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arXiv:1805.12125 · Condensed Matter Physics

Calculating spherical harmonics without derivatives

Weitzman, James K. Freericks

Published Sep 1, 2018• 7 citations
arXivHTMLPDF

Abstract

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 ).

DOI
10.5488/cmp.21.33002

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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 ).
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.
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.
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.
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.
Synopsis: Direct product spaces are encountered in the generalization to higher dimensions, and the connection to wave function separability is discussed. ƒe 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. ƒe angular momentum eigenvalues and eigenstates are determined using ladder operators and the connection to spherical harmonics is made. ƒe stage is set for understanding the Schrödinger radial equation.