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arXiv:1501.02137 · Physics Essays

A discussion on the Heisenberg uncertainty principle from the perspective of special relativity

Luca Nanni

Published Sep 14, 2016• 1 citation
arXivHTMLPDF

Abstract

In this note, we consider the implications of the Heisenberg uncertainty principle (HUP) when computing uncertainties that affect the main dynamical quantities, from the perspective of special relativity. Using the well-known formula for propagating statistical errors, we prove that the uncertainty relations between the moduli of conjugate observables are not relativistically invariant. The new relationships show that, in experiments involving relativistic particles, limitations of the precision of a quantity obtained by indirect calculations may affect the final result.

DOI
10.4006/0836-1398-29.3.309

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In this note, we consider the implications of the Heisenberg uncertainty principle (HUP) when computing uncertainties that affect the main dynamical quantities, from the perspective of special relativity. Using the well-known formula for propagating statistical errors, we prove that the uncertainty relations between the moduli of conjugate observables are not relativistically invariant. The new relationships show that, in experiments involving relativistic particles, limitations of the precision of a quantity obtained by indirect calculations may affect the final result.
One of the main objectives of quantum-gravity research is to establish how the Heisenberg position-momentum uncertainty principle is modified by Planck-scale effects. We here focus on some residual gray areas of our understanding of the interplay between the uncertainty principle and special relativity, which might have been an obstacle for the further generalization needed in order to include gravitational effects. In its original formulation the uncertainty principle is inconsistent with special relativity, and in nearly a century of investigation not much progress has been made toward a satisfactory reformulation. Some partial insight has been gained in the ultra-high-velocity regime, but a full description is still missing and in particular we have no clue about the intermediate regime of particles whose speeds are much smaller than the speed of light but still high enough for tangible departures from the Heisenberg formulation. As we stress here, that intermediate regime is also our best chance for testing experimentally our understanding of the implications of special relativity for the uncertainty principle. We introduce a new approach toward facing these challenges, based mainly on the observation that the only operative notion of position of a particle at a given time involves the crossing of the worldline of that particle with the worldline of a test particle. We find that the worldline-crossing perspective opens a path toward a special-relativistic version of the uncertainty principle, which indeed could be tested experimentally.
The derivation of the Heisenberg Uncertainty Principle (HUP) from the Uncertainty Theorem of Fourier Transform theory demonstrates that the HUP arises from the dependency of momentum on a wave number that exists at the quantum level. It also establishes that the HUP is purely a relationship between the effective widths of Fourier transform pairs of variables (i.e. conjugate variables). We note that the HUP is not a quantum mechanical measurement principle per se. We introduce the Quantum Mechanical equivalent of the Nyquist-Shannon Sampling Theorem of Fourier Transform theory, and show that it is a better principle to describe the measurement limitations of Quantum Mechanics. We show that Brillouin zones in Solid State Physics are a manifestation of the Nyquist-Shannon Sampling Theorem at the quantum level. By comparison with other fields where Fourier Transform theory is used, we propose that we need todiscern between measurement limitations and inherent limitations when interpreting the impact of the HUP on the nature of the quantum level. We further propose that while measurement limitations result in our perception of indeterminism at the quantum level, there is no evidence that there are any inherent limitations at the quantum level, based on the Nyquist-Shannon Sampling Theorem
The derivation of the Heisenberg Uncertainty Principle (HUP) from the Uncertainty Theorem of Fourier Transform theory demonstrates that the HUP arises from the dependency of momentum on wave number that exists at the quantum level. It also establishes that the HUP is purely a relationship between the effective widths of Fourier transform pairs of variables (i.e. conjugate variables). We note that the HUP is not a quantum mechanical measurement principle per se. We introduce the Quantum Mechanical equivalent of the Nyquist-Shannon Sampling Theorem of Fourier Transform theory, and show that it is a better principle to describe the measurement limitations of Quantum Mechanics. We show that Brillouin zones in Solid State physics are a manifestation of the Nyquist-Shannon Sampling Theorem at the quantum level. By comparison with other fields where Fourier Transform theory is used, we propose that we need to discern between measurement limitations and inherent limitations when interpreting the impact of the HUP on the nature of the quantum level. We further propose that while measurement limitations result in our perception of indeterminism at the quantum level, there is no evidence that there are any inherent limitations at the quantum level, based on the Nyquist-Shannon Sampling Theorem.
Does the Heisenberg uncertainty principle refer to errors of measurement, or to the spread of values of the physical variables intrinsic to a particle’s state, or to some combination of these? The most commonly quoted form of the uncertainty principle relates the spread of an ensemble of separate measurements of some variable to the analogous spread of its conjugate variable. In contrast, Heisenberg’s original argument for the uncertainty principle involved the perturbation to a particle’s state by a measurement of one variable, which affects one’s ability to predict the outcome of a subsequent measurement of the conjugate variable. The relation between these two views of the uncertainty principle is discussed in this paper. A familiar example is considered: an ensemble of identically prepared particles passing through a slit, and after further propagation being detected. From this arrangement it is possible to infer joint (although necessarily imprecise) information on both transverse position and conjugate momentum for each member of the ensemble. It is shown that in this case of joint measurement the product of standard deviations for the measurement outcomes is at least twice as large as the lower bound implied by the usual uncertainty principle. The discussion is meant to help clarify the different roles played in the various statements of the uncertainty principle by the initial state and by measurement error.
Heisenberg’s uncertainty principle (HUP) is a central concept in quantum mechanics that reveals inherent limitations in the measurement of microscopic particles and profoundly affects the understanding of the physical world. This paper reviews the physical significance of HUP and its applications and research progress in modern physics. The complexity of measurements in quantum systems is revealed by deriving the uncertainty relation and analysing the wavefunction collapse and the classicalisation of quantum states. This paper summarises the current experimental verification of the principle of measurement uncertainty in different quantum systems and discusses the research progress in improving measurement accuracy through quantum entanglement. Recent studies have shown that controlling quantum entangled states can effectively reduce measurement disturbances and thus improve the measurement accuracy of quantum systems without violating the principle of mismeasurement. Despite the many advances made, there are still challenges to further improve the measurement accuracy in complex quantum systems. Future research will continue to explore the potential for applications in this area.
Heisenberg’s uncertainty principle (HUP) is a central concept in quantum mechanics that reveals inherent limitations in the measurement of microscopic particles and profoundly affects the understanding of the physical world. This paper reviews the physical significance of HUP and its applications and research progress in modern physics. The complexity of measurements in quantum systems is revealed by deriving the uncertainty relation and analysing the wavefunction collapse and the classicalisation of quantum states. This paper summarises the current experimental verification of the principle of measurement uncertainty in different quantum systems and discusses the research progress in improving measurement accuracy through quantum entanglement. Recent studies have shown that controlling quantum entangled states can effectively reduce measurement disturbances and thus improve the measurement accuracy of quantum systems without violating the principle of mismeasurement. Despite the many advances made, there are still challenges to further improve the measurement accuracy in complex quantum systems. Future research will continue to explore the potential for applications in this area.