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Training and Research

The Research on Recurrence Formulas of Spherical Harmonics in Quanfun Mechanics

Ming Liu

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

By introducing directional operator, we can infer a se ri es of recurrence formulas of spherical harmonics by using algebraic relations an d bascial commutation relation of operator. And the recurrence formulas of spher ical harmonics is usually used in Quanfun Mechanics.

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Angular momentum theory provides a mathematical foundation for the rotation of the eigenvectors of the angular momentum operator, the spherical harmonics functions. A useful property of spherical harmonics is that their rotation in 3D space corresponds to rotation in the virtual space of their azimuthal degrees of freedom. In addition to that, spherical harmonics functions that have different orbital indices do not mix. Given that spherical harmonics functions have known rotation properties, and that any arbitrary function can be expanded in a series of spherical harmonics, this property allows the rotation of an electrostatic potential when it is expanded in the series of spherical harmonics. In this chapter, the exponential form of the rotation operator, the rotation matrix for a 3D rotation of the coordinate system, and the principles of rotation of vectors and functions are described. The chapter provides a detailed self-contained introduction to the theory of angular momentum, derives the commutative properties of the angular momentum operator, and explains shift operators, while minimizing references to quantum mechanics and entirely avoiding group theory.
The derivation of spherical harmonics is the same in nearly every quantum mechanics textbook and classroom.It is found to be difficult to follow, hard to understand, and challenging to reproduce by most students.In this work, we show how one can determine spherical harmonics in a more natural way based on operators and a powerful identity called the exponential disentangling operator identity (known in quantum optics, but little used elsewhere).This new strategy follows naturally after one has introduced Dirac notation, computed the angular momentum algebra, and determined the action of the angular momentum raising and lowering operators on the simultaneous angular momentum eigenstates (under L2 and Lz ).
Using the algebraic second quantization formalism, the general recurrence relations for some two-center harmonic oscillator integrals are obtained. The various known special cases for one-center integrals are evident in the formalism. Generating functions for these integrals are obtained and most of them are solved to find explicit expressions for polynomial, exponential, and Gaussian operator integrals. For these cases, all the generating integrals needed for utilizing the recurrence relations are also calculated in their simplest forms.
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.
In this study, Gaunt coefficients, frequently encountered in quantum mechanical calculations of atomic and molecular structures, have been algebraically derived. Firstly, the Gaunt coefficient, equal to the integral over the solid angle of the product of three spherical harmonics, is written in terms of angular momentum ladder operators. Subsequently, raising or lowering operators are applied to spherical harmonics, and the obtained integrals are solved using the recurrence and orthogonality relations of spherical harmonics. As a result, algebraic expressions for Gaunt coefficients are obtained in terms of quantum numbers.