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Fazal_Rehman
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Dr. Fazal Rehman
Government Comprehensive Higher Secondary School, KPK, Kohat, Pakistan
Email: FazalRehmankohati@gmail.com
Dhan Bir Limbu
Class 10 Student, Peljorling Higher Secondary School, Samtse, Bhutan
Email: dhanbirlimbuu@gmail.com
&
Dr. Farhat Ali Khan
Kohat University of Science and Technology, KPK, Pakistan
Email: Farhadali@kust.edu.pk
Date: January 16, 2026
Abstract
The time-dependent Schrödinger equation forms the bedrock of non-relativistic quantum mechanics, governing the temporal evolution of quantum states. This paper introduces a generalized variant incorporating a fractional-order time derivative modulated by energy uncertainty:
i hbar D_t^(1 + kappa (Delta E_psi)/(Delta E_psi + E_0)) psi(r, t) = [ - (hbar^2)/(2m) nabla^2 + V(r) ] psi(r, t).
We derive the form of this operator, analyze its implications for quantum dynamics, and explore applications in quantum fluctuations, non-equilibrium systems, and quantum gravity. Numerical simulations for a harmonic oscillator demonstrate altered wave function spreading, suggesting enhanced modeling of uncertainty-driven phenomena. This framework bridges standard quantum mechanics with fractional calculus, opening avenues for precision in complex quantum systems.
Keywords: Time-dependent Schrödinger equation, fractional derivatives, energy uncertainty, quantum fluctuations, non-equilibrium dynamics.
Government Comprehensive Higher Secondary School, KPK, Kohat, Pakistan
Email: FazalRehmankohati@gmail.com
Dhan Bir Limbu
Class 10 Student, Peljorling Higher Secondary School, Samtse, Bhutan
Email: dhanbirlimbuu@gmail.com
&
Dr. Farhat Ali Khan
Kohat University of Science and Technology, KPK, Pakistan
Email: Farhadali@kust.edu.pk
Date: January 16, 2026
Abstract
The time-dependent Schrödinger equation forms the bedrock of non-relativistic quantum mechanics, governing the temporal evolution of quantum states. This paper introduces a generalized variant incorporating a fractional-order time derivative modulated by energy uncertainty:
i hbar D_t^(1 + kappa (Delta E_psi)/(Delta E_psi + E_0)) psi(r, t) = [ - (hbar^2)/(2m) nabla^2 + V(r) ] psi(r, t).
We derive the form of this operator, analyze its implications for quantum dynamics, and explore applications in quantum fluctuations, non-equilibrium systems, and quantum gravity. Numerical simulations for a harmonic oscillator demonstrate altered wave function spreading, suggesting enhanced modeling of uncertainty-driven phenomena. This framework bridges standard quantum mechanics with fractional calculus, opening avenues for precision in complex quantum systems.
Keywords: Time-dependent Schrödinger equation, fractional derivatives, energy uncertainty, quantum fluctuations, non-equilibrium dynamics.