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American Journal of Physics

Lenz vector operations on spherical hydrogen atom eigenfunctions

Charles E. Burkhardt, Jacob J. Leventhal

Published Jul 14, 2004• 11 citations
Publisher

Abstract

We show that operators capable of converting certain spherical hydrogen atom eigenfunctions into others by raising the angular momentum quantum number ℓ can be constructed from components of the quantum mechanical Lenz vector operator. The derivation is performed using formalism no more sophisticated than that used to derive the properties of the angular momentum ladder operators in undergraduate quantum mechanics courses. The properties of these Lenz vector operators illustrate the accidental degeneracy of the hydrogen atom because they change the quantum number ℓ without affecting n.

DOI
10.1119/1.1758225

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We show that operators capable of converting certain spherical hydrogen atom eigenfunctions into others by raising the angular momentum quantum number ℓ can be constructed from components of the quantum mechanical Lenz vector operator. The derivation is performed using formalism no more sophisticated than that used to derive the properties of the angular momentum ladder operators in undergraduate quantum mechanics courses. The properties of these Lenz vector operators illustrate the accidental degeneracy of the hydrogen atom because they change the quantum number ℓ without affecting n.
The usual explicit expressions for spherical harmonics are derived by a method which does not involve the solution of differential equations or the use of angular momentum operators in spherical polar form. Instead a raising operator constructed from an arbitrary vector is used together with the usual angular momentum raising and lowering operators. The method is first presented as simply as possible in Schrödinger representation using position as the vector. Then it is generalized to an arbitrary vector and arbitrary integral angular momentum without using a particular representation. Finally it is shown how a conserved vector (other than angular momentum) of a central system can be used to reveal properties of the energy eigenstates. As an example, eigenstates of the hydrogen atom are found using the Runge-Lenz vector.
The radial equation for the hydrogen atom problem is factored to yield raising and lowering operators for the oribtal angular momentum quantum number l. This procedure, which is a direct analog for the usual treatment of the angular momentum operators, is considered to offer many pedagogical advantages for an introductory course in quantum mechanics over the usual series method of solution. The raising process is terminated by establishing the existence of an upper bound for l. The use of the hypervirial relations is advocated for displaying such an a priori upper bound.
Abstract In this chapter we introduce the concept of orbital angular momentum into the mathematical language of quantum mechanics. We discuss the commutation relations of the corresponding operators and their roles as generators of rotations in the context of the theory of Lie groups and Lie algebras, focusing on the three-dimensional rotation group SO(3). We further examine how scalar and vector-valued operators transform under spatial rotations. We then turn to the derivation of the eigenfunctions and eigenvalues of the angular-momentum operators, for which we follow a dual approach in analogy to that employed for the harmonic oscillator in Chapter 5. Using both the algebraic ladder-operator method and an analytical method using spherical polar coordinates we discover the relevant azimuthal and magnetic quantum numbers and their relation to the spherical harmonics and the Legendre polynomials
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.
Consider the problem: why does the bound state spectrum E(n)<0E(n) < 0E(n)<0, of hydrogen atom Hamiltonian HHH have more degenerate eigenstates than those required by rotational symmetry? The answer is well known and was demonstrated by Pauli. It is due to an additional conserved vector, A⃗\vec AA, of HHH, called the Laplace-Runge-Lenz vector, that was first discovered for planetary orbits. However, surprisingly, a direct link between degenerate eigenstates of HHH and the physical labels that describe them is missing. To provide such a link requires, as we show, solving a subtle problem of self adjoint operators. In our discussions we address a number of conceptual historical aspects regarding hydrogen atom that also include a careful discussion of both the classical as well as the quantum vector A⃗\vec AA.
Two vector operators aimed at shifting angular momentum quantum number l in spherical harmonics |lm>, primarily proposed by Prof. X. L. Ka in 2001, are further studied. For a given magnetic quantum number m, specific states |lm> in spherical harmonics with the lowest angular momentum quantum numbers l are obtained and the state with minimum angular momentum quantum number in whole set of the spherical harmonics is |0,0>. How to use these states to generate whole set of spherical harmonics is illustrated.