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An Exploration of Heisenberg's Uncertainty Principle through the Lens of a Calculus of Uncertainty

Jay Kumar

Published Jul 28, 2026• 0 citations
Publisher

Abstract

Heisenberg’s uncertainty principle establishes a fundamental restriction on the simultaneous resolution of certain pairs of quantum observables. In its familiar position–momentum form, the standard deviations of position and momentum satisfy a lower bound determined by the reduced Planck constant. This physical restriction, however, is distinct from the computational problem of resolving an uncertainty state from incomplete, discrete, noisy, or progressively acquired observational data. This paper explores the latter problem through the lens of a Calculus of Uncertainty, or CoU. The CoU framework considered here begins with partial data and applies a recursive reconstruction procedure consisting of rectangular approximation, trapezoidal approximation, midpoint approximation, numerical refinement, bounded logistic formulation, validation, and recursive state resolution. Applied to the uncertainty principle, this procedure does not attempt to recover simultaneously exact values of incompatible observables, nor does it modify the quantum-mechanical lower bound. Instead, it reconstructs estimates of the statistical moments of position and momentum distributions from partial observations, derives corresponding estimates of their standard deviations, evaluates those estimates relative to the uncertainty relation, and recursively determines whether the resulting state is sufficiently supported to be accepted. A distinction is therefore maintained between quantum uncertainty, which is a property of the admissible statistical structure of a quantum state, and reconstruction uncertainty, which arises from partial sampling, numerical approximation, truncation, measurement limitations, and incomplete contextual information. The proposed formulation treats the uncertainty relation as a constraint within the CoU context and validation structure. A state may be accepted only when its numerical reconstruction has converged to a specified tolerance and its estimated uncertainty product remains consistent with the quantum-mechanical lower bound. The resulting approach provides a tolerance-aware, constraint-consistent method for representing how an inferred state develops as additional evidence is acquired, approximations are refined, and prior states are recursively revised or retained. A worked minimum-uncertainty Gaussian reconstruction demonstrates that quadrature may converge accurately on a truncated domain while the reconstructed uncertainty product remains inadmissible until sufficient distributional mass is captured. **Keywords:** Calculus of Uncertainty, Heisenberg uncertainty principle, quantum uncertainty, partial data, numerical reconstruction, midpoint rule, logistic state formulation, recursive state resolution

DOI
10.5281/zenodo.21652464

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Heisenberg’s uncertainty principle establishes a fundamental restriction on the simultaneous resolution of certain pairs of quantum observables. In its familiar position–momentum form, the standard deviations of position and momentum satisfy a lower bound determined by the reduced Planck constant. This physical restriction, however, is distinct from the computational problem of resolving an uncertainty state from incomplete, discrete, noisy, or progressively acquired observational data. This paper explores the latter problem through the lens of a Calculus of Uncertainty, or CoU. The CoU framework considered here begins with partial data and applies a recursive reconstruction procedure consisting of rectangular approximation, trapezoidal approximation, midpoint approximation, numerical refinement, bounded logistic formulation, validation, and recursive state resolution. Applied to the uncertainty principle, this procedure does not attempt to recover simultaneously exact values of incompatible observables, nor does it modify the quantum-mechanical lower bound. Instead, it reconstructs estimates of the statistical moments of position and momentum distributions from partial observations, derives corresponding estimates of their standard deviations, evaluates those estimates relative to the uncertainty relation, and recursively determines whether the resulting state is sufficiently supported to be accepted. A distinction is therefore maintained between quantum uncertainty, which is a property of the admissible statistical structure of a quantum state, and reconstruction uncertainty, which arises from partial sampling, numerical approximation, truncation, measurement limitations, and incomplete contextual information. The proposed formulation treats the uncertainty relation as a constraint within the CoU context and validation structure. A state may be accepted only when its numerical reconstruction has converged to a specified tolerance and its estimated uncertainty product remains consistent with the quantum-mechanical lower bound. The resulting approach provides a tolerance-aware, constraint-consistent method for representing how an inferred state develops as additional evidence is acquired, approximations are refined, and prior states are recursively revised or retained. A worked minimum-uncertainty Gaussian reconstruction demonstrates that quadrature may converge accurately on a truncated domain while the reconstructed uncertainty product remains inadmissible until sufficient distributional mass is captured. Keywords: Calculus of Uncertainty, Heisenberg uncertainty principle, quantum uncertainty, partial data, numerical reconstruction, midpoint rule, logistic state formulation, recursive state resolution
The uncertainty principle is commonly regarded as a fundamental limitation ofquantum mechanics, often attributed directly to Heisenberg’s 1927 analysis.In this work, we revisit the original formulation of the principle and carefullydistinguish Heisenberg’s measurement–disturbance argument from the latermathematical uncertainty relations derived by Kennard, Robertson, andSchrödinger. We show that the standard variance-based formulation relies onthe prior introduction of fluctuation operators and ensemble averages, therebypresupposing statistical indeterminacy rather than deriving it from firstprinciples. From a physical perspective, Heisenberg’s argument is shown toreflect constraints imposed by invasive measurement techniques rather than auniversal prohibition on simultaneous knowledge of conjugate variables.Motivated by this distinction, we propose a conjecture that, in the absence ofmeasurement-induced energy transfer, the simultaneous determination ofposition and momentum may, in principle, be achievable. Our analysis suggeststhat quantum uncertainty, as traditionally formulated, encodes limitations ofcurrent observational methods rather than an intrinsic property of nature. \section{Heisenberg’s original observation and formulation (1927)} In his seminal 1927 paper \emph{Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik} \cite{Heisenberg1927}, Werner Heisenberg sought to clarify the physical meaning of quantum mechanics rather than to establish a rigorous mathematical theorem. His analysis was based on thought experiments concerning the act of measurement, most notably the γ\gammaγ-ray microscope. Heisenberg argued that any attempt to localize a particle within a spatial region Δx\Delta xΔx requires radiation of wavelength comparable to Δx\Delta xΔx, which inevitably transfers an uncontrollable momentum Δp\Delta pΔp to the particle. As a consequence of this measurement-induced disturbance, he proposed that the product of the inaccuracies in position and momentum measurements satisfies a relation of the order\begin{equation}\Delta x\, \Delta p \sim h .\end{equation}Here, Δx\Delta xΔx and Δp\Delta pΔp denote operational measurement imprecisions arising from the interaction between the measuring apparatus and the system, rather than statistical variances of quantum observables. Heisenberg did not introduce operators, Hilbert space, or probability distributions, nor did he provide a formal inequality with a precise lower bound. The argument was explicitly heuristic and qualitative, emphasizing limitations imposed by the measurement process itself. Consequently, the original uncertainty principle was formulated as a statement about measurement disturbance and experimental resolution, not as a universal mathematical constraint on quantum states. \section{Standard mathematical proof of the position--momentum uncertainty relation} \subsection{Setup and definitions} Let H\mathcal{H}H be a Hilbert space and let ∣ψ⟩∈H|\psi\rangle \in \mathcal{H}∣ψ⟩∈H be anormalized state, ⟨ψ∣ψ⟩=1\langle \psi | \psi \rangle = 1⟨ψ∣ψ⟩=1. Let AAA and BBB beself-adjoint operators (observables) on a common dense domain containing∣ψ⟩|\psi\rangle∣ψ⟩, such that all expressions below are well-defined. Define expectation values\cite{Kennard1927,Robertson1929,Schrodinger1930,Ballentine1998,Sakurai1994}\begin{equation}\langle A \rangle := \langle \psi | A | \psi \rangle,\qquad\langle B \rangle := \langle \psi | B | \psi \rangle .\end{equation}Define the fluctuation (centered) operators\begin{equation}\Delta A := A - \langle A \rangle, \qquad \Delta B := B - \langle B \rangle .\end{equation}The variances are\begin{equation}\sigma_A^2 := \langle (\Delta A)^2 \rangle = \langle \psi | (\Delta A)^2 | \psi \rangle,\qquad\sigma_B^2 := \langle (\Delta B)^2 \rangle = \langle \psi | (\Delta B)^2 | \psi \rangle.\end{equation} \subsection{Key inequality: Cauchy--Schwarz in Hilbert space} Define vectors\begin{equation}|\phi\rangle := \Delta A |\psi\rangle, \qquad |\chi\rangle := \Delta B |\psi\rangle .\end{equation}By the Cauchy--Schwarz inequality,\begin{equation}\langle \phi|\phi\rangle \, \langle \chi|\chi\rangle \ge |\langle \phi|\chi\rangle|^2 .\end{equation}Compute each factor:\begin{equation}\langle \phi|\phi\rangle= \langle \psi| (\Delta A)^\dagger (\Delta A) |\psi\rangle= \langle \psi| (\Delta A)^2 |\psi\rangle= \sigma_A^2,\end{equation}and similarly ⟨χ∣χ⟩=σB2\langle \chi|\chi\rangle = \sigma_B^2⟨χ∣χ⟩=σB2​. 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Hence Gaussian states saturate σxσp=ℏ/2\sigma_x \sigma_p = \hbar/2σx​σp​=ℏ/2. \subsection{Definition of fluctuation operators and their role in the uncertainty relation} In the mathematical formulation of quantum uncertainty developed after Heisenberg’s original 1927 heuristic proposal, the quantities entering the uncertainty relation were redefined in statistical terms. In particular, in his 1929 paper, Robertson introduced the operators. \noindent\textbf{Critical observation.}The introduction of the fluctuation operators \begin{center}\fcolorbox{red}{white}{ΔA=A−⟨A⟩,ΔB=B−⟨B⟩\Delta A = A - \langle A \rangle,\qquad\Delta B = B - \langle B \rangleΔA=A−⟨A⟩,ΔB=B−⟨B⟩}\end{center} \textbf{constitutes a nontrivial mathematicalassumption rather than a derived physical necessity. In the Robertson--Schrödinger formalism, the uncertainty of observables is defined \emph{a priori} in terms ofstatistical deviations from expectation values, thereby presupposing the verynotion of indeterminacy that the uncertainty principle is intended to justify.In other words, the operators ΔA\Delta AΔA and ΔB\Delta BΔB encode uncertainty byconstruction, since their variances measure ensemble spread relative to a mean.The subsequent derivation of a lower bound on σAσB\sigma_A \sigma_BσA​σB​ thereforeestablishes a constraint on these predefined statistical quantities, rather than demonstrating that physical observables must be uncertain from first principles.Consequently, the uncertainty relation obtained in this manner is logicallyconditional upon the initial choice of variance-based definitions and does not,by itself, prove the existence of intrinsic measurement indeterminacy or disturbance in individual quantum events.} \textit{They actully assumed and defined the spread and uncertainity initial in both the quantities and proved the uncertainity principle.} where AAA and BBB are self-adjoint operators representing observables and⟨A⟩=⟨ψ∣A∣ψ⟩\langle A \rangle = \langle \psi | A | \psi \rangle⟨A⟩=⟨ψ∣A∣ψ⟩ denotes the expectationvalue in a normalized quantum state ∣ψ⟩|\psi\rangle∣ψ⟩ \cite{Robertson1929}.These operators describe deviations of observables from their mean valuesand are introduced in direct analogy with classical statistical fluctuations.The corresponding variances,σA2=⟨(ΔA)2⟩\sigma_A^2 = \langle (\Delta A)^2 \rangleσA2​=⟨(ΔA)2⟩ andσB2=⟨(ΔB)2⟩\sigma_B^2 = \langle (\Delta B)^2 \rangleσB2​=⟨(ΔB)2⟩,measure the statistical spread of measurement outcomes over an ensembleof identically prepared systems
Background: There has been altercation for and against Heisenberg uncertainty principles, whose earlier mathematical form has assumed other forms. This is despite the fact that the original form ought to be for the hydrogen atom only. Methods: the method is purely theoretical and computational, involving the use of spectroscopic data in the literature. Aim and objectives: The aim of the research was to show that half of reduced Planck’s constant (ℏ/2) cannot be restricted to a lower limit to the product of the uncertainty in the momentum and the uncertainty in the position. The objectives were: 1) to undertake the derivation of an equation unifying the Heisenberg uncertainty principle and atomic principles; and 2) to give evidence that half of the reduced Planck’s constant (ℏ/2) is strictly for hydrogen. Results and discussion: With hydrogen apart from other elements, fractions and multiples of ℏ/2, 0.0053–0.0264 corresponding to the radii ranging between a0/100 and a0/20 (a0 is the Bohr’s radius), and 5-26 corresponding to the radii ranging between 4a0 and 49a0 were demonstrated; the probability of locating the position and momentum increases with decreasing radii. Conclusion: The equations for the determination of the fractions and multiples of ℏ/2 were derived. Computation gave values less than and higher than ℏ/2; the latter was found to be restricted to hydrogen only. The possibility of linking the product of uncertainties in position and momentum to Pa may be reserved for a future study.
In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.
Learning physical properties of a quantum system is essential for the developments of quantum technologies. However, Heisenberg’s uncertainty principle constrains the potential knowledge one can simultaneously have about a system in quantum theory. Aside from its fundamental significance, the mathematical characterization of this restriction, known as ‘uncertainty relation’, plays important roles in a wide range of applications, stimulating the formation of tighter uncertainty relations. In this work, we investigate the fundamental limitations of variance-based uncertainty relations, and introduce several ‘near optimal’ bounds for incompatible observables. Our results consist of two morphologically distinct phases: lower bounds that illustrate the uncertainties about measurement outcomes, and the upper bound that indicates the potential knowledge we can gain. Combining them together leads to an uncertainty interval, which captures the essence of uncertainties in quantum theory. Finally, we have detailed how to formulate lower bounds for product-form variance-based uncertainty relations by employing entropic uncertainty relations, and hence built a link between different forms of uncertainty relations.
Heisenberg's uncertainty principle was originally posed for the limit of the accuracy of simultaneous measurement of non-commuting observables as stating that canonically conjugate observables can be measured simultaneously only with the constraint that the product of their mean errors should be no less than a limit set by Planck's constant. However, Heisenberg with the subsequent completion by Kennard has long been credited only with a constraint for state preparation represented by the product of the standard deviations. Here, we show that Heisenberg actually proved the constraint for the accuracy of simultaneous measurement but assuming an obsolete postulate for quantum mechanics. This assumption, known as the repeatability hypothesis, formulated explicitly by von Neumann and Schrodinger, was broadly accepted until the 1970s, but abandoned in the 1980s, when completely general quantum measurement theory was established. We also survey the author's recent proposal for a universally valid reformulation of Heisenberg's uncertainty principle under the most general assumption on quantum measurement.
Heisenberg's uncertainty principle was originally posed for the limit of the accuracy of simultaneous measurement of non-commuting observables as stating that canonically conjugate observables can be measured simultaneously only with the constraint that the product of their mean errors should be no less than a limit set by Planck's constant. However, Heisenberg with the subsequent completion by Kennard has long been credited only with a constraint for state preparation represented by the product of the standard deviations. Here, we show that Heisenberg actually proved the constraint for the accuracy of simultaneous measurement but assuming an obsolete postulate for quantum mechanics. This assumption, known as the repeatability hypothesis, formulated explicitly by von Neumann and Schrödinger, was broadly accepted until the 1970s, but abandoned in the 1980s, when completely general quantum measurement theory was established. We also survey the author's recent proposal for a universally valid reformulation of Heisenberg's uncertainty principle under the most general assumption on quantum measurement.