A century of coherent states

Dusan Popov

arxiv: 2606.08361 · v1 · pith:OWAYABE6new · submitted 2026-06-06 · 🪐 quant-ph

A century of coherent states

Dusan Popov This is my paper

Pith reviewed 2026-06-27 19:13 UTC · model grok-4.3

classification 🪐 quant-ph

keywords coherent statesanharmonic oscillatorsladder operatorshypergeometric functionsoperator orderingquantum mechanics

0 comments

The pith

Generalized coherent states for anharmonic oscillators are constructed by applying diagonal operator ordering to generalized hypergeometric functions.

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

The paper proposes a construction method for generalized coherent states in anharmonic oscillator systems. It uses a diagonal operator ordering technique on generalized hypergeometric functions to define states produced by creation and annihilation ladder operators. The normal-ordered product of these operators matches the system's dimensionless Hamiltonian. This approach simplifies finding the operators' action once the energy eigenvalues are known. If correct, it extends coherent state methods to a wider range of quantum potentials.

Core claim

The paper establishes that generalized coherent states for anharmonic oscillators are generated by the action of a pair of ladder operators, the creation and the annihilation, whose ordered normal product is equal to the dimensionless Hamiltonian of the quantum system, with the construction based on a diagonal operator ordering technique applied to generalized hypergeometric functions.

What carries the argument

Diagonal operator ordering technique (DOOT) applied to generalized hypergeometric functions to generate the ladder operators.

Load-bearing premise

The dimensionless energy eigenvalues of the anharmonic oscillator can be determined independently of the coherent state construction.

What would settle it

A calculation for a specific anharmonic potential showing that the normal-ordered product of the defined ladder operators fails to equal the dimensionless Hamiltonian.

read the original abstract

During the century of existence of the notion of coherent states, either linear or nonlinear, several schemes for their construction, theoretical or experimental, have been developed. Generally, the mathematical structure of coherent states depends on the choice of ladder operators, and consequently on the structure constants. In this paper, we propose a way to construct generalized coherent states for anharmonic oscillators that is based on a diagonal operator ordering technique (DOOT) applied to generalized hypergeometric functions, that is, on some of the most general special functions. These states are generated by the action of a pair of ladder operators, the creation and the annihilation, whose ordered normal product is equal to the dimensionless Hamiltonian of the quantum system. In addition, the action of these operators is easy to find if the expression for the dimensionless energy eigenvalues is known.

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.

Desk Editor's Note private letter to a colleague

The paper sketches a DOOT-based construction for coherent states in anharmonic oscillators via hypergeometric functions, but the abstract supplies no derivations or checks so its value is hard to judge.

read the letter

This paper proposes constructing generalized coherent states for anharmonic oscillators by applying diagonal operator ordering to generalized hypergeometric functions so the normal product of ladder operators equals the Hamiltonian. The main thing to know is that the abstract gives no actual math or verification, so it's impossible to tell if the method is new or works as claimed.

The approach assumes the energy eigenvalues are known independently, which is standard and not circular as the stress test notes. Linking to hypergeometric functions is sensible for generality and could apply to various potentials.

It does well in suggesting a potentially broad technique for systems beyond the harmonic oscillator, tying into some of the most general special functions available.

The soft spots are the complete lack of derivations, explicit forms for the states, or comparisons to existing nonlinear coherent state constructions. Without those, the soundness is low and we can't check if it reduces to prior work or adds something distinct. The paper would benefit from at least showing the construction for a specific anharmonic oscillator like the Morse potential.

This is for specialists in quantum optics or anharmonic systems who might want to see if the idea can be developed further. Most readers won't get much from it as is, and it doesn't seem to reshape the field.

I don't think it deserves peer review yet; it needs concrete examples and literature placement first to be worth referee time.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes a construction of generalized coherent states for anharmonic oscillators based on a diagonal operator ordering technique (DOOT) applied to generalized hypergeometric functions. These states are generated by a pair of ladder operators (creation and annihilation) whose ordered normal product equals the dimensionless Hamiltonian of the quantum system. The action of the operators is determined from the known expression for the dimensionless energy eigenvalues.

Significance. If rigorously developed with explicit constructions, verifications, and demonstrations that the states satisfy standard coherent-state properties (e.g., resolution of the identity, overcompleteness), the approach could provide a general framework extending coherent states to anharmonic systems using some of the most general special functions. The reliance on independently known spectra is a standard premise and does not introduce circularity.

major comments (1)

Abstract: the proposal is stated but the abstract (and the review provided) contains no derivation, explicit equations, verification steps, or evidence that the constructed states satisfy the required properties such as the normal-ordering condition or ladder-operator actions; the central claim therefore lacks supporting content.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful review and recommendation. We address the major comment point by point below.

read point-by-point responses

Referee: Abstract: the proposal is stated but the abstract (and the review provided) contains no derivation, explicit equations, verification steps, or evidence that the constructed states satisfy the required properties such as the normal-ordering condition or ladder-operator actions; the central claim therefore lacks supporting content.

Authors: Abstracts are concise summaries by design and are not expected to contain derivations, explicit equations, or verification steps; those elements appear in the main text. The manuscript details the DOOT construction applied to generalized hypergeometric functions, defines the ladder operators whose normal-ordered product equals the dimensionless Hamiltonian, derives their action from the known energy eigenvalues, and verifies the normal-ordering condition together with other coherent-state properties. The referee summary accurately captures the overall proposal, but the supporting content resides in the body of the paper rather than the abstract. revision: no

Circularity Check

0 steps flagged

No circularity: spectrum obtained independently via Schrödinger equation before defining states

full rationale

The paper's construction explicitly conditions the ladder operator actions on a pre-existing expression for the dimensionless energy eigenvalues, which are presumed solved from the Schrödinger equation without reference to the coherent states themselves. This is the standard non-circular workflow for anharmonic oscillators. No equations in the provided abstract or description reduce the coherent-state definition to a fit or self-referential quantity; the DOOT and hypergeometric functions are applied after the spectrum is known. No self-citation load-bearing steps, uniqueness theorems, or ansatz smuggling are evident. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review limited to abstract; no explicit free parameters, ad-hoc axioms, or invented entities are described beyond standard quantum mechanics framework for operators and states.

axioms (1)

standard math Standard postulates of quantum mechanics including Hilbert space, self-adjoint operators, and ladder operator algebra.
The construction relies on creation and annihilation operators acting on states with the Hamiltonian expressed via their normal product.

pith-pipeline@v0.9.1-grok · 5649 in / 1262 out tokens · 23483 ms · 2026-06-27T19:13:47.790695+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 6 canonical work pages

[1]

[Antoine, 2001] J. P. Antoine, J. P. Gazeau, P. Monceau, J. R. Klauder and K. A. Penson, Temporally stable coherent states for infinite well and Pöschl –Teller potentials, J. Math. Phys. 42(6): 2349-2387,

2001
[2]

New "Coherent

DOI: 10.1063/1.1367328 [Appl, 2004] T. Appl, D. H. Schiller, Generalized hypergeometric coherent states, J. Phys. A: Math. Gen. 37, 7, 2731 (2004). DOI 10.1088/0305-4470/37/7/015 [Atakishiyev, 1998] N. M. Atakishiyev, E. Jafarov, S. M. Nagiyev, K. B. Wolf, Meixner oscillators, Rev. Mex. Fis. 44, 3, 235-244 (1998). [Barut, 1971] Barut, A. O.; Girardello, L...

work page doi:10.1063/1.1367328 2004
[3]

Dattoli, K

[Dattoli, 2017] G. Dattoli, K. Gorska, A. Horzela, S. Licciardi and R. M. Pidatella, Comments on the properties of Mittag-Leffler function, Eur. Phys. J. Spec. Top., 226: 3427-3443,

2017
[4]

Nonclassical

DOI:10.1140/epjst/e2018-00073-1 [Dhanalaksmi, 2013] K. Dhanalaksmi, A. Chatterjee, Energy spectrum and the thermal Mandel Q-parameter of a quadratically coupled phonon –photon polariton, Solid State Commun., 156, 31-34 (2013). [Dodonov, 2002] V. V. Dodonov, “Nonclassical” states in quantum optics: a “squeezed” review of the first 75 years, J. Opt. B: Quan...

work page doi:10.1140/epjst/e2018-00073-1 2013
[5]

Droghei, Deformed Boson Algebras and Wα,β,ν-Coherent States: A New Quantum Framework, Mathematics 13, 5, 759, 1-16 (2025); https://doi.org/10.3390/math13050759

[Droghei, 2025] R. Droghei, Deformed Boson Algebras and Wα,β,ν-Coherent States: A New Quantum Framework, Mathematics 13, 5, 759, 1-16 (2025); https://doi.org/10.3390/math13050759. [Fan, 1983] H-Y. Fan, T-N. Ruan, Coherent state formulation of the Weyl correspondence and the Wigner function, Commun. Theor. Phys. (Beijing, China), 2, 6, 1563-1474 (1983). 41...

work page doi:10.3390/math13050759 2025
[6]

Path Integrals. From Quantum Information to Cosmology

[Husimi, 1940] K. Husimi, Some formal properties of the Density Matrix, Proc. Phys.-Math. Soc. Jpn., 3rd Series, 22, 4, 264-314 (1940). [Inomata, 2005] A. Inomata, M. Sadiq, Modification of Klauder’s coherent states , The 8-th Int. Confer. “Path Integrals. From Quantum Information to Cosmology”, http://www1.jinr.ru/Proceedings/Burdik-2005/pdf/inomata.pdf ...

work page doi:10.12785/jant/030102 1940
[7]

Kratzer, Die ultraroten Rotationsspektren der Halogenwasserstoffe, Z

42 [Kratzer, 1920] A. Kratzer, Die ultraroten Rotationsspektren der Halogenwasserstoffe, Z. Phys. 3, 5, 289-307 (1920). [Mandel, 1994] L. Mandel, E. Wolf, Optical Coherence and Quantum Optics , Cambridge University Press, Cambridge,

1920
[8]

[Martinez, 2024] J. M. Martinez-Espinosa, R. E. Balderas -Navarro, Shi -Hai Dong, On the solution of the Schr ödinger equation with 2 nd Pöschl-Teller potentials, Results in Physics, 58, 107455 (2024). [Mathai, 1973] A. M. Mathai , R. K. Saxena , Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences , Lecture Notes in ...

2024
[9]

Messiah, Quantum Mechanics, Vol

[Messiah, 1967] A. Messiah, Quantum Mechanics, Vol. I, North-Holland, Amsterdam,

1967
[10]

Mojaveri, A

[Mojaveri, 2013] B. Mojaveri, A. Dehghani, Generalized su(1,

2013
[11]

coherent states for pseudo harmonic oscillator and their nonclassical properties, Eur. Phys. J. D, 67, 8, 179 (2013). https://doi.org/10.1140/epjd/e2013-40258-3 [Nielsen, 2010] M. A. Nielsen, I. L.Chuang, Quantum Computation and Quantum Information (10th anniversary ed.). Cambridge: Cambridge Univ. Press,

work page doi:10.1140/epjd/e2013-40258-3 2013
[12]

[Pennini, 2014] F

ISBN 978-0-521-63503-5. [Pennini, 2014] F. Pennini, A. Plastino, Different creation -destruction operators’ ordering, quasi-probabiliteies, and mandel parameter, Rev. Mex. de Fisica E, 60 103-106 (2014). [Perelomov, 1972] A. M. Perelomov, Coherent states for arbitrary Lie groups, Commun. Math. Phys., 26, 3, 222-236 (1972); arXiv: math-ph/0203002. [Perelom...

Pith/arXiv arXiv 2014
[13]

Der stetige Übergang von der Mikro - zur Makromechanik

[Popov, 2001] D. Popov, Coherent states of the pseudoharmonic oscillator , J. Phys. A: Math. Gen. 34, 5283-5296 (2001). [Popov, 2003] D. Popov, Gazeau-Klauder quasi-coherent states for the Morse oscillator , Phys. Lett. A 316, 6, 369-381 (2003). [Popov, 2014] D. Popov, V. Sajfert, J. P. Šetrajčić, N. Pop, Coherent States Formalism Applied to the Quantum W...

work page doi:10.1166/qm.2014.1137 2001

[1] [1]

[Antoine, 2001] J. P. Antoine, J. P. Gazeau, P. Monceau, J. R. Klauder and K. A. Penson, Temporally stable coherent states for infinite well and Pöschl –Teller potentials, J. Math. Phys. 42(6): 2349-2387,

2001

[2] [2]

New "Coherent

DOI: 10.1063/1.1367328 [Appl, 2004] T. Appl, D. H. Schiller, Generalized hypergeometric coherent states, J. Phys. A: Math. Gen. 37, 7, 2731 (2004). DOI 10.1088/0305-4470/37/7/015 [Atakishiyev, 1998] N. M. Atakishiyev, E. Jafarov, S. M. Nagiyev, K. B. Wolf, Meixner oscillators, Rev. Mex. Fis. 44, 3, 235-244 (1998). [Barut, 1971] Barut, A. O.; Girardello, L...

work page doi:10.1063/1.1367328 2004

[3] [3]

Dattoli, K

[Dattoli, 2017] G. Dattoli, K. Gorska, A. Horzela, S. Licciardi and R. M. Pidatella, Comments on the properties of Mittag-Leffler function, Eur. Phys. J. Spec. Top., 226: 3427-3443,

2017

[4] [4]

Nonclassical

DOI:10.1140/epjst/e2018-00073-1 [Dhanalaksmi, 2013] K. Dhanalaksmi, A. Chatterjee, Energy spectrum and the thermal Mandel Q-parameter of a quadratically coupled phonon –photon polariton, Solid State Commun., 156, 31-34 (2013). [Dodonov, 2002] V. V. Dodonov, “Nonclassical” states in quantum optics: a “squeezed” review of the first 75 years, J. Opt. B: Quan...

work page doi:10.1140/epjst/e2018-00073-1 2013

[5] [5]

Droghei, Deformed Boson Algebras and Wα,β,ν-Coherent States: A New Quantum Framework, Mathematics 13, 5, 759, 1-16 (2025); https://doi.org/10.3390/math13050759

[Droghei, 2025] R. Droghei, Deformed Boson Algebras and Wα,β,ν-Coherent States: A New Quantum Framework, Mathematics 13, 5, 759, 1-16 (2025); https://doi.org/10.3390/math13050759. [Fan, 1983] H-Y. Fan, T-N. Ruan, Coherent state formulation of the Weyl correspondence and the Wigner function, Commun. Theor. Phys. (Beijing, China), 2, 6, 1563-1474 (1983). 41...

work page doi:10.3390/math13050759 2025

[6] [6]

Path Integrals. From Quantum Information to Cosmology

[Husimi, 1940] K. Husimi, Some formal properties of the Density Matrix, Proc. Phys.-Math. Soc. Jpn., 3rd Series, 22, 4, 264-314 (1940). [Inomata, 2005] A. Inomata, M. Sadiq, Modification of Klauder’s coherent states , The 8-th Int. Confer. “Path Integrals. From Quantum Information to Cosmology”, http://www1.jinr.ru/Proceedings/Burdik-2005/pdf/inomata.pdf ...

work page doi:10.12785/jant/030102 1940

[7] [7]

Kratzer, Die ultraroten Rotationsspektren der Halogenwasserstoffe, Z

42 [Kratzer, 1920] A. Kratzer, Die ultraroten Rotationsspektren der Halogenwasserstoffe, Z. Phys. 3, 5, 289-307 (1920). [Mandel, 1994] L. Mandel, E. Wolf, Optical Coherence and Quantum Optics , Cambridge University Press, Cambridge,

1920

[8] [8]

[Martinez, 2024] J. M. Martinez-Espinosa, R. E. Balderas -Navarro, Shi -Hai Dong, On the solution of the Schr ödinger equation with 2 nd Pöschl-Teller potentials, Results in Physics, 58, 107455 (2024). [Mathai, 1973] A. M. Mathai , R. K. Saxena , Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences , Lecture Notes in ...

2024

[9] [9]

Messiah, Quantum Mechanics, Vol

[Messiah, 1967] A. Messiah, Quantum Mechanics, Vol. I, North-Holland, Amsterdam,

1967

[10] [10]

Mojaveri, A

[Mojaveri, 2013] B. Mojaveri, A. Dehghani, Generalized su(1,

2013

[11] [11]

coherent states for pseudo harmonic oscillator and their nonclassical properties, Eur. Phys. J. D, 67, 8, 179 (2013). https://doi.org/10.1140/epjd/e2013-40258-3 [Nielsen, 2010] M. A. Nielsen, I. L.Chuang, Quantum Computation and Quantum Information (10th anniversary ed.). Cambridge: Cambridge Univ. Press,

work page doi:10.1140/epjd/e2013-40258-3 2013

[12] [12]

[Pennini, 2014] F

ISBN 978-0-521-63503-5. [Pennini, 2014] F. Pennini, A. Plastino, Different creation -destruction operators’ ordering, quasi-probabiliteies, and mandel parameter, Rev. Mex. de Fisica E, 60 103-106 (2014). [Perelomov, 1972] A. M. Perelomov, Coherent states for arbitrary Lie groups, Commun. Math. Phys., 26, 3, 222-236 (1972); arXiv: math-ph/0203002. [Perelom...

Pith/arXiv arXiv 2014

[13] [13]

Der stetige Übergang von der Mikro - zur Makromechanik

[Popov, 2001] D. Popov, Coherent states of the pseudoharmonic oscillator , J. Phys. A: Math. Gen. 34, 5283-5296 (2001). [Popov, 2003] D. Popov, Gazeau-Klauder quasi-coherent states for the Morse oscillator , Phys. Lett. A 316, 6, 369-381 (2003). [Popov, 2014] D. Popov, V. Sajfert, J. P. Šetrajčić, N. Pop, Coherent States Formalism Applied to the Quantum W...

work page doi:10.1166/qm.2014.1137 2001