An Analytic Evolutionary Algorithm To Compute Higher-Order Solutions For Model Equation

In this work, by introducing Darboux operator in evolutionary computing frame, we propose a novel analytic evolutionary algorithm to obtain exact higher-order iteration solutions for model equation. We construct the first-, second- and third-order solutions of a generalized Schrödinger equation by applying this algorithm. The higher-order solutions not only retain the basic features of the lower-order cases, but also become more abundant than the lower-order cases.

Some applications of the most general form of the higher-order GUP with minimal length uncertainty and maximal momentum

In this paper, using the new type of D-dimensional nonperturbative Generalized Uncertainty Principle (GUP) which has predicted both a minimal length uncertainty and a maximal observable momentum,1 first, we obtain the maximally localized states and express their connections to [P. Pedram, Phys. Lett. B 714, 317 (2012)]. Then, in the context of our proposed GUP and using the generalized Schrödinger equation, we solve some important problems including particle in a box and one-dimensional hydrogen atom. Next, implying modified Bohr–Sommerfeld quantization, we obtain energy spectra of quantum harmonic oscillator and quantum bouncer. Finally, as an example, we investigate some statistical properties of a free particle, including partition function and internal energy, in the presence of the mentioned GUP.

Operator of Time and Generalized Schrödinger Equation

The equation describing the change of the state of the quantum system with respect to energy is introduced within the framework of the self-adjoint operator of time in nonrelativistic quantum mechanics. In this proposal, the operator of time appears to be the generator of the change of the energy, while the operator of energy that is conjugate to the operator of time generates the time evolution. Two examples, one with discrete time and the other with continuous one, are given and the generalization of Schrödinger equation is proposed.