering.auckland.ac.nz/pkel015/SolidMechanicsBooks/Part III/ (accessed July 2025)

    2015

  19. [19]

    M. L. Boas, Mathematical Methods in the Physical Sciences, 3rd ed. (Wiley, Hoboken, NJ, 2006)

    2006

  20. [20]

    G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed. (Academic Press, Waltham, MA, 2013)

    2013

  21. [21]

    P.Dennery and a.Krzywicki,Mathematics for Physicists, (Dover publi- cation Company, New York, 1995)

    1995

  22. [22]

    M.Shikakhwa (2025) to be published. 22

    2025