This paper is a small discussion about the uncertainty principle. With the great discoveries and researches made by predecessors, I try to answer some questions within the uncertainty principle. Using the definition of momentum, the momentum in de Broglie wave, the property of cosine function, the Euler’s formula, the basic property of plane wave, the law of derivative, the energy of quantum discovered by Max Planck, the photon energy discovered by Albert Einstein, the energy equation of a system and the Hamiltonian operator, I figure out the Schrodinger equation successfully. With the help of Schrodinger equation, the possibility density of a wave function, the concept of normalization, the average value of position and momentum, the Fourier transform, the integration by parts and the limit of normalized wave functions, I find the position operator and the momentum operator. Then using some knowledge of the standard deviation and the variance, the concept of inner product of vectors, some property of Dirac symbol, the law of calculation of complex number and the law of operator calculation, I obtain the general uncertainty relationship of abstract mathematical quantities. According to the general relationship, I get the inequality of uncertainty principle, and discuss the specific relationship between the position and the momentum. After getting the inequality, I use Heisenberg’s experiment of measuring an electron by using the γ ray to hit the electron to explain why we cannot measure the position and the momentum simultaneously. Based on Heisenberg’s experiment, I give my own explanation by discussing the earth and a spacecraft. Transforming the perspective, we can easily see how the accuracies of position and momentum change. In addition, I attempt to figure out why the uncertainty exists. However, I can only explain it by discussing the time and frequency in mathematical method and have no idea to work it out in a pure physics approach.
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
phenomena can often be analysed in a visually clear way by thought experiments. In the present work a new thought experiment is analysed, which describes the emission of a photon by a special mechanism. The analysis leads to the finding, that there is a natural connection between the energy-time uncertainty and the Heisenberg position-momentum uncertainty principle. A deeper analysis of this thought experiment, especially with regard to a justification of the Doppler effect from the uncertainty principle, leads to the conclusion, that the speed of light is invariant. Therefore it seems reasonable to assume, that the origin of relativistic laws lies in non-relativistic quantum mechanics.
One of the most striking deviations of quantum mechanics from classical mechanics is that any measurement of a microscopic system involves the interaction with the measuring instruments. The interaction cannot be neglected and in fact leads to uncontrollable disturbance on the system. In 1927, Werner Karl Heisenberg in his celebrated paper established the quantitative expression for unavoidable momentum disturbance caused by any position measurement. The crucial word here is simultaneously. The uncertainty relation gives the lower bound on the product of the uncertainties of two pairs of variables. But it says nothing about the accuracy with which individual observables can be measured. The physical meaning of energy-time uncertainty is different from that of x-px uncertainty. Position and momentum are observables to which Hermitian operators are assigned and which are measured at a particular time. An important implication of the uncertainty principle is that quantum systems do follow classical trajectories which conflict with our everyday experience.