Orthogonal Polynomials

Orthogonal polynomials are polynomial sequences $p_0,p_1,p_2,\ldots$ in which $p_n$ has degree $n$ and different degrees are orthogonal with respect to a specified weighted inner product. In quantum mechanics, they appear when separated eigenvalue problems reduce to polynomial factors multiplied by weights, endpoint powers, or exponential envelopes.

This page is the family-level canonical home. The detailed normalization and physics of the main quantum-mechanics families belong to their dedicated pages: Hermite Polynomials, Legendre Polynomials, Associated Legendre Functions, and Laguerre Polynomials.

Weighted Inner Product

Let $I$ be an interval and let $w(x)>0$ be a weight function on $I$. The weighted inner product is

$$\langle f,g\rangle_w = \int_I f(x)^*g(x)w(x)\,dx,$$

when the integral exists. A polynomial sequence $\{p_n\}_{n\ge0}$ is orthogonal with respect to $w$ if

$$\langle p_m,p_n\rangle_w = h_n\delta_{mn}, \qquad h_n>0.$$

The norm constants $h_n$ depend on the convention used for $p_n$. Changing the leading coefficient or endpoint normalization changes $h_n$ but not the orthogonal subspaces.

The weight is not decoration. It is part of the inner product. Forgetting the weight changes the problem.

Why Quantum Mechanics Uses Them

Separated quantum problems often produce a differential equation plus endpoint conditions. Normalizable solutions are not arbitrary functions; they must have the correct behavior near endpoints and at infinity. After the endpoint behavior is factored out, the remaining finite part is frequently an orthogonal polynomial.

The pattern is:

1. separate variables;
2. factor the known singular or asymptotic behavior;
3. obtain a polynomial differential equation;
4. impose regularity or square-integrability;
5. get polynomial termination and discrete labels.

For example:

• harmonic-oscillator wavefunctions contain Hermite polynomials times a Gaussian;
• hydrogen radial bound states contain generalized Laguerre polynomials times an exponential and a power of $r$;
• central-potential angular functions contain Legendre or associated Legendre functions;
• spherical harmonics combine associated Legendre functions with Fourier modes in the azimuthal angle.

The polynomial alone is usually not the physical wavefunction. The weight and envelope carry part of the Hilbert-space measure.

Classical Families

The main families used early in quantum mechanics are:

• Hermite polynomials on $(-\infty,\infty)$ with weight $e^{-x^2}$;
• Laguerre polynomials on $[0,\infty)$ with weight $x^\alpha e^{-x}$;
• Legendre polynomials on $[-1,1]$ with weight $1$;
• associated Legendre functions for fixed azimuthal label $m$, used in spherical harmonics.

Other classical families, such as Chebyshev, Gegenbauer, and Jacobi polynomials, appear in more specialized angular, approximation, and representation-theoretic settings. The common lesson is the same: the interval, weight, endpoint behavior, and normalization convention define the family.

Sturm–Liouville Origin

Orthogonal-polynomial families often arise from Sturm–Liouville Theory. In a typical self-adjoint form,

$$- \frac{d}{dx} \left[ p(x)\frac{dy}{dx} \right] +q(x)y = \lambda w(x)y,$$

with suitable endpoint conditions. If $y_m$ and $y_n$ correspond to distinct eigenvalues, self-adjointness gives

$$(\lambda_n-\lambda_m) \int_I y_m(x)^*y_n(x)w(x)\,dx =0.$$

Therefore

$$\int_I y_m(x)^*y_n(x)w(x)\,dx=0 \qquad (m\ne n).$$

This is why orthogonality is not an accident of special-function tables. It follows from the self-adjoint structure of the differential problem.

For singular endpoints or infinite intervals, the same idea persists, but the endpoint hypotheses require care. That is why rigorous statements about completeness and domains belong on the Sturm–Liouville and Hilbert-space pages.

Three-Term Recurrences

For a positive weight on an interval, multiplication by $x$ maps a degree-$n$ polynomial to degree $n+1$. Orthogonality forces a three-term recurrence. For monic orthogonal polynomials, one may write

$$xp_n(x) = p_{n+1}(x) +a_n p_n(x) +b_n p_{n-1}(x), \qquad n\ge1.$$

The reason no lower-degree terms appear is orthogonality. If $k\lt n-1$, then

$$\langle p_k,xp_n\rangle_w = \langle xp_k,p_n\rangle_w =0,$$

because $xp_k$ has degree at most $n-1$ and is orthogonal to $p_n$.

In quantum calculations, recurrence relations are practical tools for matrix elements, selection rules, numerical evaluation, and checking symbolic algebra. They should not be confused with quantum ladder operators unless an operator construction has actually been defined.

Rodrigues Formulas and Generating Functions

Many classical families have Rodrigues formulas, where the $n$th polynomial is written as an $n$th derivative of a weight-like expression. Schematically,

$$p_n(x) = \frac{1}{c_n w(x)} \frac{d^n}{dx^n} \left[ A_n(x)w(x) \right],$$

with family-dependent constants and factors. Rodrigues formulas are useful for proving orthogonality by repeated integration by parts, provided boundary terms vanish.

Generating functions package all degrees into one expression:

$$G(x,t) = \sum_{n=0}^{\infty}p_n(x)c_n t^n.$$

They are often the quickest way to derive recurrence and derivative identities. The family-specific generating functions belong on the dedicated Hermite, Legendre, and Laguerre pages.

Expansion Coefficients

If the polynomials are complete in the relevant weighted space, a suitable function can be expanded as

$$f(x) = \sum_{n=0}^{\infty} c_n p_n(x),$$

with coefficients

$$c_n = \frac{\langle p_n,f\rangle_w}{h_n}.$$

This is the same projection principle used for Fourier series and Hilbert-space bases. The convergence statement depends on the function class, the interval, and the weight. It may mean convergence in a weighted $L^2$ norm rather than pointwise convergence.

Worked Example: Expanding a Simple Polynomial

Using

$$P_0(x)=1, \qquad P_2(x)=\frac12(3x^2-1),$$

solve for $x^2$ in terms of Legendre polynomials:

$$3x^2-1=2P_2(x),$$

so

$$x^2 = \frac13 P_0(x) +\frac23 P_2(x).$$

This small example shows the general idea: an ordinary polynomial can be re-expanded in a basis adapted to a weighted inner product and boundary-value problem.

Normalization versus Physics

Mathematical special-function references often normalize polynomials for algebraic convenience:

$$P_\ell(1)=1, \qquad H_n(x)=2^n x^n+\cdots.$$

Quantum mechanics usually needs normalized wavefunctions. That requires the polynomial norm, the envelope, the coordinate measure, and any length scale. For example, a Hermite polynomial becomes a normalized oscillator wavefunction only after multiplication by a Gaussian and the correct oscillator-length factor.

This is why one should not copy a polynomial normalization into a quantum wavefunction without checking the measure. See Gamma and Beta Functions for the normalization integrals that often supply the constants.

Common Mistakes

• Omitting the weight function in the orthogonality integral.
• Treating a polynomial factor as the full wavefunction.
• Mixing normalization conventions from different references.
• Assuming pointwise convergence when only weighted $L^2$ convergence is justified.
• Using recurrence identities outside their stated family and parameter range.
• Forgetting endpoint conditions when deriving orthogonality by integration by parts.
• Confusing polynomial degree with a physical quantum number when the labels have been shifted.

Cross-Links

• Inner Products
• Eigenvalue Problems
• Sturm–Liouville Theory
• Separation of Variables
• Hermite Polynomials
• Legendre Polynomials
• Associated Legendre Functions
• Laguerre Polynomials
• Spherical Harmonics
• Hypergeometric Functions
• Gamma and Beta Functions
• Fourier Series

References

• NIST Digital Library of Mathematical Functions, Chapter 18, Orthogonal Polynomials.
• G. Szego, Orthogonal Polynomials, 4th ed., American Mathematical Society, 1975.
• T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, 1978.
• G. E. Andrews, R. Askey, and R. Roy, Special Functions, Cambridge University Press, 1999.
• F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., NIST Handbook of Mathematical Functions, Cambridge University Press, 2010.
• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.

Exercises

1. With weight $w(x)=1$ on $[-1,1]$, use Gram-Schmidt on $1,x,x^2$ to find a degree-two polynomial orthogonal to $1$ and $x$.

Solution

By parity, $x^2$ is automatically orthogonal to $x$ on $[-1,1]$. Subtract a multiple of $1$:

$$q_2(x)=x^2-a.$$

Orthogonality to $1$ requires

$$0 = \int_{-1}^{1}(x^2-a)\,dx = \frac{2}{3}-2a.$$

Thus $a=1/3$, so

$$q_2(x)=x^2-\frac13.$$

Up to normalization, this is the Legendre polynomial $P_2(x)$.

2. Explain why a three-term recurrence has no terms proportional to $p_0,\ldots,p_{n-2}$.

Solution

The polynomial $xp_n$ has degree $n+1$, so it can be expanded in $p_0,\ldots,p_{n+1}$. For $k\lt n-1$,

$$\langle p_k,xp_n\rangle_w = \langle xp_k,p_n\rangle_w.$$

But $xp_k$ has degree at most $n-1$, so it is a linear combination of $p_0,\ldots,p_{n-1}$ and is orthogonal to $p_n$. Hence those lower coefficients vanish.

3. Expand $x^2$ in Legendre polynomials $P_0$ and $P_2$.

Solution

Since

$$P_0(x)=1, \qquad P_2(x)=\frac12(3x^2-1),$$

one has

$$x^2 = \frac13P_0(x) +\frac23P_2(x).$$

4. Why is $H_n(x)$ not itself a normalizable oscillator wavefunction?

Solution

$H_n(x)$ is a polynomial and grows algebraically at infinity, so it is not square-integrable on the real line. The oscillator wavefunction includes a Gaussian envelope:

$$\psi_n(x)\propto H_n(x/\ell) e^{-x^2/(2\ell^2)}.$$

The Gaussian makes the product square-integrable, and the scale factor fixes the physical units and normalization.

5. In one sentence, what data define an orthogonal-polynomial family for quantum-mechanics use?

Solution

One must specify the interval, weight function, endpoint or boundary conditions, and normalization convention; the polynomial recurrence alone is not enough to identify the physical basis.