Quantum model of hydrogen-like atoms in hilbert space by introducing the creation and annihilation operators

TL;DR

This paper develops an algebraic quantum model for hydrogen-like atoms using creation and annihilation operators, offering a systematic approach to derive energy levels and wave functions in Hilbert space, expanding quantum system models.

Contribution

It introduces a novel algebraic method to solve hydrogen-like atom problems in Hilbert space, complementing existing differential equation approaches.

Findings

Derived energy quantization using algebraic operators

Obtained normalized radial wave functions in matrix form

Established a new fundamental model for HLA systems

Abstract

The purely algebraic technique associated with the creation and annihilation operators to resolve the radial equation of Hydrogen-like atoms (HLA) for generating the bound energy spectrum and the corresponding wave functions is suitable for many calculations in quantum physics. However, the analytical approach with series is extensively used based on wave mechanics theory in most of quantum textbooks. Indeed, much More complete than the old solution of Schr\"odinger's time-independent differential equation (TISE), one can simply earn all quantum information of a system by using the operational method. In addition to earlier two models, including the quantum harmonic oscillator and the total angular momentum, it can undoubtedly be a third fundamental model to solve Schr\"odinger's eigenvalue equation of the HLA systems in Hilbert space similar to the harmonic oscillator. We will illustrate how systematically making an appropriate groundwork to discover the coherent states can lead to providing the energy quantization and normalized radial wave functions attached to the matrix representation without additional assumptions.