Abstract In this chapter we consider the time-independent Schrödinger equation for three spatial dimensions. Making use of the orbital angular momentum observables, including the Laplacian in spherical polar coordinates, and introducing an effective potential, we recast the three-dimensional Schrödinger as a one-dimensional differential equation for the reduced radial wave function. In this context we discuss conditions for the existence of bound states and an upper bound on the ground-state energy obtained from the Rayleigh-Ritz variational principle. As an example we then consider the bound-state solutions for the spherical potential well and discuss the conditions for it admitting a ground state. We then turn to the Coulomb potential representing the hydrogen atom and discuss methods for obtaining upper and lower bounds on the ground-state energy to explain the stability of matter. Finally, obtain the Laguerre-polynomial solutions of the radial Schrödinger equation for the hydrogen atom, determine their energies and discuss their properties
Abstract In this chapter we consider the time-independent Schrödinger equation for three spatial dimensions. Making use of the orbital angular momentum observables, including the Laplacian in spherical polar coordinates, and introducing an effective potential, we recast the three-dimensional Schrödinger as a one-dimensional differential equation for the reduced radial wave function. In this context we discuss conditions for the existence of bound states and an upper bound on the ground-state energy obtained from the Rayleigh-Ritz variational principle. As an example we then consider the bound-state solutions for the spherical potential well and discuss the conditions for it admitting a ground state. We then turn to the Coulomb potential representing the hydrogen atom and discuss methods for obtaining upper and lower bounds on the ground-state energy to explain the stability of matter. Finally, obtain the Laguerre-polynomial solutions of the radial Schrödinger equation for the hydrogen atom, determine their energies and discuss their properties
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.
There are certain prime features of physical systems—notably rotation and its associated angular momentum—that simply do not appear in a one-dimensional world. This chapter introduces the three-dimensional Schrodinger equation and applies it to the analysis of a particle in a three-dimensional box and to the spherically symmetric wave functions of hydrogen. The time-independent Schrodinger equation can be viewed in a way that gives it a family resemblance to many other differential equations of physics, and which expresses its structure in very general terms. A computer solution uses dimensionless variables based on natural units. The Schrodinger theory does not have the limitations of the Bohr theory. Schrodinger theory incorporates the well-established wave nature of matter and can be generalized to analyze systems for which the Bohr theory has proven inadequate: multi-electron atoms, chemical properties of atoms, molecular and solid-state physics and chemistry, as well as more subtle effects in hydrogen itself.
This chapter begins with the simple solution of the Schrödinger equation for hydrogen atom for which the energy value has been derived. The Bohr radius of H atom has also been computed. After this, this chapter provides the generalised solution of the Schrödinger equation for H atom/hydrogen-like species in which F equation, R equation, and T equations have been derived in a simple and lucid way. This chapter also deals with the Laguerre differential equation, Laguerre polynomials, Rodrigues formula for the Laguerre polynomials, Laguerre associated equation and its solution, associated Laguerre polynomials, Rodrigues formula for the associated Laguerre polynomials, and solution of the radial equation. It also provides normalisation of the radial wave function and complete wave function; for the hydrogen atom, hydrogenic atomic orbital; electron density of 1 s electron; mathematical background of shape of orbitals; the most probable distance of electron from nucleus of hydrogen atom, and average distance of electron from the nucleus of hydrogen atom. The end of this chapter describes the references and adequate computational problems (solved problems and question on concepts).
In this chapter we solve the radial TISE for the case of a radial potential, specifically the radial potential of an electron attracted to the proton which is the nucleus of a hydrogen atom. Again, we rely on solutions to a related equation solved by a French mathematician, this time named Laguerre, whose polynomials are, in this chapter, multiplied by a simple exponential function to produce the radial wave functions. These functions contain the principle quantum number, which corresponds to the row number in the periodic table. Multiplying these radial solutions by the spherical harmonics from Chapter 14 produces the full three-dimensional solutions for the hydrogen atom, also known as the atomic orbitals. The relationship between the quantum numbers n , l , and m arise naturally from the possible combinations of radial solutions with spherical harmonics.
By using a simple procedure the general solution of the time-independent radial Schrodinger Equation for spherical symmetric potentials was made without making any approximation. The wave functions are always periodic. It appears two diffucilties: one of them is the solution of the equation E= U(r), where E and U(r) are the total an effective potential energies, respectively, and the other is the calculation of the integral of the square root of U(r). If analytical calculations are not possible, one must apply numerical methods. To find the energy wave function of the ground state, there is no need for the calculation of this integral, it is sufficient to find the classical turning points, that is to solve the equation E=U(r).
Part 1: the need for the quantum mechanics quantization of radiation the special theory of relativity wave-particle duality the nature of matter waves the uncertainty principle the Bohr atom. Part 2 Schrodinger's equation: the postulates of quantum mechanics the wave equation the wave equation in three dimensions the time-independent Schrodinger equation stationary states normalisation probability flux physical constraints on the wave-function wave packets Gaussian wave-packets and the uncertainty principle. Part 3 One-dimensional potential wells: potential wells the one-dimensional infinite square well discussion of infinite square-well results eigenvalue equations the finite square well finite square well eigenfunctions the control of the wave-function by the wave equation numerical integration of Schrodinger's equation continuum solutions of the square-well potential interpretation of unbound-particle wave functions the double potential well the quantum bouncer electric fields and the tilted well the harmonic oscillator - introduction the quantum oscillator discussion of harmonic oscillator eigenvalues and eigenfunctions the classical probability density more potential wells. Part 4 Schrodinger's equation in three dimensions: particle in a three dimensional box degeneracy Schrodingers equation in spherical polar coordinates - the hydrogen atom normalisation. Part 5 Formal development of quantum mechanics: eigenvalues equations, operators and measurements the momentum operator, eigenvalues and eigenfunctions the position operator and its eigenfunctions Hermitian operators orthogonality of eigenfunctions the expansion postulate example - the b-decay of tritium expectation values compatibility of observables - commutators non-commuting operators time-dependence of expectation values - Ehrenfest's theorem abstract operators - the parity operator, matrix mechanics. Part 6 Angular momentum: angular momentum operators angular momentum commutators angular momentum eigenvalues and eigenfunctions interpretation of angular momentum eigenvalues interpretation of angular momentum eigenfunctions the Stern-Gerlach apparatus and electron spin spin operators - the Pauli Spin matrices the Zeeman effect measurements in quantum mechanics. Part 7 Spherically symmetric potential wells: the rigid rotator the radial equation the hydrogen atom and hydrogen like ions the isotropic harmonic oscillator molecular vibrations - the Morse potential the spherical square well general properties of spherically symmetric potential wells. (Part contents)