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Schrödinger Equation and GUP of Attenuated Eigenfunction
A.S. Abdel-Rahman and Youssef A. Sabry
String theory, general relativity, quantum gravity, and the study of black holes all suggest that there could be a minimum observable length of the order of Planck’s length. This hint, together with others, led to the Generalized Uncertainty Principle (GUP), which is presented in numerous publications. The Schrödinger equation, which forms the basis of quantum mechanics, does not explicitly demonstrate this principle, but it is often used to solve problems without revealing the full range of possible answers. In this study, a particle in a one-dimensional box, one of the best-known quantum puzzles, was used to illustrate some implications of the GUP in terms of quantum physics. The study proposes a notation derived from an attenuated wavefunction that describes both the particle dimension and the wave nature of the particle in terms of a minimal length, to be introduced into the Schrödinger equation. Apart from Hawking radiation, this idea can also be used to solve problems in high-energy physics and black holes.
Keywords: Generalized uncertainty; attenuation; minimal length; solution limits