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International Journal of Science and Research Archive

QUANTUM TIME DILATION: RELATION BETWEEN TIME DILATION AND ENERGY DIFFERENCE OF QUANTUM WAVE FUNCTIONS IN GALILEAN AND LORENTZ FRAMES

Tanujit Mondal, Hriddhiman Bag, Arnab Maitra

Published Sep 18, 2026• 0 citations
Publisher

Abstract

This paper investigates the quantum mechanical relationship between time dilation and energy difference of the quantum photon and electron wave functions. The wave functions are studied under two different times and and two energy differences and . The time-dependent Schrödinger equation (TDSE) is used to analyse the wave functions and time dilation is studied in the Galilean frame of reference. A key relation linking Lorentz time dilation to the energy difference between the photon (relativistic kinetic energy) and electron (classical kinetic energy) wave functions is derived analytically. Python simulation is used to compute and visualise the wave function phase, energy ratio, total phase energy, and time dilation as functions of speed v/c.

DOI
10.30574/ijsra.2026.20.3.1779

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This paper investigates the quantum mechanical relationship between time dilation and energy difference of the quantum photon and electron wave functions. The wave functions are studied under two different times and and two energy differences and . The time-dependent Schrödinger equation (TDSE) is used to analyse the wave functions and time dilation is studied in the Galilean frame of reference. A key relation linking Lorentz time dilation to the energy difference between the photon (relativistic kinetic energy) and electron (classical kinetic energy) wave functions is derived analytically. Python simulation is used to compute and visualise the wave function phase, energy ratio, total phase energy, and time dilation as functions of speed v/c.
One of the concepts of Relativity theory that challenges conventional intuition the most is time dilation and length contraction. Usual approaches for describing relativistic effects in quantum systems merely postulate the consequences of these effects as physical constraints. Here, we propose to rebuild Special Relativity from quantum mechanical considerations. This is done by dropping one of its fundamental postulates: the universality of the speed of light. Lorentz transformations are obtained by a quantum mechanism. We use the fact that quantum states depend on the Galilean reference frame where they are defined. In other words, quantum states outside an observer's Galilean reference frame are distorted. Then, we show in a theorem the existence of time-dependent observables that are sensible to this distortion in such a way that their expectation value is a Lorentz-covariant function of time. We then postulate this mechanism as the source of the phenomena of Special Relativity. As a corollary of the main theorem, we show the existence of a Lorentz covariant momentum and mass operators which yield the relativistic momentum and mass. In this theory, the fundamental limit of the speed of light imposes a transparency condition for faster-than-light particles: they are allowed but they are not observable. The transparency effect could explain dark matter in a more general theory following this quantum formalism.
In this study, both the Klein-Gordon equation and time-dependent Schrodinger equation are directly derived from De Broglie's (\lamda=h/p). Besides that, a new formula of the Klein-Gordon equation was obtained by using two equations, i.e., the classical wave function and relativistic total energy relation [[hw] = [(h^2 k^2)/2m] + V]. Moreover, a new equation of time-dependent Schrodinger (TDSE) in three-dimension 3D was derived and evidently obtained. Thus, the previous result corresponds to (TDSE) in 3D calculated using two equations, (i.e., Einstein energy and Newton's second law of mechanics).
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The Schrodinger and de Broglie hypothesis are directly derived from classical force and blank hypothesis. Thus, depending on the quantum equations the quantum of total energy relation (E = T + V ) and de Broglie relation (? = h=p). Accordingly, new time dependent Schrodinger equation is obtained. As a result, by using the wave function and nonrelativistic wave function beside plank hypothesis. Moreover, TDSE was obtained, thus by used additional kinetic energy and potential energy (classical equation). Finally, a result corresponding to TDSE in 3D was confirmed by using De Broglie hypothesis.
Since their first introduction, Quantum Reference Frame (QRF) transformations have been extensively discussed, generalising the covariance of physical laws to the quantum domain. Despite important progress, a formulation of QRF transformations for Lorentz symmetry is still lacking. The present work aims to fill this gap. We first introduce a reformulation of relativistic quantum mechanics independent of any notion of preferred temporal slicing. Based on this, we define transformations that switch between the perspectives of different relativistic QRFs. We introduce a notion of ''quantum Lorentz transformations'' and ''superposition of Lorentz boosts'', acting on the external degrees of freedom of a quantum particle. We analyse two effects, superposition of time dilations and superposition of length contractions, that arise only if the reference frames exhibit both relativistic and quantum-mechanical features. Finally, we discuss how the effects could be observed by measuring the wave-packet extensions from relativistic QRFs.