Laguerre polynomials
More generally, a Laguerre function is a solution when n is not necessarily a non-negative integer.
The Laguerre polynomials are also used for Gauss–Laguerre quadrature to numerically compute integrals of the form
The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of the Schrödinger equation for a one-electron atom.
They also describe the static Wigner functions of oscillator systems in quantum mechanics in phase space.
They further enter in the quantum mechanics of the Morse potential and of the 3D isotropic harmonic oscillator.
Physicists sometimes use a definition for the Laguerre polynomials that is larger by a factor of n!
(Likewise, some physicists may use somewhat different definitions of the so-called associated Laguerre polynomials.)
For arbitrary real α the polynomial solutions of the differential equation[2]
Given the generating function specified above, the polynomials may be expressed in terms of a contour integral
where the contour circles the origin once in a counterclockwise direction without enclosing the essential singularity at 1 The addition formula for Laguerre polynomials:[8]
Differentiating the power series representation of a generalized Laguerre polynomial k times leads to
the shift by k sometimes causing confusion with the usual parenthesis notation for a derivative.
The derivative with respect to the second variable α has the form,[9]
The generalized Laguerre polynomials obey the differential equation
which may be compared with the equation obeyed by the kth derivative of the ordinary Laguerre polynomial,
which shows that L(α)n is an eigenvector for the eigenvalue n. The generalized Laguerre polynomials are orthogonal over [0, ∞) with respect to the measure with weighting function xα e−x:[10]
denotes the gamma distribution then the orthogonality relation can be written as
The associated, symmetric kernel polynomial has the representations (Christoffel–Darboux formula)[citation needed]
The following integral is needed in the quantum mechanical treatment of the hydrogen atom,
In quantum mechanics the Schrödinger equation for the hydrogen-like atom is exactly solvable by separation of variables in spherical coordinates.
The radial part of the wave function is a (generalized) Laguerre polynomial.
[11] Vibronic transitions in the Franck-Condon approximation can also be described using Laguerre polynomials.
where the Hn(x) are the Hermite polynomials based on the weighting function exp(−x2), the so-called "physicist's version."
Because of this, the generalized Laguerre polynomials arise in the treatment of the quantum harmonic oscillator.
is the Pochhammer symbol (which in this case represents the rising factorial).
The generalized Laguerre polynomials are used to describe the quantum wavefunction for hydrogen atom orbitals.
[16][17][18] The convention used throughout this article expresses the generalized Laguerre polynomials as [19]
In the physics literature,[18] the generalized Laguerre polynomials are instead defined as
Generalized Laguerre polynomials are linked to Umbral calculus by being Sheffer sequences for
In Umbral Calculus convention,[23] the default Laguerre polynomials are defined to be
