Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review

A Note on the Quantum Rule of the Harmonic Oscillator

L. M. Yang

Published Nov 15, 1951• 223 citations
Publisher

Abstract

The purpose of the present note is to show, with the aid of an elementary example, that the commutation rules, which are usually given the rule of postulates in quantum mechanics, are in fact not arbitrary, provided that a more stringent definition of the Hilbert space and a strict expansion theorem are adopted.
DOI
10.1103/physrev.84.788

Related papers

The Quantum-Mechanical Equations of Motion and Commutation Relations

Calvin R. Putnam

It has been shown by Wigner that the equations of motion of a quantum-mechanical system with a hamiltonian $H=\frac{1}{2}{p}^{2}+V(q)$ need not imply the commutation relation $i[p,q]=1$. Particular examples are furnished by the free particle and the harmonic oscillator. On the other hand, Wigner points out that the above commutation rule does follow from the equations of motion if $V(q)$ is of the form $V=aq+b$ or $V=a{q}^{3}$. In the present note, this latter result is generalized to the case of an arbitrary potential $V=a{q}^{n}+b$, where $n$ is an odd integer; the last-named potentials are, in fact, particular cases of those satisfying a general criterion contained in the article.

Physical Review•Sep 1, 1951•4 citations

Quantum mechanics

Peter Szekeres

Our purpose in this chapter is to present the key concepts of quantum mechanics in the language of Hilbert spaces. The reader who has not previously met the physical ideas motivating quantum mechanics, and some of the more elementary applications of Schrödinger's equation, is encouraged to read any of a number of excellent texts on the subject such as [1–4]. Otherwise, the statements given here must to a large extent be taken on trust – not an altogether easy thing to do, since the basic assertions of quantum theory are frequently counterintuitive to anyone steeped in the classical view of physics. Quantum mechanics is frequently presented in the form of several postulates, as though it were an axiomatic system such as Euclidean geometry. As often presented, these postulates may not meet the standards of mathematical rigour required for a strictly logical set of axioms, so that little is gained by such an approach. We will do things a little more informally here. For those only interested in the mathematical aspects of quantum mechanics and the role of Hilbert space see [5–8]. Many of the standard applications, such as the hydrogen atom, will be omitted here as they can be found in all standard textbooks, and we leave aside the enormous topic of measurement theory and interpretations of quantum mechanics. This is not to say that we need be totally comfortable with quantum theory as it stands. Undoubtedly, there are some philosophically disquieting features in the theory, often expressed in the form of socalled paradoxes.

Cambridge University Press eBooks•Dec 16, 2004•0 citations

The Harmonic Oscillator, a Review of Classicaland Elementary Quantum Mechanics

Carl W David

The classical harmonic oscillator and an elementary discussion of the quantum mechanical solutions for the harmonic oscillator are discussed.

•Jan 1, 2006•0 citations

From Operational Axioms, UEIR Ordering, and Conductance Geometry to Hilbert-Space Quantum Theory

Tajudeen H. Sikiru

We extend the operational uniqueness result of the ℓ2-norm sample-and-query rule [1] to the full quantum reconstruction programme. Starting from the five operational axioms A1–A5 of the companion note, we generalise them to the framework of generalised probabilistic theories (GPTs). We impose two additional structural postulates: the strict non-commutative UEIR ordering R ◦ I ◦ E ◦ U and the conductance geometry Φ = k/n derived from the Boolean hypercube Qn.We prove (Theorem 5.1) that the only GPT satisfying A1–A5 together with the UEIR ordering and the conductance geometry is complex Hilbert-space quantum mechanics (d ≥ 2). As corollaries we recover the standard Born rule p(k|ρ) = Tr(ρEk) on density operators and the emergence of Tang’s ℓ2-norm SQ access model as the unique R-layer classical realisation.We further prove (Theorem 7.1) that under Born-rule ℓ2-norm sampling, the expected conductance of Haar-random quantum states satisfies EHaar[Φeff ] = (n + 1)/(2n) → 1/2 as n → ∞. This makes the quantum-classical threshold pc = 1/2 a proved theorem derivable from quantum mechanics, not an external postulate. We identify a structural parallel between this convergence and the Riemann critical line Re(s) = 1/2, arising from the shared involution structure of L2-norm squaring, and state a precise open conjecture.

•Jun 5, 2026•0 citations

On a Quantum Action Principle

Natalya Gorobey, Alexander S. Lukyanenko

A quantum version of the action principle is formulated in terms of real parameters of a wave functional. The classical limit of the quantum action of a harmonic oscillator is obtained.

arXiv:0810.2255•Oct 13, 2008•0 citations

CXXIV. A note on the quantum dynamical principle

Julian Schwinger

It is shown that commutation relations are derived consistently from the quantum dynamical principle, for systems obeying first order equations of motion.

The London Edinburgh and Dublin Philosophical Magazine and Journal of Science•Oct 1, 1953•48 citations
It has been shown by Wigner that the equations of motion of a quantum-mechanical system with a hamiltonian H=12p2+V(q)H=\frac{1}{2}{p}^{2}+V(q)H=21​p2+V(q) need not imply the commutation relation i[p,q]=1i[p,q]=1i[p,q]=1. Particular examples are furnished by the free particle and the harmonic oscillator. On the other hand, Wigner points out that the above commutation rule does follow from the equations of motion if V(q)V(q)V(q) is of the form V=aq+bV=aq+bV=aq+b or V=aq3V=a{q}^{3}V=aq3. In the present note, this latter result is generalized to the case of an arbitrary potential V=aqn+bV=a{q}^{n}+bV=aqn+b, where nnn is an odd integer; the last-named potentials are, in fact, particular cases of those satisfying a general criterion contained in the article.
Our purpose in this chapter is to present the key concepts of quantum mechanics in the language of Hilbert spaces. The reader who has not previously met the physical ideas motivating quantum mechanics, and some of the more elementary applications of Schrödinger's equation, is encouraged to read any of a number of excellent texts on the subject such as [1–4]. Otherwise, the statements given here must to a large extent be taken on trust – not an altogether easy thing to do, since the basic assertions of quantum theory are frequently counterintuitive to anyone steeped in the classical view of physics. Quantum mechanics is frequently presented in the form of several postulates, as though it were an axiomatic system such as Euclidean geometry. As often presented, these postulates may not meet the standards of mathematical rigour required for a strictly logical set of axioms, so that little is gained by such an approach. We will do things a little more informally here. For those only interested in the mathematical aspects of quantum mechanics and the role of Hilbert space see [5–8]. Many of the standard applications, such as the hydrogen atom, will be omitted here as they can be found in all standard textbooks, and we leave aside the enormous topic of measurement theory and interpretations of quantum mechanics. This is not to say that we need be totally comfortable with quantum theory as it stands. Undoubtedly, there are some philosophically disquieting features in the theory, often expressed in the form of socalled paradoxes.
We extend the operational uniqueness result of the ℓ2-norm sample-and-query rule [1] to the full quantum reconstruction programme. Starting from the five operational axioms A1–A5 of the companion note, we generalise them to the framework of generalised probabilistic theories (GPTs). We impose two additional structural postulates: the strict non-commutative UEIR ordering R ◦ I ◦ E ◦ U and the conductance geometry Φ = k/n derived from the Boolean hypercube Qn.We prove (Theorem 5.1) that the only GPT satisfying A1–A5 together with the UEIR ordering and the conductance geometry is complex Hilbert-space quantum mechanics (d ≥ 2). As corollaries we recover the standard Born rule p(k|ρ) = Tr(ρEk) on density operators and the emergence of Tang’s ℓ2-norm SQ access model as the unique R-layer classical realisation.We further prove (Theorem 7.1) that under Born-rule ℓ2-norm sampling, the expected conductance of Haar-random quantum states satisfies EHaar[Φeff ] = (n + 1)/(2n) → 1/2 as n → ∞. This makes the quantum-classical threshold pc = 1/2 a proved theorem derivable from quantum mechanics, not an external postulate. We identify a structural parallel between this convergence and the Riemann critical line Re(s) = 1/2, arising from the shared involution structure of L2-norm squaring, and state a precise open conjecture.