Acta Physica Sinica

Coherent state of hydrogen-like atoms: an application of the pseudo-angular-momentum operator method

Wenbo Li, Li Mi-Shan, 北京交通大学物理系，北京 100044

Published Jan 1, 2008• 0 citations

Abstract

Using the pseudo-angular-momentum operator method，the radial equation for bound state of the hydrogen-like atom is solved and the analytical expression for the eigenstate is derived. The result shows that the normalization of eigenfunction should be done carefully owing to its peculiarity. The corresponding coherent state is also discussed.

DOI: 10.7498/aps.57.3973

Related papers

Solution of radial equation of Kepler’s problem by pseudo-angular-momentum method and normalization of eigenstate and coherent state

Wenbo Li, Li Ke-Xuan, 北京交通大学物理系，北京 100044

By using the pseudoangularmomentum method, the radial equation for bound state of Kepler's problem is solved and the analytic expression for eigenstate is derived. The result shows that the normalization of eigenfunction that should be done carefully is quite peculiar. The corresponding coherent state is also discussed.

Acta Physica SinicaJan 1, 20040 citations

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

Mehdi Miri

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 Schrödinger'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 Schrödinger'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.

arXiv:2308.13581Aug 25, 20230 citations

Radial coherent states for Dirac hydrogen-like atom

G. E. Drăgănescu, ANTONINO MESSINA, Anna Napoli

In this paper we use an su (2) representation of the radial eigenfunction of the Dirac hydrogen-like atom and we build the Glauber coherent states and the displacement operator coherent states. We also calculate the average values of some observables corresponding to these states.

Journal of Optics B Quantum and Semiclassical OpticsJun 10, 20027 citations

Hydrogen-like Atom and Atomic Orbitals

Caio Lima Firme

This chapter physical properties of the hydrogen atom and its Hamiltonian function as a particle in a spherically symmetric potential having a wave function as a product of the radial function and spherical harmonic function. Hereafter, it comes to two independent wave equations: the radial equation and the angular equation. After some derivation, it is shown that the angular equation is a product of the azimuthal angle equation and polar angle equation (as discussed in the previous chapter). The azimuthal angle equation is similar to the particle in a ring problem from the previous chapter and after some derivation the polar angle equation becomes an associated Legendre equation with well-known polynomials as solutions. Next, the solution to the radial equation is presented which becomes after some derivation an associated Laguerre equation with well-known polynomials as solutions and from this result is obtained the energy equation for the hydrogen-like atom. Subsequently, it is derived the normalization constant of the radial function. Afterwards, the numerical analysis of the radial equation of the hydrogen atom along with its corresponding Fortran code is presented. Finally, the hydrogen atomic orbitals equations and plots are depicted followed by the fine structure of the hydrogen (198 words).

CRC Press eBooksMar 21, 20220 citations

Further Study on the Analytical Solution of Schr dinger Equation of a Hydrogen-Like Atom

Zhong Zhang, Zhao Zhang

Based on the previous works, the Schrdinger equation of the hydrogen-like atom is analytically solved further. Three sets of analytical solutions are obtained if the factor r~ -nis not neglected. The first solution is the same as the traditional radial wave function, the second diverges, and the last one is quite different from the traditional solution. On the consideration of the finite size of the nuclear, the third wave function does not diverge when r approaches to zero. Its radial wave function has the following characteristics: (1) the angular-momentum quantum number l must be greater than the principal quantum number n; (2) l must not be 0 or 1; (3) the electron-cloud distribution differs from the traditional one; (4) the electron is closer to the nuclear than that in traditional results. On the other hand, the validity of solutions needs to be verified experimentally.

Jan 1, 20020 citations

FURTHER STUDY ON THE SOLUTION OF SCHRÖDINGER EQUATION OF HYDROGEN-LIKE ATOM

Zhong-Liang Zhang, Wu-Shou Zhang, Zhao-qun Zhang

In this work the Schrödinger equation of the hydrogen-like atom is analytically solved. Three sets of analytical solution are obtained if the factor r−l is not neglected. The first solution is the same as the traditional radial wave function; another one diverges; the last one is far different from the traditional solution. On the consideration of the finite size of the nucleus, the third wave function does not diverge while r approaches to zero. Its radial wave function has below characteristics: (1) the angular-momentum quantum number l must be greater than the principal quantum number n; (2) l must not be 0 or 1; (3) the electron-cloud distribution differs from the traditional one; (4) the electron is closer to the nucleus by comparison with that in traditional results. On the other hand, the validity of solutions needs to be verified experimentally. 1.

Aug 25, 20150 citations