Laguerre Polynomials

Laguerre polynomials are orthogonal polynomials on the half-line $[0,\infty)$ with exponential weight. In quantum mechanics, generalized Laguerre polynomials appear most prominently in the radial wavefunctions of hydrogenic Coulomb bound states.

The key point is that the polynomial is only one factor. Hydrogenic radial wavefunctions also include an exponential tail, a near-origin power, a dimensionless radial variable, and a normalization set by the radial measure.

Ordinary Laguerre Polynomials

The ordinary Laguerre polynomial of degree $n$ is

$$L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} \left( e^{-x}x^n \right), \qquad n=0,1,2,\ldots.$$

The first few are

$$\begin{aligned} L_0(x)&=1,\\ L_1(x)&=1-x,\\ L_2(x)&=\frac12(x^2-4x+2). \end{aligned}$$

They satisfy Laguerre’s differential equation

$$xy'' +(1-x)y' +ny =0.$$

Generalized Laguerre Polynomials

For $\alpha>-1$, the generalized Laguerre polynomial is

$$L_n^{(\alpha)}(x) = \frac{x^{-\alpha}e^x}{n!} \frac{d^n}{dx^n} \left( e^{-x}x^{n+\alpha} \right).$$

The ordinary polynomial is the special case

$$L_n(x)=L_n^{(0)}(x).$$

The first two generalized polynomials are

$$L_0^{(\alpha)}(x)=1, \qquad L_1^{(\alpha)}(x)=\alpha+1-x.$$

They satisfy

$$xy'' +(\alpha+1-x)y' +ny =0.$$

Some physics books write the superscript without parentheses, such as $L_n^k(x)$. This page uses $L_n^{(\alpha)}(x)$ to avoid confusing the superscript with an exponent.

Orthogonality

The generalized Laguerre polynomials obey the weighted orthogonality relation

$$\int_0^\infty e^{-x}x^\alpha L_n^{(\alpha)}(x) L_m^{(\alpha)}(x)\,dx = \frac{\Gamma(n+\alpha+1)}{n!} \delta_{nm}.$$

For integer $\alpha\ge0$ this becomes

$$\frac{\Gamma(n+\alpha+1)}{n!} = \frac{(n+\alpha)!}{n!}.$$

The gamma-function factor is the natural continuation of the factorial normalization. See Gamma and Beta Functions for the identities used in such weighted integrals.

The weight $e^{-x}x^\alpha$ is part of the inner product. Laguerre polynomials are not orthogonal with the ordinary unweighted integral on $[0,\infty)$.

Generating Function

The generating function is

$$\sum_{n=0}^{\infty} L_n^{(\alpha)}(x)t^n = \frac{1}{(1-t)^{\alpha+1}} \exp\left( - \frac{xt}{1-t} \right), \qquad \lvert t\rvert<1.$$

For $\alpha=0$, this generates the ordinary Laguerre polynomials.

Recurrence and Derivative Identities

A useful three-term recurrence is

$$(n+1)L_{n+1}^{(\alpha)}(x) = (2n+\alpha+1-x)L_n^{(\alpha)}(x) - (n+\alpha)L_{n-1}^{(\alpha)}(x).$$

The derivative identity

$$\frac{d}{dx}L_n^{(\alpha)}(x) = -L_{n-1}^{(\alpha+1)}(x)$$

is often useful in radial matrix-element calculations.

Hydrogenic Radial Functions

After separating the hydrogenic Schrödinger equation in spherical coordinates, the bound-state wavefunction has the form

$$\psi_{n\ell m}(r,\theta,\phi) = R_{n\ell}(r)Y_\ell^m(\theta,\phi).$$

The angular dependence is carried by spherical harmonics. The Coulomb radial equation produces generalized Laguerre polynomials.

For a hydrogenic Coulomb problem, introduce a dimensionless radial variable

$$\rho = \frac{2r}{na},$$

where $a$ is the appropriate Bohr-length scale for the reduced mass and nuclear charge convention being used. The radial functions have the form

$$R_{n\ell}(r) = N_{n\ell} e^{-\rho/2} \rho^\ell L_{n-\ell-1}^{(2\ell+1)}(\rho),$$

with

$$n=1,2,\ldots, \qquad \ell=0,1,\ldots,n-1.$$

The polynomial degree is

$$n_r=n-\ell-1.$$

This integer is the number of radial nodes in the bound-state radial function. It is not the same as the principal quantum number $n$.

The normalization constant $N_{n\ell}$ depends on the precise radial convention and length scale. With the usual $R_{n\ell}$ convention, normalization means

$$\int_0^\infty \lvert R_{n\ell}(r)\rvert^2 r^2\,dr =1.$$

The reduced radial function $u_{n\ell}(r)=rR_{n\ell}(r)$ instead uses the measure $dr$; see Normalization Conventions.

Why the Factors Appear

The hydrogenic radial form has three pieces with different jobs:

• $e^{-\rho/2}$ gives the bound-state exponential tail;
• $\rho^\ell$ gives the regular near-origin behavior for angular momentum $\ell$;
• $L_{n-\ell-1}^{(2\ell+1)}(\rho)$ supplies the finite polynomial factor and radial nodes.

The associated Laguerre polynomial terminates because normalizable Coulomb bound states require the radial series to stop. This is analogous in spirit to how Hermite polynomials appear in the harmonic oscillator, but the half-line domain and radial measure are different.

Equivalently, generalized Laguerre polynomials can be viewed as terminating confluent Hypergeometric Functions. The Laguerre notation is usually better for hydrogen because it keeps the radial weight and polynomial degree visible.

Common Mistakes

• Confusing the polynomial degree $n-\ell-1$ with the principal quantum number $n$.
• Forgetting the superscript parameter in $L_{n-\ell-1}^{(2\ell+1)}$.
• Treating $L_n^{(\alpha)}$ as an exponent rather than a family label.
• Omitting the weight $e^{-x}x^\alpha$ in the orthogonality integral.
• Using the dimensionful radius $r$ where the dimensionless variable $\rho$ is required.
• Normalizing $R(r)$ with $dr$ instead of $r^2\,dr$.
• Confusing ordinary Laguerre polynomials with generalized Laguerre polynomials.

Cross-Links

• Hydrogen Atom
• Radial Wavefunctions
• Separation of Variables
• Spherical Harmonics
• Associated Legendre Functions
• Orthogonal Polynomials
• Hypergeometric Functions
• Gamma and Beta Functions
• Probability Densities
• Normalization Conventions
• Sturm–Liouville Theory

References

• NIST Digital Library of Mathematical Functions, Chapter 18, Orthogonal Polynomials.
• F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., NIST Handbook of Mathematical Functions, Cambridge University Press, 2010.
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, Dover, 1965.
• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.

Exercises

1. Use Rodrigues’ formula to compute $L_0(x)$ and $L_1(x)$.

Solution

For $n=0$,

$$L_0(x)=1.$$

For $n=1$,

$$L_1(x) = e^x \frac{d}{dx} \left( e^{-x}x \right).$$

Since

$$\frac{d}{dx} \left( e^{-x}x \right) = e^{-x}(1-x),$$

one gets

$$L_1(x)=1-x.$$

2. Verify that $L_0$ and $L_1$ are orthogonal with the ordinary Laguerre weight.

Solution

For $\alpha=0$, the weight is $e^{-x}$. Thus

$$\int_0^\infty e^{-x}L_0(x)L_1(x)\,dx = \int_0^\infty e^{-x}(1-x)\,dx.$$

Using

$$\int_0^\infty e^{-x}\,dx=1, \qquad \int_0^\infty xe^{-x}\,dx=1,$$

the result is zero.

3. Show that $L_1^{(\alpha)}(x)=\alpha+1-x$ from the generalized Rodrigues formula.

Solution

For $n=1$,

$$L_1^{(\alpha)}(x) = x^{-\alpha}e^x \frac{d}{dx} \left( e^{-x}x^{\alpha+1} \right).$$

The derivative is

$$e^{-x} \left[ (\alpha+1)x^\alpha -x^{\alpha+1} \right].$$

Multiplying by $x^{-\alpha}e^x$ gives

$$L_1^{(\alpha)}(x)=\alpha+1-x.$$

4. For a hydrogenic bound state with principal quantum number $n$ and angular momentum $\ell$, how many radial nodes does the radial polynomial have?

Solution

The radial polynomial is

$$L_{n-\ell-1}^{(2\ell+1)}(\rho).$$

Its degree is

$$n_r=n-\ell-1.$$

This is the number of radial nodes in the bound-state radial function.

5. For $n=3$ and $\ell=1$, identify the generalized Laguerre polynomial appearing in the hydrogenic radial function.

Solution

The degree is

$$n-\ell-1 = 3-1-1 =1.$$

The superscript parameter is

$$2\ell+1=3.$$

Thus the polynomial is

$$L_1^{(3)}(\rho).$$

6. Why is $R_{n\ell}(r)$ not normalized using $\int_0^\infty \lvert R_{n\ell}(r)\rvert^2\,dr=1$?

Solution

The three-dimensional volume element in spherical coordinates is

$$d^3r = r^2\sin\theta\,dr\,d\theta\,d\phi.$$

When the angular factor is normalized, the radial normalization condition for $R_{n\ell}$ is

$$\int_0^\infty \lvert R_{n\ell}(r)\rvert^2 r^2\,dr =1.$$

The measure $dr$ belongs to the reduced radial function $u_{n\ell}=rR_{n\ell}$, not to $R_{n\ell}$ itself.