Our work utilizes the quantum model of the hydrogen atom which is based on the Schrödinger equation with Coulomb potential. Specifically, we concentrate on the angular components of the wave eigenfunctions derived from this model. We consider the quantum states with n ≤ 4. In order to visualize the orbital shapes of these states, we built in the spherical coordinates system their 3D geometric representations. Furthermore, we use the corresponding spherical harmonics, to calculate the θ nodal values that describe the configurations of these orbital states.
Our work utilizes the quantum model of the hydrogen atom which is based on the Schrödinger equation with Coulomb potential. Specifically, we concentrate on the angular components of the wave eigenfunctions derived from this model. We consider the quantum states with n ≤ 4. In order to visualize the orbital shapes of these states, we built in the spherical coordinates system their 3D geometric representations. Furthermore, we use the corresponding spherical harmonics, to calculate the θ nodal values that describe the configurations of these orbital states.
In the framework of the quantum model for hydrogen atom, we use the wave functions provided by the solutions of the Schrödinger equation with Coulomb potential. Considering the radial and angular densities of the localization probability, we identify the representative quantum states that play a high-probability role in transitions for H-I spectra. Besides that, we obtain some other complementary results. So, we present the numerical tables for the radial distributions, the electron density functions and some observables like θ nodal angles, nodal surfaces, along with the ordered sequences of the average radii in the sub-shells associated with this n: ⟨r⟩4f = 18a0 < ⟨r⟩4d = 21a0 < ⟨r⟩4p = 23a0 < ⟨r⟩4s = 24a0 (a0 - the first Bohr radius). Further, we have added the significant scientific 3D-visualization of the orbitals shapes and describe them via the nodal values for θ angles.
Abstract In this chapter, we go to three dimensions in space and look at the solution of the time independent Schrödinger equation for the hydrogen atom. The Hamiltonian is then the kinetic energy plus the potential energy due to the Coulomb coupling between the positively charged nucleus and the electron. We construct the angular momentum operator and find that the partial differential equation for the angular momentum eigenfunctions of the spherical coordinates θ,ϕ is the same as the angular part of the ∇2 operator in spherical coordinates. The angular momentum eigenfunctions are the spherical harmonics, with two quantum numbers, l and m, and the solution to the radial part of the Hamiltonian including the Coulomb potential are Laguerre polynomials with one quantum number, called the principle quantum number, n. The hydrogen wave function is the product of a Laguerre polynomial and a spherical harmonic with three quantum numbers. Since these are two- and three-dimensional functions for angular momentum and hydrogen respectively, they are best understood in a series of plots. The chapter concludes by giving the historical letter names to specific orbitals, since they continue to be used today.
Chemists are aware of the solution of Schroedinger's equations for the hydrogen atom in only spherical polar coordinates, but the spatial variables are separable also in three other systems - - paraboloidal, ellipsoidal and spheroconical; we report here explicit algebraic solutions directly derived in ellipsoidal and spheroconical coordinates for the first time. Our solutions progress from those previously known in spherical polar but not entirely understood, through those little known in paraboloidal, to those in systems of ellipsoidal and spheroconical coordinates unknown before the present work. Applications of these solutions include angular momenta, a quantitative calculation of the discrete absorption spectrum and accurate plots of surfaces of amplitude functions. The shape of a surface of a particular amplitude function, and even the quantum numbers in a particular set to specify such an individual function, depend on a particular chosen system of coordinates, and are therefore artefacts of that coordinate representation within wave mechanics; a choice of a coordinate system to discuss atomic or molecular properties based on the shapes of amplitude functions or their respective quantum numbers is hence arbitrary
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).
Abstract This chapter reiterates the quantum numbers for atomic orbitals, known from general chemistry, and places them into the context developed so far. It is sketched how the Schrodinger equation (SE) for the hydrogen atom hydrogen-like systems (one electron plus a nucleus of charge Z) is set up. When the nucleus is treated as a fixed point charge, the SE is only for the electron. The solutions of the SE can be obtained by switching to spherical polar coordinates, such that the variables are separable in terms of the electron distance from the nucleus, r, and two angles. The kinetic energy of the electron then has a radial component, and an angular component. The latter is associated with the angular momentum quantum number, which is codified by the letters s, p, d, f, and so forth. A step by step solution of the SE is provided later, in chapter 19.
The hydrogen atom is a system amenable to an exact treatment within Schroedinger's formulation of quantum mechanics according to coordinates in four systems -- spherical polar, paraboloidal, ellipsoidal and spheroconical coordinates; the latter solution is reported for the first time. Applications of these solutions include angular momenta, a quantitative calculation of the absorption spectrum and accurate plots of surfaces of amplitude functions. The shape of an amplitude function, and even the quantum numbers in a particular set to specify such an individual function, depend on the coordinates in a particular chosen system, and are therefore artefacts of that particular coordinate representation within wave mechanics. All discussion of atomic or molecular properties based on such shapes or quantum numbers therefore lacks general significance