Quantum Mechanics from a Heisenberg-type Equality

The usual Heisenberg uncertainty relation, ΔX ΔP ≥ ℏ/2, may be replaced by an exact equality for suitably chosen measures of position and momentum uncertainty, which is valid for all wavefunctions. This exactuncertainty relation, δX ΔPnc ≡ ℏ/2, can be generalised to other pairs of conjugate observables such as photon number and phase, and is sufficiently strong to provide the basis for moving from classical mechanics to quantum mechanics. In particular, the assumption of a nonclassical momentum fluctuation, having a strength which scales inversely with uncertainty in position, leads from the classical equations of motion to the Schrödinger equation.The usual Heisenberg uncertainty relation, ΔX ΔP ≥ ℏ/2, may be replaced by an exact equality for suitably chosen measures of position and momentum uncertainty, which is valid for all wavefunctions. This exactuncertainty relation, δX ΔPnc ≡ ℏ/2, can be generalised to other pairs of conjugate observables such as photon number and phase, and is sufficiently strong to provide the basis for moving from classical mechanics to quantum mechanics. In particular, the assumption of a nonclassical momentum fluctuation, having a strength which scales inversely with uncertainty in position, leads from the classical equations of motion to the Schrödinger equation.

arXiv:quant-ph/0201084Fortschritte der PhysikMay 1, 200220 citations

Exact uncertainty relations

Michael J. W. Hall

The Heisenberg inequality $\ensuremath{\Delta}X\ensuremath{\Delta}P>~\ensuremath{\Elzxh}/2$ can be replaced by an exact equality, for suitably chosen measures of position and momentum uncertainty, which is valid for all wave functions. The statistics of complementary observables are thus connected by an ``exact'' uncertainty relation. Results may be generalized to angular momentum and phase, photon number and phase, time and frequency, and to states described by density operators. Connections to energy bounds, entanglement, Wigner functions, and optimal estimation of an observable from the measurement of a second observable are also given.The Heisenberg inequality $\ensuremath{\Delta}X\ensuremath{\Delta}P>~\ensuremath{\Elzxh}/2$ can be replaced by an exact equality, for suitably chosen measures of position and momentum uncertainty, which is valid for all wave functions. The statistics of complementary observables are thus connected by an ``exact'' uncertainty relation. Results may be generalized to angular momentum and phase, photon number and phase, time and frequency, and to states described by density operators. Connections to energy bounds, entanglement, Wigner functions, and optimal estimation of an observable from the measurement of a second observable are also given.

arXiv:quant-ph/0107149Physical Review AOct 4, 200189 citations

Exact uncertainty relations: technical details

Michael J. W. Hall

The Heisenberg inequality ΔX ΔP \geq \hbar/2 can be replaced by an exact equality, for suitably chosen measures of position and momentum uncertainty, which is valid for all wavefunctions. The significance of this "exact" uncertainty relation is discussed, and results are generalised to angular momentum and phase, photon number and phase, time and frequency, and to states described by density operators. Connections to optimal estimation of an observable from the measurement of a second observable, Wigner functions, energy bounds, and entanglement are also given.The Heisenberg inequality ΔX ΔP \geq \hbar/2 can be replaced by an exact equality, for suitably chosen measures of position and momentum uncertainty, which is valid for all wavefunctions. The significance of this "exact" uncertainty relation is discussed, and results are generalised to angular momentum and phase, photon number and phase, time and frequency, and to states described by density operators. Connections to optimal estimation of an observable from the measurement of a second observable, Wigner functions, energy bounds, and entanglement are also given.

arXiv:quant-ph/0103072Mar 14, 20010 citations

Remarks on Uncertainty Principles

E. U. Condon

Since the publication of Heisenberg’s paper1 on the “anschaulichen Inhalt” of quantum mechanics, discussions of the fundamental limitations on the accuracy of physical measurements have been much in the foreground. According to Heisenberg, the quantum mechanics implies that it is impossible to measure simultaneous values of a coordinate and its conjugate momentum with unlimited precision. Instead, if Δp be the estimated error or uncertainty in a momentum and Δq that in the associated coordinate one must have the inequality, $\Delta p\Delta q>\frac{h}{{2\pi}}$ (1) This inequality has come to be known quite generally as Heisenberg’s uncertainty relation.Since the publication of Heisenberg’s paper1 on the “anschaulichen Inhalt” of quantum mechanics, discussions of the fundamental limitations on the accuracy of physical measurements have been much in the foreground. According to Heisenberg, the quantum mechanics implies that it is impossible to measure simultaneous values of a coordinate and its conjugate momentum with unlimited precision. Instead, if Δp be the estimated error or uncertainty in a momentum and Δq that in the associated coordinate one must have the inequality, $$ \Delta p\Delta q>\frac{h}{{2\pi}} $$ (1) This inequality has come to be known quite generally as Heisenberg’s uncertainty relation.

ScienceMay 31, 192915 citations

Mirror Geometry as the Physical Origin of Quantum Uncertainty

Andrzej Statowski

This work introduces a geometric reinterpretation of Heisenberg’s uncertainty principle using a mirror as a physical operator that separates conjugate variables into two independent information spaces. Position is encoded in the real space, while momentum is transformed into a reflection angle in the mirror space. This separation preserves quantum mechanics while allowing full state information without violating Heisenberg’s inequality. The model naturally reproduces entanglement as a correlation between the original and reflected states. ─── Conceptual Introduction (for general readers) Quantum mechanics states that position and momentum cannot be measured simultaneously with arbitrary precision. This is usually interpreted as a fundamental limit of nature. In this work, we propose a different view: A mirror creates a second information space. 1. In this space: the position of a particle remains in the real world, the momentum of the particle becomes a reflection angle. Thus: position lives in one space, momentum lives in another, and the conflict between them disappears. The uncertainty principle is not broken — its geometry is changed. This simple idea leads to a powerful reinterpretation of quantum information. ─── Formal Model (for scientific readers) 2.1 Two Hilbert Spaces We define two independent Hilbert spaces: [ \mathcal{H}R \quad \text{(real space)} ] [ \mathcal{H}{R'} \quad \text{(mirror space)} ] The full state becomes: [ |\Psi\rangle = |\psiR\rangle \otimes |\psi{R'} 2. \rangle ] Position and momentum no longer coexist in the same space. ─── 2.2 Mirror Operator The mirror is defined as a linear operator: [ \hat{M}: \mathcal{H}R \rightarrow \mathcal{H} {R'} ] acting as: [ \hat{M}|\psiR(x)\rangle = |\psi{R'}(x')\rangle ] with: [ x' = f(x) ] where (f) is the geometric reflection. ─── 2.3 Momentum Becomes Reflection Angle In mirror space, momentum is replaced by a geometric observable: [ \hat{\theta} = g(\hat{p}) ] Example mapping: [ \theta = \arctan\left(\frac{p}{p_0}\right) ] Thus: position → real space momentum → angle in mirror space And therefore: [ [\hat{x}, \hat{\theta}] = 0 ] This is the mathematical heart of the model. ─── 2.4 Modified Uncertainty Relation Standard quantum mechanics: [ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} ] In the mirror model: [ \Delta x \in \mathcal{H}R ] [ \Delta \theta \in \mathcal{H}{R'} ] Thus: [ \Delta x \cdot \Delta \theta \quad \text{is not constrained by Heisenberg} ] because (x) and (\theta) are not conjugate variables in the same space. ─── 2.5 Entanglement as Mirror Correlation The original and reflected states form a natural entangled pair: [ |\Psi\rangle = \sumi ci |xi\rangle \otimes | \thetai\rangle ] A change in position immediately changes the reflection angle: [ x \rightarrow x + \delta x ] [ \theta \rightarrow \theta + \delta\theta ] This reproduces quantum entanglement as a geometric correlation. ─── Main Conclusion A mirror acts as a geometric operator that: separates conjugate variables into two spaces, 3. 1. transforms momentum into reflection angle, allows full state information without violating Heisenberg, provides a physical model of entanglement, extends quantum mechanics without contradicting it. This is the third pillar of the broader theoretical framework developed in earlier works.This work introduces a geometric reinterpretation of Heisenberg’s uncertainty principle using a mirror as a physical operator that separates conjugate variables into two independent information spaces. Position is encoded in the real space, while momentum is transformed into a reflection angle in the mirror space. This separation preserves quantum mechanics while allowing full state information without violating Heisenberg’s inequality. The model naturally reproduces entanglement as a correlation between the original and reflected states. ─── Conceptual Introduction (for general readers) Quantum mechanics states that position and momentum cannot be measured simultaneously with arbitrary precision. This is usually interpreted as a fundamental limit of nature. In this work, we propose a different view: A mirror creates a second information space. 1. In this space: the position of a particle remains in the real world, the momentum of the particle becomes a reflection angle. Thus: position lives in one space, momentum lives in another, and the conflict between them disappears. The uncertainty principle is not broken — its geometry is changed. This simple idea leads to a powerful reinterpretation of quantum information. ─── Formal Model (for scientific readers) 2.1 Two Hilbert Spaces We define two independent Hilbert spaces: [ \mathcal{H}R \quad \text{(real space)} ] [ \mathcal{H}{R'} \quad \text{(mirror space)} ] The full state becomes: [ |\Psi\rangle = |\psiR\rangle \otimes |\psi{R'} 2. \rangle ] Position and momentum no longer coexist in the same space. ─── 2.2 Mirror Operator The mirror is defined as a linear operator: [ \hat{M}: \mathcal{H}R \rightarrow \mathcal{H} {R'} ] acting as: [ \hat{M}|\psiR(x)\rangle = |\psi{R'}(x')\rangle ] with: [ x' = f(x) ] where (f) is the geometric reflection. ─── 2.3 Momentum Becomes Reflection Angle In mirror space, momentum is replaced by a geometric observable: [ \hat{\theta} = g(\hat{p}) ] Example mapping: [ \theta = \arctan\left(\frac{p}{p_0}\right) ] Thus: position → real space momentum → angle in mirror space And therefore: [ [\hat{x}, \hat{\theta}] = 0 ] This is the mathematical heart of the model. ─── 2.4 Modified Uncertainty Relation Standard quantum mechanics: [ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} ] In the mirror model: [ \Delta x \in \mathcal{H}R ] [ \Delta \theta \in \mathcal{H}{R'} ] Thus: [ \Delta x \cdot \Delta \theta \quad \text{is not constrained by Heisenberg} ] because (x) and (\theta) are not conjugate variables in the same space. ─── 2.5 Entanglement as Mirror Correlation The original and reflected states form a natural entangled pair: [ |\Psi\rangle = \sumi ci |xi\rangle \otimes | \thetai\rangle ] A change in position immediately changes the reflection angle: [ x \rightarrow x + \delta x ] [ \theta \rightarrow \theta + \delta\theta ] This reproduces quantum entanglement as a geometric correlation. ─── Main Conclusion A mirror acts as a geometric operator that: separates conjugate variables into two spaces, 3. 1. transforms momentum into reflection angle, allows full state information without violating Heisenberg, provides a physical model of entanglement, extends quantum mechanics without contradicting it. This is the third pillar of the broader theoretical framework developed in earlier works.

Sep 10, 20260 citations

Quantum Uncertainty Principles and Classical Statistical Mechanics

Francesco R. Ruggeri

Heisenberg’s uncertainty principle for position-momentum delta x delta p >= hbar/2 is considered as an important statement regarding quantum mechanics i.e. one cannot know both p and x exactly. Delta x and delta p stand out because in classical mechanics one has an eact p and x at each t. Mathematically, however, delta x and delta p are standard deviations which may also be calculated in classical statistical mechanics, for example with the Maxwell-Boltzmann distribution. In particular, if one considers the ground state of a simple harmonic oscillator, the quantum P(x) proportional to exp(-pp/a) and P(p) proportional to exp(-xxa) match the classical statistical probabilities P(x) and P(p) for a fixed value of T proportional to sqrt(k/m) where k is the spring constant. Thus for this special T, delta x delta p for a classical statistical mechanical gas is identical to delta x delta p for the ground state quantum oscillator. The probability is normalized the same way for both. The physics of the two situations, however, is completely different. Thus the Heisenberg uncertainty principle does not necessarily describe quantum mechanical behaviour alone. If one uses entropic uncertainty, i.e. Sp + Sx where S is Shannon’s entropy, once again one uses P(p)=a*(p)a(p) and P(x)=W*(x)W(x) where W(x)=wavefunction = Sum over p a(p)exp(ipx). In principle, one could also calculate such an entropy for a classical gas i.e. use P(x)P(p) in Shannon’s entropy. The result Sp+Sx does not imply one does not know about p if one knows x i.e. the usual quantum statement.Heisenberg’s uncertainty principle for position-momentum delta x delta p >= hbar/2 is considered as an important statement regarding quantum mechanics i.e. one cannot know both p and x exactly. Delta x and delta p stand out because in classical mechanics one has an eact p and x at each t. Mathematically, however, delta x and delta p are standard deviations which may also be calculated in classical statistical mechanics, for example with the Maxwell-Boltzmann distribution. In particular, if one considers the ground state of a simple harmonic oscillator, the quantum P(x) proportional to exp(-pp/a) and P(p) proportional to exp(-xxa) match the classical statistical probabilities P(x) and P(p) for a fixed value of T proportional to sqrt(k/m) where k is the spring constant. Thus for this special T, delta x delta p for a classical statistical mechanical gas is identical to delta x delta p for the ground state quantum oscillator. The probability is normalized the same way for both. The physics of the two situations, however, is completely different. Thus the Heisenberg uncertainty principle does not necessarily describe quantum mechanical behaviour alone. If one uses entropic uncertainty, i.e. Sp + Sx where S is Shannon’s entropy, once again one uses P(p)=a*(p)a(p) and P(x)=W*(x)W(x) where W(x)=wavefunction = Sum over p a(p)exp(ipx). In principle, one could also calculate such an entropy for a classical gas i.e. use P(x)P(p) in Shannon’s entropy. The result Sp+Sx does not imply one does not know about p if one knows x i.e. the usual quantum statement.

Apr 28, 20220 citations

\Delta p\Delta q>\frac{h}{{2\pi}}

(1) This inequality has come to be known quite generally as Heisenberg’s uncertainty relation.