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Jurnal Teras Fisika

De Broglie Wave Analysis of the Heisenberg Uncertainty Minimum Limit under the Lorentz Transformation

Fima Ardianto Putra

Published Sep 20, 2018• 0 citations
Publisher

Abstract

A simple analysis using differential calculus has been done to consider the minimum limit of the Heisenberg uncertainty principle in the relativistic domain. An analysis is made by expressing the form of and based on the Lorentz transformation, and their corresponding relation according to the de Broglie wave packet modification. The result shows that in the relativistic domain, the minimum limit of the Heisenberg uncertainty is p x ?/2 and/or E t ?/2, with is the Lorentz factor which depend on the average/group velocity of relativistic de Broglie wave packet. While, the minimum limit according to p x ?/2 or E t ?/2, is the special case, which is consistent with Galilean transformation. The existence of the correction factor signifies the difference in the minimum limit of the Heisenberg uncertainty between relativistic and non-relativistic quantum. It is also shown in this work that the Heisenberg uncertainty principle is not invariant under the Lorentz transformation. The form p x ?/2 and/or E t ?/2 are properly obeyed by the Klein-Gordon and the Dirac solution. Key words: De Broglie wave packet, Heisenberg uncertainty, Lorentz transformation, and minimum limit.

DOI
10.20884/1.jtf.2018.1.2.1008

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A simple analysis using differential calculus has been done to consider the minimum limit of the Heisenberg uncertainty principle in the relativistic domain. An analysis is made by expressing the form of and based on the Lorentz transformation, and their corresponding relation according to the de Broglie wave packet modification. The result shows that in the relativistic domain, the minimum limit of the Heisenberg uncertainty is p x ?/2 and/or E t ?/2, with is the Lorentz factor which depend on the average/group velocity of relativistic de Broglie wave packet. While, the minimum limit according to p x ?/2 or E t ?/2, is the special case, which is consistent with Galilean transformation. The existence of the correction factor signifies the difference in the minimum limit of the Heisenberg uncertainty between relativistic and non-relativistic quantum. It is also shown in this work that the Heisenberg uncertainty principle is not invariant under the Lorentz transformation. The form p x ?/2 and/or E t ?/2 are properly obeyed by the Klein-Gordon and the Dirac solution. Key words: De Broglie wave packet, Heisenberg uncertainty, Lorentz transformation, and minimum limit.
The de Broglie relation is revisited in connection with an ab initio relativistic description of particles and waves, which is the same treatment that historically led to this famous relation. In the same context of the Minkowski four-vector formalism, we also discuss the phase and the group velocity of a matter wave, explicitly showing that both transform as ordinary velocities under a Lorentz transformation. We show that such a transformation rule is a necessary condition for the covariance of the de Broglie relation, and stress the pedagogical value of the Einstein-Minkowski-Lorentz relativistic context in the presentation of the de Broglie relation.
In this work, we show that it is possible to establish coordinate transformations between inertial reference frames in the theory of special relativity with a minimum universal speed of physical transmissions. The established coordinate transformations, referred to as modified Lorentz transformations because they have almost identical form to the Lorentz transformations, also comply with the requirement of invariance of the Minkowski line element. Particularly, the minimum universal speed can be associated with the phase speed of de Broglie matter wave. As application, we also discuss the possibility to formulate relativistic classical and quantum mechanics for the special relativity associated with the modified Lorentz transformations, which describes physical processes that represent an expansion or a collapsing of massive quantum particles.
The Heisenberg uncertainty principle (Δx·Δp ≥ ℏ/2) has been interpreted for nearly a century as arising from the disturbance caused by measurement. We argue that this epistemic framing is unnecessary. We demonstrate that Δx·Δp ≥ ℏ/2 follows as a logical necessity from three independently observed facts: (1) position is quantized at the Planck length lₚ ≈ 1.616 × 10⁻³⁵ m; (2) time is quantized at the Planck time tₚ ≈ 5.391 × 10⁻⁴⁴ s; (3) mass-energy is quantized. Using the discrete Fourier transform (DFT), we show that a discrete Gaussian wave packet on a Planck lattice saturates the uncertainty bound at ℏ/2 exactly, with the factor of 1/2 emerging from the minimum-uncertainty property of the Gaussian distribution, which is preserved under discretization. Furthermore, we show that the canonical commutation relation [̂x, ̂p] = iℏ is the continuum limit of a discrete difference commutator, identifying the Schrödinger equation as an effective theory of an underlying Planck-scale discrete structure. Finally, we address the relationship of our result to Lorentz invariance, arguing that the Planck-scale bounds apply as observer-independent lower limits and are consistent with available evidence for Lorentz Invariance Violation (LIV) at the Planck scale.
A Lorentz-invariant semi-relativistic formulation is developed to extend the de Broglie hypothesis and establish a unified connection between matter waves and their photonic limit. Beginning with the classical relation pc = hf, a relativistic momentum expansion p ≈ mv (1 + ½ v²/c² + 3/8 v⁴/c⁴) leads to a generalized energy–frequency relation hf = pc + (v/c) Ek + (1 − γ⁻¹) mc², where Ek = ½ mv² and γ = (1 − v²/c²)⁻½. This provides a continuous transition between the non-relativistic (v ≪ c) and photonic (v → c) domains, describing matter as “slowed-down light” in a Lorentz-consistent sense.To preserve both relativistic and quantum consistency, a modified Dirac equation is introduced:(iħ γ^μ ∂_μ + β_ν ħ² / (2m) γ^ν □) ψ = 0,where β_ν = u_ν / c and □ = ∂_μ ∂^μ.This formulation, derived from a variational Lagrangian, naturally reduces to the Schrödinger and Dirac equations in their respective limits while maintaining Lorentz covariance. Numerical analysis for electrons in the 1–100 keV range reveals wavelength deviations of approximately 0.001–1%, suitable for precision diffraction experiments, and fine-structure variations around 0.7% for light atoms. The model strengthens de Broglie’s original interpretation of matter as a localized manifestation of slowed light within a Lorentz-invariant semi-relativistic framework.