In a recent paper we introduced a model of extended electrons, which is fully compatible with quantum mechanics in the formulation of Schrödinger. However, it contradicts the current interpretation of electrons as point-particles. Here, we show by a statistical analysis of high-resolution scanning tunneling microscopy (STM) experiments, that the interpretation of electrons as point particles and, consequently, the interpretation of the density of electron charge as a statistical quantity will lead to a conflict with the Heisenberg uncertainty principle. Given the precision in these experiments we find that the uncertainty principle would be violated by close to two orders of magnitude, if this interpretation were correct. We are thus forced to conclude that the density of electron charge is a physically real, i.e. in principle precisely measurable quantity, as derived in a recent paper. Experimental evidence to the contrary, in particular high-energy scattering experiments, is briefly discussed. The finding is expected to have wide implications in condensed matter physics, chemistry, and biology, scientific disciplines which are based on the properties and interactions of electrons.
The principle of uncertainty is one of the essential principles of quantum mechanics as it outlines that there is a reflective limit to the measurement of microscopic particle motion. However it is impossible to measure at once and accurately both the position and momentum data of any microscopic particle no matter what means are used to do so. This paper starts by discussing the notion and mathematical derivation of the uncertainty principle as put forth by Heisenberg explaining the physical relevance of the principle. It introduces the natural uncertainty the principle introduces to the dynamics of particles in the microscopic world, and the significant part it plays in the theory of quantum mechanics. The article overviews the explanations of the principle on phenomena including atomic stability and scanning tunneling microscopes, and the interference of measurement actions on the states of particles and the shortcomings of measurement precision. According to the wave-particle duality of quantum mechanics, this article gives a required theoretical basis of microscopic phenomena. Likewise, the use of the research theory on the uncertainty principle by distinguished scholars in the physics fraternity helps the readers in achieving the knowledge in related subjects more easily.
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-ray microscope. Heisenberg argued that any attempt to localize a particle within a spatial region
Δx requires radiation of wavelength comparable to
Δx, which inevitably transfers an uncontrollable momentum
Δp 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 and
Δp 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 be a Hilbert space and let
∣ψ⟩∈H be anormalized state,
⟨ψ∣ψ⟩=1. Let
A and
B beself-adjoint operators (observables) on a common dense domain containing
∣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. Therefore\begin{equation}\sigma_A^2 \sigma_B^2 \ge \left| \langle \psi| \Delta A \Delta B |\psi\rangle \right|^2= |\langle \Delta A \Delta B \rangle|^2 .\label{eq:cs_step}\end{equation} \subsection{Robertson--Schr\"odinger decomposition} Decompose
ΔAΔB into commutator and anticommutator parts:\begin{equation}\Delta A \Delta B=\frac{1}{2}\bigl[\Delta A,\Delta B\bigr]+\frac{1}{2}\bigl\{\Delta A,\Delta B\bigr\},\end{equation}where
[X,Y]=XY−YX and
{X,Y}=XY+YX. Taking expectation values gives\begin{equation}\langle \Delta A \Delta B \rangle=\frac{1}{2}\langle [\Delta A,\Delta B]\rangle+\frac{1}{2}\langle \{\Delta A,\Delta B\}\rangle .\label{eq:decomp_exp}\end{equation} Now note the following facts for self-adjoint
A,B:\begin{itemize}\item The operator
[ΔA,ΔB] is anti-self-adjoint, hence
⟨[ΔA,ΔB]⟩ is purely imaginary.\item The operator
{ΔA,ΔB} is self-adjoint, hence
⟨{ΔA,ΔB}⟩ is real.\end{itemize}Therefore the two terms in \eqref{eq:decomp_exp} are orthogonal in the complex plane(real vs.\ imaginary), and we may write\begin{equation}|\langle \Delta A \Delta B \rangle|^2=\left(\frac{1}{2}|\langle [\Delta A,\Delta B]\rangle|\right)^2+\left(\frac{1}{2}\langle \{\Delta A,\Delta B\}\rangle\right)^2 .\label{eq:RS_identity}\end{equation} Substituting \eqref{eq:RS_identity} into \eqref{eq:cs_step} yields theSchr\"odinger uncertainty relation:\begin{equation}\boxed{\sigma_A^2 \sigma_B^2 \ge\left(\frac{1}{2}|\langle [\Delta A,\Delta B]\rangle|\right)^2+\left(\frac{1}{2}\langle \{\Delta A,\Delta B\}\rangle\right)^2}.\label{eq:schrodinger}\end{equation} Since constants commute with everything,\begin{equation}[\Delta A,\Delta B] = [A-\langle A\rangle,\; B-\langle B\rangle] = [A,B].\end{equation}Hence \eqref{eq:schrodinger} can be written as\begin{equation}\boxed{\sigma_A^2 \sigma_B^2 \ge\left(\frac{1}{2}|\langle [A,B]\rangle|\right)^2+\left(\frac{1}{2}\langle \{\Delta A,\Delta B\}\rangle\right)^2}.\end{equation} \subsection{Robertson inequality (weaker but widely quoted)} Dropping the nonnegative anticommutator term gives Robertson's inequality:\begin{equation}\boxed{\sigma_A \sigma_B \ge \frac{1}{2}|\langle [A,B]\rangle|}.\label{eq:robertson}\end{equation} \subsection{Specialization to position and momentum} Take
A=x and
B=p (one spatial dimension), with canonical commutation relation\begin{equation}[x,p] = i\hbar.\end{equation}Therefore,\begin{equation}|\langle [x,p]\rangle|= |\langle i\hbar \rangle|= \hbar,\end{equation}and \eqref{eq:robertson} yields the standard position--momentum uncertainty relation\begin{equation}\boxed{\sigma_x \sigma_p \ge \frac{\hbar}{2}.}\end{equation} \subsection{Condition for equality (minimum-uncertainty states)} Equality in Cauchy--Schwarz holds iff the vectors are linearly dependent:\begin{equation}\Delta A|\psi\rangle = \lambda \, \Delta B|\psi\rangle\quad \text{for some complex } \lambda.\end{equation}For
A=x and
B=p, this condition leads (in the
x-representation) to afirst-order differential equation whose normalized solutions are Gaussianwave packets. Hence Gaussian states saturate
σxσp=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⟩}\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 and
ΔB encode uncertainty byconstruction, since their variances measure ensemble spread relative to a mean.The subsequent derivation of a lower bound on
σAσB 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
A and
B are self-adjoint operators representing observables and
⟨A⟩=⟨ψ∣A∣ψ⟩ denotes the expectationvalue in a normalized quantum state
∣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⟩ and
σB2=⟨(ΔB)2⟩,measure the statistical spread of measurement outcomes over an ensembleof identically prepared systems
Compton scattering figures prominently in the introductory chapters of textbooks on quantum mechanics as providing evidence for the ‘‘particulate’’ nature of light. It also arises in discussions of ‘‘Heisenberg’s microscope,’’ in which it is shown that attempts to measure the position and the momentum of an electron simultaneously are governed by the uncertainty principle. However, recent developments in Compton scattering have not yet found their way into the textbooks. In this paper we review the history of the Compton effect and describe briefly some of the modern developments in x-ray and electron Compton scattering and their application to the study of electron momentum densities which we believe to have a pedagogical value. Finally, we return to Heisenberg’s microscope, and show how it has now become a reality. A complete analysis for all scattering angles depends crucially on the different expressions for the resolution and the depth of focus of the microscope.