A concise way to generate expressions for spherical harmonics

Abstract

This paper presents a concise way to approach spherical harmonics. Deriving expressions for spherical harmonics from basic angular-momentum algebra is natural but cumbersome in detail. The proposed short method stimulates students to “discover” needed functions instead of checking the prescribed answers. Moreover, it does not use any facts of the quantum theory of angular momentum. Many standard properties, such as orthogonality, normalization, recurrence relations, etc., follow naturally from the eigenfunction framework. The described way makes some use of spinor invariants originally proposed by H.A. Kramers. Modern trends in university education result in the shortening of basic mathematical courses. The approach presented seems preferable when students learn mathematical physics without a separate course on special functions.

Keywords:
Laplace equation; associated Legendre functions; spherical harmonics; orthonormality

1. Introduction

Students typically deal with spherical harmonics within learning quantum theory. And this approach is justifiable in general. In quantum theory, these functions appear as eigenfunctions of the orbital angular momentum. In detail, the mentioned approach is realized in section III.4 of the classic text by Condon and Shortley [¹]. At the same time, this technique turned out to be heavier than one can expect initially. Many questions rather need real spherical harmonics introduced by Laplace. Indeed, these functions are a necessary tool to solve the equation today referred to as Laplace’s one. In this sense, they allow students to learn basic classical physics before the course of quantum mechanics. The Laplace and Poisson equations are used in acoustics, electrostatics and elastostatics. Stationary Fourier’s and diffusion equations reduce to similar questions. Thus, there is a pedagogical task to provide a concise way to generate spherical harmonics. This task becomes more important due to modern trends in university education.

The desired approach can be reached within the method of spinor invariants proposed by H.A. Kramers. In fact, there is no need to learn Kramers’ method in detail. Its complete version enables the analysis of atomic spectra without the use of the irreducible representations of the rotation group [²]. It seems that Kramers’ method obtained less attention in the teachers’ community than it deserves. In particular, this method will allow us to generate expressions for spherical harmonics in a simple manner. In fact, we come across a tool to study the main properties of associated Legendre functions. Since this paper has pedagogical purposes, references to the original literature are not given. Several important works can be found in the bibliography of Brinkman’s monograph [²]. We rather aim to draw attention to Kramers’s method, even though many physical students today learn a course of group theory. Sometimes, such courses could be left out in favor of applied special courses.

2. Preliminaries

We will use the spherical coordinates: the radial distance $r$, the polar angle $\theta$, and the azimuthal angle $\phi$. The radius vector $\mathit{r}$ has components $r\sin \theta \cos \phi$, $r\sin \theta \sin \phi$, and $r\cos \theta$. A plain base point to ask for spherical harmonics is a general expression for axially symmetric harmonic functions, i.e.,

(1) $$f(r,\theta )=\sum _{\ell =0}^{\infty }{c}_{\ell }{r}^{\ell }{P}_{\ell }\left(\cos \theta \right).$$

It holds, when the domain of interest includes the origin. $P_{\ell }\left(\xi \right)$ is the $\ell$th Legendre polynomial. The properties of these polynomials can be derived via the generating function (see, e.g., item (11.4) in reference [³]),

(2) $$g(\xi ,t)=\sum _{\ell =0}^{\infty }{P}_{\ell }\left(\xi \right)t^{\ell }=\frac{1}{\sqrt{1-2\xi t+t^2}}.$$

The latter follows from the well-known integrand that typically occurs in electrostatics. In effect, the Rodrigues formula holds,

(3) $$P_{\ell }\left(\xi \right)=\frac{1}{2^{\ell }\ell !}{\left(\frac{d}{d\xi }\right)}^{\ell }{\left({\xi }^2-1\right)}^{\ell }.$$

The function (1) obeys the equation ${\mathbf{\nabla }}^{2}f=0$ without any dependence on $\phi$. We wish to extend (1) by including a dependence on the azimuthal angle.

3. Method to Build Harmonic Functions

The right-hand side of (1) is the sum of terms proportional to $r^{\ell }$. Let $\mathit{a}$ be a constant vector with components $a_x$, $a_y$, and $a_z$. It turned out that powers of the form

(4) $${\left(\mathit{a}\cdot \mathit{r}\right)}^{\ell }=r^{\ell }{\left(a_x\sin \theta \cos \phi +a_y\sin \theta \sin \phi +a_z\cos \theta \right)}^{\ell }$$

are a hint for the desired extension. This substitution was dealt with in §1 of first chapter of Brinkman’s monograph [²]. Further, we write

(5) $${\mathbf{\nabla }}^2{(\mathit{a}\cdot \mathit{r})}^{\ell }=\ell (\ell -1)(a_x^2+a_y^2+a_z^2){(\mathit{a}\cdot \mathit{r})}^{\ell -2}.$$

Assuming nontrivial solutions to Laplace’s equation for $\ell >1$, the only possibility is

(6) $$a_x^2+a_y^2+a_z^2=\left(a_x+\mathtt{i}{a}_y\right)\left(a_x-\mathtt{i}{a}_y\right)+a_z^2=0,$$

with complex $a_x$, $a_y$, and $a_z$. It follows from (6) that there are two complex numbers, $u$ and $v$, such that

(7) $$a_x+\mathtt{i}{a}_y=v^2,$$

(8) $$a_x-\mathtt{i}{a}_y=-u^2,$$

(9) $$a_z=uv,$$

(10) $$a_x=\frac{v^2-u^2}{2},$$

(11) $$a_y=\frac{u^2+v^2}{2\mathtt{i}}.$$

With slight modifications, these expressions follow Cartan’s definition of a spinor (see, e.g., section 2.4 of reference [⁴]). Substituting (9)–(11) in (4) finally leads to

$${\left(\mathit{a}\cdot \mathit{r}\right)}^{\ell }=\frac{r^{\ell }}{2^{\ell }}{\left(v^2{e}^{-\mathtt{i}\phi }\mathrm{}\sin \mathrm{}\theta -u^2{e}^{\mathtt{i}\phi }\mathrm{}\sin \mathrm{}\theta +2uv\mathrm{}\cos \mathrm{}\theta \right)}^{\ell }.$$

This is a $2\ell$-degree homogeneous function of $u$ and $v$,

(12) $${\left(\mathit{a}\cdot \mathit{r}\right)}^{\ell }=r^{\ell }\sum _{m=-\ell }^{+\ell }{Q}_{\ell m}{u}^{\ell +m}{v}^{\ell -m}.$$

It obeys the Laplace equation. Since $u$ and $v$ are arbitrary, each term $r^{\ell }{Q}_{\ell m}$ is a solution of Laplace’s equation. Up to a factor, functions of the form $Q_{\ell m}(\theta ,\phi )$ will represent spherical harmonics. They allow us to express harmonic functions similarly to (1).

4. Two Expressions for Legendre Functions

Extracting the powers of $u$ or $v$ directly is a natural way to construct the right-hand side of (12). When the factor $v^2{e}^{-\mathtt{i}\phi }$ is taken out of $\left(\mathit{a}\cdot \mathit{r}\right)$, the remainder depends on $u$, $v$ and $\phi$ via a variable $X=u{v}^{-1}{e}^{\mathtt{i}\phi }$,

$$\left(\mathit{a}\cdot \mathit{r}\right)=\frac{-r{v}^2{e}^{-\mathtt{i}\phi }}{2}\left(X^2\mathrm{}\sin \mathrm{}\theta -2X\mathrm{}\cos \mathrm{}\theta -\mathrm{}\sin \mathrm{}\theta \right).$$

Writing the Maclaurin series, we verify that

(13) $$\begin{array}{l}{\left(\mathit{a}\cdot \mathit{r}\right)}^{\ell }=\frac{{\left(-1\right)}^{\ell }{r}^{\ell }}{2^{\ell }}\frac{v^{2\ell }{e}^{-\mathtt{i}\ell \phi }}{{\left(\mathrm{}\sin \mathrm{}\theta \right)}^{\ell }}{\left[{\left(X\mathrm{}\sin \mathrm{}\theta -\mathrm{}\cos \mathrm{}\theta \right)}^2-1\right]}^{\ell }\\ =\frac{{\left(-1\right)}^{\ell }{r}^{\ell }}{2^{\ell }}\frac{v^{2\ell }{e}^{-\mathtt{i}\ell \phi }}{{\left(\mathrm{}\sin \mathrm{}\theta \right)}^{\ell }}\sum _{m=-\ell }^{+\ell }\frac{X^{\ell +m}}{\left(\ell +m\right)!}\\ \times {\{.{\left(\frac{d}{dX}\right)}^{\ell +m}{[{(X\mathrm{}\sin \mathrm{}\theta -\mathrm{}\cos \mathrm{}\theta )}^2-1]}^{\ell }|}_{X=0}\}.\end{array}$$

Since $\mathrm{}\sin \mathrm{}\theta dX=d\left(X\mathrm{}\sin \mathrm{}\theta -\mathrm{}\cos \mathrm{}\theta \right)$ at differentiating with respect to $X$, the right-hand side of (13) finally reduces to the form

(14) $${\left(\mathit{a}\cdot \mathit{r}\right)}^{\ell }=\frac{r^{\ell }}{2^{\ell }}\sum _{m=-\ell }^{+\ell }{u}^{\ell +m}{v}^{\ell -m}\frac{{\left(-1\right)}^m{\left(\mathrm{}\sin \mathrm{}\theta \right)}^m{e}^{\mathtt{i}m\phi }}{\left(\ell +m\right)!}\times {\left(\frac{d}{d\xi }\right)}^{\ell +m}{\left({\xi }^2-1\right)}^{\ell },$$

where $\xi =\mathrm{}\cos \mathrm{}\theta$. The first expression for $Q_{\ell m}$ reads as

(15) $$Q_{\ell m}=\frac{{\left(-1\right)}^m{e}^{\mathtt{i}m\phi }}{2^{\ell }\left(\ell +m\right)!}{\left(1-{\xi }^2\right)}^{m/2}{\left(\frac{d}{d\xi }\right)}^{\ell +m}{\left({\xi }^2-1\right)}^{\ell }.$$

To generate the second expression for $Q_{\ell m}$[²], we replace $X$ with $Y=X^{-1}=u^{-1}v{e}^{-\mathtt{i}\phi }$. By a parallel argument, one gets

(16) $$\begin{array}{l}{\left(\mathit{a}\cdot \mathit{r}\right)}^{\ell }=\frac{r^{\ell }}{2^{\ell }}\frac{u^{2\ell }{e}^{\mathtt{i}\ell \phi }}{{\left(\mathrm{}\sin \mathrm{}\theta \right)}^{\ell }}{\left[{\left(Y\mathrm{}\sin \mathrm{}\theta +\mathrm{}\cos \mathrm{}\theta \right)}^2-1\right]}^{\ell }\\ =\frac{r^{\ell }}{2^{\ell }}\sum _{n=-\ell }^{+\ell }{u}^{\ell -n}{v}^{\ell +n}\frac{{\left(\mathrm{}\sin \mathrm{}\theta \right)}^n{e}^{-\mathtt{i}n\phi }}{\left(\ell +n\right)!}\\ \times {\left(\frac{d}{d\xi }\right)}^{\ell +n}{\left({\xi }^2-1\right)}^{\ell }.\end{array}$$

Taking $n=\pm m$ and comparing the results with (12), we have

(17) $$Q_{\ell ,-m}=\frac{e^{-\mathtt{i}m\phi }}{2^{\ell }\left(\ell +m\right)!}{\left(1-{\xi }^2\right)}^{m/2}{\left(\frac{d}{d\xi }\right)}^{\ell +m}{\left({\xi }^2-1\right)}^{\ell },$$

(18) $$Q_{\ell m}=\frac{e^{\mathtt{i}m\phi }}{2^{\ell }\left(\ell -m\right)!}{\left(1-{\xi }^2\right)}^{-m/2}{\left(\frac{d}{d\xi }\right)}^{\ell -m}{\left({\xi }^2-1\right)}^{\ell }.$$

The latter gives the second expression for $Q_{\ell m}$. Up to a factor, the dependence of both (15) and (17) on $\xi =\mathrm{}\cos \mathrm{}\theta$ is represented by the same associated Legendre function. Diverse ways to write associated Legendre functions are related to (15) and (18). For $m=0,1,\dots ,\ell$, we have

(19) $$P_{\ell }^{\left(m\right)}\left(\xi \right)={\left(1-{\xi }^2\right)}^{m/2}{\left(\frac{d}{d\xi }\right)}^{\ell +m}\frac{{\left({\xi }^2-1\right)}^{\ell }}{2^{\ell }\ell !},$$

(20) $$=\frac{\left(\ell +m\right)!}{\left(\ell -m\right)!}\frac{{\left(-1\right)}^m}{{\left(1-{\xi }^2\right)}^{m/2}}{\left(\frac{d}{d\xi }\right)}^{\ell -m}\frac{{\left({\xi }^2-1\right)}^{\ell }}{2^{\ell }\ell !}.$$

The formula (19) actually repeats the standard definition $P_{\ell }^{\left(m\right)}\left(\xi \right)={\left(1-{\xi }^2\right)}^{m/2}\left(d^m{P}_{\ell }/d{\xi }^m\right)$ (see item (11.68) in reference [³]). The result (20) follows by the comparison of (15) with (18).

To sum up, we have arrived at the following two conclusions. First, the desired expressions for associated Legendre functions were generated in a plain and machinery way. In fact, some knowledge of power series was utilized. Moreover, the two different representations (19) and (20) simultaneously appeared. These expressions will be convenient to evaluate a normalization factor. As some experience show, more calculation efforts are required to build associated Legendre functions with the use of operators of the orbital angular momentum. Being concise in an abstract form, this technique becomes relatively cumbersome with realizing via differential operators.

5. Orthogonality and Norms

The question that should be resolved concerns orthogonality and normalization. In the spherical coordinates, the Laplace operator reads as

(21) $${\mathbf{\nabla }}^2=\frac{1}{r^2}\frac{\partial }{\partial r}{r}^2\frac{\partial }{\partial r}-\frac{{\mathbf{L}}^2}{r^2},$$

where the angular part

(22) $$-{\mathbf{L}}^2=\frac{1}{\mathrm{}\sin \mathrm{}\theta }\frac{\partial }{\partial \theta }\mathrm{}\sin \mathrm{}\theta \frac{\partial }{\partial \theta }+\frac{1}{{\sin }^2\theta }\frac{{\partial }^2}{\partial {\phi }^2}.$$

If the reader is familiar with quantum theory, he can recognize in ${\mathbf{L}}^2$ the square of the orbital angular momentum in dimensionless units (see, e.g., equation (27) in section III.4 of reference [¹]). However, this fact does not affect the subsequent conclusions. Combining (15) with

$${\mathbf{\nabla }}^2\left(r^{\ell }{Q}_{\ell m}\right)=0$$

finally leads to the associated Legendre differential equation,

(23) $$\frac{d}{d\xi }\left(1-{\xi }^2\right)\frac{d{P}_{\ell }^{\left(m\right)}}{d\xi }+\left(\ell \left(\ell +1\right)-\frac{m^2}{1-{\xi }^2}\right)P_{\ell }^{\left(m\right)}=0.$$

Thus, the associated Legendre functions are eigenfunctions of the self-adjoint differential operator of second order. One of the well-known properties is their orthogonality,

(24) $${\int }_{-1}^{+1}{P}_k^{\left(m\right)}\left(\xi \right)P_{\ell }^{\left(m\right)}\left(\xi \right)d\xi \propto {\delta }_{k\ell },$$

where ${\delta }_{k\ell }$ is the Kronecker symbol. Evaluating a normalization factor reduces to the integral

(25) $$\begin{array}{l}{\int }_{-1}^{+1}{P}_{\ell }^{\left(m\right)}\left(\xi \right)P_{\ell }^{\left(m\right)}\left(\xi \right)d\xi \\ =\frac{\left(\ell +m\right)!{\left(-1\right)}^m}{\left(\ell -m\right)!{\left(2^{\ell }\ell !\right)}^2}\\ \times {\int }_{-1}^{+1}\frac{d^{\ell +m}{\left({\xi }^2-1\right)}^{\ell }}{d{\xi }^{\ell +m}}\frac{d^{\ell -m}{\left({\xi }^2-1\right)}^{\ell }}{d{\xi }^{\ell -m}}d\xi \end{array}$$

(26) $$=\frac{\left(\ell +m\right)!\left(2\ell \right)!}{\left(\ell -m\right)!{\left(2^{\ell }\ell !\right)}^2}{\int }_{-1}^{+1}{\left(1-{\xi }^2\right)}^{\ell }d\xi .$$

Due to the formulas (19) and (20), the integrand in the right-hand side of (25) becomes a product of two derivatives of the same function. To get (26), we have integrated by parts as many times as required. There are various ways to evaluate the integral occurring in (26). For example, it can be reduced to Euler’s beta- and gamma-functions. However, the desired integral is the same for all $m=0,1,\dots ,\ell$. It will be enough to get it for a particular value of $m$, say, for $m=0$. In this sense, the question reduces to the squared norm of $\ell$th Legendre polynomial. Integrating the square of the generating function (2), one writes

$$\begin{array}{l}\sum _{k,\ell =0}^{\infty }{t}^{k+\ell }{\int }_{-1}^{+1}{P}_k\left(\xi \right)P_{\ell }\left(\xi \right)d\xi ={\int }_{-1}^{+1}\frac{d\xi }{1-2\xi t+t^2}\\ =\frac{1}{t}\ln \left(\frac{1+t}{1-t}\right)=\sum _{\ell =0}^{\infty }\frac{2t^{2\ell }}{2\ell +1},\end{array}$$

whence

(27) $${\int }_{-1}^{+1}{P}_k\left(\xi \right)P_{\ell }\left(\xi \right)d\xi =\frac{2{\delta }_{k\ell }}{2\ell +1}.$$

Combining this with (24) and (26) finally leads to

(28) $${\int }_{-1}^{+1}{P}_k^{\left(m\right)}\left(\xi \right)P_{\ell }^{\left(m\right)}\left(\xi \right)d\xi =\frac{\left(\ell +m\right)!}{\left(\ell -m\right)!}\frac{2{\delta }_{k\ell }}{2\ell +1},$$

where $m=0,1,\dots ,\ell$.

6. Spherical Harmonics

Let us finalize the expressions for complex spherical harmonics that obey the orthonormality condition

(29) $$\int d\mathrm{\Omega }{Y}_{kn}^{\ast }(\theta ,\phi )Y_{\ell m}(\theta ,\phi )={\delta }_{k\ell }{\delta }_{nm}.$$

To get $Y_{\ell m}(\theta ,\phi )$ from $Q_{\ell m}(\theta ,\phi )$, we merely multiply the latter by the normalization factor. Comparing (15) with (17), one has

(30) $$Q_{\ell ,-m}(\theta ,\phi )={\left(-1\right)}^m{Q}_{\ell m}^{\ast }(\theta ,\phi ).$$

For $m=0,1,\dots ,\ell$, the spherical harmonics appear as

(31) $$Y_{\ell m}(\theta ,\phi )={\left(-1\right)}^m\sqrt{\frac{2\ell +1}{4\pi }}\sqrt{\frac{\left(\ell -m\right)!}{\left(\ell +m\right)!}}\times e^{\mathtt{i}m\phi }{P}_{\ell }^{\left(m\right)}\left(\mathrm{}\cos \mathrm{}\theta \right),$$

(32) $$Y_{\ell ,-m}(\theta ,\phi )={\left(-1\right)}^m{Y}_{\ell m}^{\ast }(\theta ,\phi ).$$

Up to a factor, the first expression follows from (15), whereas (32) resembles (30). The additional factor $\sqrt{2\pi }$ in the denominator of the normalization constant is necessary due to integrating with respect to $\phi$. Of course, there is freedom in choosing phase factors. The formulas (31) and (32) use the Condon–Shortley phase convention (see, e.g., equation (17) in section III.4 of reference [¹]). Beyond the context of quantum theory, a real basis of spherical harmonics is more suitable. Such a basis follows from the above expressions by replacing $\exp \left(\pm \mathtt{i}m\phi \right)$ with $\mathrm{}\cos \mathrm{}m\phi$ and $\mathrm{}\sin \mathrm{}m\phi$. Then the functions $Y_{\ell 0}\left(\theta \right)$ remain unchanged; in other functions, we also replace $4\pi$ with $2\pi$ in the denominator of radicals that occur in (31). The factor ${\left(-1\right)}^m$ is typically left out. Students can themselves follow these very elementary changes.

7. Conclusions

This paper describes a concise way to generate expressions for spherical harmonics. It allows students to “discover” needed harmonics instead of checking the prescribed answers. Stimulating with suitable lines of argument is very important, with the emphasis on the independent work of students. The proposed approach can be useful within modern educational trends. These trends may result in some reduction of time devoted to basic mathematics in favor of applied special courses. Despite the existence of many textbooks, there is also a need for an independent and concise presentation of selected topics. The paper [⁵] gives an example of such guides for students. It gathered in one place basic facts on rotations and their matrix representations in quantum theory. Numerical techniques for Poisson’s equation in spherical coordinates are of separate interest [⁶]. The present author had some experience to refresh the teaching of mathematical courses for physicists. The papers [⁷, ⁸] offered simple and physically reasoned ways to derive Green’s functions for equations of mathematical physics. Continuing this trend in application to spherical harmonics, the above derivation presents facts together with a survey of how they could be obtained. It gives the reader a discovery-oriented, self-contained route to approach basic properties of spherical harmonics within limited lecture time. The proposed formulation also aims to stimulate students by performing all the necessary calculations.

Data Availability

The entire dataset supporting the results of this study is published in the article.

References

• ^[1] E.U. Condon and G.H. Shortley, The Theory of Atomic Spectra (Cambridge University Press, Cambridge, 1959).
• ^[2] H.C. Brinkman, Applications of Spinor Invariants in Atomic Physics (North Holland, Amsterdam, 1956).
• ^[3] H.J. Weber and G.B. Arfken, Essential Mathematical Methods for Physicists (Academic Press, Amsterdam, 2003).
• ^[4] L.C. Biedenharn and J.D. Louck, Angular Momentum in Quantum Physics (Addison-Wesley, London, 1981).
• ^[5] M.A. Morrison and G.A. Parker, Aust. J. Phys. 40, 465 (1987).
• ^[6] A.C. Silva, J.A.H. Neto and V.S. Costa, Rev. Bras. Ensino Fís. 43, e20210019 (2021).
• ^[7] A.E. Rastegin, Rom. Rep. Phys. 69, 903 (2017).
• ^[8] A.E. Rastegin, Rom. Rep. Phys. 77, 909 (2025).

Edited by

• Editor-in-Chief:
  Marcello Ferreira https://orcid.org/0000-0003-4945-3169

Publication Dates

• Publication in this collection
  25 May 2026
• Date of issue
  2026

History

• Received
  31 Oct 2025
• Reviewed
  23 Apr 2026
• Accepted
  24 Apr 2026