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Journal of Chemical Education

Ladder-Operator Treatment of the Radial Equation for the Hydrogenlike Atom

Donald D. Fitts

Published Dec 1, 1995• 2 citations
Publisher

Abstract

A new approach to the solution of the radial equation for the hydrogenlike atom is presented. Raising- and lowering-ladder operators are used to solve the radial differential equation for the eigenvalues and eigenfunctions, avoiding the series solution method with its problems concerning asymptotic behavior and convergence. Quantization of the energy levels and the generation of orthonormal eigenfunctions are obtained rigorously, without any arbitrary assumptions.

DOI
10.1021/ed072p1066

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A new approach to the solution of the radial equation for the hydrogenlike atom is presented. Raising- and lowering-ladder operators are used to solve the radial differential equation for the eigenvalues and eigenfunctions, avoiding the series solution method with its problems concerning asymptotic behavior and convergence. Quantization of the energy levels and the generation of orthonormal eigenfunctions are obtained rigorously, without any arbitrary assumptions.
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.
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.
We show that to obtain the ladder operators for the non-relativistic quantum 3D Coulomb problem it is enough to solve a simple linear system of first-order differential equations. We first assume the most general form for the operators compatible with problem and after, by applying the basic commutation relations they must comply with, we first deduce and then solve a differential equations system; in this way we find the usual form of the Coulomb radial ladder operators. The computations are explicitly done only for the creation operator since for the annihilation operator they are almost the same.
We add a phase variable and its corresponding operator to the description of the hydrogen atom. With the help of these additions, we device operators that act as ladder operators for the radial system. The algebra defined by the commutation relations between those operators has a Casimir operator coincident with the radial Hamiltonian of the problem. The algebra happens to be the well-known su(1,1) Lie algebra, hence the phase-dependent eigenfunctions calculated within our scheme belong in a representation of that algebra, a fact that may be useful in certain applications. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem 107: 1608 -1613, 2007
The radial form of Hylleraas’ equation for the hydrogen atom, Λl‖El〉=4ℏ4a−2‖El〉 (a=Bohr radius), is considered and it is shown that the operator Λl can be factorized. Hence ladder operators P±l are derived that are linear in the position operator r and are nonlinear functions of the momentum operator p. It is proven that P±l‖El〉 =2ℏ2a−1[1+(l+ (1)/(2) ± (1)/(2) )2 ×(2Mℏ−2a2E)]1/2‖E,l±1〉. In the momentum representation of wave mechanics the solutions to these equations are the radial momentum-space wavefunctions for the hydrogen atom. Thus a simple method of calculating these wavefunctions is obtained. The results complement the familiar operator solution for the hydrogen atom that is based on factorization of the radial Hamiltonian and yields operators that are linear in p and are nonlinear functions of r.