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BENTHAM SCIENCE PUBLISHERS eBooks

Central Force Problem I

Iraj R. Afnan

Published Apr 14, 2011• 0 citations
Publisher

Abstract

In this chapter we commence with the solution of the Schrodinger equation in three dimension and consider the case when the potential has spherical symmetry. This requires the introduction of spherical polar coordinates and the reduction of the three dimensional partial differential equation to a set of ordinary differential equations. The solution of these equations for the angular dependence will allow us to examine the rotational spectrum of diatomic molecules.

DOI
10.2174/978160805250911101010128

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In this chapter we commence with the solution of the Schrodinger equation in three dimension and consider the case when the potential has spherical symmetry. This requires the introduction of spherical polar coordinates and the reduction of the three dimensional partial differential equation to a set of ordinary differential equations. The solution of these equations for the angular dependence will allow us to examine the rotational spectrum of diatomic molecules.
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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.
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
The quantum mechanical two-body problem is analyzed. Separating the center of mass from the relative motion Hamiltonian and focusing on “central potentials,” the stationary Schrödinger equation for the relative motion in spherical coordinates is split into radial and angular equations. The universal angular equation is identified as the eigenvalue equation of the angular momentum operator, whose proper solutions are the spherical harmonics. For fixed interparticle distance, the two-body system is mapped on a “rigid rotor” Hamiltonian, whose eigenstates coincide with the angular momentum eigenstates. In diatomic molecules, timescale separation between fast vibrations (radial motion) and slow rotations (angular motion) enables one to invoke a rigid rotor approximation for interpreting rotational absorption spectrum in the microwave regime. Deviations from the predictions of the rigid rotor model and their manifestation in experiments are analyzed by explicit solution of the stationary Schrödinger equation for two particles in the presence of vibration–rotation coupling.
Chapter 12 uses the results presented in the previous chapter to deal with systems consisting of two particles interacting via a central potential. It therefore begins by transforming the Hamiltonian of the two-particle system into one that separates the center-of-mass motion from the relative motion. When the potential is spherically symmetric, the relative motion is conveniently described in spherical coordinates to take advantage of the conservation of angular momentum and the solution of the corresponding eigenvalue problem. The chapter contains a detailed discussion of the rigid rotor and the free particle, as well as the selection rules for angular momentum, in preparation for the study of the most important problem of atomic physics, the hydrogen atom. The H spectrum is discussed in detail, and the introduction of minimal coupling to the electromagnetic field allows for a discussion of the normal Zeeman effect as well as the Aharonov-Bohm effect.
Synopsis: Direct product spaces are encountered in the generalization to higher dimensions, and the connection to wave function separability is discussed. The spherical coordinate basis is introduced in preparation for discussing spherically symmetric potentials. Angular momentum operators and properties are examined, and a complete set of compatible observables is determined. The angular momentum eigenvalues and eigenstates are determined using ladder operators, and the connection to spherical harmonics is made. The stage is set for understanding the Schrödinger radial equation.