The quantum harmonic oscillator and the real Hilbert space

Sergio Giardino

arxiv: 2606.12060 · v1 · pith:QUKGDVTKnew · submitted 2026-06-10 · 🪐 quant-ph

The quantum harmonic oscillator and the real Hilbert space

Sergio Giardino This is my paper

Pith reviewed 2026-06-27 09:35 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum harmonic oscillatorreal Hilbert spacequaternionic wave functionscomplex wave functionsnon-stationary processesdamped oscillationsforced oscillationsself-interacting processes

0 comments

The pith

The quantum harmonic oscillator admits complex and quaternionic wave-function descriptions in real Hilbert space that suit non-stationary processes.

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

The paper formulates the classical harmonic oscillator with a complex position function and the quantum version with both complex and quaternionic wave functions. All quantum solutions are constructed inside the real Hilbert space formalism. The resulting descriptions are presented as appropriate for non-stationary cases such as damped oscillations, forced oscillations, and self-interacting processes that standard frameworks cannot handle adequately.

Core claim

By formulating the quantum harmonic oscillator in terms of complex and quaternionic wave functions within the real Hilbert space, the solutions reveal that these descriptions are suitable for non-stationary processes including damped oscillations, forced oscillations, and self-interacting processes that cannot be appropriately described otherwise.

What carries the argument

Real Hilbert space formalism applied to complex and quaternionic wave functions of the quantum harmonic oscillator

If this is right

Complex wave functions yield descriptions of damped and forced oscillations.
Quaternionic wave functions additionally accommodate self-interacting processes.
Both extensions remain inside the real Hilbert space formalism.
The complex and quaternionic frameworks address cases that cannot be described appropriately by other means.

Where Pith is reading between the lines

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

The same real-Hilbert-space construction could be applied to other solvable quantum systems to check whether non-stationary extensions appear systematically.
If the quaternionic solutions prove consistent, they would supply an explicit algebraic route to modeling dissipation without adding external baths or non-Hermitian terms.
The distinction between stationary and non-stationary regimes might then be re-expressed as a choice of number system rather than a change of dynamical equations.

Load-bearing premise

The real Hilbert space formalism remains consistent and physically meaningful when extended to accommodate quaternionic wave functions for the quantum harmonic oscillator.

What would settle it

Explicit derivation of an inconsistency between the quaternionic wave-function solutions and the standard energy spectrum or time-evolution operator of the stationary quantum harmonic oscillator would falsify the suitability claim.

read the original abstract

The harmonic oscillator is considered within generalized frameworks using complex and quaternionic numbers. The classical oscillator is considered in terms of a complex position function, and quantum oscillators are examined in terms of complex wave functions, and in terms of quaternionic wave functions as well. Both of the quantum solutions are obtained within the real Hilbert space formalism. The results reveal the complex and quaternionic descriptions as suitable frameworks for non-stationary processes, including damped oscillations, forced oscillations, and additionally self-interacting processes that cannot be appropriately described otherwise.

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 abstract claims quaternionic real-Hilbert-space descriptions are required for self-interacting oscillators that standard complex QM cannot handle, but supplies no derivations or counterexamples to back the exclusivity part.

read the letter

The core takeaway is that this paper positions real-Hilbert-space treatments of complex and quaternionic wave functions as useful for non-stationary quantum harmonic oscillators, specifically claiming they cover damped, forced, and self-interacting cases that supposedly cannot be handled otherwise. The abstract frames the work as an examination inside generalized frameworks rather than a broad overhaul.

What the paper actually does is extend the classical oscillator to a complex position function and then move to quantum versions using both complex and quaternionic wave functions, all inside the real Hilbert space formalism. That choice of setting is consistent with some prior work on alternative number systems in quantum mechanics, and the author at least identifies concrete physical processes (damping, forcing, self-interaction) as targets.

The main soft spot is the load-bearing claim that certain self-interacting processes cannot be described appropriately in standard complex quantum mechanics. No derivation, explicit counter-example, or comparison to time-dependent potentials, non-Hermitian terms, or open-system master equations is visible. Standard complex QM already accommodates explicit time dependence and effective dissipation, so the necessity argument remains unshown. Without the actual equations or a worked case where the complex formulation fails while the quaternionic one succeeds, the result reads as an assertion rather than a demonstrated limitation.

The paper is aimed at the small group of people working on quaternionic or real-Hilbert-space reformulations of quantum mechanics. A reader outside that niche will not find new, falsifiable predictions or reproducible calculations from the abstract alone. Given the absence of supporting math or prior-art comparison, I would not bring this to a reading group and would not recommend sending it for serious peer review in its current form. If the full manuscript contains explicit derivations that address the exclusivity claim, that could change the picture, but nothing in the available text supports that yet.

Referee Report

1 major / 0 minor

Summary. The manuscript examines the harmonic oscillator in generalized frameworks using complex and quaternionic numbers. The classical case uses a complex position function; the quantum cases use complex and quaternionic wave functions, both obtained in the real Hilbert space formalism. The central claim is that these descriptions provide suitable frameworks for non-stationary processes, including damped oscillations, forced oscillations, and self-interacting processes that cannot be appropriately described otherwise.

Significance. If the necessity claim were substantiated, the work would offer an alternative real-Hilbert-space route to certain time-dependent quantum problems. No machine-checked proofs, reproducible code, or parameter-free derivations are present. The significance remains low because the manuscript supplies no concrete demonstration that standard complex QM (time-dependent Schrödinger equation or master equations) is mathematically or physically insufficient for the self-interacting case.

major comments (1)

[Abstract] Abstract: the exclusivity claim that self-interacting processes 'cannot be appropriately described otherwise' is load-bearing for the headline result yet is unsupported; no derivation, counter-example, or explicit comparison is given showing where the standard complex formulation fails while the quaternionic real-Hilbert-space construction succeeds. Standard time-dependent potentials and non-Hermitian terms already accommodate forced and damped oscillators, so the necessity step requires explicit verification.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on the manuscript. We respond to the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the exclusivity claim that self-interacting processes 'cannot be appropriately described otherwise' is load-bearing for the headline result yet is unsupported; no derivation, counter-example, or explicit comparison is given showing where the standard complex formulation fails while the quaternionic real-Hilbert-space construction succeeds. Standard time-dependent potentials and non-Hermitian terms already accommodate forced and damped oscillators, so the necessity step requires explicit verification.

Authors: The referee is correct that the manuscript provides no explicit derivation, counter-example, or side-by-side comparison establishing that the standard complex formulation is insufficient for self-interacting processes. The abstract phrasing was intended to underscore the natural inclusion of such terms via the non-commutative quaternionic structure within the real-Hilbert-space setting, but this does not constitute a demonstrated necessity. We will revise the abstract to remove the exclusivity claim, replacing it with language indicating that the quaternionic real-Hilbert-space description supplies a suitable framework for self-interacting processes alongside damped and forced oscillations. This is a targeted clarification rather than a change to the technical results. revision: partial

Circularity Check

0 steps flagged

No circularity: no derivation chain or equations supplied to inspect

full rationale

The provided abstract and context contain no equations, no explicit derivations, and no self-citations or fitted parameters. The central claim that quaternionic descriptions are required for self-interacting processes is an unsupported assertion rather than a reduction of a result to its own inputs. Without any load-bearing mathematical steps visible, no instance of self-definitional, fitted-input, or self-citation circularity can be identified. The paper's internal consistency cannot be assessed from the given material, but the absence of any derivation chain means the circularity score is zero by the stated rules.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The abstract invokes the real Hilbert space as the setting for both complex and quaternionic quantum solutions without stating independent justification; the quaternionic wave function is presented as a new descriptive tool.

axioms (1)

domain assumption Real Hilbert space formalism can consistently host complex and quaternionic wave functions for the quantum harmonic oscillator.
Invoked to obtain the quantum solutions described in the abstract.

invented entities (1)

quaternionic wave function no independent evidence
purpose: To describe the quantum oscillator in a quaternionic framework suitable for non-stationary processes.
Introduced as part of the generalized framework; no independent evidence supplied in the abstract.

pith-pipeline@v0.9.1-grok · 5597 in / 1128 out tokens · 26677 ms · 2026-06-27T09:35:58.780326+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

86 extracted references

[1]

Rushka; J

M. Rushka; J. K. Freericks . ``A completely algebraic solution of the simple harmonic oscillator'' . Am. J. Phys. , 88:976--985, (2020)

2020
[2]

Moshinsky

M. Moshinsky . ``Transformation brackets for harmonic oscillator functions'' . Nucl. Phys. , 13 (1):104--116, (1959)

1959
[3]

Moshinski; Y

M. Moshinski; Y. F. Smirnov . The harmonic oscillator in modern physics . Harwood (1996)

1996
[4]

M. F. Bocko; R. Onofrio . ``On the measurement of a weak classical force coupled to a harmonic oscillator: experimental progress'' . Rev. Mod. Phys. , 68 :775, (1996)

1996
[5]

Blasone; P

M. Blasone; P. Jizba; G. Vitiello . ``Dissipation and quantization'' . Phys. Lett. , A287 :205--210, (2001)

2001
[6]

C-I. Um; K-H. Yeon; T. F. George . ``The quantum damped harmonic oscillator'' . Phys. Rept. , 362 :63--192, (2002)

2002
[7]

M. J. Blacker; D. L. Tilbrook . ``Alternative approach to the quantization of the damped harmonic oscillator'' . Phys. Rev. , A104 (3):032211, (2021)

2021
[8]

M\'arkus; K

F. M\'arkus; K. Gamb\'ar . ``A potential-based quantization procedure of the damped oscillator'' . Quantum Rep. , 4 (4):390--400, (2022)

2022
[9]

S. M. Barnett; J. D. Cresser; S. Croke . ``Revisiting the damped quantum harmonic oscillator'' . Phys. Scripta , 99 (2):025109, (2024)

2024
[10]

Lawrence; B

S. Lawrence; B. McPeak; D. Neill . ``Bootstrapping time-evolution in quantum mechanics'' . arXiv:2412.08721[hep-th] (2024)

arXiv 2024
[11]

Lo Chiatto; S

P. Lo Chiatto; S. Schenk; F. Yu . ``Quantum imprint of the anharmonic oscillator'' . arXiv:2308.01244[hep-th] (2023)

arXiv 2023
[12]

R. G. Lavoie; L. Marchildon; D. Rochon . ``The Bicomplex Quantum Harmonic Oscillator'' . Nuovo Cimento 125B 1173-92 , (2010)

2010
[13]

G. S. Djordjevic; B. Dragovich . ``p-adic and adelic harmonic oscillator with time dependent frequency'' . Theor. Math. Phys. , 124 :1059--1067, (2000)

2000
[14]

A. I. Arbab . ``The complex quantum harmonic oscillator model'' . Europhys. Lett , 98 :30008, (2012)

2012
[15]

Marsiglio

F. Marsiglio . ``The harmonic oscillator in quantum mechanics: A third way'' . Am. J. Phys. , 77 :253--258, (2009)

2009
[16]

D. R. M. Pimentel; A. S. Castro . ``A Laplace transform approach to the quantum harmonic oscillator'' . Eur. J. Phys. , 34 :199, (2013)

2013
[17]

Almalki; V

F. Almalki; V. V. Kisil . ``Geometric dynamics of a harmonic oscillator, arbitrary minimal uncertainty states and the smallest step 3 nilpotent Lie group'' . J. Phys. , A52 (2):025301, (2019)

2019
[18]

Deguchi; Y

S. Deguchi; Y. Fujiwara . ``Quantization of the damped harmonic oscillator based on a modified Bateman Lagrangian'' . Phys. Rev. , A101 (2):022105, (2020)

2020
[19]

Friedmann; C

T. Friedmann; C. R. Hagen . ``Group-theoretical derivation of angular momentum eigenvalues in spaces of arbitrary dimensions'' . J. Math. Phys. , 53 :122102, (2012)

2012
[20]

Marquette; C

I. Marquette; C. Quesne . ``Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators'' . J. Phys. , A46 (15):155201, (2013)

2013
[21]

Negro; S

Kuru; J. Negro; S. Salamanca . ``Demkov-Fradkin tensor for curved harmonic oscillators'' . Eur. Phys. J. Plus , 140 (2):144, (2025)

2025
[22]

F. Vega . ``Oscillators in a (2+1)- dimensional noncommutative space'' . J. Math. Phys. , 55 :032105, (2014)

2014
[23]

C. M. Bender; A. Felski; N. Hassanpour; S. P. Klevansky; A. Beygi . ``Analytic structure of eigenvalues of coupled quantum systems'' . Phys. Scripta , 92 (1):015201, (2017)

2017
[24]

D. E. Bruschi; G. S. Paraoanu; I. Fuentes; F. K. Wilhelm; A. W. Schell . ``General solution of the time evolution of two interacting harmonic oscillators'' . Phys. Rev. , A103 :023707, (2021)

2021
[25]

B. Paul; B. Bandyopadhyay; T. Banerjee . ``Attractive-repulsive interaction in coupled quantum oscillators'' . Phys. Rev. , E110 (3):034210, (2024)

2024
[26]

Izadparast; S

M. Izadparast; S. H. Mazharimousavi . ``Two dimensional non-Hermitian harmonic oscillator: coherent states'' . Phys. Scripta , 94 :115215, (2019)

2019
[27]

S. Sen; M. Dutta; S. Gangopadhyay . Lewis and Berry phases for a gravitational wave interacting with a quantum harmonic oscillator . Phys. Scripta , 99 (1):015007, (2024)

2024
[28]

Sajjad; A

M. Sajjad; A. Russo; M. Arcos; A. Grudka; J. Oppenheim . ``A quantum oscillator interacting with a classical oscillator'' . arXiv:2403.07479[quant-ph] (2024)

arXiv 2024
[29]

Coppo; A

A. Coppo; A. Cuccoli; P. Verrucchi . ``Magnetic clock for a harmonic oscillator'' . Phys. Rev. , A109 (5):052212, (2024)

2024
[30]

Pl\'avala; M

M. Pl\'avala; M. Kleinmann . ``Operational theories in phase space: toy model for the harmonic oscillator'' . Phys. Rev. Lett. , 128 (4):040405, (2022)

2022
[31]

E. E. N'Dolo . ``Non-Hermitian two-dimensional harmonic oscillator in noncommutative phase-space'' . Int. J. Geom. Meth. Mod. Phys. , 21 (04):2450085, (2024)

2024
[32]

Ghose; P

I. Ghose; P. Sen . ``The variational method applied to the harmonic oscillator in the presence of a delta function potential'' . Eur. J. Phys. , 42 :045406, (2021)

2021
[33]

Nasuda; N

Y. Nasuda; N. Sawado . ``Harmonic oscillator with a step and its isospectral properties'' . Phys. Scripta , 99 (4):045212, (2024)

2024
[34]

Avramov; M

V. Avramov; M. Radomirov; R. C. Rashkov; T. Vetsov . Complexity of quantum harmonic oscillator in external magnetic field'' . arXiv:2407.18631[quant-ph] (2024)

arXiv 2024
[35]

Ghosh; A

A. Ghosh; A. Sinha . ``Pseudo-Hermitian extensions of the harmonic and isotonic oscillators'' . J. Phys. Conf. Ser. , 2986 (1):012004, (2025)

2025
[36]

F. M. Fern\'andez; J. Garcia; N. Aquino; A. Flores-Riveros . ``On the two-dimensional harmonic oscillator with an electric field confined to a circular box'' . Phys. Scripta , 99 (12):125278, (2024)

2024
[37]

Saner; O

S. Saner; O. B a z a van; D. J. Webb; G. Araneda; D. M. Lucas; C. J. Ballance; R. Srinivas . ``Generating arbitrary superpositions of nonclassical quantum harmonic oscillator states'' . arXiv:2409.03482[quant-ph] (2024)

arXiv 2024
[38]

Chadzitaskos

G. Chadzitaskos . ``Coherent states of the asymmetric harmonic oscillator'' . arXiv:2406.03509[quant-ph] (2024)

arXiv 2024
[39]

P. Patra . ``On the two-dimensional time-dependent anisotropic harmonic oscillator in a magnetic field'' . J. Math. Phys. , 64 (4):042105, (2023)

2023
[40]

Kumar; R

R. Kumar; R. K. Yadav; A. Khare . ``Rational extension of anisotropic harmonic oscillator potentials in higher dimensions'' . Annals Phys. , 479 :170045, (2025)

2025
[41]

D. M. Tibaduiza; L. Pires; D. Szilard; C. A. D. Zarro; C. Farina; A. L. C. Rego . ``A time-dependent harmonic oscillator with two frequency jumps: an exact algebraic solution'' . Braz. J. Phys. , 50 :634--646, (2020)

2020
[42]

Soto-Eguibar; F

F. Soto-Eguibar; F. A. Asenjo; S. A. Hojman; H. M. Moya-Cessa . ``Bohm potential for the time dependent harmonic oscillator'' . J. Math. Phys. , 66 :122103, (2021)

2021
[43]

C. Yuce . ``Quantum inverted harmonic potential'' . Phys. Scripta , 96 (10):105006, (2021)

2021
[44]

't Hooft

G. 't Hooft . ``The hidden ontological variable in quantum harmonic oscillators'' . Front. Quant. Sci. Tech. , 3 :1505593, (2024)

2024
[45]

Chakraborty; A

S. Chakraborty; A. Mazumdar; R. Pradhan . ``Manifestation of quantum entanglement between harmonic oscillators in a de Sitter background'' . Phys. Rev. , D112 (8):085006, (2025)

2025
[46]

M. E. Pereira; A. G. Schmidt . ``Analogue black string in a quantum harmonic oscillator'' . Phys. Lett. , A554 :130764, (2025)

2025
[47]

Gombar; M

S. Gombar; M. Rutonjski; P. Mali; S. Rado s evi\'c; M. Panti\'c; M. Pavkov-Hrvojevi\'c . ``Infinite series involving special functions obtained using simple one-dimensional quantum mechanical problems'' . J. Phys. , A58 (11):115205, 2025

2025
[48]

E. I. Jafarov; S. M. Nagiyev; J. Van der Jeugt . ``Deformation of the Heisenberg-Weyl algebra and the Lie superalgebra osp ( 1|2 ) : exact solution for the quantum harmonic oscillator with a position-dependent mass . Eur. Phys. J. Plus , 140 (4):290, (2025)

2025
[49]

Volkoff; G

T. Volkoff; G. Gopalan . ``Length scale estimation of excited quantum oscillators'' . J. Phys. , A58 (41):41LT01, (2025)

2025
[50]

S. Alperin . ``The quantum Foucault modes'' . arXiv:2507.18420 [quant-ph] (2025)

arXiv 2025
[51]

Shapovalov; A

A. Shapovalov; A. Breev . ``Harmonic Oscillator Coherent States from the Standpoint of Orbit Theory'' . Symmetry , 15 (2):282, (2023)

2023
[52]

Giardino

S. Giardino . ``Quaternionic quantum mechanics in real Hilbert space'' . J. Geom. Phys. , 158 :103956, (2020)

2020
[53]

Giardino

S. Giardino . ``Non-anti-hermitian quaternionic quantum mechanics'' . Adv. Appl. Clifford Algebras , 28 (1):19, (2018)

2018
[54]

S. L. Adler . ``Quaternionic Quantum Mechanics and Quantum Fields'' . Oxford University Press (1995)

1995
[55]

Giardino

S. Giardino . ``Quaternionic Aharonov-Bohm Effect'' . Adv. Appl. Clifford Algebras , 27 (3):2445--2456, (2017)

2017
[56]

Giardino

S. Giardino . ``Quaternionic quantum particles'' . Adv. Appl. Clifford Algebras , 29 (4):83, (2019)

2019
[57]

Giardino

S. Giardino . ``Quaternionic quantum particles: new solutions'' . Can. J. Phys. , 99 :4, 6 (2017)

2017
[58]

Giardino

S. Giardino . ``Self-interacting quantum particles'' . Int. J. Geom. Meth. Mod. Phys. 22 , 12, 2550088 (2025)

2025
[59]

Giardino

S. Giardino . ``Virial theorem and generalized momentum in quaternionic quantum mechanics'' . Eur. Phys. J. Plus , 135 (1):114, (2020)

2020
[60]

Giardino

S. Giardino . ``Square-well potential in quaternic quantum mechanics'' . Europhys. Lett. , 132 :20007, 9 (2020)

2020
[61]

Giardino

S. Giardino . Quantum self-interaction within an infinitely deep cavity . Phys. Lett. , A578 :131474, (2026)

2026
[62]

Giardino

S. Giardino . ``Quaternionic elastic scattering'' . EPL , 132 (5):50010, (2020)

2020
[63]

Hasan; B

M. Hasan; B. P. Mandal . ``New scattering features of quaternionic point interaction in non-Hermitian quantum mechanics'' . J. Math. Phys. , 61 (3):032104, (2020)

2020
[64]

Giardino

S. Giardino . ``Quaternionic quantum harmonic oscillator'' . Eur. Phys. J. Plus , 136 (1):120, (2021)

2021
[65]

Giardino

S. Giardino . ``Deformed angular momentum algebra within the real Hilbert space'' . Int. J. Theor. Phys. , 65 (4):97, (2026)

2026
[66]

Giardino

S. Giardino . ``Spin and angular momentum in quaternionic quantum mechanics'' . EPL , 142 (1):12001, (2023)

2023
[67]

Giardino

S. Giardino . ``Self-interacting quantum particles and the Dirac delta potential'' . Braz. J. Phys. , 56 (1):46, (2026)

2026
[68]

Giardino

S. Giardino . ``Generalized imaginary units in quantum mechanics'' . Int. J. Geom. Meth. Math. Phys. (accept) arXiv:2311.14162[quant-ph] (2023)

arXiv 2023
[69]

Giardino

S. Giardino . ``Quaternionic Klein-Gordon equation'' . Eur. Phys. J. Plus , 136 (6):612, (2021)

2021
[70]

C. Rosa; S. Giardino . ``Klein-Gordon equation within the real Hilbert space formalism'' . Annals Phys. , 483 :170271, (2025)

2025
[71]

Giardino

S. Giardino . ``Quaternionic Dirac free particle'' . Int. J. Mod. Phys. , A36 (33):2150257, (2021)

2021
[72]

Giardino

S. Giardino . ``Quaternionic scalar field in the real Hilbert space'' . Int. J. Mod. Phys. , A37 (15):2250101, (2022)

2022
[73]

Giardino

S. Giardino . ``Quaternionic fermionic field'' . Int. J. Mod. Phys. , A37 (31n32):2250199, (2022)

2022
[74]

Giardino

S. Giardino . ``Expectation value dynamics within real Hilbert space quantum mechanics'' . Int. J. Theor. Phys. , 64 (10):257, (2025)

2025
[75]

Finkelstein

J. Finkelstein . ``In defense of real quantum theory'' . arXiv:2103.12740[quant-ph] (2021)

arXiv 2021
[76]

Chiribella; K

G. Chiribella; K. R. Davidson; V. I. Paulsen; M. Rahaman . ``Positive Maps and Entanglement in Real Hilbert Spaces'' . Annales Henri Poincare , 24 (12):4139--4168, (2023)

2023
[77]

h. Zhu . ``Hiding and masking quantum information in complex and real quantum mechanics'' . Phys. Rev. Res. , 3 (3):033176, (2021)

2021
[78]

C. A. Fuchs; M. Olshanii; M. B. Weiss . ``Quantum mechanics? It's all fun and games until someone loses an i '' . arXiv:2206.15343 [quant-ph] (2022)

arXiv 2022
[79]

V. Vedral . ``Quantum Mechanics with Real Numbers: Entanglement, Superselection Rules and Gauges'' . Quanta , 12 (1):164--170, (2023)

2023
[80]

Renou, D

M.-O. Renou, D. Trillo; M. Weilenmann; T. P. Le; A. Tavakoli; N. Gisin; A. Acin; M; Navascues . ``Quantum theory based on real numbers can be experimentally falsified'' . Nature , 600 (7890):625--629, (2021)

2021

Showing first 80 references.

[1] [1]

Rushka; J

M. Rushka; J. K. Freericks . ``A completely algebraic solution of the simple harmonic oscillator'' . Am. J. Phys. , 88:976--985, (2020)

2020

[2] [2]

Moshinsky

M. Moshinsky . ``Transformation brackets for harmonic oscillator functions'' . Nucl. Phys. , 13 (1):104--116, (1959)

1959

[3] [3]

Moshinski; Y

M. Moshinski; Y. F. Smirnov . The harmonic oscillator in modern physics . Harwood (1996)

1996

[4] [4]

M. F. Bocko; R. Onofrio . ``On the measurement of a weak classical force coupled to a harmonic oscillator: experimental progress'' . Rev. Mod. Phys. , 68 :775, (1996)

1996

[5] [5]

Blasone; P

M. Blasone; P. Jizba; G. Vitiello . ``Dissipation and quantization'' . Phys. Lett. , A287 :205--210, (2001)

2001

[6] [6]

C-I. Um; K-H. Yeon; T. F. George . ``The quantum damped harmonic oscillator'' . Phys. Rept. , 362 :63--192, (2002)

2002

[7] [7]

M. J. Blacker; D. L. Tilbrook . ``Alternative approach to the quantization of the damped harmonic oscillator'' . Phys. Rev. , A104 (3):032211, (2021)

2021

[8] [8]

M\'arkus; K

F. M\'arkus; K. Gamb\'ar . ``A potential-based quantization procedure of the damped oscillator'' . Quantum Rep. , 4 (4):390--400, (2022)

2022

[9] [9]

S. M. Barnett; J. D. Cresser; S. Croke . ``Revisiting the damped quantum harmonic oscillator'' . Phys. Scripta , 99 (2):025109, (2024)

2024

[10] [10]

Lawrence; B

S. Lawrence; B. McPeak; D. Neill . ``Bootstrapping time-evolution in quantum mechanics'' . arXiv:2412.08721[hep-th] (2024)

arXiv 2024

[11] [11]

Lo Chiatto; S

P. Lo Chiatto; S. Schenk; F. Yu . ``Quantum imprint of the anharmonic oscillator'' . arXiv:2308.01244[hep-th] (2023)

arXiv 2023

[12] [12]

R. G. Lavoie; L. Marchildon; D. Rochon . ``The Bicomplex Quantum Harmonic Oscillator'' . Nuovo Cimento 125B 1173-92 , (2010)

2010

[13] [13]

G. S. Djordjevic; B. Dragovich . ``p-adic and adelic harmonic oscillator with time dependent frequency'' . Theor. Math. Phys. , 124 :1059--1067, (2000)

2000

[14] [14]

A. I. Arbab . ``The complex quantum harmonic oscillator model'' . Europhys. Lett , 98 :30008, (2012)

2012

[15] [15]

Marsiglio

F. Marsiglio . ``The harmonic oscillator in quantum mechanics: A third way'' . Am. J. Phys. , 77 :253--258, (2009)

2009

[16] [16]

D. R. M. Pimentel; A. S. Castro . ``A Laplace transform approach to the quantum harmonic oscillator'' . Eur. J. Phys. , 34 :199, (2013)

2013

[17] [17]

Almalki; V

F. Almalki; V. V. Kisil . ``Geometric dynamics of a harmonic oscillator, arbitrary minimal uncertainty states and the smallest step 3 nilpotent Lie group'' . J. Phys. , A52 (2):025301, (2019)

2019

[18] [18]

Deguchi; Y

S. Deguchi; Y. Fujiwara . ``Quantization of the damped harmonic oscillator based on a modified Bateman Lagrangian'' . Phys. Rev. , A101 (2):022105, (2020)

2020

[19] [19]

Friedmann; C

T. Friedmann; C. R. Hagen . ``Group-theoretical derivation of angular momentum eigenvalues in spaces of arbitrary dimensions'' . J. Math. Phys. , 53 :122102, (2012)

2012

[20] [20]

Marquette; C

I. Marquette; C. Quesne . ``Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators'' . J. Phys. , A46 (15):155201, (2013)

2013

[21] [21]

Negro; S

Kuru; J. Negro; S. Salamanca . ``Demkov-Fradkin tensor for curved harmonic oscillators'' . Eur. Phys. J. Plus , 140 (2):144, (2025)

2025

[22] [22]

F. Vega . ``Oscillators in a (2+1)- dimensional noncommutative space'' . J. Math. Phys. , 55 :032105, (2014)

2014

[23] [23]

C. M. Bender; A. Felski; N. Hassanpour; S. P. Klevansky; A. Beygi . ``Analytic structure of eigenvalues of coupled quantum systems'' . Phys. Scripta , 92 (1):015201, (2017)

2017

[24] [24]

D. E. Bruschi; G. S. Paraoanu; I. Fuentes; F. K. Wilhelm; A. W. Schell . ``General solution of the time evolution of two interacting harmonic oscillators'' . Phys. Rev. , A103 :023707, (2021)

2021

[25] [25]

B. Paul; B. Bandyopadhyay; T. Banerjee . ``Attractive-repulsive interaction in coupled quantum oscillators'' . Phys. Rev. , E110 (3):034210, (2024)

2024

[26] [26]

Izadparast; S

M. Izadparast; S. H. Mazharimousavi . ``Two dimensional non-Hermitian harmonic oscillator: coherent states'' . Phys. Scripta , 94 :115215, (2019)

2019

[27] [27]

S. Sen; M. Dutta; S. Gangopadhyay . Lewis and Berry phases for a gravitational wave interacting with a quantum harmonic oscillator . Phys. Scripta , 99 (1):015007, (2024)

2024

[28] [28]

Sajjad; A

M. Sajjad; A. Russo; M. Arcos; A. Grudka; J. Oppenheim . ``A quantum oscillator interacting with a classical oscillator'' . arXiv:2403.07479[quant-ph] (2024)

arXiv 2024

[29] [29]

Coppo; A

A. Coppo; A. Cuccoli; P. Verrucchi . ``Magnetic clock for a harmonic oscillator'' . Phys. Rev. , A109 (5):052212, (2024)

2024

[30] [30]

Pl\'avala; M

M. Pl\'avala; M. Kleinmann . ``Operational theories in phase space: toy model for the harmonic oscillator'' . Phys. Rev. Lett. , 128 (4):040405, (2022)

2022

[31] [31]

E. E. N'Dolo . ``Non-Hermitian two-dimensional harmonic oscillator in noncommutative phase-space'' . Int. J. Geom. Meth. Mod. Phys. , 21 (04):2450085, (2024)

2024

[32] [32]

Ghose; P

I. Ghose; P. Sen . ``The variational method applied to the harmonic oscillator in the presence of a delta function potential'' . Eur. J. Phys. , 42 :045406, (2021)

2021

[33] [33]

Nasuda; N

Y. Nasuda; N. Sawado . ``Harmonic oscillator with a step and its isospectral properties'' . Phys. Scripta , 99 (4):045212, (2024)

2024

[34] [34]

Avramov; M

V. Avramov; M. Radomirov; R. C. Rashkov; T. Vetsov . Complexity of quantum harmonic oscillator in external magnetic field'' . arXiv:2407.18631[quant-ph] (2024)

arXiv 2024

[35] [35]

Ghosh; A

A. Ghosh; A. Sinha . ``Pseudo-Hermitian extensions of the harmonic and isotonic oscillators'' . J. Phys. Conf. Ser. , 2986 (1):012004, (2025)

2025

[36] [36]

F. M. Fern\'andez; J. Garcia; N. Aquino; A. Flores-Riveros . ``On the two-dimensional harmonic oscillator with an electric field confined to a circular box'' . Phys. Scripta , 99 (12):125278, (2024)

2024

[37] [37]

Saner; O

S. Saner; O. B a z a van; D. J. Webb; G. Araneda; D. M. Lucas; C. J. Ballance; R. Srinivas . ``Generating arbitrary superpositions of nonclassical quantum harmonic oscillator states'' . arXiv:2409.03482[quant-ph] (2024)

arXiv 2024

[38] [38]

Chadzitaskos

G. Chadzitaskos . ``Coherent states of the asymmetric harmonic oscillator'' . arXiv:2406.03509[quant-ph] (2024)

arXiv 2024

[39] [39]

P. Patra . ``On the two-dimensional time-dependent anisotropic harmonic oscillator in a magnetic field'' . J. Math. Phys. , 64 (4):042105, (2023)

2023

[40] [40]

Kumar; R

R. Kumar; R. K. Yadav; A. Khare . ``Rational extension of anisotropic harmonic oscillator potentials in higher dimensions'' . Annals Phys. , 479 :170045, (2025)

2025

[41] [41]

D. M. Tibaduiza; L. Pires; D. Szilard; C. A. D. Zarro; C. Farina; A. L. C. Rego . ``A time-dependent harmonic oscillator with two frequency jumps: an exact algebraic solution'' . Braz. J. Phys. , 50 :634--646, (2020)

2020

[42] [42]

Soto-Eguibar; F

F. Soto-Eguibar; F. A. Asenjo; S. A. Hojman; H. M. Moya-Cessa . ``Bohm potential for the time dependent harmonic oscillator'' . J. Math. Phys. , 66 :122103, (2021)

2021

[43] [43]

C. Yuce . ``Quantum inverted harmonic potential'' . Phys. Scripta , 96 (10):105006, (2021)

2021

[44] [44]

't Hooft

G. 't Hooft . ``The hidden ontological variable in quantum harmonic oscillators'' . Front. Quant. Sci. Tech. , 3 :1505593, (2024)

2024

[45] [45]

Chakraborty; A

S. Chakraborty; A. Mazumdar; R. Pradhan . ``Manifestation of quantum entanglement between harmonic oscillators in a de Sitter background'' . Phys. Rev. , D112 (8):085006, (2025)

2025

[46] [46]

M. E. Pereira; A. G. Schmidt . ``Analogue black string in a quantum harmonic oscillator'' . Phys. Lett. , A554 :130764, (2025)

2025

[47] [47]

Gombar; M

S. Gombar; M. Rutonjski; P. Mali; S. Rado s evi\'c; M. Panti\'c; M. Pavkov-Hrvojevi\'c . ``Infinite series involving special functions obtained using simple one-dimensional quantum mechanical problems'' . J. Phys. , A58 (11):115205, 2025

2025

[48] [48]

E. I. Jafarov; S. M. Nagiyev; J. Van der Jeugt . ``Deformation of the Heisenberg-Weyl algebra and the Lie superalgebra osp ( 1|2 ) : exact solution for the quantum harmonic oscillator with a position-dependent mass . Eur. Phys. J. Plus , 140 (4):290, (2025)

2025

[49] [49]

Volkoff; G

T. Volkoff; G. Gopalan . ``Length scale estimation of excited quantum oscillators'' . J. Phys. , A58 (41):41LT01, (2025)

2025

[50] [50]

S. Alperin . ``The quantum Foucault modes'' . arXiv:2507.18420 [quant-ph] (2025)

arXiv 2025

[51] [51]

Shapovalov; A

A. Shapovalov; A. Breev . ``Harmonic Oscillator Coherent States from the Standpoint of Orbit Theory'' . Symmetry , 15 (2):282, (2023)

2023

[52] [52]

Giardino

S. Giardino . ``Quaternionic quantum mechanics in real Hilbert space'' . J. Geom. Phys. , 158 :103956, (2020)

2020

[53] [53]

Giardino

S. Giardino . ``Non-anti-hermitian quaternionic quantum mechanics'' . Adv. Appl. Clifford Algebras , 28 (1):19, (2018)

2018

[54] [54]

S. L. Adler . ``Quaternionic Quantum Mechanics and Quantum Fields'' . Oxford University Press (1995)

1995

[55] [55]

Giardino

S. Giardino . ``Quaternionic Aharonov-Bohm Effect'' . Adv. Appl. Clifford Algebras , 27 (3):2445--2456, (2017)

2017

[56] [56]

Giardino

S. Giardino . ``Quaternionic quantum particles'' . Adv. Appl. Clifford Algebras , 29 (4):83, (2019)

2019

[57] [57]

Giardino

S. Giardino . ``Quaternionic quantum particles: new solutions'' . Can. J. Phys. , 99 :4, 6 (2017)

2017

[58] [58]

Giardino

S. Giardino . ``Self-interacting quantum particles'' . Int. J. Geom. Meth. Mod. Phys. 22 , 12, 2550088 (2025)

2025

[59] [59]

Giardino

S. Giardino . ``Virial theorem and generalized momentum in quaternionic quantum mechanics'' . Eur. Phys. J. Plus , 135 (1):114, (2020)

2020

[60] [60]

Giardino

S. Giardino . ``Square-well potential in quaternic quantum mechanics'' . Europhys. Lett. , 132 :20007, 9 (2020)

2020

[61] [61]

Giardino

S. Giardino . Quantum self-interaction within an infinitely deep cavity . Phys. Lett. , A578 :131474, (2026)

2026

[62] [62]

Giardino

S. Giardino . ``Quaternionic elastic scattering'' . EPL , 132 (5):50010, (2020)

2020

[63] [63]

Hasan; B

M. Hasan; B. P. Mandal . ``New scattering features of quaternionic point interaction in non-Hermitian quantum mechanics'' . J. Math. Phys. , 61 (3):032104, (2020)

2020

[64] [64]

Giardino

S. Giardino . ``Quaternionic quantum harmonic oscillator'' . Eur. Phys. J. Plus , 136 (1):120, (2021)

2021

[65] [65]

Giardino

S. Giardino . ``Deformed angular momentum algebra within the real Hilbert space'' . Int. J. Theor. Phys. , 65 (4):97, (2026)

2026

[66] [66]

Giardino

S. Giardino . ``Spin and angular momentum in quaternionic quantum mechanics'' . EPL , 142 (1):12001, (2023)

2023

[67] [67]

Giardino

S. Giardino . ``Self-interacting quantum particles and the Dirac delta potential'' . Braz. J. Phys. , 56 (1):46, (2026)

2026

[68] [68]

Giardino

S. Giardino . ``Generalized imaginary units in quantum mechanics'' . Int. J. Geom. Meth. Math. Phys. (accept) arXiv:2311.14162[quant-ph] (2023)

arXiv 2023

[69] [69]

Giardino

S. Giardino . ``Quaternionic Klein-Gordon equation'' . Eur. Phys. J. Plus , 136 (6):612, (2021)

2021

[70] [70]

C. Rosa; S. Giardino . ``Klein-Gordon equation within the real Hilbert space formalism'' . Annals Phys. , 483 :170271, (2025)

2025

[71] [71]

Giardino

S. Giardino . ``Quaternionic Dirac free particle'' . Int. J. Mod. Phys. , A36 (33):2150257, (2021)

2021

[72] [72]

Giardino

S. Giardino . ``Quaternionic scalar field in the real Hilbert space'' . Int. J. Mod. Phys. , A37 (15):2250101, (2022)

2022

[73] [73]

Giardino

S. Giardino . ``Quaternionic fermionic field'' . Int. J. Mod. Phys. , A37 (31n32):2250199, (2022)

2022

[74] [74]

Giardino

S. Giardino . ``Expectation value dynamics within real Hilbert space quantum mechanics'' . Int. J. Theor. Phys. , 64 (10):257, (2025)

2025

[75] [75]

Finkelstein

J. Finkelstein . ``In defense of real quantum theory'' . arXiv:2103.12740[quant-ph] (2021)

arXiv 2021

[76] [76]

Chiribella; K

G. Chiribella; K. R. Davidson; V. I. Paulsen; M. Rahaman . ``Positive Maps and Entanglement in Real Hilbert Spaces'' . Annales Henri Poincare , 24 (12):4139--4168, (2023)

2023

[77] [77]

h. Zhu . ``Hiding and masking quantum information in complex and real quantum mechanics'' . Phys. Rev. Res. , 3 (3):033176, (2021)

2021

[78] [78]

C. A. Fuchs; M. Olshanii; M. B. Weiss . ``Quantum mechanics? It's all fun and games until someone loses an i '' . arXiv:2206.15343 [quant-ph] (2022)

arXiv 2022

[79] [79]

V. Vedral . ``Quantum Mechanics with Real Numbers: Entanglement, Superselection Rules and Gauges'' . Quanta , 12 (1):164--170, (2023)

2023

[80] [80]

Renou, D

M.-O. Renou, D. Trillo; M. Weilenmann; T. P. Le; A. Tavakoli; N. Gisin; A. Acin; M; Navascues . ``Quantum theory based on real numbers can be experimentally falsified'' . Nature , 600 (7890):625--629, (2021)

2021